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- 1. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References Probabilistic Abductive Logic Programming using Possible Worlds Fulvio Rotella1 and Stefano Ferilli1,2 {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 2013 25 September 2013 ,Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 2. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References Motivation Artiﬁcial Intelligence: two approaches Numerical/statistical Relational Strengths and weaknesses Numerical/statistical + handle amount of data + handle incompleteness and uncertainty - ﬂat representations - no relationships between objects/attributes Relational + complex representations of data + comprehensibility - no incompleteness - no noise and uncertainty Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 3. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References 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] Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 4. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References 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) Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 5. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References 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 [Arvanitis et al., 2006] Implementing Probabilistic Abductive Logic Programming with Constraint Handling Rules [Christiansen, 2008] Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 6. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References 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 Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 7. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References 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? Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 8. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References 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 looks for the minimal explanation handles crisp nand-constraint PALP looks for the most probable explanation handles probabilistic typed constraint Prob, Literals, Type : Prob = [0, 1] , Type = {nand, or, xor}, Literals = l1, ...., ln Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 9. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References 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 Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 10. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References 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) Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 11. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References Logical perspective Example (Observation o1, Query and Possible Explanations) P : {printable(X) ← a4(X), text(X)} ∪ a4(o1) A = {image, text, black_white, printable, table, a4, a5, a3} I = {ic2, ic3, ic4} ic2 = 0.9, [table(X), text(X), image(X)], or ic3 = 0.3, [text(X), color(X)], nand ic4 = 0.3, [table(X), color(X)], nand ?- printable(o1) printable(o1) ← a4(o1), text(o1) ∆1 = {text(o1), table(o1)} ∆2 = {text(o1), table(o1), image(o1)} text(o1) table(o1) . table(o1) image(o1) . Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 12. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References Probabilistic perspective The chance of being true of a ground literal δj (1). The unnormalized probability of the abductive explanation (2). P(δj ) = n(δj ) n(cons)! (n(cons)−a(δj))! (1) P′ (∆i ,Ici ) = J j=1 P(δj ) ∗ K k=1 P(ick ) (2) The probability of δj is equal to 1 − P(δj ). ∆ = {P1 : (∆1, Ic1), ..., PT : (∆T , IcT )}, T consistent possible worlds for goal G ∆i = {δ1, ..., δJ }, the ground literals δj abduced in an abductive proof Ici = {ic1, ..., icK } is the set of the constraints involved in ∆i n(δj ) true groundings of the predicate used in literal δj n(cons) is total number of constants encountered in the world a(δj ) is the arity of literal δj P(ick ) is the probability of the kth-constraint. Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 13. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References Probabilistic perspective Example (Compute explanations probability ) P′ (∆1,Ic1) = P(text(o1)) ∗ P(table(o1)) ∗ P(ic2) ∗ P(ic3) ∗ P(ic4) P′ (∆1,Ic1) = 0.6 ∗ 0.1 ∗ 0.9 ∗ 0.3 ∗ 0.3 = 0.00486 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, Ic1) = 0.00486 P′(∆2, Ic2) = 0.00875 P′(printable(o1)) = max1≤i≤T P′ i : (∆i , Ici ) = P′(∆2, Ic2) = 0.00875 Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 14. