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diff --git a/js/games/nluqo.github.io/~bh/v3ch6/v3ch6.html b/js/games/nluqo.github.io/~bh/v3ch6/v3ch6.html new file mode 100644 index 0000000..631abbf --- /dev/null +++ b/js/games/nluqo.github.io/~bh/v3ch6/v3ch6.html @@ -0,0 +1,3070 @@ +<HTML> +<HEAD> +<TITLE>Computer Science Logo Style vol 3 ch 6: Artificial Intelligence</TITLE> +</HEAD> +<BODY> +<CITE>Computer Science Logo Style</CITE> volume 3: +<CITE>Beyond Programming</CITE> 2/e Copyright (C) 1997 MIT +<H1>Artificial Intelligence</H1> + +<TABLE width="100%"><TR><TD> +<IMG SRC="../csls3.jpg" ALT="cover photo"> +<TD><TABLE> +<TR><TD align="right"><CITE><A HREF="http://www.cs.berkeley.edu/~bh/">Brian +Harvey</A><BR>University of California, Berkeley</CITE> +<TR><TD align="right"><BR> +<TR><TD align="right"><A HREF="../pdf/v3ch06.pdf">Download PDF version</A> +<TR><TD align="right"><A HREF="../v3-toc2.html">Back to Table of Contents</A> +<TR><TD align="right"><A HREF="../v3ch5/v3ch5.html"><STRONG>BACK</STRONG></A> +chapter thread <A HREF="https://people.eecs.berkeley.edu/~bh/v3ch7/v3ch7.html"><STRONG>NEXT</STRONG></A> +<TR><TD align="right"><A HREF="https://mitpress.mit.edu/books/computer-science-logo-style-second-edition-volume-3">MIT +Press web page for <CITE>Computer Science Logo Style</CITE></A> +</TABLE></TABLE> + +<HR><P>Program file for this chapter: <A HREF="student.lg"><CODE>student</CODE></A> + + +<P> + +<P>Can a computer be intelligent? What would it mean for a computer to be +intelligent? John McCarthy, one of the founders of +artificial intelligence +research, once defined the field as "getting a computer to do things which, +when done by people, are said to involve intelligence." The point of the +definition was that he felt perfectly comfortable about carrying on his +research without first having to defend any particular philosophical view of +what the word "intelligence" means. + +<P>There have always been two points of view among AI researchers about what +their purpose is. One point of view is that AI programs contribute to our +understanding of <EM>human</EM> psychology; when researchers take this view +they try to make their programs reflect the actual mechanisms of intelligent +human behavior. For example, Allen Newell and +Herbert A. Simon begin their +classic AI book <EM>Human Problem Solving</EM> with the sentence, "The aim +of this book is to advance our understanding of how humans think." In one +of their research projects they studied cryptarithmetic problems, in which +digits are replaced with letters in a multi-digit addition or multiplication. +First they did a careful observation and analysis of how a human subject +attacked such a problem, then they pointed out specific problem-solving +techniques that the person used, and used those techniques as the basis for +designing a computer simulation. The other point of view +is that AI programs provide a more abstract model for intelligence in +general; just as one can learn about the properties of computers by studying +finite-state machines, even though no real computer operates precisely as a +formal finite-state machine does, we can learn about the properties of any +possible intelligent being by simulating intelligence in a computer program, +whether or not the mechanisms of that program are similar to those used by +people. + +<P>In the early days of AI research these two points of view were not sharply +divided. Sometimes the same person would switch from one to the other, +sometimes trying to model human thought processes and sometimes trying to +solve a given problem by whatever methods could be made to work. More +recently, researchers who hold one or the other point of view consistently +have begun to define two separate fields. One is <EM> +cognitive science,</EM> in which computer scientists join with +psychologists, linguists, biologists, and others to study human cognitive +psychology, using computer programs as a concrete embodiment of theories +about the human mind. The other is called <EM>expert systems</EM> +or <EM>knowledge engineering,</EM> in which programming techniques +developed by AI researchers are put to practical use in programs that solve +real-world business problems such as the diagnosis and repair of +malfunctioning equipment. + +<P><H2>Microworlds: Student</H2> + +<P>In this chapter I'm going to concentrate on one particular area of AI +research: teaching a computer to understand English. Besides its inherent +interest, this area has the advantage that it doesn't require special +equipment, as do some other parts of AI such as machine vision and the +control of robot manipulators. + +<P>In the 1950s many people were very optimistic about the use of computers to +translate from one language to another. IBM undertook a government-sponsored +project to translate scientific journals from Russian to English. At first +they thought that this translation could be done very straightforwardly, +with a Russian-English dictionary and a few little kludges to rearrange the +order of words in a sentence to account for differences in the grammatical +structure of the two languages. This simple approach was not successful. +One problem is that the same word can have different meanings, and even +different parts of speech, in different contexts. (According to one famous +anecdote, the program translated the Russian equivalent of "The spirit is +willing but the flesh is weak" into "The vodka is strong but the meat is +rotten.") + +<P>A decade later, several AI researchers had the idea that ambiguities in the +meanings of words could be resolved by trying to understand English only in +some limited context. If you know in advance that the sentence you're +trying to understand is about baseball statistics, or about relationships in +a family tree, or about telling a robot arm to move blocks on a table (these +are actual examples of work done in that period) then only certain narrowly +defined types of sentences are meaningful at all. You needn't think about +metaphors or about the many assumptions about commonsense knowledge that +people make in talking with one another. Such a limited context for a +language understanding program is called a <EM>microworld.</EM> + +<P>This chapter includes a Logo version of Student, a program written by +Daniel G. Bobrow for his 1964 Ph.D. thesis, <EM>Natural +Language Input for a Computer Problem Solving System,</EM> at MIT. Student +is a program that solves algebra word problems: + +<P><PRE>? <U>student [The price of a radio is 69.70. If this price is 15 percent</U> + <U>less than the marked price, find the marked price.]</U> + +The marked price is 82 dollars +</PRE> + +<P>(In this illustration I've left out some of Student's display of +intermediate results.) The program has two parts: one that translates the +word problem into the form of equations and another that solves the +equations. The latter part is complex (about 40 Logo procedures) but +straightforward; it doesn't seem surprising to most people that a computer +can manipulate mathematical equations. It is Student's understanding of +English sentences that furthered the cause of artificial intelligence. + +<BLOCKQUOTE> +<P><A NAME="bobrow1">The aim</A> of the +research reported here was to discover how one could build a +computer program which could communicate with people in a natural language +within some restricted problem domain. In the course of this investigation, +I wrote a set of computer programs, the Student system, which accepts as +input a comfortable but restricted subset of English which can be used to +express a wide variety of algebra story problems... + +<P>In the following discussion, I shall use phrases such as "the computer +understands English." In all such cases, the "English" is just the +restricted subset of English which is allowable as input for the computer +program under discussion. In addition, for purposes of this report I have +adopted the following operational definition of understanding. A computer +<EM>understands</EM> a subset of English if it accepts input sentences which +are members of this subset, and answers questions based on information +contained in the input. The Student system understands English in this +sense. [Bobrow, 1964.] +</BLOCKQUOTE> + +<P>How does the algebra microworld simplify the understanding problem? For one +thing, Student need not know anything about the meanings of noun phrases. +In the sample problem above, the phrase <CODE>The price of a radio</CODE> is used +as a variable name. The problem could just as well have been + +<P><PRE>The weight of a giant size detergent box is 69.70 ounces. If this weight +is 15 percent less than the weight of an enormous size box, find the +weight of an enormous size box. +</PRE> + +<P>For Student, either problem boils down to + +<P><PRE><EM>variable1</EM> = 69.70 <EM>units</EM> + +<EM>variable1</EM> = 0.85 * <EM>variable2</EM> + +Find <EM>variable2</EM>. +</PRE> + +<P>Student understands particular words only to the extent that they +have a <EM>mathematical</EM> meaning. For example, the program knows that +<CODE>15 percent less than</CODE> means the same as <CODE>0.85 times</CODE>. + +<P><H2>How Student Translates English to Algebra</H2> + +<P>Student translates a word problem into equations in several steps. In the +following paragraphs, I'll mention in parentheses the names of the Logo +procedures that carry out each step I describe, but don't read the +procedures yet. First read through the description of the process without +worrying about the programming details of each step. Later you can reread +this section while examining the complete listing at the end of the chapter. + +<P>In translating Student to Logo, I've tried not to change the capabilities of +the program in any way. The overall structure of my version is similar to +that of Bobrow's original implementation, but I've changed some details. +I've used iteration and mapping tools to make the program easier to read; +I've changed some aspects of the fine structure of the program to fit more +closely with the usual Logo programming style; in a few cases I've tried to +make exceptionally slow parts of the program run faster by finding a more +efficient algorithm to achieve the same goal. + +<P>The top-level procedure <CODE>student</CODE> takes one input, a list containing the +word problem. (The disk file that accompanies this project includes several +variables containing sample problems. For example, + +<P><PRE>? <U>student :radio</U> +</PRE> + +<P>will carry out the steps I'm about to describe.) Student begins +by printing the original problem: + +<P><PRE>? <U>student :radio</U> + +The problem to be solved is + +The price of a radio is 69.70. If this price is 15 percent less than the +marked price, find the marked price. +</PRE> + +<P>The first step is to separate punctuation characters from the +attached words. For example, the word "<CODE>price,</CODE>" in the original problem +becomes the two words "<CODE>price ,</CODE>" with the comma on its own. Then +(<CODE>student1</CODE>) certain <EM>mandatory substitutions</EM> are applied +(<CODE>idioms</CODE>). For example, the phrase <CODE>percent less than</CODE> is translated +into the single word <CODE>perless</CODE>. The result is printed: + +<P><PRE>With mandatory substitutions the problem is + +The price numof a radio is 69.70 dollars . If this price is 15 perless +the marked price , find the marked price . +</PRE> + +<P>(The word <CODE>of</CODE> in an algebra word problem can have two +different meanings. Sometimes it means "times," as in the phrase +"one half of the population." Other times, as in this problem, "of" is +just part of a noun phrase like "the price of a radio." The special word +<CODE>numof</CODE> is a flag to a later part of the program and will then be +further translated either into <CODE>times</CODE> or back into <CODE>of</CODE>. The +original implementation of Student used, instead of a special word like <CODE> +numof</CODE>, a "tagged" word represented as a list like <CODE>[of / op]</CODE>. Other +examples of tagging are <CODE>[Bill / person]</CODE> and <CODE>[has / verb]</CODE>.) + +<P> + +<P>The next step is to separate the problem into simple sentences (<CODE>bracket</CODE>): + +<P><PRE>The simple sentences are + +The price numof a radio is 69.70 dollars . + +This price is 15 perless the marked price . + +Find the marked price . +</PRE> + +<P>Usually this transformation of the problem is straightforward, but +the special case of "age problems" is recognized at this time, and special +transformations are applied so that a sentence like + +<P><PRE>Mary is 24 years old. +</PRE> + +<P>is translated into + +<P><PRE>Mary s age is 24 . +</PRE> + +<P>An age problem is one that contains any of the phrases <CODE>as old +as</CODE>, <CODE>age</CODE>, or <CODE>years old</CODE>. + +<P>The next step is to translate each simple sentence into an equation or a +variable whose value is desired as part of the solution (<CODE>senform</CODE>). + +<P><PRE>The equations to be solved are + +Equal [price of radio] [product 69.7 [dollars]] + +Equal [price of radio] [product 0.85 [marked price]] +</PRE> + +<P>The third simple sentence is translated, not into an equation, but +into a request to solve these equations for the variable <CODE>marked price</CODE>. + +<P>The translation of simple sentences into equations is the most +"intelligent" part of the program; that is, it's where the program's +knowledge of English grammar and vocabulary come into play and many special +cases must be considered. In this example, the second simple sentence +starts with the phrase <CODE>this price</CODE>. The program recognizes the word +<CODE>this</CODE> (procedure <CODE>nmtest</CODE>) and replaces the entire phrase with the +left hand side of the previous equation (procedure <CODE>this</CODE>). + +<P> + + +<P><H2>Pattern Matching</H2> + + +<P>Student analyzes a sentence by comparing it to several <EM>patterns</EM> +(<CODE>senform1</CODE>). For example, one sentence form that Student understands +is exemplified by these sentences: + +<P><PRE>Joe weighs 163 pounds . +The United States Army has 8742 officers . +</PRE> + +<P>The general pattern is + +<P><PRE><EM>something</EM> <EM>verb</EM> <EM>number</EM> <EM>unit</EM> . +</PRE> + +<P>Student treats such sentences as if they were rearranged to match + +<P><PRE>The number of <EM>unit</EM> <EM>something</EM> <EM>verb</EM> is <EM>number</EM> . +</PRE> + +<P>and so it generates the equations + +<P><PRE>Equal [number of pounds Joe weighs] 163 + +Equal [number of officers United States Army has] 8742 +</PRE> + +<P>The original version of Student was written in a pattern matching +language called Meteor, which Bobrow wrote in Lisp. In Meteor, the +instruction that handles this sentence type looks like this: + +<P><PRE>(* ( (1 / verb) (fn nmtest) 1 (1 / dlm)) 0 + (/ (*s shelf (*k equal (fn opform (*k the number of 4 1 2)) + (fn opform (*k 3 5 6))))) return) +</PRE> + +<P>The top line contains the pattern to be matched. In the pattern, +a dollar sign represents zero or more words; the notation <CODE>1</CODE> +represents a single word. The zero at the end of the line means that the +text that matches the pattern should be deleted and nothing should replace +it. The rest of the instruction pushes a new equation onto a stack named +<CODE>shelf</CODE>; that equation is formed out of the pieces of the matched +pattern according to the numbers in the instruction. That is, the number +<CODE>4</CODE> represents the fourth component of the pattern, which is <CODE>1</CODE>. + + + Here is the corresponding instruction in the Logo version: + +<P><PRE>if match [^one !verb1:verb !factor:numberp #stuff1 !:dlm] :sent + [output (list (list "equal + opform (sentence [the number of] + :stuff1 :one :verb1) + opform (list :factor) ))] +</PRE> + +<P>The pattern matcher I used for Student is the same as the one in +<EM>Advanced Techniques,</EM> the second volume of this series.