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Interpolation
Craig/Binary Interpolant A ∧ B → ⊥ I
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Tree Interpolation in Vampire R egis Blanc (EPFL) Ashutosh Gupta - - PowerPoint PPT Presentation
Tree Interpolation in Vampire R egis Blanc (EPFL) Ashutosh Gupta - - PowerPoint PPT Presentation
Tree Interpolation in Vampire R egis Blanc (EPFL) Ashutosh Gupta (IST Austria) Laura Kov acs (Chalmers) Bernhard Kragl (TU Vienna) Interpolation Craig/Binary Interpolant A B I 2 / 11 Interpolation Craig/Binary
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SLIDE 3
Interpolation
Craig/Binary Interpolant A ∧ B → ⊥ I
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SLIDE 4
Interpolation
Craig/Binary Interpolant A ∧ B → ⊥ I
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SLIDE 5
Interpolation
Craig/Binary Interpolant A ∧ B → ⊥ I
I ∈ L(A) ∩ L(B)
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SLIDE 6
Interpolation
Craig/Binary Interpolant A ∧ B → ⊥ I
I ∈ L(A) ∩ L(B)
Sequence Interpolant A1 ∧ A2 ∧ A3 ∧ . . . ∧ An → ⊥
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SLIDE 7
Interpolation
Craig/Binary Interpolant A ∧ B → ⊥ I
I ∈ L(A) ∩ L(B)
Sequence Interpolant A1 ∧ A2 ∧ A3 ∧ . . . ∧ An → ⊥ I1 I2 . . . In−1
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SLIDE 8
Interpolation
Craig/Binary Interpolant A ∧ B → ⊥ I
I ∈ L(A) ∩ L(B)
Sequence Interpolant A1 ∧ A2 ∧ A3 ∧ . . . ∧ An → ⊥ I1 I2 . . . In−1
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SLIDE 9
Interpolation
Craig/Binary Interpolant A ∧ B → ⊥ I
I ∈ L(A) ∩ L(B)
Sequence Interpolant A1 ∧ A2 ∧ A3 ∧ . . . ∧ An → ⊥ → I1 I2 . . . In−1
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SLIDE 10
Interpolation
Craig/Binary Interpolant A ∧ B → ⊥ I
I ∈ L(A) ∩ L(B)
Sequence Interpolant A1 ∧ A2 ∧ A3 ∧ . . . ∧ An → ⊥ → → I1 I2 . . . In−1
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SLIDE 11
Interpolation
Craig/Binary Interpolant A ∧ B → ⊥ I
I ∈ L(A) ∩ L(B)
Sequence Interpolant A1 ∧ A2 ∧ A3 ∧ . . . ∧ An → ⊥ → → → I1 I2 . . . In−1
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SLIDE 12
Interpolation
Craig/Binary Interpolant A ∧ B → ⊥ I
I ∈ L(A) ∩ L(B)
Sequence Interpolant A1 ∧ A2 ∧ A3 ∧ . . . ∧ An → ⊥ → → → I1 I2 . . . In−1
I1 ∈ L(A1) ∩ L(A2, . . . , An) I2 ∈ L(A1, A2) ∩ L(A3, . . . , An)In−1 ∈ L(A1, . . . , An−1) ∩ L(An)
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Tree interpolation
A0 ∧ A1 ∧ A2 ∧ A3 ∧ A4 → ⊥ A4 A3 A2 A1 A0
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Tree interpolation
A0 ∧ A1 ∧ A2 ∧ A3 ∧ A4 → ⊥ A4 I4 A3 I3 A2 I2 A1 I1 A0 ⊥
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Tree interpolation
A0 ∧ A1 ∧ A2 ∧ A3 ∧ A4 → ⊥ A4 I4 A3 I3 A2 I2 A1 I1 A0 ⊥ node ∧ child Itps → Itp A4 → I4 A3 → I3 A1 → I1 A2 ∧ I4 ∧ I3 → I2 A0 ∧ I2 ∧ I1 → ⊥
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Tree interpolation
A0 ∧ A1 ∧ A2 ∧ A3 ∧ A4 → ⊥ A4 I4 A3 I3 A2 I2 A1 I1 A0 ⊥ node ∧ child Itps → Itp A4 → I4 A3 → I3 A1 → I1 A2 ∧ I4 ∧ I3 → I2 A0 ∧ I2 ∧ I1 → ⊥ Language restrictions
I4 ∈ L(A4) ∩ L(A3, A2, A1, A0) I3 ∈ L(A3) ∩ L(A4, A2, A1, A0) I2 ∈ L(A4, A3, A2) ∩ L(A1, A0) I1 ∈ L(A1) ∩ L(A3, A2, A1, A0)
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Related Work
Solving recursion-free Horn clauses
[Gupta, Popeea, Rybalchenko POPL ’11]