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References 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(ProbLiti , c, e) < P_max(¬c, e), ∆n >← probabilistic_abductive_proof(ProbLiti , ¬c, e) compute the higher between them selects the best classiﬁcation between all concepts Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 15. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References 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 Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 16. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References 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 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.30.40.50.60.70.80.91.0 Corruption Accuracy Breast Cancer Congress Tic Tac Toe Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 17. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References Results and Discussion Dataset Corr. Abductive Reas. Deductive Reas. Prec. Rec. F1 Prec. Rec. F1 Breast 0% 0.891 0.870 0.881 0.891 0.870 0.881 10% 0.865 0.835 0.850 0.634 0.454 0.227 20% 0.853 0.411 0.556 0.571 0.118 0.195 30% 0.800 0.188 0.584 0.500 0.029 0.056 40% 1.000 0.059 0.111 —– —– —– 50% 1.000 0.035 0.068 —– —– —– 60% 1.000 0.023 0.046 —– —– —– 70% 1.000 0.012 0.023 —– —– —– Congress 0% 1.000 0.961 0.980 1.000 0.961 0.980 10% 1.000 0.961 0.981 0.971 0.793 0.873 20% 1.000 0.769 0.869 0.971 0.761 0.853 30% 1.000 0.680 0.809 0.982 0.714 0.827 40% 1.000 0.538 0.700 0.979 0.623 0.761 50% 1.000 0.500 0.667 1.000 0.425 0.596 60% 1.000 0.346 0.514 1.000 0.333 0.500 70% 1.000 0.269 0.424 1.000 0.264 0.418 TikTakToe 0% 1.000 0.983 0.992 1.000 0.983 0.992 10% 1.000 0.833 0.909 0.842 0.743 0.789 20% 1.000 0.730 0.844 0.808 0.531 0.641 30% 1.000 0.508 0.673 0.796 0.387 0.521 40% 1.000 0.302 0.463 0.829 0.261 0.397 50% 1.000 0.127 0.225 0.697 0.103 0.180 60% 1.000 0.048 0.090 0.777 0.031 0.060 70% 1.000 0.016 0.031 1.000 0.004 0.009Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 18. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References 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. Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 19. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References Thanks for attention Questions? fulvio.rotella@uniba.it Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 20. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References 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, 2006. H. Christiansen. Implementing probabilistic abductive logic programming with constraint handling rules. In T. Schrijvers and T. FrÃ1 4 hwirth, editors, Constraint Handling Rules, volume 5388 of Lecture Notes in Computer Science, pages 85–118. Springer Berlin Heidelberg, 2008. ISBN 978-3-540-92242-1. doi: 10.1007/978-3-540-92243-8_5. URL http://dx.doi.org/10.1007/978-3-540-92243-8_5. F. Esposito, G. Semeraro, N. Fanizzi, and S. Ferilli. Multistrategy theory revision: Induction and abduction in inthelex. Machine Learning, 38:133–156, 2000. ISSN 0885-6125. doi: 10.1023/A:1007638124237. URL http://dx.doi.org/10.1023/A%3A1007638124237. 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 103–120, Berlin, Heidelberg, 2005. Springer-Verlag. ISBN 3-540-28177-0, 978-3-540-28177-1. doi: 10.1007/11536314_7. URL http://dx.doi.org/10.1007/11536314_7. Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds
- 21. Introduction Probabilistic Abductive Logic Programming Experimental Evaluation Conclusions References References II L. C. Getoor. Learning statistical models from relational data. PhD thesis, Stanford, CA, USA, 2002. AAI3038093. A. C. Kakas and P. Mancarella. Generalized stable models: A semantics for abduction. In ECAI, pages 385–391, 1990. A. C. Kakas and F. Riguzzi. Abductive concept learning. New Generation Comput., 18 (3):243–294, 2000. 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 2009. URL http://www.cs.utexas.edu/users/ai-lab/?kate:pair09. D. Poole. Probabilistic horn abduction and bayesian networks. Artif. Intell., 64(1): 81–129, 1993. L. D. Raedt and K. Kersting. Probabilistic inductive logic programming. In ALT, pages 19–36, 2004. S. V. Raghavan. Bayesian abductive logic programs: A probabilistic logic for abductive reasoning. In T. Walsh, editor, IJCAI, pages 2840–2841. IJCAI/AAAI, 2011. ISBN 978-1-57735-516-8. URL http://dblp.uni-trier.de/db/conf/ijcai/ijcai2011.html#Raghavan11. Fulvio Rotella and Stefano Ferilli DIB, CILA Probabilistic Abductive Logic Programming using Possible Worlds

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