Much attention has been given recently to the mechanism of fibring of logics, allowing free mixing of the connectives and using proof rules from both logics. Fibring seems to be a rather useful and general form of combination of logics that deserves detailed study. It is now well understood at the proof-theoretic level. However, the semantics of fibring is still insufficiently understood. Herein we provide a categorial definition of both proof-theoretic and model-theoretic fibring for logics without terms. To this end, we introduce the categories of Hilbert calculi, interpretation systems and logic system presentations. By choosing appropriate notions of morphism it is possible to obtain pure fibring as a coproduct. Fibring with shared symbols is then easily obtained by cocartesian lifting from the category of signatures. Soundness is shown to be preserved by these constructions. We illustrate the constructions within propositional modal logic.

We present a new block diagram language for describing synchronous software. It coordinates the execution of synchronous, concurrent software modules, allowing real-time systems to be assembled from precompiled blocks specified in other languages. The semantics we present, based on fixed points, is deterministic even in the presence of instantaneous feedback. The execution policy develops a static schedule—a fixed order in which to execute the blocks that makes the system execution predictable. We present exact and heuristic algorithms for finding schedules that minimize system execution time, and show that good schedules can be found quickly. The scheduling algorithms are applicable to other problems where large systems of equations need to be solved.

. Java bytecode verification forms the basis for Java-based Internet security and needs a rigorous description. One important aspect of bytecode verification is to check if a Java Virtual Machine (JVM) program is statically well-typed. So far several formal specifications have been proposed to define what the static welltypedness means. This paper takes a step further and presents a chaotic fixpoint iteration, which represents a family of fixpoint computation strategies to compute a least type for each JVM program within a finite number of iteration steps. Since a transfer function in the iteration is not monotone, we choose to follow the example of a non-standard fixpoint theorem, which requires that all transfer functions are increasing, and monotone in case the bigger element is already a fixpoint. The resulting least type is the artificial top element if and only if the JVM program is not statically well-typed. The iteration is standard and close to Sun's informal specification and...

This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and type-checking, based on the equality-as-judgement presentation. We present a set-theoretic notion of model, CC-structures, and use this to give a new strong normalization proof based on a modification of the realizability interpretation. An extension of the core calculus by inductive types is investigated and we show, using the example of infinite trees, how the realizability semantics and the strong normalization argument can be extended to non-algebraic inductive types. We emphasize that our interpretation is sound for large eliminations, e.g. allows the definition of sets by recursion. Finally we apply the extended calculus to a non-trivial problem: the formalization of the strong normalization argument for Girard's System F. This formal proof has been developed and checked using the...

We study the existence and computation of extremal solutions of a system of inequations defined over lattices. Using the Knaster-Tarski fixed point theorem, we obtain sufficient conditions for the existence of supremal as well as infimal solution of a given system of inequations. Iterative techniques are presented for the computation of the extremal solutions whenever they exist, and conditions under which the termination occurs in a single iteration are provided. These results are then applied for obtaining extremal solutions of various inequations that arise in computation of maximally permissive supervisors in control of logical discrete event systems (DESs) first studied by Ramadge and Wonham. Thus our work presents a unifying approach for computation of supervisors in a variety of situations. Keywords: Fixed points, lattices, inequations, discrete event systems, supervisory control, language theory. 1 Introduction Given a set X and a function f : X ! X, x 2 X is called a fixed p...

Abstract. A significant effort has recently been made to rigorously relate the formal treatment of cryptography with the computational one. A first substantial step in this direction was taken by Abadi and Rogaway [AR02]. Considering a formal language that treats symmetric encryption, [AR02] show that an associated formal semantics is sound with respect to an associated computational semantics, under a particular, sufficient, condition on the computational encryption scheme. In this paper, we give a necessary and sufficient condition for completeness, tightly characterizing this aspect of the exposition. Our condition involves the ability to distinguish a ciphertext and the key it was encrypted with, from a ciphertext and a random key. It is shown to be strictly weaker than a previously suggested condition for completeness (confusion-freedom of Micciancio and Warinschi [MW02]), and should be of independent interest.

This paper will be most easily appreciated by the reader with some prior knowledge of Mathematical Logic [8, 19], Set Theory [11], and Denotational Semantics [9, 18, 20]. Verdi differs from its predecessor m-Verdi [4] in several significant ways:

Motivated by applications in software engineering, we propose two forms of combination of logics: synchronization on formulae and synchronization on models. We start by reviewing satisfaction systems, consequence systems, one-step derivation systems and theory spaces, as well as their functorial relationships. We define the synchronization on formulae of two consequence systems and provide a categorial characterization of the construction. For illustration we consider the synchronization of linear temporal logic and equational logic. We define the synchronization on models of two satisfaction systems and provide a categorial characterization of the construction. We illustrate the technique in two cases: linear temporal logic versus equational logic; and linear temporal logic versus branching temporal logic. Finally, we lift the synchronization on formulae to the category of logics over consequence systems. Key words: combination of logics, synchronization on formulae, sync...

This paper studies paralsg recursion. The trace speci#cationlpeci#c used in this paper incorporates sequential,j nondeterminism, reactiveness(inclvenessg,F'k traces), three forms of paral'VgJj (inclVgJjqMkEglglgl fair-interlkEglgl synchronous paralonousg and general recursion. In order to use Tarski's theorem to determine the #xpoints of recursions, we need to identify awelVjgJ,FIq partial order.Several orders are considered,incldered new order calrg the lexical order, which tends tosimulM, the execution of a recursion in asimilk manner as the EglVqgJ,E, order. A theorem of this paper shows that no appropriate order exists for the lhegIIIE Tarski's theoremalor is not enough to determine the #xpoints ofparalVI recursions. Instead of usingTarski's theoremdirectl, we reason about the #xpoints of terminatingand nonterminatingbehavioursseparateli Such reasoningis supported by the leg of a new compositioncalio partition. We propose a #xpoint techniquecalni the partitioned #xpoint, which is thelgqk #xpoint of the nonterminatingbehaviours after the terminatingbehaviours reach their greatest #xpoint. The surprisingresul is thataltg,M, a recursion may not beljV"EgJqVE' monotonic, it must have the partitioned #xpoint, which isequal to thelegj lgjIjI,gJqF' #xpoint. Since the partitioned #xpoint iswel de#ned in anycompl,q lmpl,q theresulq areappljFMgJ to various semanticmodeli Existing#xpoint techniquessimpl becomespecial cases of the partitioned #xpoint. Forexamplj an EglIIqgJq',EFglEFg recursion has itslsgj EglMMFIgJq #xpoint, which can be shown to be the same as the partitioned #xpoint. The new technique is moregeneral than thelegq EglEEkIgJq #xpoint in that the partitioned #xpoint can be determined even when a recursion is notEglVjjVgJq monotonic.Exampln of non-monotonic recur...

In this paper we present a new fixed point theorem applicable for a countable system of recursive equations over a wellfounded domain. Wellfoundedness is an essential feature of many computer science applications as it guarantees termination of the corresponding fixed point computation algorithms. Besides being a natural restriction, it marks a new area of application, where not even monotonicity is required. We demonstrate the power and versatility of our fixed point theorem, which under the wellfoundedness condition covers all the known `synchronous' versions of fixed point theorems, by means of applications in data flow analysis and program optimization. Keywords Fixed point, chaotic iteration, vector iteration, data flow analysis, program optimization, workset algorithm, partial dead code elimination. Contents 1 Introduction 1 2 Theory 2 2.1 The Main Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2.2 Vector Iterations : : : : : : : : : : : : : : : : : :...