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Logic Programming

Explore the practical applications of logic programming and automated reasoning, including first-order logic, monotonicity, resolution-based automated reasoning, and negation as finite failure.

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Logic Programming

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  1. Logic Programming Automated Reasoning in practice

  2. First order logic is monotone: T ’ |= T F2 F1 F3 G • But: people seldom reason monotonically ! Birds fly + Fred is a bird Fred flies + Fred is a penguin Motivation: monotonicity

  3. Default reasoning: • Is a form of reasoning that we’d like to use permanently • otherwise the rules are far too complex! • Is usually supported by hierarchy, inheritance and exceptions in OOP • also an early AI-formalism • CANNOT be expressed in FOL • independent on the knowledge representation !! • CAN be expressed in some FOL extensions • non-monotone logics • the simplest one of those is … …

  4. Logic Programming • Resolution-based automated reasoning: • restricted to Horn clauses • restricted to backwards linear resolution • BUT: with 3 important new extensions: • return Answer Substitutions • Least-model semanticsinstead of standard FOL model semantics • Extending Horn clause logic with Negation as Finite Failure

  5. Answer substitutions The link to programming

  6. anc(x,y)  parent(x,y) (1) anc(x,y)  parent(x,z)  anc(z,y) (2) parent(A,B) (3) parent(B,C) (4) false  anc(u,v) (2) {u/x1,v/y1} false  parent(x1,z1)  anc(z1,y1) Answer: Yes, u v anc(u,v) (3) {x1/A,z1/B} false  anc(B,y1) (1) {x2/B, y2/y1} false  parent(B,y1) (4) {y1/C} false  Answer substitutions false  anc(u,v) That is: u = A and v = C (composition of all mgu’s applied to the goal variables)

  7. anc(x,y)  parent(x,y) (1) anc(x,y)  parent(x,z)  anc(z,y) (2) parent(A,B) (3) parent(B,C) (4) false  anc(u,v) (1) {u/x1,v/y1} false  parent(x1,y1) (3) {x1/A,y1/B} false  (4) {x1/B,y1/C} false  And computes ALL answers false  anc(u,v) The third answer: u = B and v = C Another answer: u = A and v = B

  8. By computing answer substitutions Logic Programming serves as a de basis for a number of “general purpose” programming languages. Logic PROGRAMMING • Prolog, Mercury, XSB, … • with an efficiency comparable with C ! • and even faster for some programs.

  9. Example arithmetic: double_plus_1(x,y)  y is 2*x + 1 false  double_plus_1(3,z) Yes: z=7 false  double_plus_1(2,5) Yes • Example lists: append([], list, list)  append([x|list1], list2, [x|list3])  append(list1, list2, list3) false  append([1,2], [3,4,5], z) Yes: z= [1,2,3,4,5] false  append([1,2], y, [1,2,3]) Yes: y= [3] false  append(x, y, [1,2]) Yes: x = [], y = [1,2] x = [1], y = [2] … Practical programming?

  10. Least model semantics Specify in a more compact way

  11. Example: a database: celebrity(Crabé) celebrity(Jambers) celebrity(Peeters) celebrity(Lisa) celebrity(Tieleman) celebrity(Samson) • Is DeSchreye a celebrity ?? false  celebrity(DeSchreye) Least model semantics • We get no proof of inconsistency ! • FOL semantics says:celebrity(DeSchreye)is not logically entailed, thus: we do not know if it is true or not! • Least model semantics says: ~ celebrity(DeSchreye)

  12. What are the atomic consequences of theory T? model 2 T model 3 ~s ~r p model 1 q ~q s Formally: the idea • In FOL: Consequences are in the intersection: p en ~r. We do not know anything about the truth of q and s. • In LP: Consequences are in the intersection: p en ~r. Other predicates are NOT true: ~q and ~s.

  13. The logic program: celebrity(Crabé) celebrity(Jambers) celebrity(Peeters) celebrity(Lisa) celebrity(Tieleman) celebrity(Samson) • is equivalent to the infinite FOL theory: celebrity(Crabé) celebrity(Jambers) celebrity(Peeters) ~celebrity(DeSchreye) ~celebrity(Janssens) … celebrity(Lisa) celebrity(Tieleman) celebrity(Samson) ~celebrity(Cobain) ~celebrity(Dali) … • or also to: x celebrity(x)  (x = Crabé)  (x = Jambers)  (x = Peeters)  (x = Lisa)  ( x = Tieleman)  (x = Samson) Relation with FOL

  14. The “closed” assumption • Logic programming provides a compact way to express ‘complete knowledge’ on some subject. • If you do not say that something is true, than it is false. • The Closed World Assumption ! • (= everything not entailed by the theory is false) • In other words: Logic Programming supports formulating definitionsof the concepts. • Not just state what is true about these concepts (=FOL) !

