Motivation PSAT oPSAT Application Conclusion Solutions for Hard and Soft Constraints Using Optimized Probabilistic Satisfiability Marcelo Finger 1 , 2 , Ronan Le Bras 1 , Carla P. Gomes 1 , Bart Selman 1 1 Department of Computer Science, Cornell University 2 On leave from: Department of Computer Science Institute of Mathematics and Statistics University of Sao Paulo, Brazil July 2013 Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion Topics 1 Motivation 2 Probabilistic Satisfiability 3 Optimizing Probability Distributions with oPSAT 4 oPSAT and Combinatorial Materials Discovery 5 Conclusions Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion Next Topic 1 Motivation 2 Probabilistic Satisfiability 3 Optimizing Probability Distributions with oPSAT 4 oPSAT and Combinatorial Materials Discovery 5 Conclusions Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion Motivation Practical problems combine real-world hard constraints with soft constraints Soft constraints: preferences, uncertainties, flexible requirements We explore probabilistic logic as a mean of dealing with combined soft and hard constraints Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion Goals Aim: Combine Logic and Probabilistic reasoning to deal with Hard ( L ) and Soft ( P ) constraints Method: develop optimized Probabilistic Satisfiability ( oPSAT ) Application: Demonstrate effectiveness on a real-world reasoning task in the domain of Materials Discovery. Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion An Example Summer course enrollment m students and k summer courses. Potential team mates, to develop coursework. Constraints: Hard Coursework to be done alone or in pairs. Students must enroll in at least one and at most three courses. There is a limit of ℓ students per course. Soft Avoid having students with no teammate. Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion An Example Summer course enrollment m students and k summer courses. Potential team mates, to develop coursework. Constraints: Hard Coursework to be done alone or in pairs. Students must enroll in at least one and at most three courses. There is a limit of ℓ students per course. Soft Avoid having students with no teammate. In our framework: P (student with no team mate) “minimal” or bounded Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion Combining Logic and Probability Many proposals in the literature Markov Logic Networks [Richardson & Domingos 2006] Probabilistic Inductive Logic Prog [De Raedt et. al 2008] Relational Models [Friedman et al 1999], etc Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion Combining Logic and Probability Many proposals in the literature Markov Logic Networks [Richardson & Domingos 2006] Probabilistic Inductive Logic Prog [De Raedt et. al 2008] Relational Models [Friedman et al 1999], etc Our choice: Probabilistic Satisfiability (PSAT) Natural extension of Boolean Logic Desirable properties, e.g. respects Kolmogorov axioms Probabilistic reasoning free of independence presuppositions Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion Combining Logic and Probability Many proposals in the literature Markov Logic Networks [Richardson & Domingos 2006] Probabilistic Inductive Logic Prog [De Raedt et. al 2008] Relational Models [Friedman et al 1999], etc Our choice: Probabilistic Satisfiability (PSAT) Natural extension of Boolean Logic Desirable properties, e.g. respects Kolmogorov axioms Probabilistic reasoning free of independence presuppositions What is PSAT? Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion Next Topic 1 Motivation 2 Probabilistic Satisfiability 3 Optimizing Probability Distributions with oPSAT 4 oPSAT and Combinatorial Materials Discovery 5 Conclusions Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion Is PSAT a Zombie Idea? An idea that refuses to die! Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion A Brief History of PSAT Proposed by [Boole 1854], On the Laws of Thought Rediscovered several times since Boole De Finetti [1937, 1974], Good [1950], Smith [1961] Studied by Hailperin [1965] Nilsson [1986] (re)introduces PSAT to AI PSAT is NP-complete [Georgakopoulos et. al 1988] Nilsson [1993]: “complete impracticability” of PSAT computation Many other works; see Hansen & Jaumard [2000] Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion Or a Wild Amazonian Flower? Awaits special conditions to bloom! (Linear programming + SAT-based techniques) Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion The Setting Formulas α 1 , . . . , α ℓ over logical variables P = { x 1 , . . . , x n } Propositional valuation v : P → { 0 , 1 } Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion The Setting Formulas α 1 , . . . , α ℓ over logical variables P = { x 1 , . . . , x n } Propositional valuation v : P → { 0 , 1 } A probability distribution over propositional valuations π : V → [0 , 1] 2 n � π ( v i ) = 1 i =1 Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion The Setting Formulas α 1 , . . . , α ℓ over logical variables P = { x 1 , . . . , x n } Propositional valuation v : P → { 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 } Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion The PSAT Problem Consider ℓ formulas α 1 , . . . , α ℓ defined on n atoms { x 1 , . . . , x n } A PSAT problem Σ is a set of ℓ restrictions Σ = { P ( α i ) � p i | 1 ≤ i ≤ ℓ } Probabilistic Satisfiability: is there a π that satisfies Σ? Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion The PSAT Problem Consider ℓ formulas α 1 , . . . , α ℓ defined on n atoms { x 1 , . . . , x n } A PSAT problem Σ is a set of ℓ restrictions Σ = { P ( α i ) � p i | 1 ≤ i ≤ ℓ } Probabilistic Satisfiability: is there a π that satisfies Σ? In our framework, ℓ = m + k , Σ = Γ ∪ Ψ: Hard Γ = { α 1 , . . . , α m } , P ( α i ) = 1 (clauses) Soft Ψ = { P ( s i ) ≤ p i | 1 ≤ i ≤ k } s i atomic; p i given, learned or minimized Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion Example continued Only one course, three student enrollments: x , y and z Potential partnerships: p xy and p xz , mutually exclusive. Hard constraint P ( x ∧ y ∧ z ∧ ¬ ( p xy ∧ p xz )) = 1 Soft constraints P ( x ∧ ¬ p xy ∧ ¬ p xz ) ≤ 0 . 25 P ( y ∧ ¬ p xy ) ≤ 0 . 60 P ( z ∧ ¬ p xz ) ≤ 0 . 60 Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion Example continued Only one course, three student enrollments: x , y and z Potential partnerships: p xy and p xz , mutually exclusive. Hard constraint P ( x ∧ y ∧ z ∧ ¬ ( p xy ∧ p xz )) = 1 Soft constraints P ( x ∧ ¬ p xy ∧ ¬ p xz ) ≤ 0 . 25 P ( y ∧ ¬ p xy ) ≤ 0 . 60 P ( z ∧ ¬ p xz ) ≤ 0 . 60 (Small) solution: distribution π π ( x , y , z , ¬ p xy , ¬ p xz ) = 0 . 1 π ( x , y , z , p xy , ¬ p xz ) = 0 . 4 π ( x , y , z , ¬ p xy , p xz ) = 0 . 5 π ( v ) = 0 for other 29 valuations Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion Solving PSAT Algebraic formulation for Γ(¯ s , ¯ x ) ∪ { P ( s i ) = p i | 1 ≤ i ≤ k } : find A ( k +1) × 2 n a { 0 , 1 } -matrix, π 2 n × 1 ≥ 0 such that � 1 � � A π = π j = 1 , 1st line: p if π j > 0 then column A j is Γ-consistent. Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
Motivation PSAT oPSAT Application Conclusion Solving PSAT Algebraic formulation for Γ(¯ s , ¯ x ) ∪ { P ( s i ) = p i | 1 ≤ i ≤ k } : find A ( k +1) × 2 n a { 0 , 1 } -matrix, π 2 n × 1 ≥ 0 such that � 1 � � A π = π j = 1 , 1st line: p if π j > 0 then column A j is Γ-consistent. Solved by linear program (exponentially sized) c ′ π minimize A π = p and π ≥ 0 subject to c : cost vector, c j = 1 if A j is Γ-inconsistent; c j = 0 otherwise Solution when c ′ π = 0 (may not be unique) Finger, Le Bras, Gomes, Selman Cornell/USP HardSoft & PSAT
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