Probabilistic Abductive Logic Programming using Possible Worlds

Transcription

1 Probabilistic Abductive Logic Programming using Possible Worlds Fulvio Rotella 1 and Stefano Ferilli 1,2 {fulvio.rotella, stefano.ferilli}@uniba.it 1 DIB Dipartimento di Informatica Università di Bari 2 CILA Centro Interdipartimentale per la Logica e sue Applicazioni Università di Bari XXVIII Convegno Italiano di Logica Computazionale - CILC September 2013 Fulvio Rotella, and Stefano Ferilli

2 Motivation Artificial Intelligence: two approaches Numerical/statistical Relational Strengths and weaknesses Numerical/statistical Relational + handle amount of data + handle incompleteness and uncertainty - flat representations - no relationships between objects/attributes + complex representations of data + comprehensibility - no incompleteness - no noise and uncertainty

3 Motivation Problem: Real World data multi-relational, heterogeneous and semi-structured noisy and uncertain Solution: Relational Representations + Probability Logic Programming representation language and reasoning strategies Probabilistic Reasoning robustness Solutions Statistical Relational Learning (SRL) [Getoor, 2002] Probabilistic Inductive Logic Programming (PILP) [Raedt and Kersting, 2004]

4 Problems : High degree of complexity lack and incompleteness of observations deductive reasoning not enough Solution: Exploit Abduction! Abductive statement: given an observation that can not be derived in the theory, make assumptions that explain it All the beans from this bag are white.(bk) These beans (oddly) are white. (observation) These beans are from this bag.(diagnosis) Logic-based approaches multiple sets of assumptions integrity constraints Probabilistic-based approaches multiple explanations with probability (uncertainty)

5 Problems Logic-based too many logical explanations Probabilistic-based independent variables and unstructured data Some solutions Probabilistic Horn Abduction and Bayesian Networks (PHA) [Poole, 1993] Bayesian Abductive Logic Programs: A Probabilistic Logic for Abductive Reasoning (BALP) [Raghavan, 2011] Probabilistic Abduction using Markov Logic Networks (MLN) [Kate and Mooney, 2009] Abduction with stochastic logic programs based on a possible worlds semantics et al., 2006] [Arvanitis Implementing Probabilistic Abductive Logic Programming with Constraint Handling Rules [Christiansen, 2008]

6 Preliminaries: Abductive Logic Programming (ALP) Abductive Logic Program T = P, A, I [Kakas and Mancarella, 1990] P is a standard logic program A (Abducibles) is a set of predicate names IC (Integrity Constraints or domain-specific properties) Problem formulation Given an observation O and a theory T = P, A, I Find an abductive explanation s.t. P = O ( explains O) and P = IC ( is consistent). T abductively entails G (T = A O). Abductive Logic Programming [Kakas and Mancarella, 1990] extends Logic Programming: some predicates (abducibles) incompletely defined deriving hypotheses on these abducible predicates (abductive hypotheses) Goal: observations to be explained

7 Preliminaries: Abductive Logic Programming (ALP) Abductive Logic proof procedure [Kakas and Riguzzi, 2000] Two phases abductive (A) and consistency derivations (B) (A) is the standard Logic derivation extended in order to consider abducibles when an atom δ has to be proved, it is added to the current set of assumptions the addition of δ must not violate any integrity constraint (B) starts to check that all integrity constraints containing δ fails (B) calls (A) to solve each goal Considerations there are constraints that prevent an abduction? constraints verification involves: facts deductively verified true hypotheses evaluating all possible explanations constraints: classical vs typed and crisp vs soft?