<SUP>*</SUP> Student often relies on the fact that +Meteor's pattern matcher finds the <EM>first substring</EM> of the text that +matches the pattern, rather than requiring the entire text to match. Many +patterns in the Logo version therefore take the form + +<P><PRE>[^beg <EM>interesting part</EM> #end] +</PRE> + +<P>where the "interesting part" is all that appeared in the Meteor +pattern. + +<P><SMALL><BLOCKQUOTE><SMALL><SUP>*</SUP>The +version in this project is modified slightly; the <CODE>match</CODE> procedure +first does a fast test to try to reject an irrelevant pattern in O(<EM>n</EM>) +time before calling the actual pattern matcher, which could take as much +as O(2<SUP><SMALL><EM>n</EM></SMALL></SUP>) time to reject a pattern, and which has been renamed <CODE>rmatch</CODE> +(for "real match") in this project.</SMALL></BLOCKQUOTE></SMALL> + +<P>Here is a very brief summary of the Logo pattern matcher included in this +program. For a fuller description with examples, please refer to Volume 2. +<CODE>Match</CODE> is a predicate with two inputs, both lists. The first input is +the <EM>pattern</EM> and the second input is the <EM>sentence.</EM> <CODE>Match</CODE> +outputs <CODE>true</CODE> if the sentence <EM>matches</EM> the pattern. A word in +the pattern that does not begin with one of the special <EM>quantifier</EM> +characters listed below matches the identical word in the sentence. A word +in the pattern that does begin with a quantifier matches zero or more words +in the sentence, as follows: + +<P><TABLE> +<TR><TD><CODE>#</CODE><TD> zero or more +<TR><TD><CODE>&</CODE><TD> one or more +<TR><TD><CODE>?</CODE><TD> zero or one +<TR><TD><CODE>!</CODE><TD> exactly one +<TR><TD><CODE>@</CODE><TD> zero or more (test as group) +<TR><TD><CODE>^</CODE><TD> zero or more (as few as possible) +</TABLE> + +<P> +All quantifiers match as many consecutive words as possible while +still allowing the remaining portion of the pattern to be matched, except +for <CODE>^</CODE>. A quantifier may be used alone, or it can be followed by a +variable name, a predicate name, or both: + +<P><PRE># +#var +#:pred +#var:pred +</PRE> + +If a variable name is used, the word or words that match the +quantifier will be stored in that variable if the match is successful. (The +value of the variable if the match is not successful is not guaranteed.) If +a predicate is used, it must take one word as input; in order for a word in +the sentence to be accepted as (part of) a match for the quantifier, the +predicate must output <CODE>true</CODE> when given that word as input. For +example, the word + +<P><PRE>!factor:numberp +</PRE> + +<P>in the pattern above requires exactly one matching word in the +sentence; that word must be a number, and it is remembered in the variable +<CODE>factor</CODE>. If the quantifier is <CODE>@</CODE> then the predicate must take a +<EM>list</EM> as input, and it must output <CODE>true</CODE> for all the candidate +matching words taken together as a list. For example, if you define a +procedure + +<P><PRE>to threep :list +output equalp count :list 3 +end +</PRE> + +<P>then the pattern word + +<P><PRE>@:threep +</PRE> + +<P>will match exactly three words in the sentence. (Student does not +use this last feature of the pattern matcher. In fact, predicates are +applied only to the single-word quantifiers <CODE>?</CODE> and <CODE>!</CODE>.) + +<P>Pattern matching is also heavily used in converting words and phrases with +mathematical meaning into the corresponding arithmetic operations +(<CODE>opform</CODE>). An equation is a list of three members; the first member is the +word <CODE>equal</CODE> and the other two are expressions formed by applying +operations to variables and numbers. Each operation that is required is +represented as a list whose first member is the name of the Logo procedure +that carries out the operation and whose remaining members are expressions +representing the operands. For example, the equation +<P><CENTER><EM>y</EM> = 3<EM>x</EM><SUP><SMALL>2</SMALL></SUP> ++ 6<EM>x</EM> − 1</CENTER><P> +would be represented by the list + +<P><PRE>[equal [y] [sum [product 3 [square [x]]] [product 6 [x]] [minus 1]]] +</PRE> + +<P>The variables are represented by lists like <CODE>[x]</CODE> rather than +just the words because in Student a variable can be a multi-word phrase like +<CODE>price of radio</CODE>. The difference between two expressions is represented +by a <CODE>sum</CODE> of one expression and <CODE>minus</CODE> the other, rather than as +the <CODE>difference</CODE> of the expressions, because this representation turns +out to make the process of simplifying and solving the equations easier. + +<P>In word problems, as in arithmetic expressions, there is a precedence of +operations. Operations like <CODE>squared</CODE> apply to the variables right next +to them; ones like <CODE>times</CODE> are intermediate, and ones like <CODE>plus</CODE> +apply to the largest possible subexpressions. Student looks first for the +lowest-priority ones like <CODE>plus</CODE>; if one is found, the entire rest of +the clause before and after the operation word provide the operands. Those +operands are recursively processed by <CODE>opform</CODE>; when all the +low-priority operations have been found, the next level of priority will be +found by matching the pattern + +<P><PRE>[^left !op:op1 #right] +</PRE> + +<P><H2>Solving the Equations</H2> + +<P>Student uses the substitution technique to solve the equations. That is, +one equation is rearranged so that the left hand side contains only a single +variable and the right hand side does not contain that variable. Then, in +some other equation, every instance of that variable is replaced by the +right hand side of the first equation. The result is a new equation from +which one variable has been eliminated. Repeating this process enough times +should eventually yield an equation with only a single variable, which can +be solved to find the value of that variable. + +<P>When a problem gives rise to several linear equations in several variables, +the traditional technique for computer solution is to use matrix inversion; +this technique is messy for human beings because there is a lot of +arithmetic involved, but straightforward for computers because the algorithm +can be specified in a simple way that doesn't depend on the particular +equations in each problem. Bobrow chose to use the substitution method +because some problems give rise to equations that are linear in the variable +for which a solution is desired but nonlinear in other variables. Consider +this problem: + +<P><PRE>? <U>student :tom</U> + +The problem to be solved is + +If the number of customers Tom gets is twice the square of 20 per cent of +the number of advertisements he runs, and the number of advertisements he +runs is 45, what is the number of customers Tom gets? + + +With mandatory substitutions the problem is + +If the number numof customers Tom gets is 2 times the square 20 percent +numof the number numof advertisements he runs , and the number numof +advertisements he runs is 45 , what is the number numof customers +Tom gets ? + + +The simple sentences are + +The number numof customers Tom gets is 2 times the square 20 percent +numof the number numof advertisements he runs . + +The number numof advertisements he runs is 45 . + +What is the number numof customers Tom gets ? + + +The equations to be solved are + +Equal [number of customers Tom gets] + [product 2 [square [product 0.2 [number of advertisements + he runs]]]] + +Equal [number of advertisements he runs] 45 + + +The number of customers Tom gets is 162 + +The problem is solved. +</PRE> + +<P>The first equation that Student generates for this problem is +linear in the number of customers Tom gets, but nonlinear in the number of +advertisements he runs. (That is, the equation refers to the <EM>square</EM> +of the latter variable. An equation is <EM>linear</EM> in a given variable +if that variable isn't multiplied by anything other than a constant number.) +Using the substitution method, Student can solve the problem by substituting +the value 45, found in the second equation, for the number of advertisements +variable in the first equation. + +<P>(Notice, in passing, that one of the special <CODE>numof</CODE> words in this +problem was translated into a multiplication rather than back into the +original word <CODE>of</CODE>.) + +<P>The actual sequence of steps required to solve a set of equations is quite +intricate. I recommend taking that part of Student on faith the first time +you read the program, concentrating instead on the pattern matching +techniques used to translate the English sentences into equations. But here +is a rough guide to the solution process. Both <CODE>student1</CODE> and <CODE> +student2</CODE> call <CODE>trysolve</CODE> with four inputs: a list of the equations to +solve, a list of the variables for which values are wanted, and two lists of +<EM>units.</EM> A unit is a word or phrase like <CODE>dollars</CODE> or <CODE>feet</CODE> +that may be part of a solution. Student treats units like variables while +constructing the equations, so the combination of a number and a unit is +represented as a product, like + +<P><PRE>[product 69.7 [dollars]] +</PRE> + +<P>for 69.70 in the first sample problem. While constructing the +equations, Student generates two lists of units. The first, stored in the +variable <CODE>units</CODE>, contains any word or phrase that appears along with a +number in the problem statement, like the word <CODE>feet</CODE> in the phrase <CODE> +3 feet</CODE> (<CODE>nmtest</CODE>). The second, in the variable <CODE>aunits</CODE>, contains +units mentioned explicitly in the <CODE>find</CODE> or <CODE>how many</CODE> sentences +that tell Student what variables should be part of the solution +(<CODE>senform1</CODE>). If the problem includes a sentence like + +<P><PRE>How many inches is a yard? +</PRE> + +<P>then the variable <CODE>[inches]</CODE>, and <EM>only</EM> that variable, +is allowed to be part of the answer. If there are no <CODE>aunits</CODE>-type +variables in the problem, then any of the <CODE>units</CODE> variables may appear +in the solution (<CODE>trysolve</CODE>). + +<P><CODE>Trysolve</CODE> first calls <CODE>solve</CODE> to solve the equations and then uses +<CODE>pranswers</CODE> to print the results. <CODE>Solve</CODE> calls <CODE>solver</CODE> to do +most of the work and then passes its output through <CODE>solve.reduce</CODE> for +some final cleaning up. <CODE>Solver</CODE> works by picking one of the +variables from the list <CODE>:wanted</CODE> and asking <CODE>solve1</CODE> to find a +solution for that variable in terms of <EM>all</EM> the other variables--the +other wanted variables as well as the units allowed in the ultimate answer. +If <CODE>solve1</CODE> succeeds, then <CODE>solver</CODE> invokes itself, +adding the newly-found expression for one variable to an <EM> +association list</EM> (in the variable <CODE>alis</CODE>) so that, from then +on, any occurrence of that variable will be replaced with the equivalent +expression. In effect, the problem is simplified by eliminating one +variable and eliminating one equation, the one that was solved to find the +equivalent expression. + +<P><CODE>Solve1</CODE> first looks for an equation containing the variable for which +it is trying to find a solution. When it finds such an equation, the next +task is to eliminate from that equation any variables that aren't part of +the wanted-plus-units list that <CODE>solver</CODE> gave <CODE>solve1</CODE> as an input. +To eliminate these extra variables, <CODE>solve1</CODE> invokes <CODE>solver</CODE> with +the extras as the list of wanted variables. This mutual recursion between +<CODE>solver</CODE> and <CODE>solve1</CODE> makes the structure of the solution process +difficult to follow. If <CODE>solver</CODE> manages to eliminate the extra +variables by expressing them in terms of the originally wanted ones, then +<CODE>solve1</CODE> can go on to substitute those expressions into its originally +chosen equation and then use <CODE>solveq</CODE> to solve that one equation for the +one selected variable in terms of all the other allowed variables. <CODE> +Solveq</CODE> manipulates the equation more or less the way students in algebra +classes do, adding the same term to both sides, multiplying both sides by +the denominator of a polynomial fraction, and so on. + +<P>Here is how <CODE>solve</CODE> solves the radio problem. The equations, again, are + +<P><PRE>Equal [price of radio] [product 69.7 [dollars]] + +Equal [price of radio] [product 0.85 [marked price]] +</PRE> + +<P><CODE>Trysolve</CODE> evaluates the expression + +<P><PRE>(1) solve [[marked price]] + [[equal [price of radio] [product 69.7 [dollars]]] + [equal [price of radio] [product 0.85 [marked price]]] ] + [[dollars]] +</PRE> + +<P>(I'm numbering these expressions so that I can refer to them later +in the text.) The first input to <CODE>solve</CODE> is the list of variables +wanted in the