81 research outputs found
Resolution Lower Bounds for Refutation Statements
For any unsatisfiable CNF formula we give an exponential lower bound on the
size of resolution refutations of a propositional statement that the formula
has a resolution refutation. We describe three applications. (1) An open
question in (Atserias, M\"uller 2019) asks whether a certain natural
propositional encoding of the above statement is hard for Resolution. We answer
by giving an exponential size lower bound. (2) We show exponential resolution
size lower bounds for reflection principles, thereby improving a result in
(Atserias, Bonet 2004). (3) We provide new examples of CNFs that exponentially
separate Res(2) from Resolution (an exponential separation of these two proof
systems was originally proved in (Segerlind, Buss, Impagliazzo 2004))
Resolution and the binary encoding of combinatorial principles.
Res(s) is an extension of Resolution working on s-DNFs. We prove tight n (k) lower bounds for the size
of refutations of the binary version of the k-Clique Principle in Res(o(log log n)). Our result improves that of
Lauria, Pudlák et al. [27] who proved the lower bound for Res(1), i.e. Resolution. The exact complexity of
the (unary) k-Clique Principle in Resolution is unknown. To prove the lower bound we do not use any form of
the Switching Lemma [35], instead we apply a recursive argument specific for binary encodings. Since for the
k-Clique and other principles lower bounds in Resolution for the unary version follow from lower bounds in
Res(log n) for their binary version we start a systematic study of the complexity of proofs in Resolution-based
systems for families of contradictions given in the binary encoding.
We go on to consider the binary version of the weak Pigeonhole Principle Bin-PHPmn
for m > n. Using
the the same recursive approach we prove the new result that for any > 0, Bin-PHPmn
requires proofs of size
2n1− in Res(s) for s = o(log1/2 n). Our lower bound is almost optimal since for m 2
p
n log n there are
quasipolynomial size proofs of Bin-PHPmn
in Res(log n).
Finally we propose a general theory in which to compare the complexity of refuting the binary and unary
versions of large classes of combinatorial principles, namely those expressible as first order formulae in 2-form
and with no finite model
On the complexity of resolution-based proof systems
Propositional Proof Complexity is the area of Computational Complexity that studies the length of proofs in propositional logic. One of its main questions is to determine which particular propositional formulas have short proofs in a given propositional proof system. In this thesis we present several results related to this question, all on proof systems that are extensions of the well-known resolution proof system.
The first result of this thesis is that TQBF, the problem of determining if a fully-quantified propositional CNF-formula is true, is PSPACE-complete even when restricted to instances of bounded tree-width, i.e. a parameter of structures that measures their similarity to a tree. Instances of bounded tree-width of many NP-complete problems are tractable, e.g. SAT, the boolean satisfiability problem. We show that this does not scale up to TQBF. We also consider Q-resolution, a quantifier-aware version of resolution. On the negative side, our first result implies that, unless NP = PSPACE, the class of fully-quantified CNF-formulas of bounded tree-width does not have short proofs in any proof system (and in particular in Q-resolution). On the positive side, we show that instances with bounded respectful tree-width, a more restrictive condition, do have short proofs in Q-resolution. We also give a natural family of formulas with this property that have real-world applications.
The second result concerns interpretability. Informally, we say that a first-order formula can be interpreted in another if the first one can be expressed using the vocabulary of the second, plus some extra features. We show that first-order formulas whose propositional translations have short R(const)-proofs, i.e. a generalized version of resolution with DNF-formulas of constant-size terms, are closed under a weaker form of interpretability (that with no extra features), called definability. Our main result is a similar result on interpretability. Also, we show some examples of interpretations and show a systematic technique to transform some Sigma_1-definitions into quantifier-free interpretations.
The third and final result is about a relativized weak pigeonhole principle. This says that if at least 2n out of n^2 pigeons decide to fly into n holes, then some hole must be doubly occupied. We prove that the CNF encoding of this principle does not have polynomial-size DNF-refutations, i.e. refutations in the generalized version of resolution with unbounded DNF-formulas. For this proof we discuss the existence of unbalanced low-degree bipartite expanders satisfying a certain robustness condition
From Small Space to Small Width in Resolution
In 2003, Atserias and Dalmau resolved a major open question about the
resolution proof system by establishing that the space complexity of CNF
formulas is always an upper bound on the width needed to refute them. Their
proof is beautiful but somewhat mysterious in that it relies heavily on tools
from finite model theory. We give an alternative, completely elementary proof
that works by simple syntactic manipulations of resolution refutations. As a
by-product, we develop a "black-box" technique for proving space lower bounds
via a "static" complexity measure that works against any resolution
refutation---previous techniques have been inherently adaptive. We conclude by
showing that the related question for polynomial calculus (i.e., whether space
is an upper bound on degree) seems unlikely to be resolvable by similar
methods
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