This paper is a survey on logical aspects of finite automata. Central points are the connection between finite automata and monadic second-order logic, the Ehrenfeucht-Fraiss'e technique in the context of formal language theory, finite automata on !-words and their determinization, and a self-contained proof of the "Rabin Tree Theorem". Sections 5 and 6 contain material presented in a lecture series to the "Final Winter School of AMICS" (Palermo, February 1996). A modified version of the paper will be a chapter of the "Handbook of Formal Language Theory", edited by G. Rozenberg and A. Salomaa, to appear in Springer-Verlag. Keywords: Finite automata, monadic second-order logic, first-order logic, regular languages, star-free languages, tree automata, Ehrenfeucht-Fraiss'e game, !-automata, temporal logic, Buchi automata, Rabin tree automata, determinacy, decidable theories. Contents 1 Introduction 1 2 Models and Formulas 2 2.1 Words, Trees, and Graphs as Models . . . . . . . . . . ....

The purpose of this article is to introduce Monadic Second-order Logic as a practical means of specifying regularity. The logic is a highly succinct alternative to the use of regular expressions. We have built a tool MONA, which acts as a decision procedure and as a translator to finite-state automata. The tool is based on new algorithms for minimizing finitestate automata that use binary decision diagrams (BDDs) to represent transition functions in compressed form. A byproduct of this work is a new bottom-up algorithm to reduce BDDs in linear time without hashing. The potential

The MONA tool provides an implementation of the decision procedures for the logics WS1S and WS2S. It has been used for numerous applications, and it is remarkably efficient in practice, even though it faces a theoretically non-elementary worst-case complexity. The implementation has matured over a period of six years. Compared to the first naive version, the present tool is faster by several orders of magnitude. This speedup is obtained from many different contributions working on all levels of the compilation and execution of formulas. We present a selection of implementation "secrets" that have been discovered and tested over the years, including formula reductions, DAGification, guided tree automata, three-valued logic, eager minimization, BDD-based automata representations, and cache-conscious data structures. We describe these techniques and quantify their respective effects by experimenting with separate versions of the MONA tool that in turn omit each of them.

In recent years, there has been a surge of progress in automated verification methods based on state exploration. In areas like hardware design, these technologies are rapidly augmenting key phases of testing and validation. To date, one of the most successful of these methods has been symbolic model checking, in which large finite-state machines are encoded into compact data structures such as binary decision diagrams (BDDs) -- and are then checked for safety and liveness properties. However, these techniques have not realized the same success on software systems. One limitation is their inability to deal with infinite-state programs -- even those with a single unbounded integer. A second problem is that of finding efficient representations for various variable types. We recently proposed a model checker for integer-based systems that uses arithmetic constraints as the underlying state representation. While this approach easily verified some subtle, infinite-state concurrency problems...

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. We present a new approach to hardware verification based on describing circuits in Monadic Second-order Logic (M2L). We show how to use this logic to represent generic designs like n-bit adders, which are parameterized in space, and sequential circuits, where time is an unbounded parameter. M2L admits a decision procedure, implemented in the Mona tool [17], which reduces formulas to canonical automata. The decision problem for M2L is non-elementary decidable and thus unlikely to be usable in practice. However, we have used Mona to automatically verify, or find errors in, a number of circuits studied in the literature. Previously published machine proofs of the same circuits are based on deduction and may involve substantial interaction with the user. Moreover, our approach is orders of magnitude faster for the examples considered. We show why the underlying computations are feasible and how our use of Mona generalizes standard BDD-based hardware reasoning. 1. Introduction Correctnes...

Within the last few years, CPU speed has greatly overtaken memory speed. For this reason, implementation of symbolic algorithms--- with their extensive use of pointers and hashing---must be reexamined.

We integrate two concepts from programming languages into a specification language based on WS2S, namely high-level data structures such as records and recursively-defined datatypes (WS2S is the weak second-order monadic logic of two successors). Our integration is based on a new logic whose variables range over record-like trees and an algorithm for translating datatypes into tree automata. We have implemented LISA, a prototype system based on these ideas, which, when coupled with a decision procedure for WS2S like the MONA system, results in a verification tool that supports both high-level specifications and complexity estimations for the running time of the decision procedure.

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In this paper we describe an approach to constraint based syntactic theories in terms of finite tree automata. The solutions to constraints expressed in weak monadic second order (MSO) logic are represented by tree automata recognizing the assignments which make the formulas true. We show that this allows an efficient representation of knowledge about the content of constraints which can be used as a practical tool for grammatical theory verification.