426 research outputs found
Modular specifications in process algebra
In recent years a wide variety of process algebras has been proposed in the literature. Often these process algebras are closely related: they can be viewed as homomorphic images, submodels or restrictions of each other. The aim of this paper is to show how the semantical reality, consisting of a large number of closely related process algebras, can be reflected, and even used, on the level of algebraic specifications and in process verifications. This is done by means of the notion of a module. The simplest modules are building blocks of operators and axioms, each block describing a feature of concurrency in a certain semantical setting. These modules can then be combined by means of a union operator +, an export operator â–ˇ, allowing to forget some operators in a module, an operator H, changing semantics by taking homomorphic images, and an operator S which takes subalgebras. These operators enable us to combine modules in a subtle way, when the direct combination would be inconsistent. We show how auxiliary process algebra operators can be hidden when this is needed. Moreover it is demonstrated how new process combinators can be defined in terms of the more elementary ones in a clean way. As an illustration of our approach, a methodology is presented that can be used to specify FIFO-queues, and that facilitates verification of concurrent systems containing these queues
Modular specification of process algebras
AbstractThis paper proposes a modular approach to the algebraic specification of process algebras. This is done by means of the notion of a module. The simplest modules are building blocks of operators and axioms, each block describing a feature of concurrency in a certain semantical setting. These modules can then be combined by means of a union operator +, an export operator □, allowing to forget some operators in a module, an operator H, changing semantics by taking homomorphic images, and an operator S which takes subalgebras. These operators enable us to combine modules in a subtle way, when the direct combination would be inconsistent.We give a presentation of equational logic, infinitary conditional equational logic — of which we also prove the completeness — and first-order logic and show how the notion of a formal proof of a formula from a theory can be generalized to that of a proof of a formula from a module. This module logic is then applied in process algebra. We show how auxiliary process algebra operators can be hidden when this is needed. Moreover, we demonstrate how new process combinators can be defined in terms of more elementary ones in a clean way. As an illutration of our approach, we specify some FIFO-queues and verify several of their properties
On the Syntax of Logic and Set Theory
We introduce an extension of the propositional calculus to include abstracts
of predicates and quantifiers, employing a single rule along with a novel
comprehension schema and a principle of extensionality, which are substituted
for the Bernays postulates for quantifiers and the comprehension schemata of ZF
and other set theories. We prove that it is consistent in any finite Boolean
subset lattice. We investigate the antinomies of Russell, Cantor, Burali-Forti,
and others, and discuss the relationship of the system to other set theoretic
systems ZF, NBG, and NF. We discuss two methods of axiomatizing higher order
quantification and abstraction, and then very briefly discuss the application
of one of these methods to areas of mathematics outside of logic.Comment: 34 pages, accepted, to appear in the Review of Symbolic Logi
Coalgebraic Infinite Traces and Kleisli Simulations
Kleisli simulation is a categorical notion introduced by Hasuo to verify
finite trace inclusion. They allow us to give definitions of forward and
backward simulation for various types of systems. A generic categorical theory
behind Kleisli simulation has been developed and it guarantees the soundness of
those simulations with respect to finite trace semantics. Moreover, those
simulations can be aided by forward partial execution (FPE)---a categorical
transformation of systems previously introduced by the authors.
In this paper, we give Kleisli simulation a theoretical foundation that
assures its soundness also with respect to infinitary traces. There, following
Jacobs' work, infinitary trace semantics is characterized as the "largest
homomorphism." It turns out that soundness of forward simulations is rather
straightforward; that of backward simulation holds too, although it requires
certain additional conditions and its proof is more involved. We also show that
FPE can be successfully employed in the infinitary trace setting to enhance the
applicability of Kleisli simulations as witnesses of trace inclusion. Our
framework is parameterized in the monad for branching as well as in the functor
for linear-time behaviors; for the former we mainly use the powerset monad (for
nondeterminism), the sub-Giry monad (for probability), and the lift monad (for
exception).Comment: 39 pages, 1 figur
Laver's results and low-dimensional topology
In connection with his interest in selfdistributive algebra, Richard Laver
established two deep results with potential applications in low-dimensional
topology, namely the existence of what is now known as the Laver tables and the
well-foundedness of the standard ordering of positive braids. Here we present
these results and discuss the way they could be used in topological
applications
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