Four Flavors of Entailment ohle 1 , Roberto Sebastiani 2 , and Armin - - PowerPoint PPT Presentation
Four Flavors of Entailment ohle 1 , Roberto Sebastiani 2 , and Armin Biere 1 Sibylle M 2 Department of Information Engineering 1 Institute for Formal Models and Verification LIT Secure and Correct Systems Lab and Computer Science The 23rd
1Institute for Formal Models and Verification
LIT Secure and Correct Systems Lab
2Department of Information Engineering
and Computer Science The 23rd International Conference on Theory and Applications of Satisfiability Testing (SAT 2020) 3 – 10 July 2020
. . . short (partial) models model shrinking (Tibebu and Fey, DDECS’18) dual reasoning (M¨
logical entailment (Sebastiani, arXiv.org, 2020) ➤ l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models model shrinking (Tibebu and Fey, DDECS’18) dual reasoning (M¨
logical entailment (Sebastiani, arXiv.org, 2020) ➤ l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models model shrinking (Tibebu and Fey, DDECS’18) dual reasoning (M¨
logical entailment (Sebastiani, arXiv.org, 2020) ➤ l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models model shrinking (Tibebu and Fey, DDECS’18) dual reasoning (M¨
logical entailment (Sebastiani, arXiv.org, 2020) ➤ l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models model shrinking (Tibebu and Fey, DDECS’18) dual reasoning (M¨
logical entailment (Sebastiani, arXiv.org, 2020) Example F = (x ∧ y) ∨ (x ∧ ¬y) F|x = y ∨ ¬y = 1 F|xy = F|x¬y = 1 = ⇒ x | = F ➤ l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models model shrinking (Tibebu and Fey, DDECS’18) dual reasoning (M¨
logical entailment (Sebastiani, arXiv.org, 2020) Example F = (x ∧ y) ∨ (x ∧ ¬y) F|x = y ∨ ¬y = 1 F|xy = F|x¬y = 1 = ⇒ x | = F But determining logical entailment is harder than it seems! ➤ l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models . . . pairwise disjoint models add the negated models as blocking clauses variant of conflict analysis (Toda and Soh, ACM J. Exp. Algorithmics, 2016) chronological CDCL (Nadel and Ryvchin, SAT’18; M¨
➤ l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models . . . pairwise disjoint models add the negated models as blocking clauses variant of conflict analysis (Toda and Soh, ACM J. Exp. Algorithmics, 2016) chronological CDCL (Nadel and Ryvchin, SAT’18; M¨
➤ l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models . . . pairwise disjoint models add the negated models as blocking clauses variant of conflict analysis (Toda and Soh, ACM J. Exp. Algorithmics, 2016) chronological CDCL (Nadel and Ryvchin, SAT’18; M¨
➤ l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models . . . pairwise disjoint models add the negated models as blocking clauses variant of conflict analysis (Toda and Soh, ACM J. Exp. Algorithmics, 2016) chronological CDCL (Nadel and Ryvchin, SAT’18; M¨
➤ l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models . . . pairwise disjoint models l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models . . . pairwise disjoint models . . . projection F(X, Y ) where X ∩ Y = ∅ X relevant variables Y irrelevant variables ∃Y [ F(X, Y ) ] project F(X, Y ) onto X ➤ l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models . . . pairwise disjoint models . . . projection l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models . . . pairwise disjoint models . . . projection l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g
2
. . . short (partial) models . . . pairwise disjoint models . . . projection
. . . Disjoint Sum-of-Products (DSOP) ➤ l
i c a l e n t a i l m e n t c h r
i c a l C D C L p r
e c t i
d u a l r e a s
i n g DSOP
2
Formula F SAT solver Check assignment DSOP M (Partial) Assignment I Next assignment
3
Formula F SAT solver Check assignment DSOP M (Partial) Assignment I Next assignment F|I = 1 F|I ≈ 1 F|I ≡ 1 ∀X∃Y [F|I ] = 1 ➤
3
➤ Given F formula over variables in X ∪ Y I trail over variables in X ∪ Y
4
➤
F formula over variables in X ∪ Y I trail over variables in X ∪ Y
In ϕ = ∀X∀Y [ F|I ] the unassigned variables in X ∪ Y are quantified ϕ = 1: all possible total extensions of I satisfy F
4
➤
F formula over variables in X ∪ Y I trail over variables in X ∪ Y
In ϕ = ∀X∀Y [ F|I ] the unassigned variables in X ∪ Y are quantified ϕ = 1: all possible total extensions of I satisfy F
Does for each JX exist one JY such that F|I ′ = 1 where I ′ = I ∪ JX ∪ JY?