Interpolants for procedure summarization
[McMillan, Rybalchenko MSR-TR’13]
Generalized property directed reachability
[Hoder, Bjørner SAT’12]
Interpolation and Horn Clauses
[Hojjat, R¨ ummer, Kuncak CAV’13]
Nested Interpolants
[Heizmann, Hoenicke, Podelski POPL ’10] and many more ...
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Important questions
Do interpolants always exist?
Yes, in first-order logic (also with respect to a theory)
Is a logic closed under interpolation? (e.g. quantifier free fragments)
Not necessarily, consider a = 2b + 1 ∧ a = 2c over Z
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Important questions
Do interpolants always exist?
Yes, in first-order logic (also with respect to a theory)
Is a logic closed under interpolation? (e.g. quantifier free fragments)
Not necessarily, consider a = 2b + 1 ∧ a = 2c over Z
How to interpolate efficiently? How to obtain “good” interpolants?
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SLIDE 20
Proof-based interpolation
Refutations (should) capture the cause of unsatisfiability Extensive literature on interpolant extraction for various theories
☞ Notion of local proof
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SLIDE 21
Proof-based interpolation
Refutations (should) capture the cause of unsatisfiability Extensive literature on interpolant extraction for various theories
☞ Notion of local proof
Consider a = b ∧ b = c and c = d ∧ a = d
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Proof-based interpolation
Refutations (should) capture the cause of unsatisfiability Extensive literature on interpolant extraction for various theories
☞ Notion of local proof
Consider a = b ∧ b = c and c = d ∧ a = d
a = b b = c c = d b = d a = b a = d ⊥
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SLIDE 23
Proof-based interpolation
Refutations (should) capture the cause of unsatisfiability Extensive literature on interpolant extraction for various theories
☞ Notion of local proof
Consider a = b ∧ b = c and c = d ∧ a = d
a = b b = c c = d b = d a = b a = d ⊥ a = b b = c a = c c = d a = d a = d ⊥
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SLIDE 24
Proof-based interpolation
Refutations (should) capture the cause of unsatisfiability Extensive literature on interpolant extraction for various theories
☞ Notion of local proof
Consider a = b ∧ b = c and c = d ∧ a = d
a = b b = c c = d b = d a = b a = d ⊥ a = b b = c a = c c = d a = d a = d ⊥
interpolant: a = c
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Vampire
Vampire is one of the best first-order theorem provers Recent developments/extensions:
Invariant generation [FASE’09, MICAI’11] Interpolation and Symbol Elimination [CADE’09, IJCAR’10] Interpolant minimization & Theory independent proof localization [POPL ’12] Incremental tree interpolation [today]
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SLIDE 26
Incremental tree interpolation
Visit tree nodes in topological order Per node: partition the tree and compute binary interpolant Crucial: reuse previously computed interpolants
L(X1, X2, X3, X4, X5) ∩ {a, b, c, d} = ∅
a = b ∧ X1 b = c ∧ X2 X3 a = d ∧ X4 c = d ∧ X5
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SLIDE 27
Incremental tree interpolation
Visit tree nodes in topological order Per node: partition the tree and compute binary interpolant Crucial: reuse previously computed interpolants
