205 research outputs found
Tight Size-Degree Bounds for Sums-of-Squares Proofs
We exhibit families of -CNF formulas over variables that have
sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank)
but require SOS proofs of size for values of from
constant all the way up to for some universal constant.
This shows that the running time obtained by using the Lasserre
semidefinite programming relaxations to find degree- SOS proofs is optimal
up to constant factors in the exponent. We establish this result by combining
-reductions expressible as low-degree SOS derivations with the
idea of relativizing CNF formulas in [Kraj\'i\v{c}ek '04] and [Dantchev and
Riis'03], and then applying a restriction argument as in [Atserias, M\"uller,
and Oliva '13] and [Atserias, Lauria, and Nordstr\"om '14]. This yields a
generic method of amplifying SOS degree lower bounds to size lower bounds, and
also generalizes the approach in [ALN14] to obtain size lower bounds for the
proof systems resolution, polynomial calculus, and Sherali-Adams from lower
bounds on width, degree, and rank, respectively
Partial Quantifier Elimination By Certificate Clauses
We study partial quantifier elimination (PQE) for propositional CNF formulas.
In contrast to full quantifier elimination, in PQE, one can limit the set of
clauses taken out of the scope of quantifiers to a small subset of target
clauses. The appeal of PQE is twofold. First, PQE can be dramatically simpler
than full quantifier elimination. Second, it provides a language for performing
incremental computations. Many verification problems (e.g. equivalence checking
and model checking) are inherently incremental and so can be solved in terms of
PQE. Our approach is based on deriving clauses depending only on unquantified
variables that make the target clauses . Proving redundancy
of a target clause is done by construction of a ``certificate'' clause implying
the former. We describe a PQE algorithm called that employs
the approach above. We apply to generating properties of a
design implementation that are not implied by specification. The existence of
an property means that this implementation is buggy. Our
experiments with HWMCC-13 benchmarks suggest that can be used
for generating properties of real-life designs
Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation
We present an efficient proof system for Multipoint Arithmetic Circuit
Evaluation: for every arithmetic circuit of size and
degree over a field , and any inputs ,
the Prover sends the Verifier the values and a proof of length, and
the Verifier tosses coins and can check the proof in about time, with probability of error less than .
For small degree , this "Merlin-Arthur" proof system (a.k.a. MA-proof
system) runs in nearly-linear time, and has many applications. For example, we
obtain MA-proof systems that run in time (for various ) for the
Permanent, Circuit-SAT for all sublinear-depth circuits, counting
Hamiltonian cycles, and infeasibility of - linear programs. In general,
the value of any polynomial in Valiant's class can be certified
faster than "exhaustive summation" over all possible assignments. These results
strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed
by Russell Impagliazzo and others.
We also give a three-round (AMA) proof system for quantified Boolean formulas
running in time, nearly-linear time MA-proof systems for
counting orthogonal vectors in a collection and finding Closest Pairs in the
Hamming metric, and a MA-proof system running in -time for
counting -cliques in graphs.
We point to some potential future directions for refuting the
Nondeterministic Strong ETH.Comment: 17 page
A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds
We study the problem of obtaining lower bounds for polynomial calculus (PC)
and polynomial calculus resolution (PCR) on proof degree, and hence by
[Impagliazzo et al. '99] also on proof size. [Alekhnovich and Razborov '03]
established that if the clause-variable incidence graph of a CNF formula F is a
good enough expander, then proving that F is unsatisfiable requires high PC/PCR
degree. We further develop the techniques in [AR03] to show that if one can
"cluster" clauses and variables in a way that "respects the structure" of the
formula in a certain sense, then it is sufficient that the incidence graph of
this clustered version is an expander. As a corollary of this, we prove that
the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree
when restricted to constant-degree expander graphs. This answers an open
question in [Razborov '02], and also implies that the standard CNF encoding of
the FPHP formulas require exponential proof size in polynomial calculus
resolution. Thus, while Onto-FPHP formulas are easy for polynomial calculus, as
shown in [Riis '93], both FPHP and Onto-PHP formulas are hard even when
restricted to bounded-degree expanders.Comment: Full-length version of paper to appear in Proceedings of the 30th
Annual Computational Complexity Conference (CCC '15), June 201
Strongly Refuting Random CSPs Below the Spectral Threshold
Random constraint satisfaction problems (CSPs) are known to exhibit threshold
phenomena: given a uniformly random instance of a CSP with variables and
clauses, there is a value of beyond which the CSP will be
unsatisfiable with high probability. Strong refutation is the problem of
certifying that no variable assignment satisfies more than a constant fraction
of clauses; this is the natural algorithmic problem in the unsatisfiable regime
(when ).
Intuitively, strong refutation should become easier as the clause density
grows, because the contradictions introduced by the random clauses become
more locally apparent. For CSPs such as -SAT and -XOR, there is a
long-standing gap between the clause density at which efficient strong
refutation algorithms are known, , and the
clause density at which instances become unsatisfiable with high probability,
.
In this paper, we give spectral and sum-of-squares algorithms for strongly
refuting random -XOR instances with clause density in time or in
rounds of the sum-of-squares hierarchy, for any
and any integer . Our algorithms provide a smooth
transition between the clause density at which polynomial-time algorithms are
known at , and brute-force refutation at the satisfiability
threshold when . We also leverage our -XOR results to obtain
strong refutation algorithms for SAT (or any other Boolean CSP) at similar
clause densities. Our algorithms match the known sum-of-squares lower bounds
due to Grigoriev and Schonebeck, up to logarithmic factors.
Additionally, we extend our techniques to give new results for certifying
upper bounds on the injective tensor norm of random tensors
Complexity of Propositional Proofs Under a Promise
Abstract. We study – within the framework of propositional proof complexity – the problem of certifying unsatisfiability of CNF formulas under the promise that any satisfiable formula has many satisfying assignments, where “many ” stands for an explicitly specified function Λ in the number of variables n. To this end, we develop propositional proof systems under different measures of promises (that is, different Λ) as extensions of resolution. This is done by augmenting resolution with axioms that, roughly, can eliminate sets of truth assignments defined by Boolean circuits. We then investigate the complexity of such systems, obtaining an exponential separation in the average-case between resolution under different size promises: (1) Resolution has polynomial-size refutations for all unsatisfiable 3CNF formulas when the promise is ε·2n, for any constant 0 < ε < 1. (2) There are no sub-exponential size resolution refutations for random 3CNF formulas, when the promise is 2δn (and the number of clauses is o(n3/2)), for any constan
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