solution; in this case there is only one such variable. The +second input is the list of two equations. The third is the list of unit +variables that are allowed to appear in the solution; in this case only <CODE> +[dollars]</CODE> is allowed. <CODE>Solve</CODE> evaluates + +<P><PRE>(2) solver [[marked price]] [[dollars]] [] [] +</PRE> + +<P>(There is a fifth input, the word <CODE>insufficient</CODE>, but this is +used only as an error flag if the problem can't be solved. To simplify this +discussion I'm going to ignore that input for both <CODE>solver</CODE> and <CODE> +solve1</CODE>.) <CODE>Solver</CODE> picks the first (in this case, the only) wanted +variable as the major input to <CODE>solve1</CODE>: + +<P><PRE>(3) solve1 [marked price] + [[dollars]] + [] + [[equal [price of radio] [product 69.7 [dollars]]] + [equal [price of radio] [product 0.85 [marked price]]] ] + [] +</PRE> + +<P>Notice that the first input to <CODE>solve1</CODE> is a single variable, +not a list of variables. <CODE>Solve1</CODE> examines the first equation in the +list of equations making up its fourth input. The desired variable does not +appear in this equation, so <CODE>solve1</CODE> rejects that equation and invokes +itself recursively: + +<P><PRE>(4) solve1 [marked price] + [[dollars]] + [] + [[equal [price of radio] [product 0.85 [marked price]]]] + [[equal [price of radio] [product 69.7 [dollars]]]] +</PRE> + +<P>This time, the first (and now only) equation on the list of +candidates does contain the desired variable. <CODE>Solve1</CODE> removes that +equation, not from its own list of equations (<CODE>:eqns</CODE>), but from <CODE> +solve</CODE>'s overall list (<CODE>:eqt</CODE>). The equation, unfortunately, can't be +solved directly to express <CODE>[marked price]</CODE> in terms of <CODE>[dollars]</CODE>, +because it contains the extra, unwanted variable <CODE>[price of radio]</CODE>. +We must eliminate this variable by solving the remaining equations for it: + +<P><PRE>(5) solver [[price of radio]] [[marked price] [dollars]] [] [] +</PRE> + +<P>As before, <CODE>solver</CODE> picks the first (again, in this case, the +only) wanted variable and asks <CODE>solve1</CODE> to solve it: + +<P><PRE>(6) solve1 [price of radio] + [[marked price] [dollars]] + [] + [[equal [price of radio] [product 69.7 [dollars]]]] + [] +</PRE> + +<P><CODE>Solve1</CODE> does find the desired variable in the first (and +only) equation, and this time there are no extra variables. <CODE>Solve1</CODE> +can therefore ask <CODE>solveq</CODE> to solve the equation: + +<P><PRE>(7) solveq [price of radio] + [equal [price of radio] [product 69.7 [dollars]]] +</PRE> + +<P>It isn't part of <CODE>solveq</CODE>'s job to worry about which variables +may or may not be part of the solution; <CODE>solve1</CODE> doesn't call <CODE> +solveq</CODE> until it's satisfied that the equation is okay. + +<P>In this case, <CODE>solveq</CODE> has little work to do because the equation is +already in the desired form, with the chosen variable alone on the left side +and an expression not containing that variable on the right. + +<P><PRE>solveq (7) <EM>outputs</EM> [[price of radio] [product 69.7 [dollars]] + <EM>to</EM> solve1 (6) +</PRE> + +<P><CODE>Solve1</CODE> appends this result to the previously empty +association list. + +<P><PRE>solve1 (6) <EM>outputs</EM> [[[price of radio] [product 69.7 [dollars]]] + <EM>to</EM> solver (5) +</PRE> + +<P><CODE>Solver</CODE> only had one variable in its <CODE>:wanted</CODE> list, so +its job is also finished. + +<P><PRE>solver (5) <EM>outputs</EM> [[[price of radio] [product 69.7 [dollars]]] + <EM>to</EM> solve1 (4,3) +</PRE> + +<P>This outer invocation of <CODE>solve1</CODE> was trying to solve for +<CODE>[marked price]</CODE> an equation that also involved <CODE>[price of radio]</CODE>. +It is now able to use the new association list to substitute for this +unwanted variable an expression in terms of wanted variables only; this +modified equation is then passed on to <CODE>solveq</CODE>: + +<P><PRE>(8) solveq [marked price] + [equal [product 69.7 [dollars]] [product 0.85 [marked price]]] +</PRE> + +<P>This time <CODE>solveq</CODE> has to work a little harder, exchanging the +two sides of the equation and dividing by 0.85. + +<P><PRE>solveq (8) <EM>outputs</EM> [[marked price] [product 82 [dollars]]] + <EM>to</EM> solve1 (4,3) +</PRE> + +<P><CODE>Solve1</CODE> appends this result to the association list: + +<P><PRE>solve1 (4,3) <EM>outputs</EM> [[[price of radio] [product 69.7 [dollars]]] + [[marked price] [product 82 [dollars]]] ] + <EM>to</EM> solver (2) +</PRE> + +<P>Since <CODE>solver</CODE> has no other wanted variables, it outputs the +same list to <CODE>solve</CODE>, and <CODE>solve</CODE> outputs the same list to <CODE> +trysolve</CODE>. (In this example, <CODE>solve.reduce</CODE> has no effect because all +of the expressions in the association list are in terms of allowed units +only. If the equations had been different, the expression for <CODE>[price +of radio]</CODE> might have included <CODE>[marked price]</CODE> and then <CODE> +solve.reduce</CODE> would have had to substitute and simplify (<CODE>subord</CODE>).) + +<P>It'll probably take tracing a few more examples and beating your head +against the wall a bit before you really understand the structure of <CODE> +solve</CODE> and its subprocedures. Again, don't get distracted by this part of +the program until you've come to understand the language processing part, +which is our main interest in this chapter. + +<P><H2>Age Problems</H2> + +<P>The main reason why Student treats age problems specially is that the +English form of such problems is often expressed as if the variables were +people, like "Bill," whereas the real variable is "Bill's age." The +pattern matching transformations look for proper names (<CODE>personp</CODE>) and +insert the words <CODE>s age</CODE> after them (<CODE>ageify</CODE>). The first such age +variable in the problem is remembered specially so that it can be +substituted for pronouns (<CODE>agepron</CODE>). A special case is the phrase <CODE> +their ages</CODE>, which is replaced (<CODE>ageprob</CODE>) with a list of all the age +variables in the problem. + +<P><PRE>? <U>student :uncle</U> + +The problem to be solved is + +Bill's father's uncle is twice as old as Bill's father. 2 years from now +Bill's father will be 3 times as old as Bill. The sum of their ages is +92 . Find Bill's age. + + +With mandatory substitutions the problem is + +Bill s father s uncle is 2 times as old as Bill s father . 2 years +from now Bill s father will be 3 times as old as Bill . sum their +ages is 92 . Find Bill s age . + + +The simple sentences are + +Bill s father s uncle s age is 2 times Bill s father s age . + +Bill s father s age pluss 2 is 3 times Bill s age pluss 2 . + +Sum Bill s age and Bill s father s age and Bill s father s uncle s age +is 92 . + +Find Bill s age . + + +The equations to be solved are + +Equal [Bill s father s uncle s age] [product 2 [Bill s father s age]] + +Equal [sum [Bill s father s age] 2] [product 3 [sum [Bill s age] 2]] + +Equal [sum [Bill s age] + [sum [Bill s father s age] [Bill s father s uncle s age]]] 92 + + +Bill s age is 8 + +The problem is solved. +</PRE> + +<P>(Note that in the original problem statement there is a space +between the number <CODE>92</CODE> and the following period. I had to enter the +problem in that form because of an inflexibility in Logo's input parser, +which assumes that a period right after a number is part of the number, so +that "<CODE>92.</CODE>" is reformatted into <CODE>92</CODE> without the dot.) + +<P>Student represents the possessive word <CODE>Bill's</CODE> as the two words +<CODE>Bill s</CODE> because this representation allows the pattern matcher to +manipulate the possessive marker as a separate element to be matched. +A phrase like <CODE>as old as</CODE> is just deleted (<CODE>ageprob</CODE>) because the +transformation from people to ages makes it redundant. + +<P>The phrase <CODE>2 years from now</CODE> in the original problem is first +translated to <CODE>in 2 years</CODE>. This phrase is further processed according +to where it appears in a sentence. When it is attached to a particular +variable, in a phrase like <CODE>Bill s age in 2 years</CODE>, the entire phrase is +translated into the arithmetic operation <CODE>Bill s age pluss 2 years</CODE> +(<CODE>agewhen</CODE>). (The special word <CODE>pluss</CODE> is an addition operator, +just like <CODE>plus</CODE>, except for its precedence; <CODE>opform</CODE> treats it as a +tightly binding operation like <CODE>squared</CODE> instead of a loosely binding +one like the ordinary <CODE>plus</CODE>.) When a phrase like <CODE>in 2 years</CODE> +appears at the beginning of a sentence, it is remembered (<CODE>agesen</CODE>) as +an implicit modifier for <EM>every</EM> age variable in that sentence that +isn't explicitly modified. In this example, <CODE>in 2 years</CODE> modifies both +<CODE>Bill s father s age</CODE> and <CODE>Bill s age</CODE>. The special precedence of +<CODE>pluss</CODE> is needed in this example so that the equation will be based on +the grouping + +<P><PRE>3 times [ Bill s age pluss 2 ] +</PRE> + +<P>rather than + +<P><PRE>[ 3 times Bill s age ] plus 2 +</PRE> + +<P>as it would be with the ordinary <CODE>plus</CODE> operator. You can +also see how the substitution for <CODE>their ages</CODE> works in this example. + +<P>Here is a second sample age problem that illustrates a different kind of +special handling: + +<P><PRE>? <U>student :ann</U> + +The problem to be solved is + +Mary is twice as old as Ann was when Mary was as old as Ann is now. If +Mary is 24 years old, how old is Ann? + + +With mandatory substitutions the problem is + +Mary is 2 times as old as Ann was when Mary was as old as Ann is now . If +Mary is 24 years old , what is Ann ? + + +The simple sentences are + +Mary s age is 2 times Ann s age minuss g1 . + +Mary s age minuss g1 is Ann s age . + +Mary s age is 24 . + +What is Ann s age ? + + +The equations to be solved are + +Equal [Mary s age] [product 2 [sum [Ann s age] [minus [g1]]]] + +Equal [sum [Mary s age] [minus [g1]]] [Ann s age] + +Equal [Mary s age] 24 + + +Ann s age is 18 + +The problem is solved. +</PRE> + +<P>What is new in this example is Student's handling of the phrase <CODE>was +when</CODE> in the sentence + +<P><PRE>Mary is 2 times as old as Ann was when Mary was as old as Ann is now . +</PRE> + +<P>Sentences like this one often cause trouble for human algebra students +because they make <EM>implicit</EM> reference to a quantity that is not +explicitly present as a variable. The sentence says that Mary's age <EM> +now</EM> is twice Ann's age <EM>some number</EM> of years ago, but that number +is not explicit in the problem. Student makes this variable explicit by +using a <EM>generated symbol</EM> like the word <CODE>g1</CODE> in this illustration. +Student replaces the phrase <CODE>was when</CODE> with the words + +<P><PRE>was g1 years ago . g1 years ago +</PRE> + +<P>This substitution (in <CODE>ageprob</CODE>) happens <EM>before</EM> the +division of the problem statement into simple sentences (<CODE>bracket</CODE>). As +a result, this one sentence in the original problem becomes the two sentences + +<P><PRE>Mary s age is 2 times Ann s age g1 years ago . + +G1 years ago Mary s age was Ann s age now . +</PRE> + +<P>The phrase <CODE>g1 years ago</CODE> in each of these sentences is +further processed by <CODE>agesen</CODE> and <CODE>agewhen</CODE> as discussed earlier; +the final result is + +<P><PRE>Mary s age is 2 times Ann s age minuss g1 . + +Mary s age minuss g1 is Ann s age . +</PRE> + +<P>A new generated symbol is created each time this situation arises, +so there is no conflict from trying to use the same variable name for two +different purposes. The phrase <CODE>will be when</CODE> is handled similarly, +except that the translated version is + +<P><PRE>in g2 years . in g2 years +</PRE> + +<P><H2>AI and Education</H2> + +<BLOCKQUOTE> +<P><A NAME="bobrow2">These decoupling heuristics</A> are useful +not only for the Student program but +for people trying to solve age problems. The classic age problem about Mary +and Ann, given above, took an MIT graduate student over 5 minutes to solve +because he did not know this heuristic. With the heuristic he was able to +set up the appropriate equations much more rapidly. As a crude measure of +Student's relative speed, note that Student took less than one minute to +solve this problem. +</BLOCKQUOTE> + +<P>This excerpt from Bobrow's thesis illustrates the idea that +insights from artificial intelligence research can make a valuable +contribution to the education of human beings. An intellectual problem is +solved, at least in many cases, by dividing it into pieces and developing a +technique for each subproblem. The subproblems are the same whether it is a +computer or a person trying to solve the problem. If a certain technique +proves valuable for the computer, it may be helpful for a human problem +solver to be aware of the computer's methods. Bobrow's suggestion to teach +people one specific heuristic for algebra word problems is a relatively +modest example of this general theme. (A <EM>heuristic</EM> is a rule that +gives the right answer most of the time, as opposed to an <EM> +algorithm,</EM> a +rule that always works.) Some researchers in cognitive science and +education have proposed the idea of <EM>intelligent CAI</EM> +(computer assisted instruction), in which a computer would be programmed as a +"tutor" that would observe the efforts of a student in solving a problem. +The tutor would know about some of the mistaken ideas people can have about +a particular class of problem and would notice a student falling into one +of those traps. It could then offer advice tailored to the needs of that +individual student. + +<P>The development of the Logo programming language (and so also, indirectly, +this series of books) is another example of the relationship between AI and +education. Part of the idea behind Logo is that the process of programming +a computer resembles, in some ways, the process of teaching a person to do +something. (This can include teaching oneself.) For example, when a +computer program doesn't work, the experienced programmer doesn't give up in +despair, but instead <EM>debugs</EM> the program. Yet many students are +willing to give up and say "I just don't get it" if their understanding of +some problem isn't perfect on the first