79 research outputs found
Consistency of circuit lower bounds with bounded theories
Proving that there are problems in that require
boolean circuits of super-linear size is a major frontier in complexity theory.
While such lower bounds are known for larger complexity classes, existing
results only show that the corresponding problems are hard on infinitely many
input lengths. For instance, proving almost-everywhere circuit lower bounds is
open even for problems in . Giving the notorious difficulty of
proving lower bounds that hold for all large input lengths, we ask the
following question: Can we show that a large set of techniques cannot prove
that is easy infinitely often? Motivated by this and related
questions about the interaction between mathematical proofs and computations,
we investigate circuit complexity from the perspective of logic.
Among other results, we prove that for any parameter it is
consistent with theory that computational class , where is one of
the pairs: and , and , and
. In other words, these theories cannot establish
infinitely often circuit upper bounds for the corresponding problems. This is
of interest because the weaker theory already formalizes
sophisticated arguments, such as a proof of the PCP Theorem. These consistency
statements are unconditional and improve on earlier theorems of [KO17] and
[BM18] on the consistency of lower bounds with
On the proof complexity of Paris-harrington and off-diagonal ramsey tautologies
We study the proof complexity of Paris-Harrington’s Large Ramsey Theorem for bi-colorings of graphs and
of off-diagonal Ramsey’s Theorem. For Paris-Harrington, we prove a non-trivial conditional lower bound
in Resolution and a non-trivial upper bound in bounded-depth Frege. The lower bound is conditional on a
(very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in RES(2). We
show that under such an assumption, there is no refutation of the Paris-Harrington formulas of size quasipolynomial
in the number of propositional variables. The proof technique for the lower bound extends the
idea of using a combinatorial principle to blow up a counterexample for another combinatorial principle
beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced off-diagonal
Ramsey principle is established. This is obtained by adapting some constructions due to Erdos and Mills. ˝
We prove a non-trivial Resolution lower bound for a family of such off-diagonal Ramsey principles
The Cook-Reckhow definition
The Cook-Reckhow 1979 paper defined the area of research we now call Proof
Complexity. There were earlier papers which contributed to the subject as we
understand it today, the most significant being Tseitin's 1968 paper, but none
of them introduced general notions that would allow to make an explicit and
universal link between lengths-of-proofs problems and computational complexity
theory. In this note we shall highlight three particular definitions from the
paper: of proof systems, p-simulations and the pigeonhole principle formula,
and discuss their role in defining the field. We will also mention some related
developments and open problems
Monotone Proofs of the Pigeon Hole Principle
Lecture Notes in Computer Science. Geneva, Switzerland, July 9-15
Where Pigeonhole Principles meet K\"onig Lemmas
We study the pigeonhole principle for -definable injections with
domain twice as large as the codomain, and the weak K\"onig lemma for
-definable trees in which every level has at least half of the
possible nodes. We show that the latter implies the existence of -random
reals, and is conservative over the former. We also show that the former is
strictly weaker than the usual pigeonhole principle for -definable
injections.Comment: 33 page
Higher uniformity of bounded multiplicative functions in short intervals on average
Let denote the Liouville function. We show that, as ,
for all fixed and with fixed
but arbitrarily small. Previously this was only established for . We
obtain this result as a special case of the corresponding statement for
(non-pretentious) -bounded multiplicative functions that we prove. In fact,
we are able to replace the polynomial phases by degree
nilsequences . By the inverse theory for the Gowers
norms this implies the higher order asymptotic uniformity result
in the same
range of . We present applications of this result to patterns of various
types in the Liouville sequence. Firstly, we show that the number of sign
patterns of the Liouville function is superpolynomial, making progress on a
conjecture of Sarnak about the Liouville sequence having positive entropy.
Secondly, we obtain cancellation in averages of over short polynomial
progressions , which in the case of linear
polynomials yields a new averaged version of Chowla's conjecture. We are in
fact able to prove our results on polynomial phases in the wider range , thus strengthening also previous work on the
Fourier uniformity of the Liouville function.Comment: 104 page
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
Expander Construction in VNC1
We give a combinatorial analysis (using edge expansion) of a variant of the iterative expander construction due to Reingold, Vadhan, and Wigderson (2002), and show that this analysis can be formalized in the bounded arithmetic system VNC^1 (corresponding to the "NC^1 reasoning"). As a corollary, we prove the assumption made by Jerabek (2011) that a construction of certain bipartite expander graphs can be formalized in VNC^1. This in turn implies that every proof in Gentzen\u27s sequent calculus LK of a monotone sequent can be simulated in the monotone version of LK (MLK) with only polynomial blowup in proof size, strengthening the quasipolynomial simulation result of Atserias, Galesi, and Pudlak (2002)
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