Classical Problem Solving Design of Computer Problem Solvers CS 344

Transcript Classical Problem Solving Design of Computer Problem Solvers CS 344

Classical Problem Solving

Design of Computer Problem Solvers

CS 344 Winter 2000

Last time

• AI programming versus conventional programming – Emphasis on knowledge representation • Overview of systems • Homework 0 • Preview of homework 1

This week’s themes

• Problem-solvers see the world in terms of problem spaces • Describe the problem in its own terms, and let Lisp make up the difference

Components of a Problem Space

• States (conditions of the world) – Initial state (where we are) – Goal state (where we want to be) • Operators (actions that can change the current state) • Problem solving – Construct a path of states, using available operators, from initial to goal state

Problem-solvers see the world in terms of problem spaces

Sample Problem Spaces

• Subway travel – Initial state: Davis Street – Goal state: Morse Street – Operators: Take purple line, take red line • Chess – Initial state: Board setup – Goal state: Checkmate condition – Operators: Chess moves

Problem-solvers see the world in terms of problem spaces

How to characterize a problem space

• Characterizing states – Often symbols or propositional expressions • Comparing states • Choosing operators – Operator applicability (can we use it?) – Operator preference (is it the best one?) • Search strategy

Problem-solvers see the world in terms of problem spaces

– When do we backtrack? Breadth or depth-first? Use means-end analysis?

Methods for choosing operators

• When is an operator applicable?

– Preconditions • Which applicable operator is best?

– First available • May assume that operators written in order of preference (e.g., travel ops ordered by speed) – Random • Choose one--if it doesn’t work out, try other.

– Distance metric • Pick a metric that tells us which is best

Problem-solvers see the world in terms of problem spaces

Why study the classical method of problem solving?

• Historically early method in AI – GPS (Newell & Simon, 1963) – STRIPS (Fikes & Nilsson, 1971) • Descendants still viable: SOAR, Deep Blue • Provides a vocabulary for discussing problem solving – “States”, “operators”, and “problem spaces”

Problem-solvers see the world in terms of problem spaces

Classical Problem Solving

• Separation between the search method and the problem space –

Search method:

generic search methods –

Problem space:

representation of problem as set of states and operators

Search Engine Problem Space

How Problem Spaces provide modularity and explicitness

Search Engine Programming Knowledge Reasoning Engine Knowledge of the domain Representation of Domain Knowledge Computer Program Problem Space Representation

Basic search methods

• Breadth-first – Try all applicable operators before choosing secondary operators • Depth-first – On the operator chosen, continue searching until either goal is reached or search fails • Best-first – Use a distance metric to choose best operator each time

Building CPS

• Need a problem space “API” between a problem space and the search engine • Requirements: – Way to describe basic characteristics of states and operators in problem space – Should provide multiple, interchangable search engines – Should work with novel problem spaces

Describe the problem in its own terms, and let Lisp make up the difference.

Constructing a “pluggable” classical problem solver

• What’s a pluggable data structure?

• Why build a pluggable problem space obj?

– Search engines, to be general, need a consistent API for the problem space – No guarantee that all problem spaces will respect that API – Use a pluggable class, called Problem, which mediates between the two and allow an arbitrary problem space to be used with many different search engines.

Describe the problem in its own terms, and let Lisp make up the difference.

Code

PROBLEM Defstruct

• Each problem object defines a particular problem space – E.g., taking subway from MIT to aquarium • Requires following functions to define problem space: – Goal recognizer – State identicality test – Print states – Applicable operators for given state – Distance function

Once we have these, we can create “generic” search engines that will work with any problem space.

Using pluggable function slots

PROBLEM slot: Goal-recognizer Operator-applier States-identical Path-filter Distance-remaining State-printer Solution-element-printer Function description: When has goal been found?

Return set of applicable operators Are states the same?

Remove path from possible paths?

Estimate of distance to goal Print state Print solution path segment

Code

Path object structure

Code • Records a particular solution path in terms of intermediate states and operators • Has links back to the problem space, so that a path struct can be taken and “extended” easily.

State and Operator defstructs?

• Where’s the

state

defstruct?

• Where’s the

defstruct?

• CPS code doesn’t define them.

• Instead – Operator and state data structures are implicit in the comparison functions: • Operator-applier, states-identical?, goal recognizer, distance-remaining, state-printer

Describe the problem in its own terms, and let Lisp make up the difference.

Basic search methods

Bsolve function (simplified)

(defun

bsolve

(initial &aux curr-path new-paths) “Apply breadth-first search to problem space.”

