Pseudo-Boolean formulas frequently arise in planning, scheduling and optimization problems. We describe an efficient and easily verifiable decision procedure for pseudo-Boolean formulas, that is based on encoding PB formulas into the propositional satisfiability problem with the cutting-edge sequential weighted counter encoding. State-of-the-art SAT solvers that emit unsatisfiability proofs are used to solve the resulting instances. The combination of a verified translation to SAT, and certified SAT solvers leads to a verified decision procedure for PB formulas. The verification of the encoding is carried out in the Coq proof assistant.

1. A Verified Decision Procedure for
Pseudo-Boolean Formulas
Tobias Philipp Anna Tigunova
International Center for Computational Logic
Technische Universit¨at Dresden, Germany

2. The SAT approach:
easily verifiable and efficient
E´en, S¨orensson: Translating pseudo-Boolean constraints into SAT. In: JSAT, 2006.
PB F SAT solver
SAT I solution
UNSAT P drat-trim
verified encoding
verified decision procedure for PB formulas
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3. Outline
1. Normalized pseudo-Boolean formulas
2. Sequential Weighted Counter encoding
3. Mechanical verification
4. Experimental evaluation
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4. Pseudo-Boolean constraints
n
∑
i=1
wi · xi ¡ k ¡ ∈ {=, ≤, <, >, ≥}
Normal form
¡ =≤
1 ≤ k < n,
1 ≤ wi ≤ k for all i ∈ {1, . . . , n}
no variable occurs more than once.
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5. Encodings
F encodes G
1. F |= G
2. for every model I of G there is model I of F st
I (x) = I(x) for all x ∈ vars(G).
Sequential Weighted Counter (SWC)
H¨olldobler et. al: A Compact Encoding of Pseudo-Boolean
Constraints into SAT. In: KI, 2012.
less clauses in 99% in the PB benchmarks 2010 and 2011
unit propagation detects inconsistencies
unit propagation maintains generalized arc consistency
easily verifiable
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6. The Sequential Weighted Counter
x1 x2 . . . xn
. . .
...
...
...
s1,1
s1,2
s1,k+1
s2,1
s2,2
s2,k+1
sn−1,1
sn−1,2
sn−1,k+1
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7. The Sequential Weighted Counter
x1 x2 . . . xn
. . .
...
...
...
s1,1
s1,2
s1,k+1
s2,1
s2,2
s2,k+1
sn−1,1
sn−1,2
sn−1,k+1
(¬si−1,j ∨ si,j) for all 2 ≤ i < n and 1 ≤ j ≤ k
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8. The Sequential Weighted Counter
x1 x2 . . . xn
. . .
...
...
...
s1,1
s1,2
s1,k+1
s2,1
s2,2
s2,k+1
sn−1,1
sn−1,2
sn−1,k+1
(¬xi ∨ si,j) for all 1 ≤ i < n, and 1 ≤ j ≤ wi
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9. The Sequential Weighted Counter
x1 x2 . . . xn
. . .
...
...
...
s1,1
s1,2
s1,k+1
s2,1
s2,2
s2,k+1
sn−1,1
sn−1,2
sn−1,k+1
(¬si−1,j ∨ ¬xi ∨ si,j+wi
) for all 2 ≤ i < n, 1 ≤ j < k − wi + 1
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10. The Sequential Weighted Counter
x1 x2 . . . xn
. . .
...
...
...
s1,1
s1,2
s1,k+1
s2,1
s2,2
s2,k+1
sn−1,1
sn−1,2
sn−1,k+1
(¬si−1,k+1−wi
∨ ¬xi) for all 2 ≤ i ≤ n
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11. Mechanical Verification
Coq as a language for formalization, specification and proofs
Theorem MainTheorem:
∀ wf : nat → nat, ∀ n k : nat,
normalized n k wf → correctly encodes (pbc wf n k) (SWC n k wf).
Extraction to Haskell
SWC and the sequential counter: 3800 lines of Coq code
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12. Evaluation: Setting
Benchmark
all small integer, linear instances from the PB competition 2016,
total: 777
Hardware
cluster with Intel E5-2670 CPUs and 2.5 GB RAM
Limits 6h wall clock time
verified normalization, lingeling, drat-trim
pbsolver more encodings, MiniSAT
sat4j SAT-based with special PB reasoning procedures
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13. Evaluation: Results
350
400
450
500
550
600
0 5000 10000 15000 20000
numberofsolvedinstances
timeout in seconds
pbsolver
sat4j
verified
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14. Conclusion
efficient and easily-verifiable decision procedure for PB formulas
using certifying SAT approach
Future work
mechanical verification of normalization
adaption to maximum satisfiability, planning
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15. A Verified Decision Procedure for
Pseudo-Boolean Formulas
Tobias Philipp Anna Tigunova
International Center for Computational Logic
Technische Universit¨at Dresden, Germany
Thank you for your attention.
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16. The Sequential Weighted Counter: Example
Consider the PB-constraint 3x1 + 2x2 + 4x3 ≤ 5, where wf is a
function giving the corresponding weights. Then
SWC(3, 5, wf) = F1 ∧ F2 ∧ F3 ∧ F4, where
F1 = (¬s1,1 ∨ s2,1) ∧ (¬s1,2 ∨ s2,2) ∧ (¬s1,3 ∨ s2,3)∧
(¬s1,4 ∨ s2,4) ∧ (¬s1,5 ∨ s2,5)
F2 = (¬x1 ∨ s1,1) ∧ (¬x1 ∨ s1,2) ∧ (¬x1 ∨ s1,3)∧
(¬x2 ∨ s2,1) ∧ (¬x2 ∨ s2,2)
F3 = (¬s1,1 ∨ ¬x2 ∨ s2,3) ∧ (¬s1,2 ∨ ¬x2 ∨ s2,4)∧
(¬s1,3 ∨ ¬x2 ∨ s2,5)
F4 = (¬s1,4 ∨ ¬x2) ∧ (¬s2,2 ∨ ¬x3)
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