try. + +<BLOCKQUOTE> +<P><A NAME="papert">The critic</A> is afraid +that children will adopt the computer as model and +eventually come to "think mechanically" themselves. Following the +opposite tack, I have invented ways to take educational advantage of the +opportunities to master the art of <EM>deliberately</EM> thinking like a +computer, according, for example, to the stereotype of a computer program +that proceeds in a step-by-step, literal, mechanical fashion. There are +situations where this style of thinking is appropriate and useful. Some +children's difficulties in learning formal subjects such as grammar or +mathematics derive from their inability to see the point of such a style. + +<P> +A second educational advantage is indirect but ultimately more important. +By deliberately learning to imitate mechanical thinking, the learner becomes +able to articulate what mechanical thinking is and what it is not. The +exercise can lead to greater confidence about the ability to choose a +cognitive style that suits the problem. Analysis of "mechanical thinking" +and how it is different from other kinds and practice with problem analysis +can result in a new degree of intellectual sophistication. By providing a +very concrete, down-to-earth model of a particular style of thinking, work +with the computer can make it easier to understand that there is such a +thing as a "style of thinking." And giving children the opportunity to +choose one style or another provides an opportunity to develop the skill +necessary to choose between styles. Thus instead of inducing mechanical +thinking, contact with computers could turn out to be the best conceivable +antidote to it. And for me what is most important in this is that through +these experiences these children would be serving their apprenticeships as +epistemologists, that is to say learning to think articulately about +thinking. [Seymour Papert, <EM>Mindstorms</EM>, Basic Books, 1980, p. 27.] +</BLOCKQUOTE> + +<P><H2>Combining Sentences Into One Equation</H2> + +<P>In age problems, as we've just seen, a single sentence may give rise to two +equations. Here is an example of the opposite, several sentences that +together contribute a single equation. + +<P><PRE>? <U>student :nums</U> + +The problem to be solved is + +A number is multiplied by 6 . This product is increased by 44 . This +result is 68 . Find the number. + + +With mandatory substitutions the problem is + +A number ismulby 6 . This product is increased by 44 . This result is +68 . Find the number . + + +The simple sentences are + +A number ismulby 6 . + +This product is increased by 44 . + +This result is 68 . + +Find the number . + + +The equations to be solved are + +Equal [sum [product [number] 6] 44] 68 + + +The number is 4 + +The problem is solved. +</PRE> + +<P>Student recognizes problems like this by recognizing the phrases "is +multiplied by," "is divided by," and "is increased by" (<CODE>senform1</CODE>). +A sentence containing one of these phrases is not translated into an +equation; instead, a <EM>partial</EM> equation is saved until the next +sentence is read. That next sentence is expected to start with a phrase +like "this result" or "this product." The same procedure (<CODE>this</CODE>) +that in other situations uses the left hand side of the last equation as the +expression for the <CODE>this</CODE>-phrase notices that there is a remembered +partial equation and uses that instead. In this example, the sentence + +<P><PRE>A number ismulby 6 . +</PRE> + +<P>remembers the algebraic expression + +<P><PRE>[product [number] 6] +</PRE> + +<P>The second sentence uses that remembered expression as part of a +new, larger expression to be remembered: + +<P><PRE>[sum [product [number] 6] 44] +</PRE> + +<P>The third sentence does not contain one of the special "is +increased by" phrases, but is instead a standard "A is B" sentence. +That sentence, therefore, does give rise to an equation, as shown above. + +<P>Perhaps the most interesting thing to notice about this category of word +problem is how narrowly defined Student's criterion for recognizing the +category is. Student gets away with it because algebra word problems are +highly <EM>stereotyped;</EM> there are just a few categories, with +traditional, standard wordings. In principle there could be a word problem +starting + +<P><PRE>Robert has a certain number of jelly beans. This number is twice the +number of jelly beans Linda has. +</PRE> + +<P>These two sentences are together equivalent to + +<P><PRE>The number of jelly beans Robert has is twice the number of jelly beans +Linda has. +</PRE> + +<P>But Student would not recognize the situation because the first +sentence doesn't talk about "is increased by." We could teach Student to +understand a word problem in this form by adding the instruction + +<P><PRE>if match [^one !verb1:verb a certain number of #stuff1 !:dlm] :sent + [push "ref opform (se [the number of] :stuff1 :one :verb1) + op []] +</PRE> + +<P>along with the other known sentence forms in <CODE>senform1</CODE>. +(Compare this to the pattern matching instruction shown earlier for a +similar sentence but with an explicitly specified number.) + +<P>Taking advantage of the stereotyped nature of word problems is an example of +how the microworld strategy helped make the early AI programs possible. If +word problems were expressed with all the flexibility of language in general, +Student would need many more sentence patterns than it actually has. (How +many different ways can you think of to express the same idea about Robert +and Linda? How many of those ways can Student handle?) + +<P><H2>Allowing Flexible Phrasing</H2> + +<P>In the examples we've seen so far, Student has relied on the repetition of +identical or near-identical phrases such as "the marked price" or "the +number of advertisements he runs." (The requirement is not quite strictly +identical phrases because articles are removed from the noun phrases to make +variable names.) In real writing, though, such phrases are often +abbreviated when they appear for a second time. Student will translate such +a problem into a system of equations that can't be solved, because what +should be one variable is instead a different variable in each equation. +But Student can recognize this situation and apply heuristic rules to guess +that two similar variable names are meant, in fact, to represent the same +variable. (Some early writers on AI considered the use of heuristic methods +one of the defining characteristics of the field. Computer scientists +outside of AI were more likely to insist on fully reliable algorithms. This +distinction still has some truth to it, but it isn't emphasized so much as a +critical issue these days.) Student doesn't try to equate different +variables until it has first tried to solve the equations as they are +originally generated. If the first attempt at solution fails, Student has +recourse to less certain techniques (<CODE>student2</CODE> calls <CODE>vartest</CODE>). + +<P><PRE>? <U>student :sally</U> + +The problem to be solved is + +The sum of Sally's share of some money and Frank's share is 4.50. +Sally's share is twice Frank's. Find Frank's and Sally's share. + + +With mandatory substitutions the problem is + +sum Sally s share numof some money and Frank s share is 4.50 dollars . +Sally s share is 2 times Frank s . Find Frank s and Sally s share . + + +The simple sentences are + +Sum Sally s share numof some money and Frank s share is 4.50 dollars . + +Sally s share is 2 times Frank s . + +Find Frank s and Sally s share . + + +The equations to be solved are + +Equal [sum [Sally s share of some money] [Frank s share]] + [product 4.50 [dollars]] + +Equal [Sally s share] [product 2 [Frank s]] + + +The equations were insufficient to find a solution. + +Assuming that +[Frank s] is equal to [Frank s share] + +Assuming that +[Sally s share] is equal to [Sally s share of some money] + +Frank s is 1.5 dollars + +Sally s share is 3 dollars + +The problem is solved. +</PRE> + +<P>In this problem Student has found two pairs of similar variable names. +When it finds such a pair, Student adds an equation of the form + +<P><PRE>[equal <EM>variable1</EM> <EM>variable2</EM>] +</PRE> + +<P>to the previous set of equations. In both of the pairs in this +example, the variable that appears later in the problem statement is +entirely contained within the one that appears earlier. + +<P>Another point of interest in this example is that the variable <CODE>[dollars]</CODE> +is included in the list of units that may be part of the answer. The word +problem does not explicitly ask "How many dollars is Sally's share," but +because one of the sentences sets an expression equal to "4.50 dollars" +Student takes that as implicit permission to express the answer in dollars. + +<P>The only other condition under which Student will consider two variables +equal is if their names are identical except that some phrase in the one +that appears earlier is replaced with a pronoun in the one that appears +later. That is, a variable like <CODE>[the number of ice cream cones the +children eat]</CODE> will be considered equal to a later variable <CODE>[the number +of ice cream cones they eat]</CODE>. Here is a problem in which this rule is +applied: + +<P><PRE>? <U>student :guns</U> + +The problem to be solved is + +The number of soldiers the Russians have is one half of the number of +guns they have. They have 7000 guns. How many soldiers do they have? + + +With mandatory substitutions the problem is + +The number numof soldiers the Russians have is 0.5 numof the number numof +guns they have . They have 7000 guns . howm soldiers do they have ? + + +The simple sentences are + +The number numof soldiers the Russians have is 0.5 numof the number numof +guns they have . + +They have 7000 guns . + +Howm soldiers do they have ? + + +The equations to be solved are + +Equal [number of soldiers Russians have] + [product 0.5 [number of guns they have]] + +Equal [number of guns they have] 7000 + + +The equations were insufficient to find a solution. + +Assuming that +[number of soldiers they have] is equal to + [number of soldiers Russians have] + +The number of soldiers they have is 3500 + +The problem is solved. +</PRE> + +<P><H2>Using Background Knowledge</H2> + +<P>In some word problems, not all of the necessary information is contained +within the problem statement itself. The problem requires the student to +supply some piece of general knowledge about the world in order to determine +the appropriate equations. This knowledge may be about unit conversions +(one foot is 12 inches) or about relationships among physical quantities +(distance equals speed times time). Student "knows" some of this <EM> +background</EM> information and can apply it (<CODE>geteqns</CODE>) if the equations +determined by the problem statement are insufficient. + +<P><PRE>? <U>student :jet</U> + +The problem to be solved is + +The distance from New York to Los Angeles is 3000 miles. If the average +speed of a jet plane is 600 miles per hour, find the time it takes to +travel from New York to Los Angeles by jet. + + +With mandatory substitutions the problem is + +The distance from New York to Los Angeles is 3000 miles . If the average +speed numof a jet plane is 600 miles per hour , find the time it takes to +travel from New York to Los Angeles by jet . + + +The simple sentences are + +The distance from New York to Los Angeles is 3000 miles . + +The average speed numof a jet plane is 600 miles per hour . + +Find the time it takes to travel from New York to Los Angeles by jet . + + +The equations to be solved are + +Equal [distance from New York to Los Angeles] [product 3000 [miles]] + +Equal [average speed of jet plane] + [quotient [product 600 [miles]] [product 1 [hours]]] + + +The equations were insufficient to find a solution. + +Using the following known relationships + +Equal [distance] [product [speed] [time]] + +Equal [distance] [product [gas consumption] + [number of gallons of gas used]] + + +Assuming that +[speed] is equal to [average speed of jet plane] + +Assuming that +[time] is equal to [time it takes to travel + from New York to Los Angeles by jet] + +Assuming that +[distance] is equal to [distance from New York to Los Angeles] + +The time it takes to travel from New York + to Los Angeles by jet is 5 hours + +The problem is solved. +</PRE> + +<P>Student's library of known relationships is indexed according to the first +word of the name of each variable involved in the relationship. (If a +variable starts with the words <CODE>number of</CODE> it is indexed under the +following word.) The relationships, in the form of equations, are stored in +the property lists of these index words. + +<P>Property lists are also used to keep track of irregular plurals and the +corresponding singulars. Student tries to keep all units in plural form +internally, so that if a problem refers to both <CODE>1 foot</CODE> and <CODE> +2 feet</CODE> the same variable name will be used for both. (That is, the first +of these will be translated into + +<P><PRE>[product 1 [feet]] +</PRE> + +<P>in Student's internal representation. Then the opposite +translation is needed if the product of <CODE>1</CODE> and some unit appears in an +answer to be printed. + +<P>The original Student also used property lists to remember the parts of +speech of words and the precedence of operators, but because of differences +in the syntax of the Meteor pattern matcher and my Logo pattern matcher I've +found it easier to use predicate operations for that purpose. + +<P>The original Student system included a separately invoked <CODE>remember</CODE> +procedure that allowed all these kinds of global information to be entered +in the form of English sentences. You'd say + +<P><PRE>Feet is the plural of foot +</PRE> + +<P>or + +<P><PRE>Distance equals speed times time +</PRE> + +<P>and <CODE>remember</CODE> would use patterns much like those used in +understanding word problems to translate these sentences into <CODE>pprop</CODE> +instructions. Since Lisp programs, like Logo programs, can themselves be +manipulated as lists, <CODE>remember</CODE> could even accept information of a +kind that's stored in +the Student program itself, such as the wording transformations in <CODE> +idioms</CODE>, and modify the program to reflect this information. I haven't +bothered to implement that part of the Student system because it takes up +extra memory and doesn't exhibit any new