(do ((queue (list (list initial initial)) (append (cdr queue) new-paths))) ((null queue)) (setq curr-path (car queue)) (when (goal-recognized? (car curr-path) (return curr-path)) (setq new-paths ( extend-path curr-path ))

Extend-path (simplified)

(defun extend-path (path &aux new-paths new-path pr) “Extend the given path, and return a list of possible paths+1” (setq pr (path-pr path)) (dolist (op (pr-operators pr) new-paths) (dolist (op-pair (apply-op path op pr)) (setq new-path (make-path (add-to-path path (cdr op-pair) (car op-pair)))) (unless (path-has-loop? new-path) ;avoid loops (unless (and (pr-path-filter pr) (funcall (pr-path-filter pr) new-path)) (push new-path new-paths))))))

Extend-path (simplified)

Apply each op to path’s end state.

(make-path (add-to-path path (cdr op-pair) (car op-pair)))) (unless (path-has-loop? new-path) ;avoid loops (unless (and (pr-path-filter pr) (funcall (pr-path-filter pr) new-path)) (push new-path new-paths))))))

Extend-path (simplified)

Apply domain specific filter.

(unless (path-has-loop? new-path) ;avoid loops (unless (and (pr-path-filter pr) (funcall (pr-path-filter pr) new-path)) (push new-path new-paths))))))

Bsolve function (simplified)

(defun

bsolve

(initial &aux curr-path new-paths) “Apply breadth-first search to problem space.”

Bsolve function (simplified)

(defun

bsolve

(initial &aux curr-path new-paths) “Apply breadth-first search to problem space.”

Code

Bsolve function (simplified)

Check current state to see if goal (defun

bsolve

(initial &aux curr-path new-paths) “Apply breadth-first search to problem space.”

reached:

Fifo queue

Dsolve

Bsolve function (simplified)

(defun

bsolve

(initial &aux curr-path new-paths) “Apply breadth-first search to problem space.”

(do ((queue (list (list initial initial)) (append (cdr queue) new-paths))) (append new-paths (cdr queue))) ((null queue)) (setq curr-path (car queue)) (when (goal-recognized? (car curr-path) (return curr-path)) (setq new-paths (extend-path curr-path)) LIFO Fifo queue

The Agenda type determines the search characteristics

FIFO == Breadth-first 0 1 2 3 4 5 6 7 8 9 LIFO == Depth-first 0 1 6 2 5 7 3 4 8 9

More search variants (variants.lsp)

• Best-first search – Estimate distance from goal for each solution path – Sort queue so that closest solution paths tried first – How good must the distance metric be?

• Beam search – Same as best-first search, but limit size of queue to a fixed “beam width”

Example 1: Subway navigation

Let’s use our search engine to tackle a commonly used search space: the Boston “T”

Building the representation

• • • Representation – Stations: the lines they are on, x-y map coordinates – Lines: the stations on that line • Problem space mapping – State = station – Operator: Take the line to another station Definition mechanism T definitions

Describe the problem in its own terms, and let Lisp make up the difference.

Stations

(defstruct ( subway-station print-procedure)) (:PRINT-FUNCTION subway-station "Data structure representating a single subway station." (name nil) ;; Name of station.

(lines nil) ;; Subways lines it is on.

(coordinates nil)) ;; For advanced CPS versions which use a distance metric.

(defun subway-station-print-procedure "Print name of station." (declare (ignore ignore)) (pr str ignore) (format str "" (subway-station-name pr)))

Defining a station

(defvar KENDALL-SQUARE) (setq KENDALL-SQUARE (make-subway-station :name ‘Kendall-square :lines ‘(RED-LINE) :coordinates ‘(1 . 0)) (push ‘Kendall-square (subway-line-stations RED-LINE)) (push ‘Kendall-square *stations*)

Defining a station

(defmacro defstation (name lines &optional (x 0) (y 0)) "Define a subway station." `(progn (defvar ,name) (setq ,name (make-subway-station :NAME ',name :LINES ',lines :COORDINATES (cons ,x ,y))) ,@ (mapcar #'(lambda (line) `(push ',name (subway-line-stations ,line))) lines) (push ',name *stations*)))

Defining a station

Describe the problem in its own terms, and let Lisp make up the difference.

