Results 1 
7 of
7
Modelling Concurrency with Partial Orders
, 1986
"... Concurrency has been expressed variously in terms of formal languages (typically via the shuffle operator), partial orders, and temporal logic, inter alia. In this paper we extract from these three approaches a single hybrid approach having a rich language that mixes algebra and logic and having a n ..."
Abstract

Cited by 236 (18 self)
 Add to MetaCart
Concurrency has been expressed variously in terms of formal languages (typically via the shuffle operator), partial orders, and temporal logic, inter alia. In this paper we extract from these three approaches a single hybrid approach having a rich language that mixes algebra and logic and having a natural class of models of concurrent processes. The heart of the approach is a notion of partial string derived from the view of a string as a linearly ordered multiset by relaxing the linearity constraint, thereby permitting partially ordered multisets or pomsets. Just as sets of strings form languages, so do sets of pomsets form processes. We introduce a number of operations useful for specifying concurrent processes and demonstrate their utility on some basic examples. Although none of the operations is particularly oriented to nets it is nevertheless possible to use them to express processes constructed as a net of subprocesses, and more generally as a system consisting of components. Th...
Event Spaces and Their Linear Logic
 In AMAST’91: Algebraic Methodology and Software Technology, Workshops in Computing
, 1991
"... Boolean logic treats disjunction and conjunction symmetrically and algebraically. The corresponding operations for computation are respectively nondeterminism (choice) and concurrency. Petri nets treat these symmetrically but not algebraically, while event structures treat them algebraically but not ..."
Abstract

Cited by 22 (9 self)
 Add to MetaCart
Boolean logic treats disjunction and conjunction symmetrically and algebraically. The corresponding operations for computation are respectively nondeterminism (choice) and concurrency. Petri nets treat these symmetrically but not algebraically, while event structures treat them algebraically but not symmetrically. Here we achieve both via the notion of an event space as a poset with all nonempty joins representing concurrence and a top representing the unreachable event. The symmetry is with the dual notion of state space, a poset with all nonempty meets representing choice and a bottom representing the start state. The algebra is that of a parallel programming language expanded to the language of full linear logic, Girard's axiomatization of which is satisfied by the event space interpretation of this language. Event spaces resemble finite dimensional vector spaces in distinguishing tensor product from direct product and in being isomorphic to their double dual, but differ from them i...
Chu spaces: Complementarity and Uncertainty in Rational Mechanics
, 1994
"... this paper will be realizations. The category of Boolean operations and their propertypreserving renamings is not selfdual since nonT 0 Chu spaces transpose to nonextensional ones. By the same reasoning the full subcategory consisting of T 0 operations, those with no properties a j b for distinct ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
this paper will be realizations. The category of Boolean operations and their propertypreserving renamings is not selfdual since nonT 0 Chu spaces transpose to nonextensional ones. By the same reasoning the full subcategory consisting of T 0 operations, those with no properties a j b for distinct variables a; b, is selfdual. This is a very important fact: it means that to every full subcategory C of this selfdual category we may associate its dual as the image of C under the selfduality. This associates sets to complete atomic Boolean algebras, Boolean algebras to Stone spaces, distributive lattices to StonePriestley posets, semilattices to algebraic lattices, complete semilattices to themselves, and so on for many other familiar [Joh82] and not so familiar (selfduality of finitedimensional vector spaces over GF (2)) instances of Stone duality We now illustrate the general idea with some examples.