8 Probabilistic Abductive Logic Programming (PALP) A new approach using Possible Worlds each time one assumes something he hypothesizes that situation in a specific world each abductive explanation can be seen as a possible world likelihood assessed considering what we have seen and what we should expect to see typed probabilistic constraints: personal belief in the likelihood of whole constraint {nand, or, xor}-constraints Classical vs Probabilistic ALP ALP PALP looks for the minimal explanation handles crisp nand-constraint looks for the most probable explanation handles probabilistic typed constraint Prob, Literals, Type : Prob = [0, 1], Type = {nand, or, xor}, Literals = l 1,..., l n

9 Probabilistic Abductive Logic Programming (PALP) New probabilistic proof procedure Two perspectives: Logical exploits ALP to generate many logical explanations extends ALP to handle typed constraints Probabilistic rank all explanations according to their chance of being true

10 Logical perspective New Logical Proof Procedure extends Abductive and Consistency Derivation: Classical: when an atom δ has to be proved, it is added to the current set of assumption New: when an atom δ has to be proved, two sets of assumptions are considered: one where it holds and another where it does not. extends Consistency Derivation: integrity checking on constraints NAND,OR,XOR NAND satisfied when: at least one condition is false OR satisfied when: at least one condition is true XOR satisfied when: only one condition is true each conclusion is a possible consistent world New Approach Classical + (new rules and backtracking on each choice point)

11 Logical perspective Example (Observation o 1, Query and Possible Explanations) P : {printable(x) a4(x), text(x)} a4(o 1 ) A = {image, text, black_white, printable, table, a4, a5, a3} I = {ic 2, ic 3, ic 4 } ic 2 = 0.9,[table(X), text(x), image(x)], or ic 3 = 0.3,[text(X), color(x)], nand ic 4 = 0.3,[table(X), color(x)], nand table(o 1 ) text(o 1 ) table(o 1 )?- printable(o 1 ) printable(o 1 ) a4(o 1 ), text(o 1 ). image(o 1 ) 1 = {text(o 1 ), table(o 1 )} 2 = {text(o 1 ), table(o 1 ), image(o 1 )}.

12 Probabilistic perspective The chance of being true of a ground literal δ j (1). The unnormalized probability of the abductive explanation (2). P(δ j ) = n(δ j ) J K (1) P n(cons)! ( i,ic i ) = P(δ j ) P(ic k ) (2) (n(cons) a(δ j ))! j=1 k=1 The probability of δ j is equal to 1 P(δ j ). = {P 1 : ( 1, Ic 1 ),..., P T : ( T, Ic T )}, T consistent possible worlds for goal G i = {δ 1,...,δ J }, the ground literals δ j abduced in an abductive proof Ic i = {ic 1,..., ic K } is the set of the constraints involved in i n(δ j ) true groundings of the predicate used in literal δ j n(cons) is total number of constants encountered in the world a(δ j ) is the arity of literal δ j P(ic k ) is the probability of the kth-constraint.

13 Probabilistic perspective Example (Compute explanations probability ) P ( 1,Ic 1 ) = P(text(o 1)) P(table(o 1 )) P(ic 2 ) P(ic 3 ) P(ic 4 ) P ( 1,Ic 1 ) = = Example (Probability assessment of the Abductive Explanations) A = {0.2:image, 0.4:text, 0.1:black_white, 0.6:printable, 0.1:table, 0.9:a4, 0.1:a5, 0.1:a3} P ( 1, Ic 1 ) = P ( 2, Ic 2 ) = P (printable(o 1 )) = max 1 i T P i : ( i, Ic i ) = P ( 2, Ic 2 ) =

14 Improving Classification Exploiting Probabilistic Abductive Reasoning Exploiting our probabilistic abductive logic proof procedure learns the model (i.e. the Abductive Logic Program < P, A, IC >) and the parameters (i.e. literals probabilities) classify never-seen instances Solution: A new system for classification tasks given a Training set and a abducibles set A (possibly empty), it learns: the corresponding theory T by INTHELEX [Esposito et al., 2000] the integrity constraints nand, xor by [Ferilli et al., 2005] given a Test set, tries to cover the example considering both as positive and as negative for the class c < P_max(c, e), p > probabilistic_abductive_proof(problit i, c, e) < P_max( c, e), n > probabilistic_abductive_proof(problit i, c, e) compute the higher between them selects the best classification between all concepts