Defining a station

(defstation South-Station (Red-Line) 3.0 -1.0) (defstation Washington (Red-Line Orange-Line) 2.75 0.75) (defstation Kendall-Square (Red-Line) 1.0 0.0) (defstation Central-Square (Red-Line) -1.0 0.0) (defstation Harvard-Square (Red-Line) -2.0 1.0) (defline Red-Line) (defline Green-Line) (defline Orange-Line) (defline Blue-Line)

Describe the problem in its own terms, and let Lisp make up the difference.

Setting up Subway problem

PROBLEM: Goal-recognizer Operator-applier States-identical Path-filter Distance-remaining State-printer Solution-element-printer

Setting up Subway problem

PROBLEM: Goal-recognizer Operator-applier States-identical Path-filter Distance-remaining State-printer Solution-element-printer

Subway-states-identical?

Subway-operator-finder Subway-states-identical?

Prune-subway-path Subway-distance Format Print-subway-path

RUN

Test run

Next time: Algebra!

Brady’s example:

log (x+1) + log(x-1) = c.

Classical Problem Solving, II

Design of Computer Problem Solvers

CS 344 Winter 2000

Lessons from last time

• Constructing problem spaces – See a problem in terms of a problem space – How to construct a “pluggable” problem space – Represent the problem in its own terms • Search engines – Breadth, Depth, and Best-First • Fitting the representation to the task – DefLine, DefStation.

Today: Solving algebraic expressions

• Algebra as a problem space – Alan Bundy’s model for heuristic-based simplification of algebra problems.

• Building the system – Representing states and operators – Pattern matching • How to build it • Making canonical expressions • Homework 1 • Preview of homework 2 (Natural Deduction)

A quick example: Bundy’s challenge

log e (x+1) + log e (x-1) = c.

2. log e [(x+1) * (x-1)] = c.

log e

[x2-1] = c.

x

-1=e

.

x

= e

+1.

x = +/- sqrt(e

+ 1).

Algebra as a Problem Space

• The problem space is the transformational grammar – States: valid sentences within that grammar – Operators: transformation operators • Solving algebraic expressions – 3X = 5X - 2. What is the value for X?

– Significant expert-novice differences – What do expects know that novices don’t?

Algebra as a Problem Space

• States: Algebraic equations • Initial state: Starting equation .

• Goal state: Version of equation with X (unknown) on the left.

• Operators: Transformations of the equation using various algebraic laws.

• Path: Sets of transformations.

• Distance metric: ??

Standard Lisp expressions for states...

…named functions for operators (take equation, returns transformed equation).

Data structures?

Bundy’s Claim

Note: There’s a • Experts have

control knowledge

that lets them hiding in here that avoid many false paths we can use later...

–This knowledge limits the needed search • Transform equations to reduce the depth and frequency of unknown variable.

–Each transformation either reduces the number of unknowns, or brings them closer together.

–Three kinds of methods: isolation methods, collection methods, and attraction methods.

–This is best understood by graphing the equation.

Isolation methods

• Reduces the depth of the occurrences of the unknown • Examples – U - W = Y  U = Y + W – log( U ,w) = Y  U = W Y • Assumptions – U contains

, Y & W do not.

Reduce the depth of unknowns.

Isolation methods

log

e c

expt

log e [x2-1] = c.

x 2 -1=e c .

x 2

Depth=4.

expt -

x 2

Depth=3.

1 = e

expt

Collection methods

• Reduce the number of occurrences of the unknown • Examples – U W + U Y  U (W+Y) – ( U + V )( U V )  U 2 V 2 • Assumptions – U and V contain

, W & Y do not.

Collect together multiple instances of the unknown:

Collection methods

* log 2. log e [(x+1) * (x-1)] = c.

log e [x2-1] = c.

expt

+ 2 x 1 1 e x =

One less X .

1 c

Attraction methods

• Bring occurrences of the unknown “closer together” • Examples: – W U + W V  W( U + V ) – log( U ,w) + log( V ,w)  log( UV ,w) • Assumptions: – U and V expressions contain

– W expression does not.

Attraction methods

Bring occurrences of the unknown “closer together”

log

+

U W V log

Distance=4.

log e (x+1) + log e (x-1) = c.

2. log e [(x+1) * (x-1)] = c.

log

Distance=2.

* W U V

A quick derivation, revisited.

log e (x+1) + log e (x-1) = c.

2. log e [(x+1) * (x-1)] = c.

log e

[x2-1] = c.

x

-1=e

.

x

= e

+1.

Initial 2. Attract log sum.

Collect prod diffs 4.

Isolate log.

Isolate difference.

x = +/- sqrt(e

+ 1).