4
➤
F formula over variables in X ∪ Y I trail over variables in X ∪ Y
In ϕ = ∀X∀Y [ F|I ] the unassigned variables in X ∪ Y are quantified ϕ = 1: all possible total extensions of I satisfy F
Does for each JX exist one JY such that F|I ′ = 1 where I ′ = I ∪ JX ∪ JY? QBF(ϕ) = 1 where ϕ = ∀X∃Y [F|I ] = 1?
4
➤ 1) F|I = 1 (syntactic check) F = (x1 ∨ y ∨ x2) X = {x1, x2} Y = {y} I = x1: F|I = 1 = ⇒ I | = F
5
➤ 1) F|I = 1 (syntactic check) 2) F|I ≈ 1 (incomplete check in P) F = x1y ∨ yx2 X = {x1, x2} Y = {y} I = x1x2: F|I = y ∨ y = 1 but is valid I = x1x2y: 0 ∈ BCP(¬F, I) = ⇒ x1x2 | = F
5
➤ 1) F|I = 1 (syntactic check) 2) F|I ≈ 1 (incomplete check in P) 3) F|I ≡ 1 (semantic check in coNP) F = x1(x2 y ∨ x2y ∨ x2y ∨ x2y) X = {x1, x2} Y = {y} I = x1: I(F) = x2 y ∨ x2y ∨ x2y ∨ x2y = 1 but is valid P = CNF(F) N = CNF(¬F): P|I and N|I are non-constant and contain no units N|I = (x2 ∨ y)(x2 ∨ y)(x2 ∨ y)(x2 ∨ y): SAT(N ∧ I) = 0 = ⇒ I | = F
5
➤ 1) F|I = 1 (syntactic check) 2) F|I ≈ 1 (incomplete check in P) 3) F|I ≡ 1 (semantic check in coNP) 4) ∀X∃Y [F|I ] = 1 (check in ΠP
2 )
F = x1(x2 ↔ y2) X = {x1, x2} Y = {y2} P = CNF(F) and N = CNF(¬F): P = (x1)(s1 ∨ s2)(s1 ∨ x2)(s1 ∨ y2)(s2 ∨ x2)(s2 ∨ y2) where S = {s1, s2} N = (x1 ∨ t1 ∨ t2)(t1 ∨ x2)(t1 ∨ y2)(t2 ∨ x2)(t2 ∨ y2) where T = {t1, t2} I = x1: P|I and N|I are non-constant and contain no units I = x1t2t1y2 : N|I = 1 ϕ = ∀X∃Y [ x2y2 ∨ x2 y2 ]: QBF(ϕ) = 1 = ⇒ x1 | = F
5
Input: formula F(X, Y ) over variables X ∪ Y such that X ∩ Y = ∅, trail I, decision level function δ Output: DNF M consisting of models of F projected onto X
Enumerate( F )
1 I := ε; δ := ∞; M := 0 2 forever do 3
C := PropagateUnits( F, I, δ )
4
if C = 0 then
5
c := δ(C)
6
if c = 0 then return M
7
AnalyzeConflict( F, I, C, c )
8
else if all variables in X ∪ Y are assigned then
9
if V (decs(I)) ∩ X = ∅ then return M ∨ π(I, X)
10
M := M ∨ π(I, X)
11
b := δ(decs(π(I, X)))
12
Backtrack( I, b − 1 )
13
else if Entails( I, F ) then
14
if V (decs(I)) ∩ X = ∅ then return M ∨ π(I, X)
14
M := M ∨ π(I, X)
15
b := δ(decs(π(I, X)))
16
Backtrack( I, b − 1 )
17
else Decide( I, δ )
6
7
Method for computing partial assignments entailing the formula on-the-fly Inspired by the interaction of theory and SAT solvers in SMT Combines dual reasoning and chronological CDCL Algorithm (in the paper) Formalization (in the paper) Entailment test in four flavors of increasing strength F|I = 1 (syntactic check) F|I ≈ 1 (incomplete check in P) F|I ≡ 1 (semantic check in coNP) ∀X∃Y [F|I ] = 1 (check in ΠP
2 )
8
Method for computing partial assignments entailing the formula on-the-fly Inspired by the interaction of theory and SAT solvers in SMT Combines dual reasoning and chronological CDCL Algorithm (in the paper) Formalization (in the paper) Entailment test in four flavors of increasing strength F|I = 1 (syntactic check) F|I ≈ 1 (incomplete check in P) F|I ≡ 1 (semantic check in coNP) ∀X∃Y [F|I ] = 1 (check in ΠP
2 )
Implement and validate our method Target weighted model integration and model counting with or without projection Investigate methods concerning the implementation of QBF oracles
Dependency schemes (Samer and Szeider, JAR, 2009) Incremental QBF (Lonsing and Egly, CP’14)
Combine with decomposition-based approaches and generate d-DNNF
8