186 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
Generalising unit-refutation completeness and SLUR via nested input resolution
We introduce two hierarchies of clause-sets, SLUR_k and UC_k, based on the
classes SLUR (Single Lookahead Unit Refutation), introduced in 1995, and UC
(Unit refutation Complete), introduced in 1994.
The class SLUR, introduced in [Annexstein et al, 1995], is the class of
clause-sets for which unit-clause-propagation (denoted by r_1) detects
unsatisfiability, or where otherwise iterative assignment, avoiding obviously
false assignments by look-ahead, always yields a satisfying assignment. It is
natural to consider how to form a hierarchy based on SLUR. Such investigations
were started in [Cepek et al, 2012] and [Balyo et al, 2012]. We present what we
consider the "limit hierarchy" SLUR_k, based on generalising r_1 by r_k, that
is, using generalised unit-clause-propagation introduced in [Kullmann, 1999,
2004].
The class UC, studied in [Del Val, 1994], is the class of Unit refutation
Complete clause-sets, that is, those clause-sets for which unsatisfiability is
decidable by r_1 under any falsifying assignment. For unsatisfiable clause-sets
F, the minimum k such that r_k determines unsatisfiability of F is exactly the
"hardness" of F, as introduced in [Ku 99, 04]. For satisfiable F we use now an
extension mentioned in [Ansotegui et al, 2008]: The hardness is the minimum k
such that after application of any falsifying partial assignments, r_k
determines unsatisfiability. The class UC_k is given by the clause-sets which
have hardness <= k. We observe that UC_1 is exactly UC.
UC_k has a proof-theoretic character, due to the relations between hardness
and tree-resolution, while SLUR_k has an algorithmic character. The
correspondence between r_k and k-times nested input resolution (or tree
resolution using clause-space k+1) means that r_k has a dual nature: both
algorithmic and proof theoretic. This corresponds to a basic result of this
paper, namely SLUR_k = UC_k.Comment: 41 pages; second version improved formulations and added examples,
and more details regarding future directions, third version further examples,
improved and extended explanations, and more on SLUR, fourth version various
additional remarks and editorial improvements, fifth version more
explanations and references, typos corrected, improved wordin
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
Recommended from our members
Proof Complexity and Beyond
Proof complexity is a multi-disciplinary intellectual endeavor that addresses questions of the general form “how difficult is it to prove certain mathematical facts?” The current workshop focused on recent advances in our understanding of logic-based proof systems and on connections to algorithms, geometry and combinatorics research, such as the analysis of approximation algorithms, or the size of linear or semidefinite programming formulations of combinatorial optimization problems, to name just two important examples
Regular resolution for CNF of bounded incidence treewidth with few long clauses
We demonstrate that Regular Resolution is FPT for two restricted families of
CNFs of bounded incidence treewidth. The first includes CNFs having at most
clauses whose removal results in a CNF of primal treewidth at most . The
parameters we use in this case are and . The second class includes CNFs
of bounded one-sided (incidence) treewdth, a new parameter generalizing both
primal treewidth and incidence pathwidth. The parameter we use in this case is
the one-sided treewidth
On SAT representations of XOR constraints
We study the representation of systems S of linear equations over the
two-element field (aka xor- or parity-constraints) via conjunctive normal forms
F (boolean clause-sets). First we consider the problem of finding an
"arc-consistent" representation ("AC"), meaning that unit-clause propagation
will fix all forced assignments for all possible instantiations of the
xor-variables. Our main negative result is that there is no polysize
AC-representation in general. On the positive side we show that finding such an
AC-representation is fixed-parameter tractable (fpt) in the number of
equations. Then we turn to a stronger criterion of representation, namely
propagation completeness ("PC") --- while AC only covers the variables of S,
now all the variables in F (the variables in S plus auxiliary variables) are
considered for PC. We show that the standard translation actually yields a PC
representation for one equation, but fails so for two equations (in fact
arbitrarily badly). We show that with a more intelligent translation we can
also easily compute a translation to PC for two equations. We conjecture that
computing a representation in PC is fpt in the number of equations.Comment: 39 pages; 2nd v. improved handling of acyclic systems, free-standing
proof of the transformation from AC-representations to monotone circuits,
improved wording and literature review; 3rd v. updated literature,
strengthened treatment of monotonisation, improved discussions; 4th v. update
of literature, discussions and formulations, more details and examples;
conference v. to appear LATA 201
Parameterized complexity of DPLL search procedures
We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game which models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a k-clique requires n steps for a non-trivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked in [11] of understanding the Resolution complexity of this family of formulas
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