CSC3315 (Spring 2009) 1

CSC 3315CSC 3315Programming Programming ParadigmsParadigmsProlog LanguageProlog Language

Hamid HarroudHamid HarroudSchool of Science and Engineering, Akhawayn School of Science and Engineering, Akhawayn

UniversityUniversityhttp://www.aui.ma/~H.Harroud/csc3315/

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Logic Programming Logic Programming LanguagesLanguages

Introduction A Brief Introduction to Predicate Calculus Predicate Calculus and Proving Theorems An Overview of Logic Programming The Origins of Prolog The Basic Elements of Prolog Deficiencies of Prolog Applications of Logic Programming

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IntroductionIntroduction Logic programming language or declarative

programming language Express programs in a form of symbolic logic Use a logical inferencing process to produce

results Declarative rather that procedural:

Only specification of results are stated (not detailed procedures for producing them)

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PropositionProposition A logical statement that may or may not be

true Consists of objects and relationships of objects to

each other

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Symbolic LogicSymbolic Logic Logic which can be used for the basic needs of

formal logic: Express propositions Express relationships between propositions Describe how new propositions can be inferred

from other propositions Particular form of symbolic logic used for logic

programming called predicate calculus

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Object RepresentationObject Representation Objects in propositions are represented by

simple terms: either constants or variables Constant: a symbol that represents an object Variable: a symbol that can represent

different objects at different times Different from variables in imperative languages

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Compound TermsCompound Terms Atomic propositions consist of compound

terms Compound term: one element of a

mathematical relation, written like a mathematical function Mathematical function is a mapping Can be written as a table

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Parts of a Compound TermParts of a Compound Term Compound term composed of two parts

Functor: function symbol that names the relationship

Ordered list of parameters (tuple) Examples:

student(john)

like(nick, windows)

like(jim, linux)

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Forms of a PropositionForms of a Proposition Propositions can be stated in two forms:

Fact: proposition is assumed to be true Query: truth of proposition is to be determined

Compound proposition: Have two or more atomic propositions Propositions are connected by operators

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Logical OperatorsLogical OperatorsName Symbol Example Meaning

negation a not a

conjunction a b a and b

disjunction a b a or b

equivalence a b a is equivalent to b

implication

a ba b

a implies bb implies a

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QuantifiersQuantifiers

Name Example Meaning

universal X.P For all X, P is true

existential X.P There exists a value of X such that P is true

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Clausal FormClausal FormToo many ways to state the same thingUse a standard form for propositionsClausal form: B1 B2 … Bn A1 A2 … Am

means if all the As are true, then at least one B is true

Antecedent: right sideConsequent: left side

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Predicate Calculus and Predicate Calculus and Proving TheoremsProving Theorems A use of propositions is to discover new

theorems that can be inferred from known axioms and theorems

Resolution: an inference principle that allows inferred propositions to be computed from given propositions

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ResolutionResolution Unification: finding values for variables in

propositions that allows matching process to succeed

Instantiation: assigning temporary values to variables to allow unification to succeed

After instantiating a variable with a value, if matching fails, may need to backtrack and instantiate with a different value

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Proof by ContradictionProof by Contradiction Hypotheses: a set of pertinent propositions Goal: negation of theorem stated as a

proposition Theorem is proved by finding an

inconsistency

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Theorem ProvingTheorem Proving Basis for logic programming When propositions used for resolution, only

restricted form can be used Horn clause - can have only two forms

Headed: single atomic proposition on left side Headless: empty left side (used to state facts)

Most propositions can be stated as Horn clauses

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Overview of Logic Overview of Logic ProgrammingProgramming

Declarative semantics There is a simple way to determine the meaning

of each statement Simpler than the semantics of imperative

languages Programming is nonprocedural

Programs do not state now a result is to be computed, but rather the form of the result

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Example: Sorting a ListExample: Sorting a List Describe the characteristics of a sorted list,

not the process of rearranging a list

sort(old_list, new_list) permute (old_list, new_list) sorted (new_list)

sorted (list) j such that 1 j < n, list(j) list (j+1)

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The Origins of PrologThe Origins of Prolog University of Aix-Marseille

Natural language processing University of Edinburgh

Automated theorem proving

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TermsTerms Edinburgh Syntax Term: a constant, variable, or structure Constant: an atom or an integer Atom: symbolic value of Prolog Atom consists of either:

a string of letters, digits, and underscores beginning with a lowercase letter

a string of printable ASCII characters delimited by apostrophes

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Terms: Variables and Terms: Variables and StructuresStructures

Variable: any string of letters, digits, and underscores beginning with an uppercase letter

Instantiation: binding of a variable to a value Lasts only as long as it takes to satisfy one

complete goal Structure: represents atomic proposition

functor(parameter list)

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Fact StatementsFact Statements Used for the hypotheses Headless Horn clauses

female(shelley).

male(bill).

father(bill, jake).

