Analyzing refutations of the well known
pebbling formulas we prove some new strong connections between pebble games and algebraic proof system, showing that
there is a parallelism between the reversible, black and black-white pebbling games on one side, and
the three algebraic proof systems NS, MC and ... more >>>

The width complexity measure plays a central role in Resolution and other propositional proof systems like Polynomial Calculus (under the name of degree). The study of width lower bounds is the most extended method for proving size lower bounds, and it is known that for these systems, proofs with small ... more >>>

We show that the number of variables and the quantifier depth needed to distinguish a pair of graphs by first-order logic sentences exactly match the complexity measures of clause width and positive depth needed to refute the corresponding graph isomorphism formula in propositional narrow resolution.

Using this connection, we ... more >>>

We show a new connection between the space measure in tree-like resolution and the reversible pebble game in graphs. Using this connection we provide several formula classes for which there is a logarithmic factor separation between the space complexity measure in tree-like and general resolution. We show that these separations ... more >>>

We characterize several complexity measures for the resolution of Tseitin formulas in terms of a two person cop-robber game. Our game is a slight variation of the one Seymour and Thomas used in order to characterize the tree-width parameter. For any undirected graph, by counting the number of cops needed ... more >>>

We consider the PermCode problem to decide, given a representation of a permutation group G and a parameter k, whether there is a non-trivial element of G with support at most k. This problem generalizes several problems in the literature. We introduce a new method that allows to reduce the ... more >>>

The solution graph of a Boolean formula on n variables is the subgraph of the hypercube Hn induced by the satisfying assignments of the formula. The structure of solution graphs has been the object of much research in recent years since it is important for the performance of SAT-solving procedures ... more >>>

It is well known that problems encoded with circuits or formulas generally gain an exponential complexity blow-up compared to their original complexity.

We introduce a new way for encoding graph problems, based on $\textrm{CNF}$ or $\textrm{DNF}$ formulas. We show that contrary to the other existing succinct models, there are ... more >>>

Given a system of linear equations $Ax=b$ over the binary field $\mathbb{F}_2$ and an integer $t\ge 1$, we study the following three algorithmic problems:
1. Does $Ax=b$ have a solution of weight at most $t$?
2. Does $Ax=b$ have a solution of weight exactly $t$?
3. Does $Ax=b$ have a ... more >>>

We give a new upper bound for the Group and Quasigroup
Isomorphism problems when the input structures
are given explicitly by multiplication tables. We show that these problems can be computed by polynomial size nondeterministic circuits of unbounded fan-in with $O(\log\log n)$ depth and $O(\log^2 n)$ nondeterministic bits, ... more >>>

The Graph Isomorphism problem restricted to graphs of bounded treewidth or bounded tree distance width
are known to be solvable in polynomial time \cite{Bo90},\cite{YBFT}.
We give restricted space algorithms for these problems proving the following results:

Isomorphism for bounded tree distance width graphs is in L and thus complete ... more >>>

We show that several reducibility notions coincide when applied to the
Graph Isomorphism (GI) problem. In particular we show that if a set is
many-one logspace reducible to GI, then it is in fact many-one AC^0
reducible to GI. For the case of Turing reducibilities we show that ... more >>>

The Group Isomorphism problem consists in deciding whether two input
groups $G_1$ and $G_2$ given by their multiplication tables are
isomorphic. We first give a 2-round Arthur-Merlin protocol for the
Group Non-Isomorphism problem such that on input groups $(G_1,G_2)$
of size $n$, Arthur uses ... more >>>

We show that the Player-Adversary game from a paper
by Pudlak and Impagliazzo played over
CNF propositional formulas gives
an exact characterization of the space needed
in treelike resolution refutations. This
characterization is purely combinatorial
and independent of the notion of resolution.
We use this characterization to give ... more >>>

We relate the existence of disjunctive hard sets for NP to
other well studied hypotheses in complexity theory showing
that if an NP-complete set or a coNP-complete set is
polynomial-time disjunctively reducible
to a sparse set then FP$^{\rm NP}_{||}$ = FP$^{\rm NP[log]}$. Using
a similar argument we obtain also that ... more >>>

A polynomial time computable function $h:\Sigma^*\to\Sigma^*$ whose range
is the set of tautologies in Propositional Logic (TAUT), is called
a proof system. Cook and Reckhow defined this concept
and in order to compare the relative strenth of different proof systems,
they considered the notion ... more >>>