techniques. + +<P>As the above example shows, it's important that Student's search for +relevant known relationships comes before the attempt to equate variables +with similar names. The general relationship that uses a variable named +simply <CODE>[distance]</CODE> doesn't help unless Student can identify it as +relevant to the variable named <CODE>[distance from New York to Los Angeles]</CODE> +in the specific problem under consideration. + +<P>Here is another example in which known relationships are used: + +<P><PRE>? <U>student :span</U> + +The problem to be solved is + +If 1 span is 9 inches, and 1 fathom is 6 feet, + how many spans is 1 fathom? + + +With mandatory substitutions the problem is + +If 1 span is 9 inches , and 1 fathom is 6 feet , howm spans is 1 fathom ? + + +The simple sentences are + +1 span is 9 inches . + +1 fathom is 6 feet . + +Howm spans is 1 fathom ? + + +The equations to be solved are + +Equal [product 1 [spans]] [product 9 [inches]] + +Equal [product 1 [fathoms]] [product 6 [feet]] + +Equal g2 [product 1 [fathoms]] + + +The equations were insufficient to find a solution. + +Using the following known relationships + +Equal [product 1 [yards]] [product 3 [feet]] + +Equal [product 1 [feet]] [product 12 [inches]] + + +1 fathom is 8 spans + +The problem is solved. +</PRE> + +<P>Besides the use of known relationships, this example illustrates two other +features of Student. One is the use of an explicitly requested unit in the +answer. Since the problem asks + +<P><PRE>How many spans is 1 fathom? +</PRE> + +<P>Student knows that the answer must be expressed in <CODE>spans</CODE>. +Had there been no explicit request for a particular unit, all the units that +appear in phrases along with a number would be eligible to appear in the +answer: <CODE>inches</CODE>, <CODE>feet</CODE>, and <CODE>fathoms</CODE>. Student might then +blithely inform us that + +<P><PRE>1 fathom is 1 fathom + +The problem is solved. +</PRE> + +<P>The other new feature demonstrated by this example is the use of a generated +symbol to represent the desired answer. In the statement of this problem, +there is no explicit variable representing the unknown. <CODE>[Fathoms]</CODE> is +a <EM>unit,</EM> not a variable for which a value could be found. The problem +asks for the value of the expression + +<P><PRE>[product 1 [fathoms]] +</PRE> + +<P>in terms of spans. Student generates a variable name (<CODE>g2</CODE>) +to represent the unknown and produces an equation + +<P><PRE>[equal g2 [product 1 [fathoms]] +</PRE> + +<P>to add to the list of equations. A generated symbol will be +needed whenever the "Find" or "What is" sentence asks for an expression +rather than a simple variable name. For example, an age problem that asks +"What is the sum of their ages" would require the use of a generated +symbol. (The original Student <EM>always</EM> used a generated symbol for +the unknowns, even if there was already a single variable in the problem +representing an unknown. It therefore had equations like + +<P><PRE>[equal g3 [marked price]] +</PRE> + +<P>in its list, declaring one variable equal to another. I chose to +check for this case and avoid the use of a generated symbol because the time +spent in the actual solution of the equations increases quadratically with +the number of equations.) + +<P><H2>Optional Substitutions</H2> + +<P>We have seen many cases in which Student replaces a phrase in the statement +of a problem with a different word or phrase that fits better with the later +stages of processing, like the substitution of <CODE>2 times</CODE> for <CODE>twice</CODE> +or a special keyword like <CODE>perless</CODE> for <CODE>percent less than</CODE>. +Student also has a few cases of <EM>optional</EM> substitutions that may or +may not be made (<CODE>tryidiom</CODE>). + +<P>There are two ways in which optional substitutions can happen. One is +exemplified by the phrase <CODE>the perimeter of the rectangle</CODE>. Student +first attempts the problem without any special processing of this phrase. +If a solution is not found, Student then replaces the phrase with <CODE>twice +the sum of the length and width of the rectangle</CODE> and processes the +resulting new problem from the beginning. Unlike the use of known +relationships or similarity of variable names, which Student handles by +adding to the already-determined equations, this optional substitution +requires the entire translation process to begin again. For example, the +word <CODE>twice</CODE> that begins the replacement phrase will be further +translated to <CODE>2 times</CODE>. + +<P>The second category of optional substitution is triggered by the phrase <CODE> +two numbers</CODE>. This phrase must always be translated to something, because +it indicates that two different variables are needed. But the precise +translation depends on the wording of the rest of the problem. Student +tries two alternative translations: <CODE>one of the numbers and the other +number</CODE> and <CODE>one number and the other number</CODE>. Here is an example in +which the necessary translation is the one Student tries second: + +<P><PRE>? <U>student :sumtwo</U> + +The problem to be solved is + +The sum of two numbers is 96, and one number is 16 larger than the other +number. Find the two numbers. + + +The problem with an idiomatic substitution is + +The sum of one of the numbers and the other number is 96 , and one +number is16 larger than the other number . Find the one of the numbers +and the other number . + + +With mandatory substitutions the problem is + +sum one numof the numbers and the other number is 96 , and one number +is 16 plus the other number . Find the one numof the numbers and the +other number . + + +The simple sentences are + +Sum one numof the numbers and the other number is 96 . + +One number is 16 plus the other number . + +Find the one numof the numbers and the other number . + + +The equations to be solved are + +Equal [sum [one of numbers] [other number]] 96 + +Equal [one number] [sum 16 [other number]] + + +The equations were insufficient to find a solution. + +The problem with an idiomatic substitution is + +The sum of one number and the other number is 96 , and one number is 16 +larger than the other number . Find the one number and the other number . + + +With mandatory substitutions the problem is + +sum one number and the other number is 96 , and one number is 16 plus the +other number . Find the one number and the other number . + + +The simple sentences are + +Sum one number and the other number is 96 . + +One number is 16 plus the other number . + +Find the one number and the other number . + + +The equations to be solved are + +Equal [sum [one number] [other number]] 96 + +Equal [one number] [sum 16 [other number]] + + +The one number is 56 + +The other number is 40 + +The problem is solved. +</PRE> + +<P>There is no essential reason why Student uses one mechanism rather than +another to deal with a particular problematic situation. The difficulties +about perimeters and about the phrase "two numbers" might have been solved +using mechanisms other than this optional substitution one. For example, +the equation + +<P><PRE>[equal [perimeter] [product 2 [sum [length] [width]]]] +</PRE> + +<P>might have been added to the library of known relationships. The +difficulty about alternate phrasings for "two numbers" could be solved by +adding + +<P><PRE>[[one of the !word:pluralp] ["one singular :word]] +</PRE> + +<P>to the list of idiomatic substitutions in <CODE>idiom</CODE>. + +<P>Not all the mechanisms are equivalent, however. The "two numbers" problem +couldn't be solved by adding equations to the library of known relationships, +because that phrase appears as part of a larger phrase like "the sum of two +numbers," and Student's understanding of the word <CODE>sum</CODE> doesn't allow +it to be part of a variable name. The word <CODE>sum</CODE> only makes sense to +Student in the context of a phrase like <CODE>the sum of <CODE><EM> +something</EM></CODE> and <CODE><EM>something else</EM></CODE></CODE>. (See procedure <CODE>tst.sum</CODE>.) + +<P><H2>If All Else Fails</H2> + +<P>Sometimes Student fails to solve a problem because the problem is beyond +either its linguistic capability or its algebraic capability. For example, +Student doesn't know how to solve quadratic equations. But sometimes a +problem that Student could solve in principle stumps it because it happens +to lack a particular piece of common knowledge. When a situation like that +arises, Student is capable of asking the user for help (<CODE>student2</CODE>). + +<P><PRE>? <U>student :ship</U> + +The problem to be solved is + +The gross weight of a ship is 20000 tons. If its net weight is 15000 +tons, what is the weight of the ships cargo? + + +With mandatory substitutions the problem is + +The gross weight numof a ship is 20000 tons . If its net weight is 15000 +tons , what is the weight numof the ships cargo ? + + +The simple sentences are + +The gross weight numof a ship is 20000 tons . + +Its net weight is 15000 tons . + +What is the weight numof the ships cargo ? + + +The equations to be solved are + +Equal [gross weight of ship] [product 20000 [tons]] + +Equal [its net weight] [product 15000 [tons]] + + +The equations were insufficient to find a solution. + +Do you know any more relationships among these variables? + +Weight of ships cargo + +Its net weight + +Tons + +Gross weight of ship + +<U>The weight of a ships cargo is the gross weight minus the net weight</U> + +Assuming that +[net weight] is equal to [its net weight] + +Assuming that +[gross weight] is equal to [gross weight of ship] + +The weight of the ships cargo is 5000 tons + +The problem is solved. +</PRE> + +<P><H2>Limitations of Pattern Matching</H2> + + +<P>Student relies on certain stereotyped forms of sentences in the problems it +solves. It's easy to make up problems that will completely bewilder it: + +<P><PRE>Suppose you have 14 jelly beans. You give 2 each to Tom, Dick, and +Harry. How many do you have left? +</PRE> + +<P>The first mistake Student makes is that it thinks the word <CODE> +and</CODE> following a comma separates two clauses; it generates simple sentences + +<P><PRE>You give 2 each to Tom , Dick . + +Harry . +</PRE> + +<P>This is quite a fundamental problem; Student's understanding of +the difference between a phrase and a clause is extremely primitive and +prone to error. Adding another pattern won't solve this one; the trouble +is that Student pays no attention to the words in between the key words like +<CODE>and</CODE>. + +<P>There are several other difficulties with this problem, some worse than +others. Student doesn't recognize the word <CODE>suppose</CODE> as having a +special function in the sentence, so it makes up a noun phrase <CODE>suppose +you</CODE> just like <CODE>the russians</CODE>. This could be fixed with an idiomatic +substitution that just ignored <CODE>suppose</CODE>. Another relatively small +problem is that the sentence starting <CODE>how many</CODE> doesn't say how many of +what; Student needs a way to understand that the relevant noun phrase is +<CODE>jelly beans</CODE> and not, for example, <CODE>Tom</CODE>. The words <CODE>give</CODE> +(representing subtraction) and <CODE>each</CODE> (representing counting a set and +then multiplying) have special mathematical meanings comparable to <CODE> +percent less</CODE>. A much more subtle problem +in knowledge representation is +that in this problem there are two different quantities that could be called +<CODE>the number of jelly beans you have</CODE>: the number you have at the +beginning of the problem and the number you have at the end. Student has a +limited understanding of this passage-of-time difficulty when it's doing an +age problem, but not in general. + +<P>How many more difficulties can you find in this problem? For how many of +them can you invent improvements to Student to get around them? + +<P>Some difficulties seem to require a "more of the same" strategy: adding +some new patterns to Student that are similar to the ones already there. +Other difficulties seem to require a more fundamental redesign. Can that +redesign be done using a pattern matcher as the central tool, or are more +powerful tools needed? How powerful <EM>is</EM> pattern matching, anyway? + +<P>Answering questions like these is the job of automata theory. From that +point of view, the answer is that it depends exactly what you mean by +"pattern matching." The pattern matcher used in Student is equivalent +to a finite-state machine. The important thing to note about the patterns +used in Student is that they only apply predicates to one word at a time, +not to groups of words. In other words, they don't use the <CODE>@</CODE> +quantifier. Here is a typical <CODE>student</CODE> pattern: + +<P><PRE>[^ what !:in [is are] #one !:dlm] +</PRE> + +<P>For the purposes of this discussion, you can ignore the fact that +the pattern matcher can set variables to remember which words matched each +part of the pattern. In comparing a pattern matcher to a finite-state +machine, the question we're asking is what categories of strings can the +pattern matcher accept. This particular pattern is equivalent to the +following machine: + +<P><CENTER><IMG SRC="pattern.gif" ALT="figure: pattern"></CENTER> + +<P>The arrow that I've labeled <EM>dlm</EM> is actually several arrows +connecting the same states, one for each symbol that the predicate <CODE>dlm</CODE> +accepts, i.e., period, question mark, and semicolon. Similarly, the arrows +labeled <EM>any</EM> are followed for any symbol at all. This machine is +nondeterministic, but you'll recall that that doesn't matter; we can turn it +into a deterministic one if necessary. + +<P>To be sure you understand the equivalence of patterns and finite-state +machines, see if you can draw a machine equivalent to this pattern: + +<P><PRE>[I see !:in [the a an] ?:numberp &:adjective !:noun #:adverb] +</PRE> + +<P>This pattern uses all the quantifiers that test words one at a +time. + +<P>If these patterns are equivalent to finite-state machines, you'd expect them +to have trouble recognizing sentences that involve <EM>embedding</EM> of +clauses within clauses, since these pose the same problem as keeping track +of balancing of parentheses. For example, a sentence like "The book that +the boy whom I saw yesterday was reading is interesting" would strain the +capabilities of a finite-state machine. (As in the case of parentheses, we +could design a FSM that could