984 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
Space complexity in polynomial calculus
During the last decade, an active line of research in proof complexity has been to study space
complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of
intrinsic interest, space is also an important issue in SAT solving, and so research has mostly focused
on weak systems that are used by SAT solvers.
There has been a relatively long sequence of papers on space in resolution, which is now reasonably
well understood from this point of view. For other natural candidates to study, however, such as
polynomial calculus or cutting planes, very little has been known. We are not aware of any nontrivial
space lower bounds for cutting planes, and for polynomial calculus the only lower bound has been
for CNF formulas of unbounded width in [Alekhnovich et al. ’02], where the space lower bound is
smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent
with current knowledge that polynomial calculus could be able to refute any k-CNF formula in
constant space.
In this paper, we prove several new results on space in polynomial calculus (PC), and in the
extended proof system polynomial calculus resolution (PCR) studied in [Alekhnovich et al. ’02]:
1. We prove an Ω(n) space lower bound in PC for the canonical 3-CNF version of the pigeonhole
principle formulas PHPm
n with m pigeons and n holes, and show that this is tight.
2. For PCR, we prove an Ω(n) space lower bound for a bitwise encoding of the functional pigeonhole
principle. These formulas have width O(log n), and hence this is an exponential
improvement over [Alekhnovich et al. ’02] measured in the width of the formulas.
3. We then present another encoding of the pigeonhole principle that has constant width, and
prove an Ω(n) space lower bound in PCR for these formulas as well.
4. Finally, we prove that any k-CNF formula can be refuted in PC in simultaneous exponential
size and linear space (which holds for resolution and thus for PCR, but was not obviously
the case for PC). We also characterize a natural class of CNF formulas for which the space
complexity in resolution and PCR does not change when the formula is transformed into 3-CNF
in the canonical way, something that we believe can be useful when proving PCR space lower
bounds for other well-studied formula families in proof complexity
Trading inference effort versus size in CNF Knowledge Compilation
Knowledge Compilation (KC) studies compilation of boolean functions f into
some formalism F, which allows to answer all queries of a certain kind in
polynomial time. Due to its relevance for SAT solving, we concentrate on the
query type "clausal entailment" (CE), i.e., whether a clause C follows from f
or not, and we consider subclasses of CNF, i.e., clause-sets F with special
properties. In this report we do not allow auxiliary variables (except of the
Outlook), and thus F needs to be equivalent to f.
We consider the hierarchies UC_k <= WC_k, which were introduced by the
authors in 2012. Each level allows CE queries. The first two levels are
well-known classes for KC. Namely UC_0 = WC_0 is the same as PI as studied in
KC, that is, f is represented by the set of all prime implicates, while UC_1 =
WC_1 is the same as UC, the class of unit-refutation complete clause-sets
introduced by del Val 1994. We show that for each k there are (sequences of)
boolean functions with polysize representations in UC_{k+1}, but with an
exponential lower bound on representations in WC_k. Such a separation was
previously only know for k=0. We also consider PC < UC, the class of
propagation-complete clause-sets. We show that there are (sequences of) boolean
functions with polysize representations in UC, while there is an exponential
lower bound for representations in PC. These separations are steps towards a
general conjecture determining the representation power of the hierarchies PC_k
< UC_k <= WC_k. The strong form of this conjecture also allows auxiliary
variables, as discussed in depth in the Outlook.Comment: 43 pages, second version with literature updates. Proceeds with the
separation results from the discontinued arXiv:1302.442
MaxSAT Resolution and Subcube Sums
We study the MaxRes rule in the context of certifying unsatisfiability. We
show that it can be exponentially more powerful than tree-like resolution, and
when augmented with weakening (the system MaxResW), p-simulates tree-like
resolution. In devising a lower bound technique specific to MaxRes (and not
merely inheriting lower bounds from Res), we define a new proof system called
the SubCubeSums proof system. This system, which p-simulates MaxResW, can be
viewed as a special case of the semialgebraic Sherali-Adams proof system. In
expressivity, it is the integral restriction of conical juntas studied in the
contexts of communication complexity and extension complexity. We show that it
is not simulated by Res. Using a proof technique qualitatively different from
the lower bounds that MaxResW inherits from Res, we show that Tseitin
contradictions on expander graphs are hard to refute in SubCubeSums. We also
establish a lower bound technique via lifting: for formulas requiring large
degree in SubCubeSums, their XOR-ification requires large size in SubCubeSums
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