47 research outputs found
Bounded-Depth Frege Complexity of Tseitin Formulas for All Graphs
We prove that there is a constant K such that Tseitin formulas for an undirected graph G requires proofs of size 2tw(G)Ω(1/d) in depth-d Frege systems for d < (Formula presented.) where tw(G) is the treewidth of G. This extends HÄstad recent lower bound for the grid graph to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2tw(G)O(1/d)poly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution
Bounded-depth Frege complexity of Tseitin formulas for all graphs
We prove that there is a constant K such that Tseitin formulas for a connected graph G requires proofs of size 2tw(G)javax.xml.bind.JAXBElement@531a834b in depth-d Frege systems for [Formula presented], where tw(G) is the treewidth of G. This extends HĂ„stad's recent lower bound from grid graphs to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2tw(G)javax.xml.bind.JAXBElement@25a4b51fpoly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution
Satisfiable Tseitin Formulas Are Hard for Nondeterministic Read-Once Branching Programs
We consider satisfiable Tseitin formulas TS_{G,c} based on d-regular expanders G with the absolute value of the second largest eigenvalue less than d/3. We prove that any nondeterministic read-once branching program (1-NBP) representing TS_{G,c} has size 2^{Omega(n)}, where n is the number of vertices in G. It extends the recent result by Itsykson at el. [STACS 2017] from OBDD to 1-NBP.
On the other hand it is easy to see that TS_{G,c} can be represented as a read-2 branching program (2-BP) of size O(n), as the negation of a nondeterministic read-once branching program (1-coNBP) of size O(n) and as a CNF formula of size O(n). Thus TS_{G,c} gives the best possible separations (up to a constant in the exponent) between
1-NBP and 2-BP, 1-NBP and 1-coNBP and between 1-NBP and CNF
Recommended from our members
Proof Complexity and Beyond
Proof complexity is a multi-disciplinary intellectual endeavor that addresses questions of the general form âhow difficult is it to prove certain mathematical facts?â 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
Subspace-Invariant AC Formulas
We consider the action of a linear subspace of on the set of
AC formulas with inputs labeled by literals in the set , where an element acts on formulas by
transposing the th pair of literals for all such that . A
formula is {\em -invariant} if it is fixed by this action. For example,
there is a well-known recursive construction of depth formulas of size
computing the -variable PARITY function; these
formulas are easily seen to be -invariant where is the subspace of
even-weight elements of . In this paper we establish a nearly
matching lower bound on the -invariant depth
formula size of PARITY. Quantitatively this improves the best known
lower bound for {\em unrestricted} depth
formulas, while avoiding the use of the switching lemma. More generally,
for any linear subspaces , we show that if a Boolean function is
-invariant and non-constant over , then its -invariant depth
formula size is at least where is the minimum Hamming
weight of a vector in
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
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