handle such sentences up to some fixed depth +of embedding, but not one that could handle arbitrarily deep embedding.) + +<P><H2>Context-Free Languages</H2> + + +<P>If we allow the use of the <CODE>@</CODE> quantifier in patterns, and if the +predicates used to test substrings of the sentences are true functions +without side effects, then the pattern matcher is equivalent to an RTN or a +production rule grammar. What makes an RTN different from a finite-state +machine is that the former can include arrows that match several symbols +against another (or the same) RTN. Equivalently, the <CODE>@</CODE> quantifier +matches several symbols against another (or the same) pattern. + +<P>A language that can be represented by an RTN is called a <EM> +context-free</EM> language. The reason for the name is that in such a +language a given string consistently matches or doesn't match a given +predicate regardless of the rest of the sentence. That's the point of what +I said just above about side effects; the output from a test predicate +can't depend on anything other than its input. Pascal is a context-free +language because + +<P><PRE>this := that +</PRE> + +<P>is always an assignment statement regardless of what other statements +might be in the program with it. + +<P>What <EM>isn't</EM> a context-free language? The classic example in automata +theory is the language consisting of the strings + +<P><PRE>abc +aabbcc +aaabbbccc +aaaabbbbcccc +</PRE> + +<P>and so on, with the requirement that the number of <CODE>a</CODE>s be +equal to the number of <CODE>b</CODE>s and also equal to the number of <CODE>c</CODE>s. +That language can't be represented as RTNs or production rules. (Try it. +Don't confuse it with the language that accepts any number of <CODE>a</CODE>s +followed by any number of <CODE>b</CODE>s and so on; even a finite-state machine +can represent that one. The equal number requirement is important.) + +<P>The classic formal system that can represent <EM>any</EM> language for which +there are precise rules is the Turing machine. Its advantage over +the RTN is precisely that it can "jump around" in its memory, looking at +one part while making decisions about another part. + +<P>There is a sharp theoretical boundary between context-free and +context-sensitive languages, but in practice the boundary is sometimes fuzzy. +Consider again the case of Pascal and that assignment statement. I said +that it's recognizably an assignment statement because it matches a +production rule like + +<P><PRE>assignment : identifier := expression +</PRE> + +<P>(along with a bunch of other rules that determine what qualifies +as an expression). But that production rule doesn't really express <EM> +all</EM> the requirements for a legal Pascal assignment statement. For +example, the identifier and the expression must be of the same type. The +actual Pascal compiler (any Pascal compiler, not just mine) includes +instructions that represent the formal grammar plus extra instructions that +represent the additional requirements. + +<P>The type agreement rule is an example of context sensitivity. The types of +the relevant identifiers were determined in <CODE>var</CODE> declarations earlier +in the program; those declarations are part of what determines whether the +given string of symbols is a legal assignment. + +<P><H2>Augmented Transition Networks</H2> + +<P>One could create a clean formal description of Pascal, type agreement rules +and all, by designing a Turing machine to accept Pascal programs. However, +Turing machines aren't easy to work with for any practical problem. It's +much easier to set up a context-free grammar for Pascal and then throw in a +few side effects to handle the context-sensitive aspects of the language. + +<P>Much the same is true of English. It's possible to set up an RTN (or a +production rule grammar) for noun phrases, for example, and another one for +verb phrases. It's tempting then to set up an RTN for a sentence like this: + +<P><CENTER><IMG SRC="sentence.gif" ALT="figure: sentence"></CENTER> + +<P>This machine captures some, but not all, of the rules of English. +It's true that a sentence requires a noun phrase (the subject) and a verb +phrase (the predicate). But there are <EM>agreement</EM> rules for person +and number (I <EM>run</EM> but he <EM>runs</EM>) analogous to the type +agreement rules of Pascal. + +<P>Some artificial intelligence researchers, understanding all this, parse +English sentences using a formal description called an <EM> +augmented transition network</EM> (ATN). An ATN is just +like an RTN except +that each transition arrow can have associated with it not only the name of +a symbol or another RTN but also some <EM>conditions</EM> that must be met in +order to follow the arrow and some <EM>actions</EM> that the program should +take if the arrow is followed. For example, we could turn the RTN just +above into an ATN by adding an action to the first arrow saying "store the +number (singular or plural) of the noun phrase in the variable <CODE> +number</CODE>" and adding a condition to the second arrow saying "the number of +the verb phrase must be equal to the variable <CODE>number</CODE>." + +<P>Subject-predicate agreement is not the only rule in English grammar best +expressed as a side effect in a transition network. Below is an ATN +for noun phrases taken from <EM>Language as a Cognitive Process, Volume 1: +Syntax</EM> by Terry Winograd (page 598). I'm not going to attempt to explain +the notation or the detailed rules here, but just to give one example, the +condition labeled "h16p" says that the transition for apostrophe-s can be +followed if the head of the phrase is an ordinary noun ("the book's") but +not if it's a pronoun ("you's"). + +<P> +<CENTER><A NAME="winograd"><IMG SRC="atn.gif" ALT="figure: atn"></A></CENTER> + +<P>The ATN is equivalent in power to a Turing machine; there is no known +mechanism that is more flexible in carrying out algorithms. The flexibility +has a cost, though. The time required to parse a string with an ATN is not +bounded by a polynomial function. (Remember, the time for an RTN is +O(<EM>n</EM><SUP><SMALL>3</SMALL></SUP>).) It can easily be exponential, O(2<SUP><SMALL><EM>n</EM></SMALL></SUP>). One reason is that a +context-sensitive procedure can't be subject to memoization. If two +invocations of the same procedure with the same inputs can give different +results because of side effects, it does no good to remember what result we +got the last time. Turning an ATN into a practical program is often +possible, but not a trivial task. + +<P>In thinking about ATNs we've brought together most of the topics in this +book: formal systems, algorithms, language parsing, and artificial +intelligence. Perhaps that's a good place to stop. + +<P> +<TABLE width="100%"><TR><TD><A HREF="../v3-toc2.html">(back to Table of Contents)</A> +<TD align="right"><A HREF="../v3ch5/v3ch5.html"><STRONG>BACK</STRONG></A> +chapter thread <A HREF="https://people.eecs.berkeley.edu/~bh/v3ch7/v3ch7.html"><STRONG>NEXT</STRONG></A> +</TABLE> + +<H2>Program Listing</H2> + +<P><PRE> +to student :prob +say [The problem to be solved is] :prob +make "prob map.se [depunct ?] :prob +localmake "orgprob :prob +student1 :prob ~ + [[[the perimeter of ! rectangle] + [twice the sum of the length and width of the rectangle]] + [[two numbers] [one of the numbers and the other number]] + [[two numbers] [one number and the other number]]] +end + +to student1 :prob :idioms +local [simsen shelf aunits units wanted ans var lasteqn + ref eqt1 beg end idiom reply] +make "prob idioms :prob +if match [^ two numbers #] :prob ~ + [make "idiom find [match (sentence "^beg first ? "#end) :orgprob] :idioms ~ + tryidiom stop] +while [match [^beg the the #end] :prob] [make "prob (sentence :beg "the :end)] +say [With mandatory substitutions the problem is] :prob +ifelse match [# @:in [[as old as] [age] [years old]] #] :prob ~ + [ageprob] [make "simsen bracket :prob] +lsay [The simple sentences are] :simsen +foreach [aunits wanted ans var lasteqn ref units] [make ? []] +make "shelf filter [not emptyp ?] map.se [senform ?] :simsen +lsay [The equations to be solved are] :shelf +make "units remdup :units +if trysolve :shelf :wanted :units :aunits [print [The problem is solved.] stop] +make "eqt1 remdup geteqns :var +if not emptyp :eqt1 [lsay [Using the following known relationships] :eqt1] +student2 :eqt1 +end + +to student2 :eqt1 +make "var remdup sentence (map.se [varterms ?] :eqt1) :var +make "eqt1 sentence :eqt1 vartest :var +if not emptyp :eqt1 ~ + [if trysolve (sentence :shelf :eqt1) :wanted :units :aunits + [print [The problem is solved.] stop]] +make "idiom find [match (sentence "^beg first ? "#end) :orgprob] :idioms +if not emptyp :idiom [tryidiom stop] +lsay [Do you know any more relationships among these variables?] :var +make "reply readlist +if equalp :reply [yes] [print [Tell me.] make "reply readlist] +if equalp :reply [no] [print [] print [I can't solve this problem.] stop] +make "reply map.se [depunct ?] :reply +if dlm last :reply [make "reply butlast :reply] +if not match [^beg is #end] :reply [print [I don't understand that.] stop] +make "shelf sentence :shelf :eqt1 +student2 (list (list "equal opform :beg opform :end)) +end + +;; Mandatory substitutions + +to depunct :word +if emptyp :word [output []] +if equalp first :word "$ [output sentence "$ depunct butfirst :word] +if equalp last :word "% [output sentence depunct butlast :word "percent] +if memberp last :word [. ? |;| ,] [output sentence depunct butlast :word last :word] +if emptyp butfirst :word [output :word] +if equalp last2 :word "'s [output sentence depunct butlast butlast :word "s] +output :word +end + +to last2 :word +output word (last butlast :word) (last :word) +end + +to idioms :sent +local "number +output changes :sent ~ + [[[the sum of] ["sum]] [[square of] ["square]] [[of] ["numof]] + [[how old] ["what]] [[is equal to] ["is]] + [[years younger than] [[less than]]] [[years older than] ["plus]] + [[percent less than] ["perless]] [[less than] ["lessthan]] + [[these] ["the]] [[more than] ["plus]] + [[first two numbers] [[the first number and the second number]]] + [[three numbers] + [[the first number and the second number and the third number]]] + [[one half] [0.5]] [[twice] [[2 times]]] + [[$ !number] [sentence :number "dollars]] [[consecutive to] [[1 plus]]] + [[larger than] ["plus]] [[per cent] ["percent]] [[how many] ["howm]] + [[is multiplied by] ["ismulby]] [[is divided by] ["isdivby]] + [[multiplied by] ["times]] [[divided by] ["divby]]] +end + +to changes :sent :list +localmake "keywords map.se [findkey first ?] :list +output changes1 :sent :list :keywords +end + +to findkey :pattern +if equalp first :pattern "!:in [output first butfirst :pattern] +if equalp first :pattern "?:in [output sentence (item 2 :pattern) (item 3 :pattern)] +output first :pattern +end + +to changes1 :sent :list :keywords +if emptyp :sent [output []] +if memberp first :sent :keywords [output changes2 :sent :list :keywords] +output fput first :sent changes1 butfirst :sent :list :keywords +end + +to changes2 :sent :list :keywords +changes3 :list :list +output fput first :sent changes1 butfirst :sent :list :keywords +end + +to changes3 :biglist :nowlist +if emptyp :nowlist [stop] +if changeone first :nowlist [changes3 :biglist :biglist stop] +changes3 :biglist butfirst :nowlist +end + +to changeone :change +local "end +if not match (sentence first :change [#end]) :sent [output "false] +make "sent run (sentence "sentence last :change ":end) +output "true +end + +;; Division into simple sentences + +to bracket :prob +output bkt1 finddelim :prob +end + +to finddelim :sent +output finddelim1 :sent [] [] +end + +to finddelim1 :in :out :simples +if emptyp :in ~ + [ifelse emptyp :out [output :simples] [output lput (sentence :out ".) :simples]] +if dlm first :in ~ + [output finddelim1 (nocap butfirst :in) [] + (lput (sentence :out first :in) :simples)] +output finddelim1 (butfirst :in) (sentence :out first :in) :simples +end + +to nocap :words +if emptyp :words [output []] +if personp first :words [output :words] +output sentence (lowercase first :words) butfirst :words +end + +to bkt1 :problist +local [first word rest] +if emptyp :problist [output []] +if not memberp ", first :problist ~ + [output fput first :problist bkt1 butfirst :problist] +if match [if ^first , !word:qword #rest] first :problist ~ + [output bkt1 fput (sentence :first ".) + fput (sentence :word :rest) butfirst :problist] +if match [^first , and #rest] first :problist ~ + [output fput (sentence :first ".) (bkt1 fput :rest butfirst :problist)] +output fput first :problist bkt1 butfirst :problist +end + +;; Age problems + +to ageprob +local [beg end sym who num subj ages] +while [match [^beg as old as #end] :prob] [make "prob sentence :beg :end] +while [match [^beg years old #end] :prob] [make "prob sentence :beg :end] +while [match [^beg will be when #end] :prob] ~ + [make "sym gensym + make "prob (sentence :beg "in :sym [years . in] :sym "years :end)] +while [match [^beg was when #end] :prob] ~ + [make "sym gensym + make "prob (sentence :beg :sym [years ago .] :sym [years ago] :end)] +while [match [^beg !who:personp will be in !num years #end] :prob] ~ + [make "prob (sentence :beg :who [s age in] :num "years #end)] +while [match [^beg was #end] :prob] [make "prob (sentence :beg "is :end)] +while [match [^beg will be #end] :prob] [make "prob (sentence :beg "is :end)] +while [match [^beg !who:personp is now #end] :prob] ~ + [make "prob (sentence :beg :who [s age now] :end)] +while [match [^beg !num years from now #end] :prob] ~ + [make "prob (sentence :beg "in :num "years :end)] +make "prob ageify :prob +ifelse match [^ !who:personp ^end s age #] :prob ~ + [make "subj sentence :who :end] [make "subj "someone] +make "prob agepron :prob +make "end :prob +make "ages [] +while [match [^ !who:personp ^beg age #end] :end] ~ + [push "ages (sentence "and :who :beg "age)] +make "ages butfirst reduce "sentence remdup :ages +while [match [^beg their ages #end] :prob] [make "prob (sentence :beg :ages :end)] +make "simsen map [agesen ?] bracket :prob +end + +to ageify :sent +if emptyp :sent [output []] +if not personp first :sent [output fput first :sent ageify butfirst :sent] +catch "error [if equalp first butfirst :sent "s + [output fput first :sent ageify butfirst :sent]] +output (sentence first :sent [s age] ageify butfirst :sent) +end + +to agepron :sent +if emptyp :sent [output []] +if not pronoun first :sent [output