L(X1, X2, X3, X4, X5) ∩ {a, b, c, d} = ∅
a = b ∧ X1 b = c ∧ X2 X3 a = d ∧ X4 c = d ∧ X5
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SLIDE 28
Incremental tree interpolation
Visit tree nodes in topological order Per node: partition the tree and compute binary interpolant Crucial: reuse previously computed interpolants
L(X1, X2, X3, X4, X5) ∩ {a, b, c, d} = ∅
a = b ∧ X1 a = b b = c ∧ X2 X3 a = d ∧ X4 c = d ∧ X5
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SLIDE 29
Incremental tree interpolation
Visit tree nodes in topological order Per node: partition the tree and compute binary interpolant Crucial: reuse previously computed interpolants
L(X1, X2, X3, X4, X5) ∩ {a, b, c, d} = ∅
a = b ∧ X1 a = b b = c ∧ X2 X3 a = d ∧ X4 c = d ∧ X5
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SLIDE 30
Incremental tree interpolation
Visit tree nodes in topological order Per node: partition the tree and compute binary interpolant Crucial: reuse previously computed interpolants
L(X1, X2, X3, X4, X5) ∩ {a, b, c, d} = ∅
a = b ∧ X1 a = b b = c ∧ X2 b = c X3 a = d ∧ X4 c = d ∧ X5
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SLIDE 31
Incremental tree interpolation
Visit tree nodes in topological order Per node: partition the tree and compute binary interpolant Crucial: reuse previously computed interpolants
L(X1, X2, X3, X4, X5) ∩ {a, b, c, d} = ∅
a = b ∧ X1 a = b b = c ∧ X2 b = c X3 a = d ∧ X4 c = d ∧ X5
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SLIDE 32
Incremental tree interpolation
Visit tree nodes in topological order Per node: partition the tree and compute binary interpolant Crucial: reuse previously computed interpolants
L(X1, X2, X3, X4, X5) ∩ {a, b, c, d} = ∅
a = b ∧ X1 a = b b = c ∧ X2 b = c X3 a = c a = d ∧ X4 c = d ∧ X5
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SLIDE 33
Incremental tree interpolation
Visit tree nodes in topological order Per node: partition the tree and compute binary interpolant Crucial: reuse previously computed interpolants
L(X1, X2, X3, X4, X5) ∩ {a, b, c, d} = ∅
a = b ∧ X1 a = b b = c ∧ X2 b = c X3 a = c a = d ∧ X4 c = d ∧ X5
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SLIDE 34
Incremental tree interpolation
Visit tree nodes in topological order Per node: partition the tree and compute binary interpolant Crucial: reuse previously computed interpolants
L(X1, X2, X3, X4, X5) ∩ {a, b, c, d} = ∅
a = b ∧ X1 a = b b = c ∧ X2 b = c X3 a = c a = d ∧ X4 a = d c = d ∧ X5
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SLIDE 35
Incremental tree interpolation
Visit tree nodes in topological order Per node: partition the tree and compute binary interpolant Crucial: reuse previously computed interpolants
L(X1, X2, X3, X4, X5) ∩ {a, b, c, d} = ∅
a = b ∧ X1 a = b b = c ∧ X2 b = c X3 a = c a = d ∧ X4 a = d c = d ∧ X5 ⊥
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SLIDE 36
Tool usage
n1 : a = b n2 : b = c n3 : ⊤ n4 : a = d n5 : c = d
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SLIDE 37
Tool usage
n1 : a = b n2 : b = c n3 : ⊤ n4 : a = d n5 : c = d Tree interpolation problem in SMT-LIB 1.2 syntax using iZ3 convention ... :assumption (implies (and (= a b) ) n1 ) :assumption (implies (and (= b c) ) n2 ) :assumption (implies (and n1 n2 true ) n3 ) :assumption (implies (and (not (= a d)) ) n4 ) :formula (implies (and n3 n4 (= c d) ) false )