Isolate square

Algebra as a problem space

• States: Algebraic equations • Initial state: Starting equation .

Standard Lisp expressions for states...

• Goal state: Version of equation with X (unknown) on the left.

• Operators: Transformations of the equation Isolation methods using various algebraic laws.

• Path: Sets of transformations.

• Distance metric: ??

Measure number and depths of unknowns in equation...

…named functions for operators (take equation, returns transformed equation).

Implementing Bundy’s idea

• States: equations in Lisp form –

(= (- U W) Y)

for U - W = Y.

• Operators: – Functions which perform individual isolation, collection and attraction methods • Distance metric: – Function that measures depth and frequence of X. • Glue: – Flexible pattern matcher for operators and simplifier rules – Simplifier rules run after each operation – Canonicalize expressions

“Pluggable” problem space

PROBLEM slot: Goal-recognizer Operator-applier States-identical Path-filter Distance-remaining State-printer Solution-element-printer Operators Function description: When has goal been found?

Return set of applicable operators Are states the same?

Remove path from possible paths?

Estimate of distance to goal Print state Print solution path segment Operator list

“Pluggable” problem space

• • •

Search Engine: bsolve dsolve best-solve Goal-recognizer Operator-applier States-identical Path-filter Distance-remaining State-printer Solution-element-printer Operators

Problem space Algebra domain States: Lisp expression Operators: Isolation, collection, & attraction methods

Setting up Algebra problem solver

PROBLEM: Goal-recognizer Operator-applier States-identical Path-filter Distance-remaining State-printer Solution-element-printer

Setting up Algebra problem solver

PROBLEM: Goal-recognizer Operator-applier States-identical Path-filter Distance-remaining State-printer Solution-element-printer Operators

Got-algebra-goal?

Find-algebra-operator Equal NIL Algebra-distance Format Print-derivative-step

Some of these functions are easy

• Got-algebra-goal?

• Find-algebra-operator (misnamed, btw).

• States-identical = EQUAL – But wait, what about: • (* (+ x y) z) vs. (* z (+ y x))?

• This leaves: – The operators themselves – The pattern matcher – The distance metric

Got algebra goal

(defun

occurs-in?

(exp1 exp2) "True if expression 1 is contained somewhere in expression 2." (cond ((equal exp1 exp2) t) ((null exp2) nil) ((listp exp2) (or (occurs-in? exp1 (car exp2)) (occurs-in? exp1 (cdr exp2)))))) (defun

has-unknown?

(exp) (occurs-in? 'x exp))

(defun got-algebra-goal? (state) "Has goal of algebra problem been met?" (and (eq (cadr state) 'x) ;; LHS=X (no-unknown? (rhs state))))

(defun

no-unknown?

(exp) "True if expression contains no unknown values." (not (occurs-in? 'x exp)))

Canonicalization

• Equations must be canonicalized to allow for better matching of expressions – Get rid of trivial cases: • (+ x 0) => x.

• (+ 3 7) => 10.

– Flatten trees: • (+ (x y) z) => (+ x y z).

– Sort the terms: • (+ z x b) => (+ b x z).

Building the operators

• Requires ability to match particular patterns and then transform them.

– We’ll need a pattern matcher • Operators: – Isolation, collection & attraction – Also a “canonicalization” operator • Will canonicalize the expression, if needed.

• Will throw in a bunch of “trivial” simplifications.

• Also needs a pattern matcher

Creating a pattern matcher

• For matching – (? x) : Matches symbol, binding to pattern var X – (?? x) : Matches sequence, binding to segment var X – (? x test) : Matches single symbol x when passed to function test .

that also returns true • Matcher returns either a set of bindings or :FAIL – (match ‘(+ (? A) (? B) (? C)) ‘(+ 1 2 3)) returns ((c 3) (b 2) (a 1)) • For simplification – subst function for instantiating matched patterns.

– (:eval (+ (? X) (? Y))) : Matches X and Y, and then adds them and returns that value. – : splice act similarly, but splices result into list.

Tree-walking for dummies

Tree-walking : Apply Foo to every element in tree

thing

Function foo-treewalk (

thing

) Null

thing

? Return nil.

