Yechezkel Zalcstein

The complexity of Grigorchuk groups with application to cryptography

Abstract The Turing complexity of the word problems of a class of groups introduced by Grigorchuk (1985) is examined. In particular, it is shown that such problems of permutation groups of the infinite complete binary tree yield natural complete sets that separate time and space complexity classes if they are distinct. A refinement of Savitchs translation theorem as well as a similar result restricted for time complexity follow. New families of nonfinitely presented groups are shown to have word problems uniformly solvable in simultaneous logspace and quadratic time. A new family of public- key cryptosystems based on these word problems is constructed.

Alternative methods for the reconstruction of trees from their traversals

It is well-known that given the inorder traversal of a binary trees nodes, along with either one of its preorder or postorder traversals, the original binary tree can be reconstructed using a recursive algorithm. In this short note we provide a short, elegent, iterative solution to this classical problem.

On isomorphism testing of a class of 2-Nilpotent groups

Abstract A polynomial time isomorphism test for a class of groups, properly containing the class of abelian groups, given either by multiplication tables or by generators and relators, is described. It is also shown that graph isomorphism testing is uniformly reducible to a word problem of a finitely presented group.

On star-free events

It is an open problem, suggested by Papert and McNaughton, to find a decision procedure for determining whether a regular event is locally testable. In this paper we provide a partial solution, giving two effectively decidable conditions, one necessary and one sufficient, for local testability. Our proofs are for the most part algebraic, using machine decompositions and semigroup theory.

On permutation properties in groups and semigroups

A semigroupS satisfiesPPn, thepermutation property of degree n (n≥2) if every product ofn elements inS remains invariant under some nontrivial permutation of its factors. It is shown that a semigroup satisfiesPP3 if and only if it contains at most one nontrivial commutator. Further a regular semigroup is a semilattice ofPP3 right or left groups, and a subdirect product ofPP3 semigroups of a simple type. A negative answer to a question posed by Restivo and Reutenauer is provided by a suitablePP3 group.

Bounds on threshold dimension and disjoint threshold coverings

Bn threshold graphs, with B = 1.5A. Thus the difference between these two covering numbers can grow linearly in the number of vertices .

The size of chordal, interval and threshold subgraphs

Given a graphG withn vertices andm edges, how many edges must be in the largest chordal subgraph ofG? Form=n2/4+1, the answer is 3n/2−1. Form=n2/3, it is 2n−3. Form=n2/3+1, it is at least 7n/3−6 and at most 8n/3−4. Similar questions are studied, with less complete results, for threshold graphs, interval graphs, and the stars on edges, triangles, andK4s.

Testing homotopy equivalence is isomorphism complete

Abstract It is shown that homotopy equivalence of finite topological spaces is polynomially equivalent to testing graph isomorphism.

The complexity of isomorphism testing

A polynomial time isomorphism test for a class of groups, properly containing the class of abelian groups, is presented. Isomorphism testing of group presentations for (a subclass of) the same class of groups is shown to be (graph) isomorphism complete. These seem to be the first known isomorphism complete problems in group theory. Subexponential tests are presented as well for rings and algebras.

Bounds on threshold dimension and disjoint threshold coverings (abstract only)

The threshold dimension<supscrpt>1</supscrpt> of a graph G is the smallest number of threshold graphs needed to cover the edges of G. If t(n) is the greatest threshold dimension of any graph of n vertices, we show that for some constant c, n-c √n log n < t(n) < n- √n + 1 We establish the same bounds for edge-disjoint coverings of graphs by threshold graphs. The results have applications to manipulating systems of simultaneous linear inequalities and to space bounds for synchronization problems<supscrpt>2</supscrpt>.