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Rule StatementsRule Statements Used for the hypotheses Headed Horn clause Right side: antecedent (if part)

May be single term or conjunction Left side: consequent (then part)

Must be single term Conjunction: multiple terms separated by

logical AND operations (implied)

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Example RulesExample Rulesancestor(mary,shelley):- mother(mary,shelley).

Can use variables (universal objects) to generalize meaning:parent(X,Y):- mother(X,Y).parent(X,Y):- father(X,Y).grandparent(X,Z):- parent(X,Y), parent(Y,Z).sibling(X,Y):- mother(M,X), mother(M,Y),

father(F,X), father(F,Y).

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Goal StatementsGoal Statements For theorem proving, theorem is in form of

proposition that we want system to prove or disprove – goal statement

Same format as headless Hornman(fred)

Conjunctive propositions and propositions with variables also legal goalsfather(X,mike)

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Inferencing Process of Inferencing Process of PrologProlog

Queries are called goals If a goal is a compound proposition, each of the facts is a

subgoal To prove a goal is true, must find a chain of inference rules

and/or facts. For goal Q:B :- A

C :- B

…Q :- P

Process of proving a subgoal called matching, satisfying, or resolution

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ApproachesApproaches Bottom-up resolution, forward chaining

Begin with facts and rules of database and attempt to find sequence that leads to goal

Works well with a large set of possibly correct answers

Top-down resolution, backward chaining Begin with goal and attempt to find sequence that leads to set of

facts in database Works well with a small set of possibly correct answers

Prolog implementations use backward chaining

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Subgoal StrategiesSubgoal Strategies When goal has more than one subgoal, can

use either Depth-first search: find a complete proof for the

first subgoal before working on others Breadth-first search: work on all subgoals in

parallel Prolog uses depth-first search

Can be done with fewer computer resources

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BacktrackingBacktracking With a goal with multiple subgoals, if fail to

show truth of one of subgoals, reconsider previous subgoal to find an alternative solution: backtracking

Begin search where previous search left off Can take lots of time and space because may

find all possible proofs to every subgoal

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Simple ArithmeticSimple Arithmetic Prolog supports integer variables and integer

arithmetic is operator: takes an arithmetic expression

as right operand and variable as left operand

A is B / 17 + C Not the same as an assignment statement!

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ExampleExamplespeed(ford,100).speed(chevy,105).speed(dodge,95).speed(volvo,80).time(ford,20).time(chevy,21).time(dodge,24).time(volvo,24).distance(X,Y) :- speed(X,Speed),

time(X,Time), Y is Speed * Time.

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List StructuresList Structures Other basic data structure (besides atomic

propositions we have already seen): list List is a sequence of any number of elements Elements can be atoms, atomic propositions, or

other terms (including other lists)

[apple, prune, grape, kiwi]

[] (empty list)

[X | Y] (head X and tail Y)

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Append ExampleAppend Exampleappend([], List, List).

append([Head | List_1], List_2, [Head | List_3]) :-

append (List_1, List_2, List_3).

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Reverse ExampleReverse Examplereverse([], []).

reverse([Head | Tail], List) :-

reverse (Tail, Result),

append (Result, [Head], List).

Example: mylengthmylength([],0).mylength([_|Tail], Len) :- mylength(Tail, TailLen), Len is TailLen + 1.

?- mylength([a,b,c],X).

X = 3

Yes?- mylength(X,3).

X = [_G266, _G269, _G272]

Yes

Counterexample: mylengthmylength([],0).mylength([_|Tail], Len) :- mylength(Tail, TailLen), Len = TailLen + 1.

?- mylength([1,2,3,4,5],X).

X = 0+1+1+1+1+1

Yes

Example: sumsum([],0).sum([Head|Tail],X) :- sum(Tail,TailSum), X is Head + TailSum.

?- sum([1,2,3],X).

X = 6

Yes?- sum([1,2.5,3],X).

X = 6.5

Yes

Example: gcd

gcd(X,Y,Z) :- X =:= Y, Z is X.gcd(X,Y,Denom) :- X < Y, NewY is Y - X, gcd(X,NewY,Denom).gcd(X,Y,Denom) :- X > Y, NewX is X - Y, gcd(NewX,Y,Denom).