15 Experimental Settings Goal: assessing the quality of the results in presence of incomplete and noisy data comparing with deductive-reasoning with increasing levels of data corruption Methodology: 10-fold split to obtain < Train, Test > replace each test-set by corrupted versions: removed at random K% of each example (K varying from 10% to 70% with step 10) 5 runs to randomize (35 test-sets for each fold) assume learned constraints true with probability 1.0 (no prev. knowledge) Dataset: Breast-Cancer Congressional Voting Records Tic-Tac-Toe

16 Results and Discussion Breast-Cancer (#Pos = 201; #Neg: 85) Each instance: 9 literals Theory: 30 clauses; 6 lits/clause Learned IC: 1784 nand-constraints (55% -> 4, 35% -> 3 and 10% -> 2); 9 type-domain Congressional Voting Records (#Republicans = 267; #Democrats: 168) Each instance: 16 literals Theory: 35 clauses; 4.5 lits/clause Learned IC: 4173 nand-constraints (16% -> 4, 37% -> 3 and 47% -> 2); 16 type-domain Tic-Tac-Toe (#Pos = 626; #Neg: 332) Each instance: 8 literals Theory: 18 clauses; 4 lits/clause Learned IC: 1863 nand-constraints (99% -> 4, 1% -> 3); 16 type-domain Accuracy Breast Cancer Congress Tic Tac Toe Corruption

17 Results and Discussion Dataset Breast Congress TikTakToe Corr. Abductive Reas. Deductive Reas. Prec. Rec. F 1 Prec. Rec. F 1 0% % % % % % % % % % % % % % % % % % % % % % % %

18 Probabilistic Abductive Logic Approach Reasoning in complex contexts deduction is not enough. Abduction might help it should be logical + probabilistic. Our approach: Abductive Logic Programming generates multiple explanations; Probabilistic assessment of each explanation. Our strategy to classification works correctly in presence of noisy and corruption. Current and Future works Learning the probabilistic constraints. Enriching the probabilistic model of literal distribution. Test our procedure on other tasks such as: NLU and plan recognition.

19 Thanks for attention Questions?

20 References I A. Arvanitis, S. H. Muggleton, J. Chen, and H. Watanabe. Abduction with stochastic logic programs based on a possible worlds semantics. In In Short Paper Proc. of 16th ILP, H. Christiansen. Implementing probabilistic abductive logic programming with constraint handling rules. In T. Schrijvers and T. Frà 1 hwirth, editors, Constraint 4 Handling Rules, volume 5388 of Lecture Notes in Computer Science, pages Springer Berlin Heidelberg, ISBN doi: / _5. URL F. Esposito, G. Semeraro, N. Fanizzi, and S. Ferilli. Multistrategy theory revision: Induction and abduction in inthelex. Machine Learning, 38: , ISSN doi: /A: URL S. Ferilli, T. M. A. Basile, N. Di Mauro, and F. Esposito. Automatic induction of abduction and abstraction theories from observations. In Proc. of the 15th ILP, ILP 05, pages , Berlin, Heidelberg, Springer-Verlag. ISBN , doi: / _7. URL

21 References II L. C. Getoor. Learning statistical models from relational data. PhD thesis, Stanford, CA, USA, AAI A. C. Kakas and P. Mancarella. Generalized stable models: A semantics for abduction. In ECAI, pages , A. C. Kakas and F. Riguzzi. Abductive concept learning. New Generation Comput., 18 (3): , R. J. Kate and R. J. Mooney. Probabilistic abduction using markov logic networks. In Proceedings of the IJCAI-09 Workshop on Plan, Activity, and Intent Recognition (PAIR-09), Pasadena, CA, July URL D. Poole. Probabilistic horn abduction and bayesian networks. Artif. Intell., 64(1): , L. D. Raedt and K. Kersting. Probabilistic inductive logic programming. In ALT, pages 19 36, S. V. Raghavan. Bayesian abductive logic programs: A probabilistic logic for abductive reasoning. In T. Walsh, editor, IJCAI, pages IJCAI/AAAI, ISBN URL