Rational mechanics and natural mathematics
 In TAPSOFT'95
, 1995
"... Chu spaces have found applications in computer science, mathematics, and physics. They enjoy a useful categorical duality analogous to that of lattice theory and projective geometry. As natural mathematics Chu spaces borrow ideas from the natural sciences, particularly physics, while as rational mec ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Chu spaces have found applications in computer science, mathematics, and physics. They enjoy a useful categorical duality analogous to that of lattice theory and projective geometry. As natural mathematics Chu spaces borrow ideas from the natural sciences, particularly physics, while as rational mechanics they cast Hamiltonian mechanics in terms of the interaction of body and mind. This paper addresses the chief stumbling block for Descartes ’ 17thcentury philosophy of mindbody dualism, how can the fundamentally dissimilar mental and physical planes causally interact with each other? We apply Cartesian logic to reject not only divine intervention, preordained synchronization, and the eventual mass retreat to monism, but also an assumption Descartes himself somehow neglected to reject, that causal interaction within these planes is an easier problem than between. We use Chu spaces and residuation to derive all causal interaction, both between and within the two planes, from a uniform and algebraically rich theory of betweenplane interaction alone. Lifting the twovalued Boolean logic of binary relations to the complexvalued fuzzy logic of quantum mechanics transforms residuation into a natural generalization of the inner product operation of a Hilbert space and demonstrates that this account of causal interaction is of essentially the same form as the HeisenbergSchrödinger quantummechanical solution to analogous problems of causal interaction in physics. 1 Cartesian Dualism The Chu construction [Bar79] strikes us as extraordinarily useful, more so with every passing month. Elsewhere we have described the application of Chu spaces to process algebra [GP93], metamathematics [Pra93, Pra94a], and physics [Pra94b]. Here we make a first attempt at applying them to philosophy. It might seem that traditional philosophical questions would be beyond the scope of TAPSOFT. Bear in mind however that Boolean logic as the basis for
Types as Processes, via Chu spaces
 EXRESS'97 Proceedings
, 1997
"... We match up types and processes by putting values in correspondence with events, coproduct with (noninteracting) parallel composition, and tensor product with orthocurrence. We then bring types and processes into closer correspondence by broadening and unifying the semantics of both using Chu spaces ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We match up types and processes by putting values in correspondence with events, coproduct with (noninteracting) parallel composition, and tensor product with orthocurrence. We then bring types and processes into closer correspondence by broadening and unifying the semantics of both using Chu spaces and their transformational logic. Beyond this point the connection appears to break down; we pose the question of whether the failures of the corrrespondence are intrinsic or cultural. 1 Introduction Typesasprocesses modernizes dataasprograms. It is the CurryHoward propositionsastypes correspondence with propositions replaced by processes. To the extent that types and processes are both part of the working programmer 's toolkit, even more than propositions, the typesasprocesses correspondence is more central to the practice of programming than propositionsastypes. Moreover the connection works out very well mathematically, at least up to a point. The similarities and differences ...
NEWCASTLE UNIVERSITY
, 1378
"... Noninterleaving semantics of concurrent systems is often expressed using posets, where causally related events are ordered and concurrent events are unordered. Each causal poset describes a unique concurrent history which is a set of executions, expressed as sequences or step sequences, consistent ..."
Abstract
 Add to MetaCart
Noninterleaving semantics of concurrent systems is often expressed using posets, where causally related events are ordered and concurrent events are unordered. Each causal poset describes a unique concurrent history which is a set of executions, expressed as sequences or step sequences, consistent with it. Moreover, such a poset captures all precedencebased invariant relationships between the events in the executions belonging to the concurrent history. Causal poset semantics underpins efficient verification techniques based on unfoldings of safe Petri nets and concurrent automata models. However, when one considers extensions of these standard models, such as nets with inhibitor arcs, concurrent histories become too intricate to be described solely in terms of causal posets. In this paper, we introduce and investigate generalised mutex order structures which can capture the invariant causal relationships in any concurrent history consisting of step sequence executions. Each such structure comprises two relations, viz. interleaving/mutex and weak causality. As our main result we prove that each generalised mutex order structure is the intersection of step sequence executions which are consistent with it.
Causal Structures for General Concurrent
"... Abstract. Noninterleaving semantics of concurrent systems is often expressed using posets, where causally related events are ordered and concurrent events are unordered. Each causal poset describes a unique concurrent history, i.e., a set of executions, expressed as sequences or step sequences, tha ..."
Abstract
 Add to MetaCart
Abstract. Noninterleaving semantics of concurrent systems is often expressed using posets, where causally related events are ordered and concurrent events are unordered. Each causal poset describes a unique concurrent history, i.e., a set of executions, expressed as sequences or step sequences, that are consistent with it. Moreover, a poset captures all precedencebased invariant relationships between the events in the executions belonging to its concurrent history. However, concurrent histories in general may be too intricate to be described solely in terms of causal posets. In this paper, we introduce and investigate generalised mutex order structures which can capture the invariant causal relationships in any concurrent history consisting of step sequence executions. Each such structure comprises two relations, viz. interleaving/mutex and weak causality. As our main result we prove that each generalised mutex order structure is the intersection of the step sequence executions which are consistent with it.