Note: not just

gcd(X,X,X)

The gcd Predicate At Work

?- gcd(5,5,X).X = 5 Yes?- gcd(12,21,X).X = 3 Yes?- gcd(91,105,X).X = 7 Yes?- gcd(91,X,7).ERROR: Arguments are not sufficiently instantiated

Example: factorialfactorial(X,1) :- X =:= 1.factorial(X,Fact) :- X > 1, NewX is X - 1, factorial(NewX,NF), Fact is X * NF.

?- factorial(5,X).X = 120 Yes?- factorial(20,X).X = 2.4329e+018 Yes?- factorial(-2,X).No

Problem Space Search Prolog’s strength is (obviously) not numeric

computation The kinds of problems it does best on are

those that involve problem space search You give a logical definition of the solution Then let Prolog find it

The Knapsack Problem You are packing for a camping trip Your pantry contains these items:

Your knapsack holds 4 kg. What choice <= 4 kg. maximizes calories?

Item Weight in kilograms Calories

bread 4 9200

pasta 2 4600

peanut butter 1 6700

baby food 3 6900

Greedy Methods Do Not Work

Most calories first: bread only, 9200 Lightest first: peanut butter + pasta, 11300 (Best choice: peanut butter + baby food,

13600)

Item Weight in kilograms Calories

bread 4 9200

pasta 2 4600

peanut butter 1 6700

baby food 3 6900

Search No algorithm for this problem is known that

Always gives the best answer, and Takes less than exponential time

So brute-force search is used here That’s good, since search is something

Prolog does really well

Representation We will represent each food item as a term food(N,W,C)

Pantry in our example is[food(bread,4,9200), food(pasta,2,4500),

food(peanutButter,1,6700), food(babyFood,3,6900)]

Same representation for knapsack contents

/*weight(L,N) takes a list L of food terms, N is the sum of all the Weights. */

weight([],0).weight([food(_,W,_) | Rest], X) :- weight(Rest,RestW), X is W + RestW.

/*calories(L,N) takes a list L of food terms, N is the sum of all the Calories. */

calories([],0).calories([food(_,_,C) | Rest], X) :- calories(Rest,RestC), X is C + RestC.

The Knapsack Problem

/* subseq(X,Y) succeeds when list X is the same as list Y, but with zero or more elements omitted.*/

subseq([],[]).subseq([Item | RestX], [Item | RestY]) :- subseq(RestX,RestY).subseq(X, [_ | RestY]) :- subseq(X,RestY).

A subsequence of a list is a copy of the list with any number of elements omitted

(Knapsacks are subsequences of the pantry)

The Knapsack Problem

?- subseq([1,3],[1,2,3,4]).Yes

?- subseq(X,[1,2,3]).

X = [1, 2, 3] ;X = [1, 2] ;X = [1, 3] ;X = [1] ;X = [2, 3] ;X = [2] ;X = [3] ;X = [] ;

No

Note that subseq can do more than just test whether one list is a subsequence of another; it can generate subsequences, which is how we will use it for the knapsack problem.

The Knapsack Problem

/* knapsackDec(Pantry,Capacity,Goal,Knapsack) takes a list Pantry of food terms, a positive number Capacity, and a positive number Goal. We unify Knapsack with a subsequence of Pantry representing a knapsack with total calories >= goal, subject to the constraint that the total weight is =< Capacity.*/knapsackDec(Pantry,Capacity,Goal,Knapsack) :- subseq(Knapsack,Pantry), weight(Knapsack,Weight), Weight =< Capacity, calories(Knapsack,Calories), Calories >= Goal.

The Knapsack Problem

This decides whether there is a solution that meets the given calorie goal

?- knapsackDec(| [food(bread,4,9200),| food(pasta,2,4500),| food(peanutButter,1,6700),| food(babyFood,3,6900)],| 4,| 10000,| X).

