191 research outputs found
Resolution over Linear Equations and Multilinear Proofs
We develop and study the complexity of propositional proof systems of varying
strength extending resolution by allowing it to operate with disjunctions of
linear equations instead of clauses. We demonstrate polynomial-size refutations
for hard tautologies like the pigeonhole principle, Tseitin graph tautologies
and the clique-coloring tautologies in these proof systems. Using the
(monotone) interpolation by a communication game technique we establish an
exponential-size lower bound on refutations in a certain, considerably strong,
fragment of resolution over linear equations, as well as a general polynomial
upper bound on (non-monotone) interpolants in this fragment.
We then apply these results to extend and improve previous results on
multilinear proofs (over fields of characteristic 0), as studied in
[RazTzameret06]. Specifically, we show the following:
1. Proofs operating with depth-3 multilinear formulas polynomially simulate a
certain, considerably strong, fragment of resolution over linear equations.
2. Proofs operating with depth-3 multilinear formulas admit polynomial-size
refutations of the pigeonhole principle and Tseitin graph tautologies. The
former improve over a previous result that established small multilinear proofs
only for the \emph{functional} pigeonhole principle. The latter are different
than previous proofs, and apply to multilinear proofs of Tseitin mod p graph
tautologies over any field of characteristic 0.
We conclude by connecting resolution over linear equations with extensions of
the cutting planes proof system.Comment: 44 page
Extending SMTCoq, a Certified Checker for SMT (Extended Abstract)
This extended abstract reports on current progress of SMTCoq, a communication
tool between the Coq proof assistant and external SAT and SMT solvers. Based on
a checker for generic first-order certificates implemented and proved correct
in Coq, SMTCoq offers facilities both to check external SAT and SMT answers and
to improve Coq's automation using such solvers, in a safe way. Currently
supporting the SAT solver zChaff, and the SMT solver veriT for the combination
of the theories of congruence closure and linear integer arithmetic, SMTCoq is
meant to be extendable with a reasonable amount of effort: we present work in
progress to support the SMT solver CVC4 and the theory of bit vectors.Comment: In Proceedings HaTT 2016, arXiv:1606.0542
Splitting Proofs for Interpolation
We study interpolant extraction from local first-order refutations. We
present a new theoretical perspective on interpolation based on clearly
separating the condition on logical strength of the formula from the
requirement on the com- mon signature. This allows us to highlight the space of
all interpolants that can be extracted from a refutation as a space of simple
choices on how to split the refuta- tion into two parts. We use this new
insight to develop an algorithm for extracting interpolants which are linear in
the size of the input refutation and can be further optimized using metrics
such as number of non-logical symbols or quantifiers. We implemented the new
algorithm in first-order theorem prover VAMPIRE and evaluated it on a large
number of examples coming from the first-order proving community. Our
experiments give practical evidence that our work improves the state-of-the-art
in first-order interpolation.Comment: 26th Conference on Automated Deduction, 201
New Results on Cutting Plane Proofs for Horn Constraint Systems
In this paper, we investigate properties of cutting plane based refutations for a class of integer programs called Horn constraint systems (HCS). Briefly, a system of linear inequalities A * x >= b is called a Horn constraint system, if each entry in A belongs to the set {0,1,-1} and furthermore there is at most one positive entry per row. Our focus is on deriving refutations i.e., proofs of unsatisfiability of such programs using cutting planes as a proof system. We also look at several properties of these refutations. Horn constraint systems can be considered as a more general form of propositional Horn formulas, i.e., CNF formulas with at most one positive literal per clause. Cutting plane calculus (CP) is a well-known calculus for deciding the unsatisfiability of propositional CNF formulas and integer programs. Usually, CP consists of a pair of inference rules. These are called the addition rule (ADD) and the division rule (DIV). In this paper, we show that cutting plane calculus is still complete for Horn constraints when every intermediate constraint is required to be Horn. We also investigate the lengths of cutting plane proofs for Horn constraint systems
Narrow Proofs May Be Maximally Long
We prove that there are 3-CNF formulas over n variables that can be refuted
in resolution in width w but require resolution proofs of size n^Omega(w). This
shows that the simple counting argument that any formula refutable in width w
must have a proof in size n^O(w) is essentially tight. Moreover, our lower
bound generalizes to polynomial calculus resolution (PCR) and Sherali-Adams,
implying that the corresponding size upper bounds in terms of degree and rank
are tight as well. Our results do not extend all the way to Lasserre, however,
where the formulas we study have proofs of constant rank and size polynomial in
both n and w
Verified AIG Algorithms in ACL2
And-Inverter Graphs (AIGs) are a popular way to represent Boolean functions
(like circuits). AIG simplification algorithms can dramatically reduce an AIG,
and play an important role in modern hardware verification tools like
equivalence checkers. In practice, these tricky algorithms are implemented with
optimized C or C++ routines with no guarantee of correctness. Meanwhile, many
interactive theorem provers can now employ SAT or SMT solvers to automatically
solve finite goals, but no theorem prover makes use of these advanced,
AIG-based approaches.
