We give a general complexity classification scheme for monotone computation, including monotone space-bounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic log-space) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = co-NL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for st-connectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by Barrington's result [Bar89] that BWBP = NC 1 . Although we cannot answer t...

this paper we show that nondeterministic space s(n) is closed under complementation, for s(n) greater than or equal to log n. It immediately follows that the context-sensitive languages are closed under complementation, thus settling a question raised by Kuroda in 1964 [9]. See Hartmanis and Hunt [4] for a discussion of the history and importance of this problem, and Hopcroft and Ullman [5] for all relevant background material and definitions. The history behind the proof is as follows. In 1981 we showed that the set of first-order inductive definitions over finite structures is closed under complementation [6]. This holds with or without an ordering relation on the structure. If an ordering is present the resulting class is P. Many people expected that the result was false in the absence of an ordering. In 1983 we studied first-order logic, with ordering, with a transitive closure operator. We showed that NSPACE[log n] is equal to (FO + pos TC), i.e. first-order logic with ordering, plus a transitive closure operation, in which the transitive closure operator does not appear within any negation symbols [7]. Now we have returned to the issue of complementation in the light of recent results on the collapse of the log space hierarchies [10, 2, 14]. We have shown that the class (FO + pos TC) is closed under complementation. Our

We survey lower bounds established for the complexity of computing explicitly given Boolean functions by switching-and-rectifier networks, branching programs and switching networks. We first consider the unrestricted case and then proceed to various restricted models. Among these are monotone networks, bounded-width devices , oblivious devices and read-k times only devices. 1 Introduction The main goal of the Boolean complexity theory is to prove lower bounds on the complexity of computing "explicitly given" Boolean functions in interesting computational models. By "explicitly given" researchers usually mean "belonging to the class NP ". This is a very plausible interpretation since on the one hand this class contains the overwhelming majority of interesting Boolean functions and on the other hand it is small enough to prevent us from the necessity to take into account counting arguments. To illustrate the second point, let me remind the reader that already the class \Delta p 2 ,...

Computational complexity began with the natural physical notions of time and space. Given a property, S, an important issue is the computational complexity of checking whether or not an input satisfies S. For a long time, the notion of complexity referred to the time or space used in the computation. A mathematician might ask, "What is the complexity of expressing the property S?" It should not be surprising that these two questions -- that of checking and that of expressing -- are related. However it is startling how closely tied they are when the second question refers to expressing the property in first-order logic. Many complexity classes originally defined in terms of time or space resources have precise definitions as classes in first-order logic. In 1974 Fagin gave a characterization of nondeterministic polynomial time (NP) as the set of properties expressible in second-order existential logic

In this paper we survey the major developments in understanding the complexity of the graph connectivity problem in several computational models, and highlight some challenging open problems. 1

A method schema is a simple programming formalism for object-oriented databases with features such as classes, methods, inheritance, name overloading, and late binding. An important problem is to check whether a given method schema can lead to an inconsistency in some interpretation. This consistency question is shown to be undecidable in general. Decidability is obtained for monadic and/or recursion-free method schemas. In particular, consistency of monadic method schemas is shown to be decidable in O(nc 3 ) time, where n is the size of the method definitions and c is the size of the class hierarchy; also, it is logspace-complete in PTIME, even for monadic, recursion-free schemas. Method signature covariance is shown to simplify the computational complexity of key decidable cases. For example: one coded method in the context of base methods with covariant signatures can be tested for consistency in O(n+c) time for the monadic case (without covariance this problem is in O(nc 2 ) t...

In this work a complete problem for an unambiguous logspace class is presented. This is surprising since unambiguity is a `promise' or `semantic' concept. These usually lead to classes apparently without complete problems. 1 Introduction One of the most central questions of complexity theory is to compare determinism with nondeterminism. Our inability to exhibit the precise relationship between these two features motivates the investigation of intermediate features such as symmetry or unambiguity. In this paper we will concentrate on the notion of unambiguity. Unfortunately, unambiguity of a device or of a language is in general an undecidable property. Unambiguous classes are not defined by a `syntactical' machine property but rather by a `semantical' restriction. A nasty consequence is the apparent lack of complete sets. In the case of time bounded computations there are relativizations of unambiguity which provably have no complete problem ([10]). For space bounded computations t...

In this report, we emphasize the differences of grammar families and their properties versus language families and their properties. To this end, we investigate grammar families from an abstract standpoint, developping a new framework of reasoning. In particular when considering decidability questions, special care must be taken when trying to use decidability results (which are, in the first place, properties of grammar families) in order to establish results (e.g. hierarchy results) on language families. We illustrate this by inspecting some theorems and their proofs in the field of regulated rewriting. In this way, we also correct the formulation of an important theorem of Hinz and Dassow. As an exercise, we show that there is no `effective' grammatical characterization of the family of recursive languages. Moreover, we show how to prove the strictness of the Chomsky hierarchy using decidability properties only.

Directed s-t connectivity is the problem of detecting whether there is a path from a distinguished vertex s to a distinguished vertex t in a directed graph. We prove time-space lower bounds of ST = \Omega\Gamma n 2 = log n) and S 1=2 T = \Omega\Gamma mn 1=2 ) for Cook and Rackoff's JAG model [8], where n is the number of vertices and m the number of edges in the input graph, and S is the space and T the time used by the JAG. We also prove a timespace lower bound of S 1=3 T = \Omega\Gamma m 2=3 n 2=3 ) on the more powerful node-named JAG model of Poon [13]. These bounds approach the known upper bound of T = O(m) when S = \Theta(n log n).

We consider some Bounded Set Theories (BST), which are analogues to Bounded Arithmetic. Corresponding provably-recursive operations over sets are characterized in terms of explicit definability and PTIME- or LOGSPACE-computability. We also present some conservativity results and describe a relation between BST, possibly with Anti-Foundation Axiom, and a Logic of Inductive Definitions (LID) and Finite Model Theory.