  15. How relevant is the change of semantics? • In FOL: • {smart(Kelly)}implies neitherstrong(Kelly)nor~strong(Kelly) • In LP: {smart(Kelly)}implies~strong(Kelly) • In particular: LP is a non-monotone logic !! • In {smart(Kelly), strong(Kelly)},~strong(Kelly) is no longer entailed. • Knowledge is differently represented in these 2 formalisms. • Also: some concepts can be completely axiomatized in LP but not in FOL. • Ex.: the natural numbers !

  16. Negation as finite failure

  17. Here: Introduce negations in bodies ! Negation as finite failure • The basic idea: • extending the representation power of Logic Programming beyond the Horn Clauses logic • How? • equivalent: • allow disjunctions in the heads • allow negation before the body atoms • both give complete predicate logic ! • (but: with the least model semantics we will get something different from FOL!)

  18. not(B) means: • If all attempts to prove Busing linear LP-resolution, • fail in a finite time, conclude than not(B) Meaning of negation as finite failure • Not the meaning of standard negation • This is meaningful only with the least model semantics (where everything that is not proven to be ‘true’, is considered to be ‘false’)

  19. anc(x,y)  parent(x,y) (1) anc(x,y)  parent(x,z)  anc(z,y) (2) parent(A,B) (3) parent(B,C) (4) false  anc(u,v) false  anc(John,B) (2) {x/John,y/B} (1) {x/John,y/B} false  parent(John,B) false  parent(John,z)  anc(z,B) fails fails The ancestor example • Try to prove that “anc(John,B)” holds! • Conclusion: not anc(John,B)

  20. false  odd(s(s(s(0)))) false  even(s(s(s(0)))) false  not even(s(s(s(0)))) {x/s(0)} false  even(s(0)) Proof foreven(s(s(s(0)))) fails: conclusion not even(s(s(s(0)))) fails false  Another example even(0) even(s(s(x)))  even(x) odd(y)  not even(y) false  odd(s(s(s(0))))

  21. false  p false  q false  not q false  q Proof forq goes into infinite derivation: no conclusion for q … no answer And another example q  q p  not q false  p • But ~q is true according to the least model semantics!

  22. Also known: isa(Fred,Bird), Prove that:x locomotion(Fred,x) false <- locomotion(Fred,x) {x/Fly} false <- abnormal1(Fred) false <- Default reasoning in LP (1): locomotion(x,Fly) isa(x,Bird), not abnormal1(x) locomotion(x,Walk)  isa(x,Ostrich), not abnormal2(x) isa(x,Bird)  isa(x,Ostrich) abnormal1(x)  isa(x,Ostrich) false <- isa(Fred,Bird), not abnormal1(Fred) false <- not abnormal1(Fred) false <- isa(Fred,Ostrich) fails

  23. Also known: isa(Fred,Ostrich), Prove that:x locomotion(Fred,x) false <- locomotion(Fred,x) {x/Fly} false <- abnormal1(Fred) fails (for this branch) backtracking: the 2nd branch Default reasoning in LP (2): locomotion(x,Fly) isa(x,Bird), not abnormal1(x) locomotion(x,Walk)  isa(x,Ostrich), not abnormal2(x) isa(x,Bird)  isa(x,Ostrich) abnormal1(x)  isa(x,Ostrich) isa(Fred,Bird) false <- isa(Fred,Bird), not abnormal1(Fred) false <- not abnormal1(Fred) false <- isa(Fred,Ostrich) false <-

  24. locomotion(x,Fly) isa(x,Bird), not abnormal1(x) locomotion(x,Walk)  isa(x,Ostrich), not abnormal2(x) isa(x,Bird)  isa(x,Ostrich) abnormal1(x)  isa(x,Ostrich) isa(Fred,Bird) Also known: isa(Fred,Ostrich), Prove that:x locomotion(Fred,x) false <- locomotion(Fred,x) {x/Walk} false <- abnormal2(Fred) false <- Default reasoning (3): false <- isa(Fred,Ostrich), not abnormal2(Fred) false <- not abnormal2(Fred) fails

  25. Prolog • A specific programming language based on LP. • Uses a depth-first strategy to search linear resolution proofs. • incomplete • can get stuck in infinite branches • Has a bunch of built-in predicates (sometimes without logical meaning) for: • Numerical computations, input-output, changing the search strategy, meta-programming, etc. • More recent LP languages: Goedel, Mercury, Hal, ..

  26. Beyond FOL and Logic Programming • Logic Programming is very useful if you have a COMPLETE knowledge over your predicates • FOL is very useful if your knowledge is INCOMPLETE • Combine ! • Open Logic Programming • LP-definitions for the part for which you have a complete knowledge, • FOL formulae for the rest.

  27. Constraint Logic Programming • Integrate constraint processing techniques (consistency, forward checking, looking ahead, …) with Logic Programming. • Advantages of Logic for knowledge representation • Advantages of Constraint solving for efficient problem solving • A number of languages: CHIP, Eclipse, Sicsus, etc.

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