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SLIDE 38
Tool usage
n1 : a = b n2 : b = c n3 : ⊤ n4 : a = d n5 : c = d Tree interpolation problem in SMT-LIB 1.2 syntax using iZ3 convention ... :assumption (implies (and (= a b) ) n1 ) :assumption (implies (and (= b c) ) n2 ) :assumption (implies (and n1 n2 true ) n3 ) :assumption (implies (and (not (= a d)) ) n4 ) :formula (implies (and n3 n4 (= c d) ) false )
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Tool usage
n1 : a = b n2 : b = c n3 : ⊤ n4 : a = d n5 : c = d > vampire --show_interpolant tree x.smt Parsing SMTLIB file: x.smt Parsing terminated. Building Tree. Building Tree terminated. n1: (= a b) n2: (= b c) n3: (= a c) n4: (not (= a d)) Tree interpolation problem in SMT-LIB 1.2 syntax using iZ3 convention ... :assumption (implies (and (= a b) ) n1 ) :assumption (implies (and (= b c) ) n2 ) :assumption (implies (and n1 n2 true ) n3 ) :assumption (implies (and (not (= a d)) ) n4 ) :formula (implies (and n3 n4 (= c d) ) false )
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SLIDE 40
Evaluation
Quantifier-free benchmarks
175 QF AUFLIA problems from model checking Windows device drivers (90 nodes on average)
Quantified benchmarks
4 small AUFLIA problems
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Evaluation
Quantifier-free benchmarks
175 QF AUFLIA problems from model checking Windows device drivers (90 nodes on average)
Quantified benchmarks
4 small AUFLIA problems
Vampire/Vampire 101 141 Vampire/Z3 113 iZ3 175
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SLIDE 42
Evaluation
Quantifier-free benchmarks
175 QF AUFLIA problems from model checking Windows device drivers (90 nodes on average)
Quantified benchmarks
4 small AUFLIA problems
Vampire/Vampire 101 141 Vampire/Z3 113 iZ3 175
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SLIDE 43
Evaluation
Quantifier-free benchmarks
175 QF AUFLIA problems from model checking Windows device drivers (90 nodes on average)
Quantified benchmarks
4 small AUFLIA problems
Vampire/Vampire 101 141 Vampire/Z3 113 iZ3 175 b > a c > b ∧ A[c] = 5 sorted array(A) ∧ A[a] = 10
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SLIDE 44
Evaluation
Quantifier-free benchmarks
175 QF AUFLIA problems from model checking Windows device drivers (90 nodes on average)
Quantified benchmarks
4 small AUFLIA problems
Vampire/Vampire 101 141 Vampire/Z3 113 iZ3 175 b > a b > a c > b ∧ A[c] = 5 ¬∀i A[i] ≤ 10 → i ≤ b sorted array(A) ∧ A[a] = 10 ⊥
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SLIDE 45
Evaluation
Quantifier-free benchmarks
175 QF AUFLIA problems from model checking Windows device drivers (90 nodes on average)
Quantified benchmarks
4 small AUFLIA problems
Vampire/Vampire 101 141 Vampire/Z3 113 iZ3 175 Vampire/Vampire 4 iZ3 1
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SLIDE 46
Conclusion
Tree interpolation in Vampire Strength: reasoning with quantifiers Challenges: Theory specific reasoning Visit, try, utilize! http://vprover.org/tree_itp
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Conclusion
Tree interpolation in Vampire Strength: reasoning with quantifiers Challenges: Theory specific reasoning Visit, try, utilize! http://vprover.org/tree_itp
Thank you!
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