Motivation ASP PSAT Probabilistic Answer Set Programming Resolution Methods Approximations Future work Approximated Probabilistic Answer Set Programming Eduardo Menezes de Morais Department of Computer Science Universidade de S˜ ao Paulo S˜ ao Paulo, Brazil 2014 Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT Probabilistic Answer Set Programming Resolution Methods Approximations Future work Toy Example 6 4 5 2 3 1 Limit the percentage of time the edge (1 , 3) is used to 50% and the edge (3 , 4) to 40% Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT Probabilistic Answer Set Programming Resolution Methods Approximations Future work Outline Motivation 1 Answer Set Programming 2 Probabilistic Satisfiability 3 Probabilistic Answer Set Programming 4 Resolution Methods 5 Approximations 6 Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming Future work 7
Motivation ASP PSAT Probabilistic Answer Set Programming Resolution Methods Approximations Future work Next Topic Motivation 1 Answer Set Programming 2 Probabilistic Satisfiability 3 History Definition Applications Probabilistic Answer Set Programming 4 Definition Resolution Methods 5 Approximations 6 Future work 7 Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT Probabilistic Answer Set Programming Resolution Methods Approximations Future work What is ASP? Non-monotonic, declarative programming paradigm for hard combinatorial problems A ASP Program is a set of rules h ← L 1 , . . . , L m , not L m +1 , . . . , not L n The symbol not represents default negation or negation as a failure to prove Programs may have variables and functions, but must be grounded before solving Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT Probabilistic Answer Set Programming Resolution Methods Approximations Future work Constraints and Weight Rules Other types of rules: Restrictions : Rules without heads ← L 1 , . . . , L m , not L m +1 , . . . , not L n Weight rules : Rules made from weight constraints C 0 ← C 1 , . . . , C n Weight Constraints : L ≤ { h 1 = w 1 , . . . , not h n = w n } ≤ U . Weight rules make ASP Σ P 2 -complete. Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT Probabilistic Answer Set Programming Resolution Methods Approximations Future work Answer Sets For programs without default negation, the Answer Set is the minimal model that satisfies all rules For programs with default negation, maybe there is not a unique minimal model We must first assume a set a literals and them verify if this set is a minimal model of the resulting rules Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT Probabilistic Answer Set Programming Resolution Methods Approximations Future work Answer Sets Definition Let M be a finite set of atoms of P , the program P M , obtained from P by removing: all the rules that have a literal A in their negative body if A ∈ M ; the negative body of the remaining rules is called reduction of P by M . Definition Let P M be the reduction of the program P by M , M is an Answer Set of P if the minimal model of P M is M . Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT Probabilistic Answer Set Programming Resolution Methods Approximations Future work Toy Example 6 4 5 2 3 1 An ASP Program that finds all the paths from vertex 1 to n . 1 ≤ { visited ( X , Y ) = 1 for each edge ( X , Y ) } ≤ 1 ← vertex ( X ) , pathTo ( X ) . pathTo (1) . pathTo ( Y ) ← pathTo ( X ) , visited ( X , Y ) , edge ( X , Y ) . ← not pathTo ( n ) . Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT History Probabilistic Answer Set Programming Definition Resolution Methods Applications Approximations Future work Next Topic Motivation 1 Answer Set Programming 2 Probabilistic Satisfiability 3 History Definition Applications Probabilistic Answer Set Programming 4 Definition Resolution Methods 5 Approximations 6 Future work 7 Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT History Probabilistic Answer Set Programming Definition Resolution Methods Applications Approximations Future work A Brief History of PSAT Probabilistic logic was proposed in On the Laws of Thought [Boole 1854] Classical probability and classical logic No assumption of a priori statistical independence Rediscovered several times since Boole De Finetti [1937, 1974], Good [1950], Smith [1961] Studied by Hailperin [1965] Nilsson [1986] (re)introduces PSAT to AI Papadimitriou et al [1988]: NP-complete Many other works; see Hansen & Jaumard [2000] Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT History Probabilistic Answer Set Programming Definition Resolution Methods Applications Approximations Future work The Setting: the language Logical variables or atoms: P = { x 1 , . . . , x n } Connectives: ∧ , ∨ , ¬ , → , ↔ . Formulas ( L ) are inductively composed form atoms using connectives Formulas can be brought to clausal form, but need not be. Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT History Probabilistic Answer Set Programming Definition Resolution Methods Applications Approximations Future work Semantics Propositional valuation v : P → { 0 , 1 } Generalized for any propositional formula (clausal or not) v : L → { 0 , 1 } Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT History Probabilistic Answer Set Programming Definition Resolution Methods Applications Approximations Future work Semantics Propositional valuation v : P → { 0 , 1 } Generalized for any propositional formula (clausal or not) v : L → { 0 , 1 } A probability distribution over propositional valuations π : V → [0 , 1] 2 n � π ( v i ) = 1 i =1 Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT History Probabilistic Answer Set Programming Definition Resolution Methods Applications Approximations Future work Semantics Propositional valuation v : P → { 0 , 1 } Generalized for any propositional formula (clausal or not) v : L → { 0 , 1 } A probability distribution over propositional valuations π : V → [0 , 1] 2 n � π ( v i ) = 1 i =1 Probability of a formula α according to π � P π ( α ) = { π ( v i ) | v i ( α ) = 1 } Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT History Probabilistic Answer Set Programming Definition Resolution Methods Applications Approximations Future work The PSAT Problem Consider k formulas α 1 , . . . , α k defined on n atoms { x 1 , . . . , x n } A PSAT problem Σ is a set of k restrictions Σ = { P ( α i ) = p i | 1 ≤ i ≤ k } Probabilistic Satisfiability: are these restrictions consistent ? Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT History Probabilistic Answer Set Programming Definition Resolution Methods Applications Approximations Future work The PSAT Problem Consider k formulas α 1 , . . . , α k defined on n atoms { x 1 , . . . , x n } A PSAT problem Σ is a set of k restrictions Σ = { P ( α i ) = p i | 1 ≤ i ≤ k } Probabilistic Satisfiability: are these restrictions consistent ? Given Σ = { P ( φ i ) = p i | φ i ∈ L PL , 1 ≤ i ≤ q } . Is there a π such that P π ( φ i ) = p i , for 1 ≤ i ≤ q ? Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT History Probabilistic Answer Set Programming Definition Resolution Methods Applications Approximations Future work A PSAT example Is the Hypothesis Consistent with the Data The problem: how to fit precise theories with an imprecise world? Doctor investigating disease D Examine role of genes G 1 , G 2 , G 3 Hypothesis At least two genes have to be present for D to develop Data Gene occurrence in D -patients G 1 60% 60% G 2 G 3 60% Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT History Probabilistic Answer Set Programming Definition Resolution Methods Applications Approximations Future work A PSAT example � P ( a ∨ b ) = P ( a ∨ c ) = P ( b ∨ c ) = 1 � Σ = P ( a ) = P ( b ) = P ( c ) = 0 . 6 Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
Motivation ASP PSAT History Probabilistic Answer Set Programming Definition Resolution Methods Applications Approximations Future work A PSAT example � P ( a ∨ b ) = P ( a ∨ c ) = P ( b ∨ c ) = 1 � Σ = P ( a ) = P ( b ) = P ( c ) = 0 . 6 v 1 = { a = b = c = 1 } , v 2 = { a = b = 1; c = 0 } , v 3 = { a = c = 1; b = 0 } and v 4 = { a = 0; b = c = 1 } . Eduardo Menezes de Morais Approximated Probabilistic Answer Set Programming
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