ed to be true under the valuation u iff there exists an a 2 N such that the formula x = y is true under the valuation u[x=a], where u[x=a] agrees with u everywhere except x, on which it takes the value a. This definition involves a metalogical operation that produces u[x=a] from u for all possible values a 2 N. This operation becomes explicit in DL in the form of the program x := ?, called a nondeterministic or wildcard assignment. This is a rather unconventional program, since it is not effective; however, it is quite useful as a descriptive tool. A more conventional way to obtain a square root of y, if it exists, would be the program x := 0 ; while x < y do x := x + 1: (1) In DL, such programs are first-class objects on a par with formulas, complete with a collection of operators for forming compound programs inductively from a basis of primitive programs. To discuss the effect of the execution of a program on the truth of a formula ', DL uses a modal construct <>', which

Abstract. The automata-theoretic approach to linear temporal logic uses the theory of automata as a unifying paradigm for program specification, verification, and synthesis. Both programs and specifications are in essence descriptions of computations. These computations can be viewed as words over some alphabet. Thus,programs and specificationscan be viewed as descriptions of languagesover some alphabet. The automata-theoretic perspective considers the relationships between programs and their specifications as relationships between languages.By translating programs and specifications to automata, questions about programs and their specifications can be reduced to questions about automata. More specifically, questions such as satisfiability of specifications and correctness of programs with respect to their specifications can be reduced to questions such as nonemptiness and containment of automata. Unlike classical automata theory, which focused on automata on finite words, the applications to program specification, verification, and synthesis, use automata on infinite words, since the computations in which we are interested are typically infinite. This paper provides an introduction to the theory of automata on infinite words and demonstrates its applications to program specification, verification, and synthesis. 1

We show that the propositional Mu-Calculus is equivalent in expressive power to finite automata on infinite trees. Since complementation is trivial in the Mu-Calculus, our equivalence provides a radically simplified, alternative proof of Rabin's complementation lemma for tree automata, which is the heart of one of the deepest decidability results. We also show how Mu-Calculus can be used to establish determinacy of infinite games used in earlier proofs of complementation lemma, and certain games used in the theory of on-line algorithms.

Recent proposals to improve the quality of interaction with the World Wide Web suggest considering the Web as a huge semistructured database, so that retrieving information can be supported by the task of database querying. Under this view, it is important to represent the form of both the network, and the documents placed in the nodes of the network. However, the current proposals do not pay sufficient attention to represent document structures and reasoning about them. In this paper, we address these problems by providing a framework where Document Type Definitions (DTDs) expressed in the eXtensible Markup Language (XML) are formalized in an expressive Description Logic equipped with sound and complete inference algorithms. We provide methods for verifying conformance of a document to a DTD in polynomial time, and structural equivalence of DTDs in worst case deterministic exponential time, improving known algorithms for this problem which were double exponential. We also deal with parametric versions of conformance and structural equivalence, and investigate other forms of reasoning on DTDs. Finally, we show how to take advantage of the reasoning capabilities of our formalism in order to perform several optimization steps in answering queries posed to a document base.

Abstract. One distinctive characteristic of object-oriented data models over traditional database systems is that they provide more expressive power in schema de nition. Nevertheless, the de ning power of objectoriented models is still somewhat limited, mainly because it is commonly accepted that part of the semantics of the application can be represented within methods. The research work reported in this paper explores the possibility of enhancing the power of object-oriented data models in schema de nition, thus o ering more possibilities to reason about the intension of the database and better supporting data management. We demonstrate our approach by presenting a new data model, called CVL, that extends the usual object-oriented data models with several aspects, including view de nition, recursive structure modeling, navigation of the schema through forward and backward traversal of links (attributes and relations), subsetting of attributes, and cardinality ratio constraints on links. CVL is equipped with sound, complete, and terminating inference procedures, that allow various forms of reasoning to be carried out on the intensional level of the database. 1

Given a formula of the propositional µ-calculus, we construct a tableau of the formula and define an infinite game of two players of which one wants to show that the formula is satisfiable, and the other seeks the opposite. The strategy for the first player can be further transformed into a model of the formula while the strategy for the second forms what we call a refutation of the formula. Using Martin's Determinacy Theorem, we prove that any formula has either a model or a refutation. This completeness result is a starting point for the completeness theorem for the µ-calculus to be presented elsewhere. However, we argue that refutations have some advantages of their own. They are generated by a natural system of sound logical rules and can be presented as regular trees of the size exponential in the size of a refuted formula. This last aspect completes the small model theorem for the µ-calculus established by Emerson and Jutla [3]. Thus, on a more practical side, refutations can be...

. There is a growing need for reliable methods of designing correct reactive systems such as computer operating systems and air traffic control systems. It is widely agreed that certain formalisms such as temporal logic, when coupled with automated reasoning support, provide the most effective and reliable means of specifying and ensuring correct behavior of such systems. This paper discusses known complexity and expressiveness results for a number of such logics in common use and describes key technical tools for obtaining essentially optimal mechanical reasoning algorithms. However, the emphasis is on underlying intuitions and broad themes rather than technical intricacies. 1 Introduction There is a growing need for reliable methods of designing correct reactive systems. These systems are characterized by ongoing, typically nonterminating and highly nondeterministic behavior. Examples include operating systems, network protocols, and air traffic control systems. There is w...

For the basic Description Logics reasoning with respect to finite models amounts to reasoning with respect to arbitrary ones, but finiteness of the domain needs to be considered if expressivity is increased and the finite model property fails. Procedures for reasoning with respect to arbitrary models in very expressive Description Logics have been developed, but these are not directly applicable in the finite case. We first show that we can nevertheless capture a restricted form of finiteness and represent finite modeling structures such as lists and trees, while still reasoning with respect to arbitrary models. The main result of this paper is a procedure to reason with respect to finite models in an expressive Description Logic equipped with inverse roles, cardinality constraints, and in which arbitrary inclusions between concepts can be specified without any restriction. This provides the necessary expressivity to go beyond most semantic and object-oriented Database models, and capture several useful extensions. 1

Infinite behavior of nondeterministic finite state automata running over infinite trees or more generally over elements of an arbitrary algebraic structure is characterized by a calculus of fixed point terms interpreted in powerset algebras. These terms involve the least and greatest fixed point operators and disjunction as the only logical operation. A tight correspondence is established between a hierarchy of Rabin indices of automata and a hierarchy induced by alternation of the least and greatest fixed point operators. It is shown that, in the powerset algebra of trees constructed from a set of functional symbols, the fixed point hierarchy is infinite unless all the symbols are unary (i.e. trees are words). It is also shown that an interpretation of a closed fixed point term in any powerset algebra can be factorized through the interpretation of this term in the powerset algebra of trees, from which it is deduced that the question whether a term denotes always ; can be answered in ...