Finite Automata, Formal Logic, and Circuit Complexity by Howard Straubing

By Howard Straubing

The research of the connections among mathematical automata and for­ mal good judgment is as previous as theoretical laptop technology itself. within the founding paper of the topic, released in 1936, Turing confirmed tips to describe the habit of a common computing laptop with a formulation of first­ order predicate good judgment, and thereby concluded that there's no set of rules for finding out the validity of sentences during this common sense. learn at the log­ ical elements of the idea of finite-state automata, that's the topic of this publication, all started within the early 1960's with the paintings of J. Richard Biichi on monadic second-order good judgment. Biichi's investigations have been prolonged in different instructions. this kind of, explored by way of McNaughton and Papert of their 1971 monograph Counter-free Automata, used to be the characterization of automata that admit first-order behavioral descriptions, when it comes to the semigroup­ theoretic method of automata that had lately been constructed within the paintings of Krohn and Rhodes and of Schiitzenberger. within the greater than 20 years that experience handed because the visual appeal of McNaughton and Papert's e-book, the underlying semigroup thought has grown enor­ mously, allowing a substantial extension in their effects. in the course of the similar interval, although, basic investigations within the concept of finite automata traditionally fell out of favor within the theoretical com­ puter technological know-how neighborhood, which moved to different concerns.

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Because disjunction and conjunction are idempotent, we can eliminate any repetition of a conjunct, and any repetition of a disjunct. This will constitute our normal form for quantifier-free formulas. 3x, or , where c( in normal form, and eliminate repeated conjuncts and disjuncts. This constitutes our normal form for formulas of complexity c+ 1. Ll Proposition. Let V be a finite set of first-order variables, and let c ~ o. There are only finitely many normal-form formulas of complexity c in which every variable belongs to V.

Lob Example. Now consider the language L = {w E {O, 1}* : Iwlt =- 0 (mod 2)}. Thus L consists of the set of words in which the number of of occurrences of 1 is even. Once thegain we find that M (L) is a group of order 2; the identity is the class of words with an even number of 1'so Observe that in the preceding example, 'f/L maps both 0 and 1 to the generator of the group, while in the present example, 'f/L maps 0 to the identity and 1 to the generator. The syntactic monoid is defined for every subset L of A *.

This is no accident. If we restrict the available numerical predicates appropriately, then the language defined by a monadic secondorder sentence is a regular language. 1 below, every regular language can be defined in this fashion. We will thus obtain a characterization of the regular languages in terms of logic. We will consider monadic second-order sentences in which the only numerical predicates are R~ and ~, where R~(x, y) is interpreted as x = y, and ~(x, y) is interpreted as y = x + 1. We will write "x = y" and "y = x + 1" explicitly in our formulas, rather than use the symbols R~ and ~.

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