Better complexity bounds for cost register automata

Theoretical Computer Science, 2003

For any q > 1, let MOD q be a quantum gate that determines if the number of 1's in the input is divisible by q. We show that for any q, t > 1, MOD q is equivalent to MOD t (up to constant depth). Based on the case q = 2, Moore [8] has shown that quantum analogs of AC (0) , ACC[q], and ACC, denoted QAC (0) wf , QACC[2], QACC respectively, define the same class of operators, leaving q > 2 as an open question. Our result resolves this question, proving that QAC

2020

We revisit the complexity of procedures on SFAs (such as intersection, emptiness, etc.) and analyze them according to the measures we find suitable for symbolic automata: the number of states (n), the maximal number of transitions exiting a state (m) and the size of the most complex transition predicate (l). We pay attention to the special forms of SFAs: normalized SFAs and neat SFAs, as well as to SFAs over a monotonic effective Boolean algebra.

Transactions of the American Mathematical Society, 2012

The question of computing the group complexity of finite semigroups and automata was first posed in K. Krohn and J. Rhodes, Complexity of finite semigroups, Annals of Mathematics (2) 88 (1968), 128-160, motivated by the Prime Decomposition Theorem of K. Krohn and J. Rhodes, Algebraic theory of machines, I: Prime decomposition theorem for finite semigroups and machines, Transactions of the American Mathematical Society 116 (1965), 450-464. Here we provide an effective lower bound for group complexity.

Information and Computation, 2011

Finite automata are probably best known for being equivalent to right-linear context-free grammars and, thus, for capturing the lowest level of the Chomsky-hierarchy, the family of regular languages. Over the last half century, a vast literature documenting the importance of deterministic, nondeterministic, and alternating finite automata as an enormously valuable concept has been developed. In the present paper, we tour a fragment of this literature. Mostly, we discuss developments relevant to finite automata related problems like, for example, (i) simulation of and by several types of finite automata, (ii) standard automata problems such as fixed and general membership, emptiness, universality, equivalence, and related problems, and (iii) minimization and approximation. We thus come across descriptional and computational complexity issues of finite automata. We do not prove these results but we merely draw attention to the big picture and some of the main ideas involved.

SIAM Journal on Computing, 1999

The complexity of testing nonemptiness of finite state automata on infinite trees is investigated. It is shown that for tree automata with the pairs (or complemented pairs) acceptance condition having m states and n pairs, nonemptiness can be tested in deterministic time (mn) O(n) ; however, it is shown that the problem is in general NP-complete (or co-NP-complete, respectively). The new nonemptiness algorithm yields exponentially improved, essentially tight upper bounds for numerous important modal logics of programs, interpreted with the usual semantics over structures generated by binary relations. For example, it follows that satisfiability for the full branching time logic CTL * can be tested in deterministic double exponential time. Another consequence is that satisfiability for propositional dynamic logic (PDL) with a repetition construct (PDL-delta) and for the propositional Mu-calculus (Lµ) can be tested in deterministic single exponential time.

Mathematical Systems Theory, 1994

We introduce a natural set of arithmetic expressions and define the complexity class AE to consist of all those arithmetic functions (over the fields F2n) that are described by these expressions. We show that AE coincides with the class of functions that are computable with constant depth and polynomial size unbounded fan-in arithmetic circuits satisfying a natural uniformity constraint (DLOGTIME-uniformity). A 1-input and 1-output arithmetic function over the fields F 2 n may be identified with an n-input and n-output Boolean function when field elements are represented as bit strings. We prove that if some such representation is X-uniform (where X is P or DLOGTIME) then the arithmetic complexity of a function (measured with X-uniform unbounded fan-in arithmetic circuits) is identical to the Boolean complexity of this function (measured with X-uniform threshold circuits). We show the existence of a P-uniform representation and we give partial results concerning the existence of representations with more restrictive uniformity properties.

RAIRO - Theoretical Informatics and Applications, 2006

We investigate the complexity of several problems concerning Las Vegas finite automata. Our results are as follows. (1) The membership problem for Las Vegas finite automata is in NL. (2) The nonemptiness and inequivalence problems for Las Vegas finite automata are NL-complete. (3) Constructing for a given Las Vegas finite automaton a minimum state deterministic finite automaton is in NP. These results provide partial answers to some open problems posed by Hromkovič and Schnitger [Theoret. Comput. Sci. 262 (2001) 1-24)].

RAIRO - Theoretical Informatics and Applications, 2014

Some of the most interesting and important results concerning quantum finite automata are those showing that they can recognize certain languages with (much) less resources than corresponding classical finite automata. This paper shows three results of such a type that are stronger in some sense than other ones because (a) they deal with models of quantum finite automata with very little quantumness (so-called semi-quantum one-and two-way finite automata); (b) differences, even comparing with probabilistic classical automata, are bigger than expected; (c) a trade-off between the number of classical and quantum basis states needed is demonstrated in one case and (d) languages (or the promise problem) used to show main results are very simple and often explored ones in automata theory or in communication complexity, with seemingly little structure that could be utilized.

1998

We define the counting classes #NC¹, GapNC¹, PNC¹ and C=NC¹. We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that #NC¹ ` #L, that PNC¹ ` L, and that C=NC¹ ` L. Then we exploit our finite automaton model and extend the padding techniques used to investigate leaf languages. Finally, we draw some consequences from the resulting body of leaf language characterizations of complexity classes, including the unconditional separations of ACC⁰ from MOD-PH and that of TC⁰ from the counting hierarchy. Moreover we obtain that if dlogtime-uniformity and logspace-uniformity for AC⁰ coincide then the polynomial time hierarchy equals PSPACE.