Pierre McKenzie

Problems complete for deterministic logarithmic space

Abstract We exhibit several problems complete for deterministic logarithmic space under NC1 (i.e., log depth) reducibility. The list includes breadth-first search and depth-first search of an undirected tree, connectivity of undirected graphs known to be made up of one or more disjoint cycles, undirected graph acyclicity, and several problems related to representing and to operating with permutations of a finite set.

Separation of the monotone NC hierarchy

, for the monotone depth of functions in monotone-P. As a result we achieve the separation of the following classes. 1. monotone-NC ≠ monotone-P. 2. For every i≥1, monotone-≠ monotone-. 3. More generally: For any integer function D(n), up to (for some ε>0), we give an explicit example of a monotone Boolean function, that can be computed by polynomial size monotone Boolean circuits of depth D(n), but that cannot be computed by any (fan-in 2) monotone Boolean circuits of depth less than Const·D(n) (for some constant Const).Only a separation of monotone- from monotone- was previously known. Our argument is more general: we define a new class of communication complexity search problems, referred to below as DART games, and we prove a tight lower bound for the communication complexity of every member of this class. As a result we get lower bounds for the monotone depth of many functions. In particular, we get the following bounds: 1.  For st-connectivity, we get a tight lower bound of . That is, we get a new proof for Karchmer–Wigdersons theorem, as an immediate corollary of our general result. 2.  For the k-clique function, with , we get a tight lower bound of Ω(k log n). This lower bound was previously known for k≤ log n [1]. For larger k, however, only a bound of Ω(k) was previously known.

Reversible space equals deterministic space

This paper describes the simulation of an S(n) space-bounded deterministic Turing machine by a reversible Turing machine operating in space S(n). It thus answers a question posed by C. Bennett (1989) and refutes the conjecture, made by M. Li and P. Vitanyi (1996), that any reversible simulation of an irreversible computation must obey Bennetts reversible pebble game rules.

Completeness results for graph isomorphism

We prove that the graph isomorphism problem restricted to trees and to colored graphs with color multiplicities 2 and 3 is many-one complete for several complexity classes within NC2. In particular we show that tree isomorphism, when trees are encoded as strings, is NC1-hard under AC0-reductions. NC1- completeness thus follows from Busss NC1 upper bound. By contrast, we prove that testing isomorphism of two trees encoded as pointer lists is L-complete. Concerning colored graphs we show that the isomorphism problem for graphs with color multiplicities 2 and 3 is complete for symmetric logarithmic space SL under many-one reductions. This result improves the existing upper bounds for the problem. We also show that the graph automorphism problem for colored graphs with color classes of size 2 is equivalent to deciding whether a graph has more than a single connected component and we prove that for color classes of size 3 the graph automorphism problem is contained in SL.

A well-structured framework for analysing petri net extensions

Transition systems defined from recursive functions IN^p->IN^p are introduced and named WSNs, or well-structured nets. Such nets sit conveniently between Petri net extensions and general transition systems. In the first part of this paper, we study decidability properties of WSN classes obtained by imposing natural restrictions on their defining functions, with respect to termination, coverability, and four variants of the boundedness problem. We are able to precisely answer almost all the questions which arise, thus gaining much insight into old and new generalized Petri net decidability results. In the second part, we specialize our analysis to WSNs defined from affine functions, which elegantly encompass most Petri net extensions studied in the literature. Again, we study decidability properties of natural classes of affine WSN with respect to the above six computational problems. In particular, we develop an algorithm computing limits of iterated nonnegative affine functions, in order to decide the path-place variant of the boundedness problem for non-negative affine WSN.

The parallel complexity of Abelian permutation group problems

We classify Abelian permutation group problems with respect to their parallel complexity. For such groups specified by generating permutations we show that testing membership, computing order and testing isomorphism are

NC1: The automata-theoretic viewpoint

NC^1

Logspace and logtime leaf languages

-equivalent to (and therefore have essentially the same parallel complexity as) determining solvability of a system of linear equations modulo a product of small prime powers; we show that intersecting two such groups is

Finite Monoids: From Word to Circuit Evaluation

NC^1

The parallel complexity of the abelian permutation group membership problem

-equivalent to computing setwise stabilizers; we show that each of these problems is