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56
Languages, Automata, and Logic
 Handbook of Formal Languages
, 1996
"... This paper is a survey on logical aspects of finite automata. Central points are the connection between finite automata and monadic secondorder logic, the EhrenfeuchtFraiss'e technique in the context of formal language theory, finite automata on !words and their determinization, and a selfcontai ..."
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Cited by 182 (4 self)
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This paper is a survey on logical aspects of finite automata. Central points are the connection between finite automata and monadic secondorder logic, the EhrenfeuchtFraiss'e technique in the context of formal language theory, finite automata on !words and their determinization, and a selfcontained 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 SpringerVerlag. Keywords: Finite automata, monadic secondorder logic, firstorder logic, regular languages, starfree languages, tree automata, EhrenfeuchtFraiss'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 . . . . . . . . . . ....
Automatic Structures
 IN PROC. 15TH IEEE SYMP. ON LOGIC IN COMPUTER SCIENCE
, 1999
"... We study definability and complexity issues for automatic and wautomatic structures. These are, in general, infinite structures but they can be finitely presented by a collection of automata. Moreover, they admit effective (in fact automatic) evaluation of all firstorder queries. Therefore, automa ..."
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Cited by 89 (7 self)
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We study definability and complexity issues for automatic and wautomatic structures. These are, in general, infinite structures but they can be finitely presented by a collection of automata. Moreover, they admit effective (in fact automatic) evaluation of all firstorder queries. Therefore, automatic structures provide an interesting framework for extending many algorithmic and logical methods from finite structures to infinite ones. We explain the notion of (w)automatic structures, give examples, and discuss the relationship to automatic groups. We determine the complexity of model checking and query evaluation on automatic structures for fragments of firstorder logic. Further, we study closure properties and definability issues on automatic structures and present a technique for proving that a structure is not automatic. We give modeltheoretic characterisations for automatic structures via interpretations. Finally we discuss the composition theory of automatic structures and pro...
An AutomataTheoretic Approach to Presburger Arithmetic Constraints (Extended Abstract)
 In Proc. Static Analysis Symposium, LNCS 983
, 1995
"... This paper introduces a finiteautomata based representation of Presburger arithmetic definable sets of integer vectors. The representation consists of concurrent automata operating on the binary encodings of the elements of the represented sets. This representation has several advantages. First, be ..."
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Cited by 46 (4 self)
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This paper introduces a finiteautomata based representation of Presburger arithmetic definable sets of integer vectors. The representation consists of concurrent automata operating on the binary encodings of the elements of the represented sets. This representation has several advantages. First, being automatabased it is operational in nature and hence leads directly to algorithms, for instance all usual operations on sets of integer vectors translate naturally to operations on automata. Second, the use of concurrent automata makes it compact. Third, it is insensitive to the representation size of integers. Our representation can be used whenever arithmetic constraints are needed. To il...
Finite Presentations of Infinite Structures: Automata and Interpretations
 Theory of Computing Systems
, 2002
"... We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by modeltheoretic interpretations. ..."
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Cited by 42 (3 self)
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We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by modeltheoretic interpretations.
Modular Data Structure Verification
 EECS DEPARTMENT, MASSACHUSETTS INSTITUTE OF TECHNOLOGY
, 2007
"... This dissertation describes an approach for automatically verifying data structures, focusing on techniques for automatically proving formulas that arise in such verification. I have implemented this approach with my colleagues in a verification system called Jahob. Jahob verifies properties of Java ..."
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Cited by 36 (21 self)
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This dissertation describes an approach for automatically verifying data structures, focusing on techniques for automatically proving formulas that arise in such verification. I have implemented this approach with my colleagues in a verification system called Jahob. Jahob verifies properties of Java programs with dynamically allocated data structures. Developers write Jahob specifications in classical higherorder logic (HOL); Jahob reduces the verification problem to deciding the validity of HOL formulas. I present a new method for proving HOL formulas by combining automated reasoning techniques. My method consists of 1) splitting formulas into individual HOL conjuncts, 2) soundly approximating each HOL conjunct with a formula in a more tractable fragment and 3) proving the resulting approximation using a decision procedure or a theorem prover. I present three concrete logics; for each logic I show how to use it to approximate HOL formulas, and how to decide the validity of formulas in this logic. First, I present an approximation of HOL based on a translation to firstorder logic, which enables the use of existing resolutionbased theorem provers. Second, I present an approximation of HOL based on field constraint analysis, a new technique that enables
Deciding Boolean Algebra with Presburger Arithmetic
 J. of Automated Reasoning
"... Abstract. We describe an algorithm for deciding the firstorder multisorted theory BAPA, which combines 1) Boolean algebras of sets of uninterpreted elements (BA) and 2) Presburger arithmetic operations (PA). BAPA can express the relationship between integer variables and cardinalities of unbounded ..."
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Cited by 31 (26 self)
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Abstract. We describe an algorithm for deciding the firstorder multisorted theory BAPA, which combines 1) Boolean algebras of sets of uninterpreted elements (BA) and 2) Presburger arithmetic operations (PA). BAPA can express the relationship between integer variables and cardinalities of unbounded finite sets, and supports arbitrary quantification over sets and integers. Our original motivation for BAPA is deciding verification conditions that arise in the static analysis of data structure consistency properties. Data structures often use an integer variable to keep track of the number of elements they store; an invariant of such a data structure is that the value of the integer variable is equal to the number of elements stored in the data structure. When the data structure content is represented by a set, the resulting constraints can be captured in BAPA. BAPA formulas with quantifier alternations arise when verifying programs with annotations containing quantifiers, or when proving simulation relation conditions for refinement and equivalence of program fragments. Furthermore, BAPA constraints can be used for proving the termination of programs that manipulate data structures, as well as
Expressive Languages for Path Queries over GraphStructured Data
, 2010
"... For many problems arising in the setting of graph querying (such as finding semantic associations in RDF graphs, exact and approximate pattern matching, sequence alignment, etc.), the power of standard languages such as the widely studied conjunctive regular path queries (CRPQs) is insufficient in a ..."