Atom

thing

? Then Foo

Thing

List

thing

? Foo-treewalk the car(

thing

Foo-treewalk the cdr(

thing

Creating a pattern matcher

(defun

match

(pat dat &optional (dict nil)) "Take a single pattern and data, and return any matches based on the pattern. Dictionary contains binding list." (cond ((eq dict :FAIL) :FAIL) ;; Propagate lossage ((eq pat dat) dict) ((element-var? pat) ( match-element-var pat dat dict)) ((not (consp pat)) (if (equal? pat dat) dict :FAIL)) ((segment-var? (car pat)) ( match-segment-var pat dat dict)) ((not (consp dat)) :FAIL) (t ( match (cdr pat) (cdr dat) ( match (car pat) (car dat) dict)))))

Creating a pattern matcher

(defun

match-element-var

(pat dat dict &aux entry pred) "Match single element pattern variable to given data, using the bindings in dictionary. Returns either :FAIL or updated binding dictionary." (setq entry ( lookup-var pat dict)) (cond (entry (if (equal? (cadr entry) dat) dict :FAIL)) (t (setq pred ( var-restriction pat)) (cond ((or (not pred) (funcall pred dat)) ( bind-element-var (t :FAIL))))) (var-name pat) dat dict))

Creating a pattern matcher

(defun

match-segment-var

(pat dat dict &aux entry rest) "Given sequence pattern variable, attempt matching. Returns either bindings or :FAIL." (setq entry (lookup-var (car pat) dict)) (cond (entry ;; check for match (setq rest (check-segment dat (segment-beg entry) (segment-end entry))) (if (eq rest :FAIL) :FAIL (match (cdr pat) rest dict))) (t ;; Search for alternate segment bindings (try-segment-bindings (car pat) (cdr pat) dat dict))))

Uses of the pattern matcher

• Two uses: – Bundy’s method operators – Simplifier rules for canonicalization operator

Simplifier rules

• Simplifier rules are used by the canonicalization operator – Identities: x + 0 = x; x * 1 = x.

– Canonicalization: Ordering operands for consistant matching – Evaluation: computing the function of constants • Canonicalization operator runs all the simplifier rules each time it is called.

Simplifier rules

• Rules have simple before /after pattern.

;; Flush degenerate cases ((? op +/*?) (? e)) (? e)) ((+ (? zero zero?) (?? e)) (+ (?? e))) ;; Combine numerical constants (((? op +/*?) (? e1 numberp) (? e2 numberp) (?? e3)) ((? op) (:EVAL ((? op) (? e1) (? e2))) (?? e3))) ((- (? e1 numberp) (? e2 numberp)) (:EVAL (- (? e1) (? e2)))) ((- (? e1 numberp)) ;; Flatten +,* (:EVAL (- (? e1)))) (((? op +/*?) (?? e1) ((? op) (?? e2) (?? e3))) ((? op) (?? e1) (?? e2) (?? e3))) ;; Canonicalize +,* (((? op +/*?) (?? terms,#'(lambda (terms) (not (sorted? terms #'alg<))))) ((? op) (:SPLICE (:EVAL (sort (quote (? terms)) #'alg<)))))

Canonicalization operator

TRY-CANONICALIZATION +-> SIMPLIFY +-> SIMPLIFY-IT +-> TRY-MATCHER-RULES (defun try-matcher-rules (exp rules) ;; Return the original expression by default (dolist (rule rules exp) (let ((bindings (match (rule-pattern rule) exp nil))) (unless (eq bindings :FAIL) (return-from try-matcher-rules (substitute-in (rule-result rule) bindings))))))

Setting up Algebra problem solver

PROBLEM: Goal-recognizer Operator-applier States-identical Path-filter Distance-remaining State-printer Solution-element-printer Operators

Got-algebra-goal?

Find-algebra-operator Equal NIL Algebra-distance Format Print-derivative-step

Setting up the operators in the problem space

(defun setup-algebra-problem () "Create an generic algebra 'problem space'." (make-problem :NAME 'Algebra :GOAL-RECOGNIZER 'got-algebra-goal?

:OPERATOR-APPLIER 'find-algebra-operator :STATE-PRINTER #'(lambda (f) (format nil "~A" f)) :SOLUTION-ELEMENT-PRINTER #'print-derivation-step :STATES-IDENTICAL? 'equal (defun find-algebra-operator (state operator) ;; Operators take the form ( ) :DISTANCE-REMAINING 'algebra-distance :OPERATORS '((Isolate-Log try-isolate-log) (Isolate-Sum try-isolate-sum) (Isolate-Difference try-isolate-difference) (Isolate-Square try-isolate-square) (Collect-Product-Difference try-collect-prod-diff) (Attract-Log-Sum try-attract-log-sum) (Canonicalize try-canonicalization))))

Building the rest of the operators

• Attempt to match the current expression • If it fails, return :FAIL • If it succeeds, do the substitution and simplify.