fput first :sent agepron butfirst :sent] +if posspro first :sent [output (sentence :subj "s agepron butfirst :sent)] +output (sentence :subj [s age] agepron butfirst :sent) +end + +to agesen :sent +local [when rest num] +make "when [] +if match [in !num years #rest] :sent ~ + [make "when sentence "pluss :num make "sent :rest] +if match [!num years ago #rest] :sent ~ + [make "when sentence "minuss :num make "sent :rest] +output agewhen :sent +end + +to agewhen :sent +if emptyp :sent [output []] +if not equalp first :sent "age [output fput first :sent agewhen butfirst :sent] +if match [in !num years #rest] butfirst :sent ~ + [output (sentence [age pluss] :num agewhen :rest)] +if match [!num years ago #rest] butfirst :sent ~ + [output (sentence [age minuss] :num agewhen :rest)] +if equalp "now first butfirst :sent ~ + [output sentence "age agewhen butfirst butfirst :sent] +output (sentence "age :when agewhen butfirst :sent) +end + +;; Translation from sentences into equations + +to senform :sent +make "lasteqn senform1 :sent +output :lasteqn +end + +to senform1 :sent +local [one two verb1 verb2 stuff1 stuff2 factor] +if emptyp :sent [output []] +if match [^ what are ^one and ^two !:dlm] :sent ~ + [output fput (qset :one) (senform (sentence [what are] :two "?))] +if match [^ what !:in [is are] #one !:dlm] :sent ~ + [output (list qset :one)] +if match [^ howm !one is #two !:dlm] :sent ~ + [push "aunits (list :one) output (list qset :two)] +if match [^ howm ^one do ^two have !:dlm] :sent ~ + [output (list qset (sentence [the number of] :one :two "have))] +if match [^ howm ^one does ^two have !:dlm] :sent ~ + [output (list qset (sentence [the number of] :one :two "has))] +if match [^ find ^one and #two] :sent ~ + [output fput (qset :one) (senform sentence "find :two)] +if match [^ find #one !:dlm] :sent [output (list qset :one)] +make "sent filter [not article ?] :sent +if match [^one ismulby #two] :sent ~ + [push "ref (list "product opform :one opform :two) output []] +if match [^one isdivby #two] :sent ~ + [push "ref (list "quotient opform :one opform :two) output []] +if match [^one is increased by #two] :sent ~ + [push "ref (list "sum opform :one opform :two) output []] +if match [^one is #two] :sent ~ + [output (list (list "equal opform :one opform :two))] +if match [^one !verb1:verb ^factor as many ^stuff1 as + ^two !verb2:verb ^stuff2 !:dlm] ~ + :sent ~ + [if emptyp :stuff2 [make "stuff2 :stuff1] + output (list (list "equal ~ + opform (sentence [the number of] :stuff1 :one :verb1) ~ + opform (sentence :factor [the number of] :stuff2 :two :verb2)))] +if match [^one !verb1:verb !factor:numberp #stuff1 !:dlm] :sent ~ + [output (list (list "equal ~ + opform (sentence [the number of] :stuff1 :one :verb1) ~ + opform (list :factor)))] +say [This sentence form is not recognized:] :sent +throw "error +end + +to qset :sent +localmake "opform opform filter [not article ?] :sent +if not operatorp first :opform ~ + [queue "wanted :opform queue "ans list :opform oprem :sent output []] +localmake "gensym gensym +queue "wanted :gensym +queue "ans list :gensym oprem :sent +output (list "equal :gensym opform (filter [not article ?] :sent)) +end + +to oprem :sent +output map [ifelse equalp ? "numof ["of] [?]] :sent +end + +to opform :expr +local [left right op] +if match [^left !op:op2 #right] :expr [output optest :op :left :right] +if match [^left !op:op1 #right] :expr [output optest :op :left :right] +if match [^left !op:op0 #right] :expr [output optest :op :left :right] +if match [#left !:dlm] :expr [make "expr :left] +output nmtest filter [not article ?] :expr +end + +to optest :op :left :right +output run (list (word "tst. :op) :left :right) +end + +to tst.numof :left :right +if numberp last :left [output (list "product opform :left opform :right)] +output opform (sentence :left "of :right) +end + +to tst.divby :left :right +output (list "quotient opform :left opform :right) +end + +to tst.tothepower :left :right +output (list "expt opform :left opform :right) +end + +to expt :num :pow +if :pow < 1 [output 1] +output :num * expt :num :pow - 1 +end + +to tst.per :left :right +output (list "quotient ~ + opform :left ~ + opform (ifelse numberp first :right [:right] [fput 1 :right])) +end + +to tst.lessthan :left :right +output opdiff opform :right opform :left +end + +to opdiff :left :right +output (list "sum :left (list "minus :right)) +end + +to tst.minus :left :right +if emptyp :left [output list "minus opform :right] +output opdiff opform :left opform :right +end + +to tst.minuss :left :right +output tst.minus :left :right +end + +to tst.sum :left :right +local [one two three] +if match [^one and ^two and #three] :right ~ + [output (list "sum opform :one opform (sentence "sum :two "and :three))] +if match [^one and #two] :right ~ + [output (list "sum opform :one opform :two)] +say [sum used wrong:] :right +throw "error +end + +to tst.squared :left :right +output list "square opform :left +end + +to tst.difference :left :right +local [one two] +if match [between ^one and #two] :right [output opdiff opform :one opform :two] +say [Incorrect use of difference:] :right +throw "error +end + +to tst.plus :left :right +output (list "sum opform :left opform :right) +end + +to tst.pluss :left :right +output tst.plus :left :right +end + +to square :x +output :x * :x +end + +to tst.square :left :right +output list "square opform :right +end + +to tst.percent :left :right +if not numberp last :left ~ + [say [Incorrect use of percent:] :left throw "error] +output opform (sentence butlast :left ((last :left) / 100) :right) +end + +to tst.perless :left :right +if not numberp last :left ~ + [say [Incorrect use of percent:] :left throw "error] +output (list "product ~ + (opform sentence butlast :left ((100 - (last :left)) / 100)) ~ + opform :right) +end + +to tst.times :left :right +if emptyp :left [say [Incorrect use of times:] :right throw "error] +output (list "product opform :left opform :right) +end + +to nmtest :expr +if match [& !:numberp #] :expr [say [argument error:] :expr throw "error] +if and (equalp first :expr 1) (1 < count :expr) ~ + [make "expr (sentence 1 plural (first butfirst :expr) (butfirst butfirst :expr))] +if and (numberp first :expr) (1 < count :expr) ~ + [push "units (list first butfirst :expr) ~ + output (list "product (first :expr) (opform butfirst :expr))] +if numberp first :expr [output first :expr] +if memberp "this :expr [output this :expr] +if not memberp :expr :var [push "var :expr] +output :expr +end + +to this :expr +if not emptyp :ref [output pop "ref] +if not emptyp :lasteqn [output first butfirst last :lasteqn] +if equalp first :expr "this [make "expr butfirst :expr] +push "var :expr +output :expr +end + +;; Solving the equations + +to trysolve :shelf :wanted :units :aunits +local "solution +make "solution solve :wanted :shelf (ifelse emptyp :aunits [:units] [:aunits]) +output pranswers :ans :solution +end + +to solve :wanted :eqt :terms +output solve.reduce solver :wanted :terms [] [] "insufficient +end + +to solve.reduce :soln +if emptyp :soln [output []] +if wordp :soln [output :soln] +if emptyp butfirst :soln [output :soln] +local "part +make "part solve.reduce butfirst :soln +output fput (list (first first :soln) (subord last first :soln :part)) :part +end + +to solver :wanted :terms :alis :failed :err +local [one result restwant] +if emptyp :wanted [output :err] +make "one solve1 (first :wanted) ~ + (sentence butfirst :wanted :failed :terms) ~ + :alis :eqt [] "insufficient +if wordp :one ~ + [output solver (butfirst :wanted) :terms :alis (fput first :wanted :failed) :one] +make "restwant (sentence :failed butfirst :wanted) +if emptyp :restwant [output :one] +make "result solver :restwant :terms :one [] "insufficient +if listp :result [output :result] +output solver (butfirst :wanted) :terms :alis (fput first :wanted :failed) :one +end + +to solve1 :x :terms :alis :eqns :failed :err +local [thiseq vars extras xterms others result] +if emptyp :eqns [output :err] +make "thiseq subord (first :eqns) :alis +make "vars varterms :thiseq +if not memberp :x :vars ~ + [output solve1 :x :terms :alis (butfirst :eqns) (fput first :eqns :failed) :err] +make "xterms fput :x :terms +make "extras setminus :vars :xterms +make "eqt remove (first :eqns) :eqt +if not emptyp :extras ~ + [make "others solver :extras :xterms :alis [] "insufficient + ifelse wordp :others + [make "eqt sentence :failed :eqns + output solve1 :x :terms :alis (butfirst :eqns) + (fput first :eqns :failed) :others] + [make "alis :others + make "thiseq subord (first :eqns) :alis]] +make "result solveq :x :thiseq +if listp :result [output lput :result :alis] +make "eqt sentence :failed :eqns +output solve1 :x :terms :alis (butfirst :eqns) (fput first :eqns :failed) :result +end + +to solveq :var :eqn +local [left right] +make "left first butfirst :eqn +ifelse occvar :var :left ~ + [make "right last :eqn] [make "right :left make "left last :eqn] +output solveq1 :left :right "true +end + +to solveq1 :left :right :bothtest +if :bothtest [if occvar :var :right [output solveqboth :left :right]] +if equalp :left :var [output list :var :right] +if wordp :left [output "unsolvable] +local "oper +make "oper first :left +if memberp :oper [sum product minus quotient] [output run (list word "solveq. :oper)] +output "unsolvable +end + +to solveqboth :left :right +if not equalp first :right "sum [output solveq1 (subterm :left :right) 0 "false] +output solveq.rplus :left butfirst :right [] +end + +to solveq.rplus :left :right :newright +if emptyp :right [output solveq1 :left (simone "sum :newright) "false] +if occvar :var first :right ~ + [output solveq.rplus (subterm :left first :right) butfirst :right :newright] +output solveq.rplus :left butfirst :right (fput first :right :newright) +end + +to solveq.sum +if emptyp butfirst butfirst :left [output solveq1 first butfirst :left :right "true] +output solveq.sum1 butfirst :left :right [] +end + +to solveq.sum1 :left :right :newleft +if emptyp :left [output solveq.sum2] +if occvar :var first :left ~ + [output solveq.sum1 butfirst :left :right fput first :left :newleft] +output solveq.sum1 butfirst :left (subterm :right first :left) :newleft +end + +to solveq.sum2 +if emptyp butfirst :newleft [output solveq1 first :newleft :right "true] +localmake "factor factor :newleft :var +if equalp first :factor "unknown [output "unsolvable] +if equalp last :factor 0 [output "unsolvable] +output solveq1 first :factor (divterm :right last :factor) "true +end + +to solveq.minus +output solveq1 (first butfirst :left) (minusin :right) "false +end + +to solveq.product +output solveq.product1 :left :right +end + +to solveq.product1 :left :right +if emptyp butfirst butfirst :left [output solveq1 (first butfirst :left) :right "true] +if not occvar :var first butfirst :left ~ + [output solveq.product1 (fput "product butfirst butfirst :left) + (divterm :right first butfirst :left)] +localmake "rest simone "product butfirst butfirst :left +if occvar :var :rest [output "unsolvable] +output solveq1 (first butfirst :left) (divterm :right :rest) "false +end + +to solveq.quotient +if occvar :var first butfirst :left ~ + [output solveq1 (first butfirst :left) (simtimes list :right last :left) "true] +output solveq1 (simtimes list :right last :left) (first butfirst :left) "true +end + +to denom :fract :addends +make "addends simplus :addends +localmake "den last :fract +if not equalp first :addends "quotient ~ + [output simdiv list (simone "sum + (remop "sum list (distribtimes (list :addends) :den) + first butfirst :fract)) + :den] +if equalp :den last :addends ~ + [output simdiv (simplus list (first butfirst :fract) (first butfirst :addends)) + :den] +localmake "lowterms simdiv list :den last :addends +output simdiv list (simplus (simtimes list first butfirst :fract last :lowterms) + (simtimes list first butfirst :addends + first butfirst :lowterms)) ~ + (simtimes list first butfirst :lowterms last :addends) +end + +to distribtimes :trms :multiplier +output simplus map [simtimes (list ? :multiplier)] :trms +end + +to distribx :expr +local [oper args] +if emptyp :expr [output :expr] +make "oper first :expr +if not operatorp :oper [output :expr] +make "args map [distribx ?] butfirst :expr +if reduce "and map [numberp ?] :args [output run (sentence [(] :oper :args [)])] +if equalp :oper "sum [output simplus :args] +if equalp :oper "minus [output minusin first :args] +if equalp :oper "product [output simtimes :args] +if equalp :oper "quotient [output simdiv :args] +output fput :oper :args +end + +to divterm :dividend :divisor +if equalp :dividend 0 [output 0] +output simdiv list :dividend :divisor +end + +to factor :exprs :var +local "trms +make "trms map [factor1 :var ?] :exprs +if memberp "unknown :trms [output fput "unknown :exprs] +output list :var simplus :trms +end + +to factor1 :var :expr +localmake "negvar minusin :var +if equalp :var :expr [output 1] +if equalp :negvar :expr [output -1] +if emptyp :expr [output "unknown] +if equalp first :expr "product [output factor2 butfirst :expr] +if not equalp first :expr "quotient [output "unknown] +localmake "dividend first butfirst :expr +if equalp :var :dividend [output (list "quotient 1 last :expr)] +if not equalp first :dividend "product [output "unknown] +localmake "result factor2 butfirst :dividend +if equalp :result "unknown [output "unknown] +output (list "quotient :result last :expr) +end + +to factor2 :trms +if memberp :var :trms [output simone "product (remove :var :trms)] +if memberp :negvar :trms [output minusin simone "product (remove :negvar :trms)] +output "unknown +end + +to maybeadd :num :rest +if equalp :num 0 [output :rest] +output fput :num :rest +end + +to maybemul :num :rest +if equalp :num 1 [output :rest] +output fput :num :rest +end + +to minusin :expr +if emptyp :expr [output -1] +if equalp first :expr "sum [output fput "sum map [minusin ?] butfirst :expr] +if equalp first :expr "minus [output last :expr] +if memberp first :expr [product quotient] ~ + [output fput first :expr + (fput (minusin first butfirst :expr) butfirst butfirst :expr)] +if numberp :expr [output minus :expr] +output list "minus :expr +end + +to occvar :var :expr +if emptyp :expr [output "false] +if wordp :expr [output equalp :var :expr] +if operatorp first :expr [output not emptyp find [occvar :var ?] butfirst :expr] +output equalp :var :expr +end + +to remfactor :num :den +foreach butfirst :num [remfactor1 ?] +output (list "quotient (simone "product butfirst :num) (simone "product butfirst :den)) +end + +to remfactor1 :expr +local "neg +if memberp :expr :den ~ + [make "num remove :expr :num make "den remove :expr :den stop] +make "neg minusin :expr +if not memberp :neg :den [stop] +make "num remove :expr :num +make "den minusin remove :neg :den +end + +to remop :oper :exprs +output map.se [ifelse equalp first ? :oper [butfirst ?] [(list ?)]] :exprs +end + +to simdiv :list +local [num den numop denop] +make "num first :list +make "den last :list +if equalp :num :den [output 1] +if numberp :den [output simtimes (list (quotient 1 :den) :num)] +make "numop first :num +make "denop first :den +if equalp :numop "quotient ~ + [output simdiv list (first butfirst :num) (simtimes list last :num :den)] +if equalp :denop "quotient ~ + [output simdiv list (simtimes list :num last :den) (first butfirst :den)] +if and equalp :numop "product equalp :denop "product [output remfactor :num :den] +if and equalp :numop "product memberp :den :num [output remove :den :num] +output fput "quotient :list +end + +to simone :oper :trms +if emptyp :trms [output ifelse equalp :oper "product [1] [0]] +if emptyp butfirst :trms [output first :trms] +output fput :oper :trms +end + +to simplus :exprs +make "exprs remop "sum :exprs +localmake "factor [unknown] +catch "simplus ~ + [foreach :terms ~ + [make "factor (factor :exprs ?) ~ + if not equalp first :factor "unknown [throw "simplus]]] +if not equalp first :factor "unknown [output fput "product remop "product :factor] +localmake "nums 0 +localmake "nonnums [] +localmake "quick [] +catch "simplus [simplus1 :exprs] +if not emptyp :quick [output :quick] +if not equalp :nums 0 [push "nonnums :nums] +output simone "sum :nonnums +end + +to simplus1 :exprs +if emptyp :exprs [stop] +simplus2 first :exprs +simplus1 butfirst :exprs +end + +to simplus2 :pos +local "neg +make "neg minusin :pos +if numberp :pos [make "nums sum :pos :nums stop] +if memberp :neg butfirst :exprs [make "exprs remove :neg :exprs stop] +if equalp first :pos "quotient ~ + [make "quick (denom :pos (maybeadd :nums sentence :nonnums butfirst :exprs)) ~ + throw "simplus] +push "nonnums :pos +end + +to simtimes :exprs +local [nums nonnums quick] +make "nums 1 +make "nonnums [] +make "quick [] +catch "simtimes [foreach remop "product :exprs [simtimes1 ?]] +if not emptyp :quick [output :quick] +if equalp :nums 0 [output 0] +if not equalp :nums 1 [push "nonnums :nums] +output simone "product :nonnums +end + +to simtimes1 :expr +if equalp :expr 0 [make "nums 0 throw "simtimes] +if numberp :expr [make "nums product :expr :nums stop] +if equalp first :expr "sum ~ + [make "quick distribtimes (butfirst :expr) + (simone "product maybemul :nums sentence :nonnums ?rest) + throw "simtimes] +if equalp first :expr "quotient ~ + [make "quick + simdiv (list (simtimes (list (first butfirst :expr) + (simone "product + maybemul :nums + sentence :nonnums ?rest))) + (last :expr)) + throw "simtimes] +push "nonnums :expr +end + +to subord :expr :alist +output distribx subord1 :expr :alist +end + +to subord1 :expr :alist +if emptyp :alist [output :expr] +output subord (substop (last first :alist) (first first :alist) :expr) ~ + (butfirst :alist) +end + +to substop :val :var :expr +if emptyp :expr [output []] +if equalp :expr :var [output :val] +if not operatorp first :expr [output :expr] +output fput first :expr map [substop :val :var ?] butfirst :expr +end + +to subterm :minuend :subtrahend +if equalp :minuend 0 [output minusin :subtrahend] +if equalp :minuend :subtrahend [output 0] +output simplus (list :minuend minusin :subtrahend) +end + +to varterms :expr +if emptyp :expr [output []] +if numberp :expr [output []] +if wordp :expr [output (list :expr)] +if operatorp first :expr [output map.se [varterms ?] butfirst :expr] +output (list :expr) +end + +;; Printing the solutions + +to pranswers :ans :solution +print [] +if equalp :solution "unsolvable ~ + [print [Unable to solve this set of equations.] output "false] +if equalp :solution "insufficient ~ + [print [The equations were insufficient to find a solution.] output "false] +localmake "gotall "true +foreach :ans [if prans ? :solution [make "gotall "false]] +if not :gotall [print [] print [Unable to solve this set of equations.]] +output :gotall +end + +to prans :ans :solution +localmake "result find [equalp first ? first :ans] :solution +if emptyp :result [output "true] +print (sentence cap last :ans "is unitstring last :result) +print [] +output "false +end + +to unitstring :expr +if numberp :expr [output roundoff :expr] +if equalp first :expr "product ~ + [output sentence (unitstring first butfirst :expr) + (reduce "sentence butfirst butfirst :expr)] +if (and (listp :expr) + (not numberp first :expr) + (not operatorp first :expr)) ~ + [output (sentence 1 (singular first :expr) (butfirst :expr))] +output :expr +end + +to roundoff :num +if (abs (:num - round :num)) < 0.0001 [output round :num] +output :num +end + +to abs :num +output ifelse (:num < 0) [-:num] [:num] +end + +;; Using known relationships + +to geteqns :vars +output map.se [gprop varkey ? "eqns] :vars +end + +to varkey :var +local "word +if match [number of !word #] :var [output :word] +output first :var +end + +;; Assuming equality of similar variables + +to vartest :vars +if emptyp :vars [output []] +local [var beg end] +make "var first :vars +output (sentence (ifelse match [^beg !:pronoun #end] :var + [vartest1 :var (sentence :beg "& :end) butfirst :vars] + [[]]) + (vartest1 :var (sentence "# :var "#) butfirst :vars) + (vartest butfirst :vars)) +end + +to vartest1 :target :pat :vars +output map [varequal :target ?] filter [match :pat ?] :vars +end + +to varequal :target :var +print [] +print [Assuming that] +print (sentence (list :target) [is equal to] (list :var)) +output (list "equal :target :var) +end + +;; Optional substitutions + +to tryidiom +make "prob (sentence :beg last :idiom :end) +while [match (sentence "^beg first :idiom "#end) :prob] ~ + [make "prob (sentence :beg last :idiom :end)] +say [The problem with an idiomatic substitution is] :prob +student1 :prob (remove :idiom :idioms) +end + +;; Utility procedures + +to qword :word +output memberp :word [find what howm how] +end + +to dlm :word +output memberp :word [. ? |;|] +end + +to article :word +output memberp :word [a an the] +end + +to verb :word +output memberp :word [have has get gets weigh weighs] +end + +to personp :word +output memberp :word [Mary Ann Bill Tom Sally Frank father uncle] +end + +to pronoun :word +output memberp :word [he she it him her they them his her its] +end + +to posspro :word +output memberp :word [his her its] +end + +to op0 :word +output memberp :word [pluss minuss squared tothepower per sum difference numof] +end + +to op1 :word +output memberp :word [times divby square] +end + +to op2 :word +output memberp :word [plus minus lessthan percent perless] +end + +to operatorp :word +output memberp :word [sum minus product quotient expt square equal] +end + +to plural :word +localmake "plural gprop :word "plural +if not emptyp :plural [output :plural] +if not emptyp gprop :word "sing [output :word] +if equalp last :word "s [output :word] +output word :word "s +end + +to singular :word +localmake "sing gprop :word "sing +if not emptyp :sing [output :sing] +if not emptyp gprop :word "plural [output :word] +if equalp last :word "s [output butlast :word] +output :word +end + +to setminus :big :little +output filter [not memberp ? :little] :big +end + +to say :herald :text +print [] +print :herald +print [] +print :text +print [] +end + +to lsay :herald :text +print [] +print :herald +print [] +foreach :text [print cap ? print []] +end + +to cap :sent +if emptyp :sent [output []] +output sentence (word uppercase first first :sent butfirst first :sent) ~ + butfirst :sent +end + +;; The pattern matcher + +to match :pat :sen +if prematch :pat :sen [output rmatch :pat :sen] +output "false +end + +to prematch :pat :sen +if emptyp :pat [output "true] +if listp first :pat [output prematch butfirst :pat :sen] +if memberp first first :pat [! @ # ^ & ?] [output prematch butfirst :pat :sen] +if emptyp :sen [output "false] +localmake "rest member first :pat :sen +if not emptyp :rest [output prematch butfirst :pat :rest] +output "false +end + +to rmatch :pat :sen +local [special.var special.pred special.buffer in.list] +if or wordp :pat wordp :sen [output "false] +if emptyp :pat [output emptyp :sen] +if listp first :pat [output special fput "!: :pat :sen] +if memberp first first :pat [? # ! & @ ^] [output special :pat :sen] +if emptyp :sen [output "false] +if equalp first :pat first :sen [output rmatch butfirst :pat butfirst :sen] +output "false +end + +to special :pat :sen +set.special parse.special butfirst first :pat " +output run word "match first first :pat +end + +to parse.special :word :var +if emptyp :word [output list :var "always] +if equalp first :word ": [output list :var butfirst :word] +output parse.special butfirst :word word :var first :word +end + +to set.special :list +make "special.var first :list +make "special.pred last :list +if emptyp :special.var [make "special.var "special.buffer] +if memberp :special.pred [in anyof] [set.in] +if not emptyp :special.pred [stop] +make "special.pred first butfirst :pat +make "pat fput first :pat butfirst butfirst :pat +end + +to set.in +make "in.list first butfirst :pat +make "pat fput first :pat butfirst butfirst :pat +end + +to match! +if emptyp :sen [output "false] +if not try.pred [output "false] +make :special.var first :sen +output rmatch butfirst :pat butfirst :sen +end + +to match? +make :special.var [] +if emptyp :sen [output rmatch butfirst :pat :sen] +if not try.pred [output rmatch butfirst :pat :sen] +make :special.var first :sen +if rmatch butfirst :pat butfirst :sen [output "true] +make :special.var [] +output rmatch butfirst :pat :sen +end + +to match# +make :special.var [] +output #test #gather :sen +end + +to #gather :sen +if emptyp :sen [output :sen] +if not try.pred [output :sen] +make :special.var lput first :sen thing :special.var +output #gather butfirst :sen +end + +to #test :sen +if rmatch butfirst :pat :sen [output "true] +if emptyp thing :special.var [output "false] +output #test2 fput last thing :special.var :sen +end + +to #test2 :sen +make :special.var butlast thing :special.var +output #test :sen +end + +to match& +output &test match# +end + +to &test :tf +if emptyp thing :special.var [output "false] +output :tf +end + +to match^ +make :special.var [] +output ^test :sen +end + +to ^test :sen +if rmatch butfirst :pat :sen [output "true] +if emptyp :sen [output "false] +if not try.pred [output "false] +make :special.var lput first :sen thing :special.var +output ^test butfirst :sen +end + +to match@ +make :special.var :sen +output @test [] +end + +to @test :sen +if @try.pred [if rmatch butfirst :pat :sen [output "true]] +if emptyp thing :special.var [output "false] +output @test2 fput last thing :special.var :sen +end + +to @test2 :sen +make :special.var butlast thing :special.var +output @test :sen +end + +to try.pred +if listp :special.pred [output rmatch :special.pred first :sen] +output run list :special.pred quoted first :sen +end + +to quoted :thing +if listp :thing [output :thing] +output word "" :thing +end + +to @try.pred +if listp :special.pred [output rmatch :special.pred thing :special.var] +output run list :special.pred thing :special.var +end + +to always :x +output "true +end + +to in :word +output memberp :word :in.list +end + +to anyof :sen +output anyof1 :sen :in.list +end + +to anyof1 :sen :pats +if emptyp :pats [output "false] +if rmatch first :pats :sen [output "true] +output anyof1 :sen butfirst :pats +end + +;; Sample word problems + +make "ann [Mary is twice as old as Ann was when Mary was as old as Ann is now. + If Mary is 24 years old, how old is Ann?] +make "guns [The number of soldiers the Russians have is + one half of the number of guns they have. They have 7000 guns. + How many soldiers do they have?] +make "jet [The distance from New York to Los Angeles is 3000 miles. + If the average speed of a jet plane is 600 miles per hour, + find the time it takes to travel from New York to Los Angeles by jet.] +make "nums [A number is multiplied by 6 . This product is increased by 44 . + This result is 68 . Find the number.] +make "radio [The price of a radio is $69.70. + If this price is 15 percent less than the marked price, find the marked price.] +make "sally [The sum of Sally's share of some money and Frank's share is $4.50. + Sally's share is twice Frank's. Find Frank's and Sally's share.] +make "ship [The gross weight of a ship is 20000 tons. + If its net weight is 15000 tons, what is the weight of the ships cargo?] +make "span [If 1 span is 9 inches, and 1 fathom is 6 feet, + how many spans is 1 fathom?] +make "sumtwo [The sum of two numbers is 96, + and one number is 16 larger than the other number. Find the two numbers.] +make "tom [If the number of customers Tom gets is + twice the square of 20 per cent of the number of advertisements he runs, + and the number of advertisements he runs is 45, + what is the number of customers Tom gets?] +make "uncle [Bill's father's uncle is twice as old as Bill's father. + 2 years from now Bill's father will be 3 times as old as Bill. + The sum of their ages is 92 . Find Bill's age.] + +;; Initial data base + +pprop "distance "eqns ~ + [[equal [distance] [product [speed] [time]]] + [equal [distance] [product [gas consumtion] [number of gallons of gas used]]]] +pprop "feet "eqns ~ + [[equal [product 1 [feet]] [product 12 [inches]]] + [equal [product 1 [yards]] [product 3 [feet]]]] +pprop "feet "sing "foot +pprop "foot "plural "feet +pprop "gallons "eqns ~ + [[equal [distance] [product [gas consumtion] [number of gallons of gas used]]]] +pprop "gas "eqns ~ + [[equal [distance] [product [gas consumtion] [number of gallons of gas used]]]] +pprop "inch "plural "inches +pprop "inches "eqns [[equal [product 1 [feet]] [product 12 [inches]]]] +pprop "people "sing "person +pprop "person "plural "people +pprop "speed "eqns [[equal [distance] [product [speed] [time]]]] +pprop "time "eqns [[equal [distance] [product [speed] [time]]]] +pprop "yards "eqns [[equal [product 1 [yards]] [product 3 [feet]]]] +</PRE> + +<P><A HREF="../v3-toc2.html">(back to Table of Contents)</A> +<P><A HREF="../v3ch5/v3ch5.html"><STRONG>BACK</STRONG></A> +chapter thread <A HREF="https://people.eecs.berkeley.edu/~bh/v3ch7/v3ch7.html"><STRONG>NEXT</STRONG></A> + +<P> +<ADDRESS> +<A HREF="../index.html">Brian Harvey</A>, +<CODE>bh@cs.berkeley.edu</CODE> +</ADDRESS> +</BODY> +</HTML> |