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Logic Programming Automated Reasoning in practice

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

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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 … …

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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 semantics instead of standard FOL model semantics Extending Horn clause logic with Negation as Finite Failure

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Answer substitutions The link to programming

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

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

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

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

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Least model semantics Specify in a more compact way

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11 Least model semantics Example: a database: celebrity(Crabé)celebrity(Jambers)celebrity(Peeters) celebrity(Lisa)celebrity(Tieleman)celebrity(Samson) Is DeSchreye a celebrity ?? false celebrity(DeSchreye) 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)

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

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

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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. In other words: Logic Programming supports formulating definitions of the concepts. Not just state what is true about these concepts (=FOL) ! The Closed World Assumption ! (= everything not entailed by the theory is false)

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15 How relevant is the change of semantics? In FOL: {smart(Kelly)} implies neither strong(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 !

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Negation as finite failure

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17 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!) Here: Introduce negations in bodies !

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

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19 Try to prove that “anc(John,B)” holds! false anc(John,B) The ancestor example 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) Conclusion: not anc(John,B) false parent(John,B) (1) {x/John,y/B}fails false parent(John,z) anc(z,B) (2) {x/John,y/B}fails

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

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

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22 {x/Fly} false <- locomotion(Fred,x) false <- abnormal1(Fred) 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) Also known: isa(Fred,Bird), Prove that: x locomotion(Fred,x) false <- isa(Fred,Bird), not abnormal1(Fred) false <- not abnormal1(Fred) false <- fails false <- isa(Fred,Ostrich)

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

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

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25 A specific programming language based on LP. Prolog 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,..

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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.

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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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