We have developed two ways to represent AIGs within the ACL2 theorem prover.
One representation, Hons-AIGs, is especially convenient to use and reason
about. The other, Aignet, is the opposite; it is styled after modern AIG
packages and allows for efficient algorithms. We have implemented functions for
converting between these representations, random vector simulation, conversion
to CNF, etc., and developed reasoning strategies for verifying these
algorithms.
Aside from these contributions towards verifying AIG algorithms, this work
has an immediate, practical benefit for ACL2 users who are using GL to
bit-blast finite ACL2 theorems: they can now optionally trust an off-the-shelf
SAT solver to carry out the proof, instead of using the built-in BDD package.
Looking to the future, it is a first step toward implementing verified AIG
simplification algorithms that might further improve GL performance.Comment: In Proceedings ACL2 2013, arXiv:1304.712
ADCL: Acceleration Driven Clause Learning for Constrained Horn Clauses
Constrained Horn Clauses (CHCs) are often used in automated program
verification. Thus, techniques for (dis-)proving satisfiability of CHCs are a
very active field of research. On the other hand, acceleration techniques for
computing formulas that characterize the N-fold closure of loops have
successfully been used for static program analysis. We show how to use
acceleration to avoid repeated derivations with recursive CHCs in resolution
proofs, which reduces the length of the proofs drastically. This idea gives
rise to a novel calculus for (dis)proving satisfiability of CHCs, called
Acceleration Driven Clause Learning (ADCL). We implemented this new calculus in
our tool LoAT and evaluate it empirically in comparison to other
state-of-the-art tools
On the Relative Strength of Pebbling and Resolution
The last decade has seen a revival of interest in pebble games in the context
of proof complexity. Pebbling has proven a useful tool for studying
resolution-based proof systems when comparing the strength of different
subsystems, showing bounds on proof space, and establishing size-space
trade-offs. The typical approach has been to encode the pebble game played on a
graph as a CNF formula and then argue that proofs of this formula must inherit
(various aspects of) the pebbling properties of the underlying graph.
Unfortunately, the reductions used here are not tight. To simulate resolution
proofs by pebblings, the full strength of nondeterministic black-white pebbling
is needed, whereas resolution is only known to be able to simulate
deterministic black pebbling. To obtain strong results, one therefore needs to
find specific graph families which either have essentially the same properties
for black and black-white pebbling (not at all true in general) or which admit
simulations of black-white pebblings in resolution. This paper contributes to
both these approaches. First, we design a restricted form of black-white
pebbling that can be simulated in resolution and show that there are graph
families for which such restricted pebblings can be asymptotically better than
black pebblings. This proves that, perhaps somewhat unexpectedly, resolution
can strictly beat black-only pebbling, and in particular that the space lower
bounds on pebbling formulas in [Ben-Sasson and Nordstrom 2008] are tight.
Second, we present a versatile parametrized graph family with essentially the
same properties for black and black-white pebbling, which gives sharp
simultaneous trade-offs for black and black-white pebbling for various
parameter settings. Both of our contributions have been instrumental in
obtaining the time-space trade-off results for resolution-based proof systems
in [Ben-Sasson and Nordstrom 2009].Comment: Full-length version of paper to appear in Proceedings of the 25th
Annual IEEE Conference on Computational Complexity (CCC '10), June 201
Small Proofs from Congruence Closure
Satisfiability Modulo Theory (SMT) solvers and equality saturation engines
must generate proof certificates from e-graph-based congruence closure
procedures to enable verification and conflict clause generation. Smaller proof
certificates speed up these activities. Though the problem of generating proofs
of minimal size is known to be NP-complete, existing proof minimization
algorithms for congruence closure generate unnecessarily large proofs and
introduce asymptotic overhead over the core congruence closure procedure. In
this paper, we introduce an O(n^5) time algorithm which generates optimal
proofs under a new relaxed "proof tree size" metric that directly bounds proof
size. We then relax this approach further to a practical O(n \log(n)) greedy
algorithm which generates small proofs with no asymptotic overhead. We
implemented our techniques in the egg equality saturation toolkit, yielding the
first certifying equality saturation engine. We show that our greedy approach
in egg quickly generates substantially smaller proofs than the state-of-the-art
Z3 SMT solver on a corpus of 3760 benchmarks
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