Operators

• Example :

(defun try-isolate-square (form &aux bindings) (setq bindings (match '(= (sqr (? arg has-unknown?)) (? rhs no-unknown?)) form)) (unless (eq bindings :FAIL) `(,(cons `(isolate-square ,form) (simplify (substitute-in `(= (? arg) (sqrt (? rhs))) bindings))))))

• Example : Attempt to

(defun try-isolate-square (setq bindings (form &aux bindings) (match '(= (sqr (? arg has-unknown?)) (? rhs no-unknown?)) form)) (unless (eq bindings :FAIL)

match X^2 = expression.

`(,(cons `(isolate-square ,form) (simplify (substitute-in `(= (? arg) (sqrt (? rhs))) bindings))))))

• Example : Attempt to

(defun try-isolate-square (setq bindings (form &aux bindings) (match '(= (sqr (? arg has-unknown?)) (? rhs no-unknown?)) form)) (unless (eq bindings :FAIL)

match X^2 = expression.

`(,(cons `(isolate-square ,form) (simplify (substitute-in `(= (? arg) (sqrt (? rhs))) bindings))))))

If success, substitute bindings into new expression

Attraction and Collection Ops

(defun try-collect-prod-diff (form &aux bindings results) (dolist (ldt (find-least-dominating-terms form) results) (setq bindings (match '(* (+ (? v no-unknown?) (? u has-unknown?)) (- (? u) (? v))) ldt)) (unless (eq bindings :FAIL) (push (cons `(collect-product-sum ,ldt) (simplify (subst (substitute-in `(- (sqr (? U)) (sqr (? V))) bindings) ldt form))) results))))

Setting up Algebra problem solver

PROBLEM: Goal-recognizer Operator-applier States-identical Path-filter Distance-remaining State-printer Solution-element-printer Operators

Got-algebra-goal?

Find-algebra-operator Equal NIL Algebra-distance Format Print-derivative-step

Algebra-distance

(defun algebra-distance (expr) "Estimate how close this expression is to solution, return number." (labels (( sum-tree-depth (exp depth) (cond ((null exp) 0) ((eq exp 'X) depth) ((not (listp exp)) 0) (t (+ (sum-tree-depth (car exp) (1+ depth)) (sum-tree-depth (cdr exp) depth)))))) (+ ( sum-tree-depth ( sum-tree-depth (lhs expr) 1) (rhs expr) 1))))

Running the system

Moral: For working through a problem space, control knowledge is essential

• Bundy’s insights: – It is not enough to know what the rules are – Control knowledge needed to know when to apply particular rules • Careful analysis of a domain can avoid a lot of search • More leverage in domain-specific knowledge than in domain-general search techniques • Imposing a problem-space on a problem • Represent the problem in its own terms – Not done here as much, but in Subway problem.

– Homework: fix this.

Homework 1:

Build a better algebra system.

Due:

January 21 by midnight!

Chapter 3, Problem 5 (parts a & b), then extend system to solve 10 problems given on web page.

Better operator definitions

• Before: (defun try-collect-prod-diff (form &aux bindings results) (dolist (ldt (find-least-dominating-terms form) results) (setq bindings (match '(* (+ (? v no-unknown?) (? u has-unknown?)) (- (? u) (? v))) ldt)) (unless (eq bindings :FAIL) (push (cons `(collect-product-sum ,ldt) (simplify (subst (substitute-in `(- (sqr (? U)) (sqr (? V))) results)))) bindings) ldt form))) • After: (defAlgebraOperator collect-prod-diff COLLECTION ;;; (U+v)*(U-v) => (U^2 - V^2).

(* (+ (? v no-unknown?) (? u has-unknown?)) (- (? u) (? v))) (- (sqr (? U)) (sqr (? V))))

Reading for next week

• Chapters 4-5 on Pattern-Directed Rule Systems • Download code for TRE and FTRE • Note: – Natural deduction system implemented for FTRE, but not TRE.

Homework 2

Preview

Extend the Natural Deduction system to use premises and goals.

 BPS, pp. 148-149: • Exercise 17, (a) and (b)

Due:

  • Exercise 18, (a), (b), and (c)

Hints:

Start early if possible. carefully before modifying the rule set. January 28 by midnight!

Run the ND examples with the debug flag on, and contemplate the output Building a driver loop that automates the testing process (and turns the debug flag on and off cleverly) will make your life easier.