Advanced Topics in LP and FP

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1 Lecture 3: Logic Programming in Prolog

2 Outline The control of execution

3 Logic Programming Declarative programming style: Emphasis on What is the solution? instead of How to find the solution? Symbolic representation of knowledge (= axioms and derivation rules) Clear separation between data and the inference process, which is built into the execution environment Uniform representation for axioms and inference rules. Modular representation of knowledge The possibility to dinamically modify programs, by adding and removing axioms and rules in the executed program.

4 Logic Programming Execution Based on a restricted part of First-Order Predicate Logic Calculus = satisfiability of goals, using reduction to contradiction Inference rule = resolution with unification Control strategy during proof search backward chaining: from goal to axioms depth traversal of the derivation tree danger of infinite descent along a path which has no solution incomplete search strategy increased efficiency in the usage of memory space

5 Prolog Programs Program = collection of Horn clauses: A 1... A n A true B (rule) (axiom) Lack of explicit negations Closed word assumption = what can not be proved is assumed to be false In contrast the open word assumption considers that we can not say anything about things that can not be proved.

6 Proof steps (1) 1 Initialize the stack of goals with the initial goal (specified by the user) 2 Initialize the answer substitution (used during unification) to be empty (that is, without variable bindings) 3 Extract the goal from the top of the stack of goals and find the first program clause whose conclusion can be unified with the extracted goal 4 Extend the answer substitution with the bindings of the unifier from the previous step, and add the premises of the program clause to the stack of goals. 5 Go to step 3.

7 Proof steps (2) We say that the goal from the top of the stack of goals can not be satisfied if there is no program clause whose head unifies with it. When this happens, the proof process backtracks to the situation when the previous goal was on the top of the stack, and tries to find the next program clause whose head unifies with it. The proof search reports success when the stack of goals gets empty. failure when it can not satisfy the goal from the top of the stack of goals.

8 Example of Prolog program Genealogic example parent(allen,bob). parent(allen,bianca). parent(bob,chris). grandparent(x,y):-parent(x,z), parent(z,y). stands for the following set of Horn clauses: true parent(allen, bob). true parent(allen, bianca). true parent(bob, chris). X. Y. Z.(parent(X, Z) parent(z, Y ) grandparent(x, Y )).

9 Genealogic example: Proof steps Initial goal: Find X, Y such that X is grandparent of Y σ = G = {gp(x, Y )} gp(x 1, Y 1 ) : p(x 1, Z 1 ), p(z 1, Y 1 ) σ = {X X 1, Y Y 1 } G = {p(x 1, Z 1 ), p(z 1, Y 1 )} p(allen, bob) p(allen, bianca) p(bob, chris) σ = {X X 1, Y Y 1, X 1 allen, Z 1 bob} G = {p(bob, Y 1 )} σ = {X X 1, Y Y 1, X 1 allen, Z 1 bob, Y 1 chris} G = {} node(bob, chris) σ = {X X 1, Y Y 1, X 1 allen, Z 1 bianca} G = {p(bianca, Y 1 )} failure σ = {X X 1, Y Y 1, X 1 bob, Z 1 chris} G = {p(chris, Y 1 )} failure success

10 Genealogic example Remarks Order of evaluation = order of trying to prove goals depends on order of clauses in the program order of premises in the bodies of rules. Rule of thumb: place first the premises which are easier to satisfy and are more specific (e.g., those provable directly from axioms)

11 Proofs Control strategies Forward chaining (data-driven): All possible conclusions are proved starting from the initial data The process stops when we reach all goals Backward chaining (goal-driven): Based on using only the rules which can contribute effectively to the satisfaction of a goal: Detect the rules whose conclusion unifies with the goal Attempt to satisfy the premises of this rule, a.s.o.

12 Control strategies The backward chaining algorithm 1. BackwardChaining(rules, G, σ) rules : list of program rules, G : stack of goals, σ : the answer substitution returns the satisfiability of G 2. if G = then 3. return success 4. goal head(g) 5. G tail(g) 6. for-each rule rules do // in program order 7. if unify(goal, conclusion(rule), σ) σ 8. newg premises(rule) G // depth traversal 9. σ σ σ 10. if BackwardChaining(rules, newg, σ )=success 11. return success 12. return failure

13 Example Computing the minimum of two numbers Example Minimum of two numbers: Prolog code 1 min (X, Y, M) :- X =< Y, M is X. 2 min (X, Y, M) :- X > Y, M is Y. 3 4 min2 (X, Y, M) :- X =< Y, M = X. 5 min2 (X, Y, M) :- X > Y, M = Y. 6 7 % definition equivalent with min2 8 min3 (X, Y, X) :- X =< Y. 9 min3 (X, Y, Y) :- X > Y.

14 Computing the minimum of two numbers Usage After consult-ing the text file in which the program was stored:?- min(1+2,3+4,m). M = 3 ; false.?- min(3+4,1+2,m). M = 3.?-min2(1+2,3+4,M). M = 1+2 ; false.?- min2(3+4,1+2,m). M = 1+2.

15 Computing the minimum of two numbers Remarks The conditions X =< Y and X > Y are mutual exclusive How can we eliminate the redundancy? Attempt 1 min4 (X,Y,X) :- X =< Y. min4 (X,Y,Y). Then?- min4(1+2,3+4,m). M = 1+2 ; M = 3+4. the last answer is wrong

16 Computing the minimum of two numbers Improvement The search for a minimum can be interrupted (or cut) after detecting the first satisfiability of a goal. Example min5 (X,Y,X) :- X =< Y,!. min5 (X,Y,Y). Then?- min5(1+2,3+4,m). M = 1+2.

17 The cut operator (!) When encountered the first time it is satisfied. when encountered the second time (during backtracking) failure and blocking of all further attempts to satisfy the goal which was unified with the rule whose body contained the cut operator. Extremely useful to make the execution of logic programs more efficient.

18 The cut operator (!) Example Consider the following Prolog program: 1 girl(mary). 2 girl(ann). 3 4 boy(john). 5 boy(bill). 6 7 pair (X, Y) :- girl (X), boy(y). 8 pair (bella, harry) pair2(x, Y) :- girl (X),!, boy(y). 11 pair2(bella, harry).

19 The cut operator (!) Usage?- pair(x, Y).?- pair2(x, Y). X = mary, X = mary. Y = john ; Y = john ; X = mary, X = mary, Y = bill ; Y = bill. X = ann, Y = john ; X = ann, Y = bill ; X = bella, Y = harry.

20 Negation as failure nott (P) :- P,!, fail. nott (P). Assume P is an atom, for example boy(john) If P is satisfiable: The application of the first rule fails, because of fail in its body. The application of the second Horn clause (which is a fact) is abandoned, because! was encountered before. Outcome: nott(p) is unsatisfiable. If P is unsatisfiable: The application first rule fails before reaching! in its body. The application of the rule succeeds. Outcome: nott(p) is satisfiable.

21 Summary In this lecture we learned The structure of Prolog programs The functionality of a proof The order of evaluation The control algorithm of the proof search Techniques to control the program execution

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