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Cited by 28 (10 self)
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For many problems arising in the setting of graph querying (such as finding semantic associations in RDF graphs, exact and approximate pattern matching, sequence alignment, etc.), the power of standard languages such as the widely studied conjunctive regular path queries (CRPQs) is insufficient in at least two ways. First, they cannot output paths and second, more crucially, they cannot express relations among paths. We thus propose a class of extended CRPQs, called ECRPQs, which add regular relations on tuples of paths, and allow path variables in the heads of queries. We provide several examples of their usefulness in querying graph structured data, and study their properties. We analyze query evaluation and representation of tuples of paths in the output by means of automata. We present a detailed analysis of data and combined complexity of queries, and consider restrictions that lower the complexity of ECRPQs to that of relational conjunctive queries. We study the containment problem, and look at further extensions with firstorder features, and with nonregular relations that express arithmetic properties of paths, based on the lengths and numbers of occurrences of labels.
An algorithm for deciding BAPA: Boolean Algebra with Presburger Arithmetic
 In 20th International Conference on Automated Deduction, CADE20
, 2005
"... Abstract. We describe an algorithm for deciding the firstorder multisorted theory BAPA, which combines 1) Boolean algebras of sets of uninterpreted elements (BA) and 2) Presburger arithmetic operations (PA). BAPA can express the relationship between integer variables and cardinalities of a priory u ..."
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Cited by 26 (13 self)
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Abstract. We describe an algorithm for deciding the firstorder multisorted theory BAPA, which combines 1) Boolean algebras of sets of uninterpreted elements (BA) and 2) Presburger arithmetic operations (PA). BAPA can express the relationship between integer variables and cardinalities of a priory unbounded finite sets, and supports arbitrary quantification over sets and integers. Our motivation for BAPA is deciding verification conditions that arise in the static analysis of data structure consistency properties. Data structures often use an integer variable to keep track of the number of elements they store; an invariant of such a data structure is that the value of the integer variable is equal to the number of elements stored in the data structure. When the data structure content is represented by a set, the resulting constraints can be captured in BAPA. BAPA formulas with quantifier alternations arise when verifying programs with annotations containing quantifiers, or when proving simulation relation conditions for refinement and equivalence of program fragments. Furthermore, BAPA constraints can be used for proving the termination of programs that manipulate data structures, and have applications in constraint databases. We give a formal description of a decision procedure for BAPA, which implies the decidability of BAPA. We analyze our algorithm and obtain an elementary upper bound on the running time, thereby giving the first complexity bound for BAPA. Because it works by a reduction to PA, our algorithm yields the decidability of a combination of sets of uninterpreted elements with any decidable extension of PA. Our algorithm can also be used to yield an optimal decision procedure for BA through a reduction to PA with bounded quantifiers. We have implemented our algorithm and used it to discharge verification conditions in the Jahob system for data structure consistency checking of Java programs; our experience with the algorithm is promising. 1
An effective decision procedure for linear arithmetic with integer and real variables
 ACM Transactions on Computational Logic (TOCL
, 2005
"... This article considers finiteautomatabased algorithms for handling linear arithmetic with both real and integer variables. Previous work has shown that this theory can be dealt with by using finite automata on infinite words, but this involves some difficult and delicate to implement algorithms. T ..."
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Cited by 26 (6 self)
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This article considers finiteautomatabased algorithms for handling linear arithmetic with both real and integer variables. Previous work has shown that this theory can be dealt with by using finite automata on infinite words, but this involves some difficult and delicate to implement algorithms. The contribution of this article is to show, using topological arguments, that only a restricted class of automata on infinite words are necessary for handling real and integer linear arithmetic. This allows the use of substantially simpler algorithms, which have been successfully implemented.
On the Expressiveness of Real and Integer Arithmetic Automata (Extended Abstract)
 ICALP'98, LNCS 1443
, 1998
"... If read digit by digit, a ndimensional vector of integers represented in base r can be viewed as a word over the alphabet r n . It has been known for some time that, under this encoding, the sets of integer vectors recognizable by finite automata are exactly those de nable in Presburger arithmetic ..."
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Cited by 23 (6 self)
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If read digit by digit, a ndimensional vector of integers represented in base r can be viewed as a word over the alphabet r n . It has been known for some time that, under this encoding, the sets of integer vectors recognizable by finite automata are exactly those de nable in Presburger arithmetic if independence with respect to the base is required, and those de nable in a slight extension of Presburger arithmetic if only a specific base is considered. Using the same encoding idea, but moving to infinite words, finite automata on infinite words can recognize sets of real vectors. This leads to the question of which sets of real vectors are recognizable by finite automata, which is the topic of this paper. We show that the recognizable sets of real vectors are those definable in the theory of reals and integers with addition and order, extended with a special basedependent predicate that tests the value of a specified digit of a number. Furthermore, in the course of proving that sets of vectors de ned in this theory are recognizable by finite automata, we show that linear equations and inequations have surprisingly compact representations by automata, which leads us to believe that automata accepting sets of real vectors can be of more than theoretical interest.