X = [food(pasta, 2, 4500), food(peanutButter, 1, 6700)]

Yes

The Knapsack Problem

The 8-Queens Problem Chess background:

Played on an 8-by-8 grid Queen can move any number of spaces

vertically, horizontally or diagonally Two queens are in check if they are in the same

row, column or diagonal, so that one could move to the other’s square

The problem: place 8 queens on an empty chess board so that no queen is in check

Representation We could represent a queen in column 2, row

5 with the term queen(2,5) But it will be more readable if we use

something more compact Since there will be no other pieces, we can

just use a term of the form X/Y (We won’t evaluate it as a quotient)

Example

Our 8-queens solution [8/4, 7/2, 6/7, 5/3, 4/6, 3/8, 2/5, 1/1]

8

7

6

5

4

3

2

1

2 1 4 3 6 5 8 7

Q

Q

Q

Q

Q

Q

Q

Q

/* nocheck(X/Y,L) takes a queen X/Y and a list of queens. We succeed if and only if the X/Y queen holds none of the others in check.*/

nocheck(_, []).nocheck(X/Y, [X1/Y1 | Rest]) :- X =\= X1, Y =\= Y1, abs(Y1-Y) =\= abs(X1-X), nocheck(X/Y, Rest).

The 8-Queens Problem

/* legal(L) succeeds if L is a legal placement of queens: all coordinates in range and no queen in check.*/

legal([]).legal([X/Y | Rest]) :- legal(Rest), member(X,[1,2,3,4,5,6,7,8]), member(Y,[1,2,3,4,5,6,7,8]), nocheck(X/Y, Rest).

The 8-Queens Problem

Adequate This is already enough to solve the problem:

the query legal(X) will find all legal configurations:

?- legal(X).

X = [] ;

X = [1/1] ;

X = [1/2] ;

X = [1/3]

8-Queens Solution Of course that will take too long: it finds all

64 solutions with one queen, then starts on those with two, and so on

To make it concentrate right away on eight queens, we can give a different query:

?- X = [_,_,_,_,_,_,_,_], legal(X).

X = [8/4, 7/2, 6/7, 5/3, 4/6, 3/8, 2/5, 1/1]

Yes

Room For Improvement Slow Finds trivial permutations after the first:

?- X = [_,_,_,_,_,_,_,_], legal(X).

X = [8/4, 7/2, 6/7, 5/3, 4/6, 3/8, 2/5, 1/1] ;

X = [7/2, 8/4, 6/7, 5/3, 4/6, 3/8, 2/5, 1/1] ;

X = [8/4, 6/7, 7/2, 5/3, 4/6, 3/8, 2/5, 1/1] ;

X = [6/7, 8/4, 7/2, 5/3, 4/6, 3/8, 2/5, 1/1]

An Improvement Clearly every solution has 1 queen in each

column So every solution can be written in a fixed

order, like this:X=[1/_,2/_,3/_,4/_,5/_,6/_,7/_,8/_]

Starting with a goal term of that form will restrict the search (speeding it up) and avoid those trivial permutations

/* eightqueens(X) succeeds if X is a legal placement of eight queens, listed in order of their X coordinates.*/

eightqueens(X) :- X = [1/_,2/_,3/_,4/_,5/_,6/_,7/_,8/_], legal(X).

The 8-Queens Problem

nocheck(_, []).nocheck(X/Y, [X1/Y1 | Rest]) :- % X =\= X1, assume the X's are distinct Y =\= Y1, abs(Y1-Y) =\= abs(X1-X), nocheck(X/Y, Rest).

legal([]).legal([X/Y | Rest]) :- legal(Rest), % member(X,[1,2,3,4,5,6,7,8]), assume X in range member(Y,[1,2,3,4,5,6,7,8]), nocheck(X/Y, Rest).

Since all X-coordinates are already known to be in range and distinct, these can be optimized a little

The 8-Queens Problem

Improved 8-Queens Solution

Now much faster Does not bother with permutations

?- eightqueens(X).

X = [1/4, 2/2, 3/7, 4/3, 5/6, 6/8, 7/5, 8/1] ;

X = [1/5, 2/2, 3/4, 4/7, 5/3, 6/8, 7/6, 8/1]

Cut & NegationCut & Negation

Cut (Minimum): mini1(X, Y, X) :- X < Y.

mini1(X, Y, Y) :- X >= Y.

mini2(X, Y, X) :- X < Y, !.mini2(_, Y, Y).

mini3(X, Y, Z) :- X < Y, !, Z = X.mini3(X, Y, Y).

Negation: not(P):- P, ! ,fail. not(P). %sinon

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Applications of Logic Applications of Logic ProgrammingProgramming

Relational database management systems Expert systems Natural language processing

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SummarySummary Symbolic logic provides basis for logic

programming Logic programs should be nonprocedural Prolog statements are facts, rules, or goals Resolution is the primary activity of a Prolog

interpreter Although there are a number of drawbacks

with the current state of logic programming it has been used in a number of areas