Descriptional Complexity of Finite Automata -- Selected Highlights

The state complexity, respectively, nondeterministic state complexity of a regular language $L$ is the number of states of the minimal deterministic, respectively, of a minimal nondeterministic finite automaton for $L$. Some of the most studied state complexity questions deal with size comparisons of nondeterministic finite automata of differing degree of ambiguity. More generally, if for a regular language we compare the size of description by a finite automaton and by a more powerful language definition mechanism, such as a context-free grammar, we encounter non-recursive trade-offs. Operational state complexity studies the state complexity of the language resulting from a regularity preserving operation as a function of the complexity of the argument languages. Determining the state complexity of combined operations is generally challenging and for general combinations of operations that include intersection and marked concatenation it is uncomputable

A SURVEY OF LIMITED NONDETERMINISM IN COMPUTATIONAL COMPLEXITY THEORY

Nondeterminism is typically used as an inherent part of the computational models used incomputational complexity. However, much work has been done looking at nondeterminism asa separate resource added to deterministic machines. This survey examines several differentapproaches to limiting the amount of nondeterminism, including Kintala and Fischer\u27s βhierarchy, and Cai and Chen\u27s guess-and-check model

A linear time extension of deterministic pushdown automata

Proceedings of the 17th Nordic Conference of Computational Linguistics NODALIDA 2009. Editors: Kristiina Jokinen and Eckhard Bick. NEALT Proceedings Series, Vol. 4 (2009), 182-189. © 2009 The editors and contributors. Published by Northern European Association for Language Technology (NEALT) http://omilia.uio.no/nealt . Electronically published at Tartu University Library (Estonia) http://hdl.handle.net/10062/9206

Pushdown Compression

The pressing need for eficient compression schemes for XML documents has recently been focused on stack computation [6, 9], and in particular calls for a formulation of information-lossless stack or pushdown compressors that allows a formal analysis of their performance and a more ambitious use of the stack in XML compression, where so far it is mainly connected to parsing mechanisms. In this paper we introduce the model of pushdown compressor, based on pushdown transducers that compute a single injective function while keeping the widest generality regarding stack computation. The celebrated Lempel-Ziv algorithm LZ78 [10] was introduced as a general purpose compression algorithm that outperforms finite-state compressors on all sequences. We compare the performance of the Lempel-Ziv algorithm with that of the pushdown compressors, or compression algorithms that can be implemented with a pushdown transducer. This comparison is made without any a priori assumption on the data's source and considering the asymptotic compression ratio for infinite sequences. We prove that Lempel-Ziv is incomparable with pushdown compressors

Computations and interaction

We enhance the notion of a computation of the classical theory of computing with the notion of interaction. In this way, we enhance a Turing machine as a model of computation to a Reactive Turing Machine that is an abstract model of a computer as it is used nowadays, always interacting with the user and the world

On non-recursive trade-offs between finite-turn pushdown automata

It is shown that between one-turn pushdown automata (1-turn PDAs) and deterministic finite automata (DFAs) there will be savings concerning the size of description not bounded by any recursive function, so-called non-recursive tradeoffs. Considering the number of turns of the stack height as a consumable resource of PDAs, we can show the existence of non-recursive trade-offs between PDAs performing k+ 1 turns and k turns for k >= 1. Furthermore, non-recursive trade-offs are shown between arbitrary PDAs and PDAs which perform only a finite number of turns. Finally, several decidability questions are shown to be undecidable and not semidecidable

On Measuring Non-Recursive Trade-Offs

We investigate the phenomenon of non-recursive trade-offs between descriptional systems in an abstract fashion. We aim at categorizing non-recursive trade-offs by bounds on their growth rate, and show how to deduce such bounds in general. We also identify criteria which, in the spirit of abstract language theory, allow us to deduce non-recursive tradeoffs from effective closure properties of language families on the one hand, and differences in the decidability status of basic decision problems on the other. We develop a qualitative classification of non-recursive trade-offs in order to obtain a better understanding of this very fundamental behaviour of descriptional systems