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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re

We describe three orthogonal complexity measures: parallel time, amount of hardware, and degree of nonuniformity, which together parametrize most complexity classes. We show that the descriptive complexity framework neatly captures these measures using the parameters: quantifier depth, number of variable bits, and type of numeric predicates respectively. A fairly simple picture arises in which the basic questions in complexity theory -- solved and unsolved -- can be understood as questions about tradeoffs among these three dimensions. 1 Introduction An initial presentation of complexity theory usually makes the implicit assumption that problems, and hence complexity classes, are linearly ordered by "difficulty ". In the Chomsky Hierarchy each new type of automaton can decide more languages, and the Time Hierarchy Theorem tells us adding more time allows a Turing machine to decide more languages. Indeed the word "complexity" is often used (e.g., in the study of algorithms) to mean "wo...

Cai and Furst introduced the notion of bottleneck Turing machines and showed that the languages recognized by width-5 bottleneck Turing machines are exactly those in PSPACE. Computational power of bottleneck Turing machines with width fewer than 5 is investigated. It is shown that width-2 bottleneck Turing machines capture polynomial-time many-one closure of nearly near-testable sets. For languages recognized by bottleneck Turing machines with intermediate width 3 and 4, some lower- and upper-bounds are shown. 1 Introduction Branching program is one of the most interesting topics in complexity theory. For k 2, a width-k branching program for n-bit inputs is a sequence of instructions f(p i ; f i ; g i )g m i=1 such that for each i; 1 i m, 1 p i n and f i ; g i 2 F k , where F k is the monoid consisting of all mappings of [k] = f1; \Delta \Delta \Delta ; kg to itself. Given an input x 2 \Sigma = f0; 1g of length n, for each i, let h i = f i if the p i -th bit of x is a 1 ...

this paper we show that the Integer Circuit problem is PSPACE-complete, resolving an open problem posed by McKenzie, Vollmer and Wagner [7]

In this paper, we present stronger results in the theory of succinct problem representation and establish a close relationship between succinct problems and leaf languages. As a major tool, we use projection reductions from descriptive complexity theory. A succinct problem is a problem whose high complexity stems from the fact that its instances are not given straightforward, but are themselves encoded by boolean circuits. In [Balc'azar et al. 92, Papa, Yann 85] there have been developed methods to quantify the complexity leap obtained this way. We prove a strictly stronger version of this result which allows iterative application and completeness results under projection reductions. A leaf language [Bovet et al 92, Bovet et al 91] is the language of words accepted by a nondeterministic Turing Machine where for acceptance the word obtained by concatenating the bits at the leaves of the computation graph must fit in a certain pattern. Complexity bounds on the leaf pattern allow to unifo...

Identify a string x over f0; 1g with the positive integer whose binary representation is 1x. We say that a self-reduction is k-local if on input x all queries belong to fx \Gamma 1; : : : ; x \Gamma kg. We show that all k-locally selfreducible sets belong to PSPACE. However, the power of k-local self-reductions changes drastically between k = 2 and k = 3. Although all 2-locally self-reducible sets belong to MOD 6 PH, some 3-locally self-reducible sets are PSPACE-complete. Furthermore, there exists a 6-locally self-reducible PSPACE-complete set whose self-reduction is an m-reduction (in fact, a permutation) . We prove all these results by showing that such languages are equivalent in complexity to the problem of multiplying an exponentially long sequence of uniformly generated elements in a finite monoid, and then exploiting the algebraic structure of the monoid. 1. Introduction In this paper we identify a string x over f0; 1g with the positive integer whose binary representation is 1...

Cai and Furst [CF91] proved that every PSPACE language can be solved via a large number of identical, simple tasks, each of which is provided with the original input, its own unique task number, and at most three bits of output from the previous task. In the Cai-Furst model, the tasks are required to be run in the order specified by the task numbers. To study the extent to which the Cai-Furst PSPACE result is due to this strict scheduling, we remove their ordering restriction, allowing tasks to execute in any serial order. That is, we study the extent to which complex tasks can be decomposed into large numbers of simple tasks that can be scheduled arbitrarily. We provide upper bounds on the complexity of the sets thus accepted. Our bounds suggest that Cai and Furst's surprising PSPACE result is due in large part to the fixed order of their task execution. In fact, our bounds suggest the possibility that even relatively low levels of the polynomial hierarchy cannot be accepted via large...

For integer k 0, let srm(n O(1) ; k) denote the collection of relations computable by a stack register machine with stack registers bounded by a polynomial p(n) in the input n, and work registers bounded by k. Let nsrm(n O(1) ; k) denote the analogous class accepted by nondeterministic stack register machines. In this paper, nondeterminism is shown to provide no additional power. Specifically, nsrm(n O(1) ; 0) = srm(n O(1) ; 0) nsrm(n O(1) ; 1) = srm(n O(1) ; 1) nsrm(n O(1) ; k) = srm(n O(1) ; k); for k 4 srm(n O(1) ; k) = alintime ; for k 4: