Abstract Particular focus in this paper is on quantum information protocols, which exploit quantu... more Abstract Particular focus in this paper is on quantum information protocols, which exploit quantum-mechanical effects in an essential way. The particular examples we shall use to illustrate our approach will be teleportation (Benett et al., 1993), logic-gate teleportation (Gottesman and Chuang, 1999), and entanglement swapping (Zukowski et al., 1993).
Abstract Particular focus in this paper is on quantum information protocols, which exploit quantu... more Abstract Particular focus in this paper is on quantum information protocols, which exploit quantum-mechanical effects in an essential way. The particular examples we shall use to illustrate our approach will be teleportation (Benett et al., 1993), logic-gate teleportation (Gottesman and Chuang, 1999), and entanglement swapping (Zukowski et al., 1993).
Abstract Some basic topics in the theory of concurrency are studied from the point of view of den... more Abstract Some basic topics in the theory of concurrency are studied from the point of view of denotational semantics, and particularly the “domain theory in logical form” developed by the author. A domain of synchronization trees is defined by means of a recursive domain equation involving the Plotkin powerdomain. The logical counterpart of this domain is described, and shown to be related to it by Stone duality.
The very existence of the CONCUR conference bears witness to the fact that "concurrency theory" h... more The very existence of the CONCUR conference bears witness to the fact that "concurrency theory" has developed into a subject unto itself, with substantiaUy different emphases and techniques to those prominent elsewhere in the semantics of computation. Whatever the past merits of this separate development, it seems timely to look for some convergence and unification. In addressing these issues, I have found it instructive to trace some of the received ideas in concurrency back to their origins in the early 1970's.
Abstract We propose Interaction Categories as a new paradigm for the semantics of computation. Th... more Abstract We propose Interaction Categories as a new paradigm for the semantics of computation. The categories standardly used for denotational semantics have structured sets as objects and functions as morphisms. This limits the really e ective use of denotational methods to the sphere of functional computation. Interaction categories have speci cations as objects, processes as morphisms, and interaction as composition.
Abstract We use the mathematical language of sheaf theory to give a unified treatment of non-loca... more Abstract We use the mathematical language of sheaf theory to give a unified treatment of non-locality and contextuality, in a setting that generalizes the familiar probability tables used in non-locality theory to arbitrary measurement covers; this includes Kochen–Specker configurations and more. We show that contextuality, and non-locality as a special case, correspond exactly to obstructions to the existence of global sections.
For the 1st-generation models, logic could be taken\ as it was"| static and timeless For our 2nd-... more For the 1st-generation models, logic could be taken\ as it was"| static and timeless For our 2nd-generation models, getting an adequate account| a genuine\ logic of interaction"| may require a fundamental reconceptualization of logic itself. This radical revision of our view of logic is happening anyway| prompted partly by the applications, and partly from ideas arising within pure logic.
Genericity is the idea that the same program can work at many different data types. Longo, Milste... more Genericity is the idea that the same program can work at many different data types. Longo, Milsted and Soloviev proposed to capture the inability of generic programs to probe the structure of their instances by the following equational principle: if two generic programs, viewed as terms of type.∀ XA [X], are equal at any given instance A [T], then they are equal at all instances.
Abstract This paper defines the categorical notions of relators and transformations and shows tha... more Abstract This paper defines the categorical notions of relators and transformations and shows that these concepts enable us to give a semantics for polymorphic, higher order functional programs. We demonstrate the pertinence of this semantics to the analysis of polymorphic programs by proving that strictness analysis is a polymorphic invariant.
Abstract: We present a novel coalgebraic formulation of infinite extensive games. We define both ... more Abstract: We present a novel coalgebraic formulation of infinite extensive games. We define both the game trees and the strategy profiles by possibly infinite systems of corecursive equations. Certain strategy profiles are proved to be subgame perfect equilibria using a novel proof principle of predicate coinduction which is shown to be sound by reducing it to Kozen's metric coinduction. We characterize all subgame perfect equilibria for the dollar auction game.
The manipulation of objects with state which changes over time is all-pervasive in computing. Per... more The manipulation of objects with state which changes over time is all-pervasive in computing. Perhaps the simplest example of such objects are the program variables of classical imperative languages. An important strand of work within the study of such languages, pioneered by John Reynolds, focuses on “Idealized Algol”, an elegant synthesis of imperative and functional features. We present a novel semantics for Idealized Algol using games, which is quite unlike traditional denotational models of state.
An intensional model for the programming language PCF is described in which the types of PCF are ... more An intensional model for the programming language PCF is described in which the types of PCF are interpreted by games and the terms by certain history-free strategies. This model is shown to capture definability in PCF. More precisely, every compact strategy in the model is definable in a certain simple extension of PCF. We then introduce an intrinsic preorder on strategies and show that it satisfies some striking properties such that the intrinsic preorder on function types coincides with the pointwise preorder.
Abstract: We define a strongly normalising proof-net calculus corresponding to the logic of stron... more Abstract: We define a strongly normalising proof-net calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with biproducts on a given category with an involution. This syntax can be used to represent and reason about quantum processes.
Our general motivation is to answer the question:“What is a model of concurrent computation?”. As... more Our general motivation is to answer the question:“What is a model of concurrent computation?”. As a preliminary exercise, we study dataflow networks. We develop a very general notion of model for asynchronous networks. The “Kahn Principle”, which states that a network built from functional nodes is the least fixpoint of a system of equations associated with the network, has become a benchmark for the formal study of dataflow networks.
The aim of these notes is to provide a succinct, accessible introduction to some of the basic ide... more The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain numerous exercises, and hopefully will prove useful for self-study by those seeking a first introduction to the subject, with fairly minimal prerequisites. The coverage is by no means comprehensive, but should provide a good basis for further study; a guide to further reading is included.
Diagonal arguments lie at the root of many fundamental phenomena in the foundations of logic and ... more Diagonal arguments lie at the root of many fundamental phenomena in the foundations of logic and mathematics. Recently, a striking form of diagonal argument has appeared in the foundations of epistemic game theory, in a paper by Adam Brandenburger and H. Jerome Keisler [11]. The core Brandenburger-Keisler result can be seen, as they observe, as a two-person or interactive version of Russell's Paradox.
Abstract: We use a simple relational framework to develop the key notions and results on hidden v... more Abstract: We use a simple relational framework to develop the key notions and results on hidden variables and non-locality. The extensive literature on these topics in the foundations of quantum mechanics is couched in terms of probabilistic models, and properties such as locality and no-signalling are formulated probabilistically. We show that to a remarkable extent, the main structure of the theory, through the major No-Go theorems and beyond, survives intact under the replacement of probability distributions by mere relations.
Game Semantics has emerged as a powerful paradigm for giving semantics to a variety of programmin... more Game Semantics has emerged as a powerful paradigm for giving semantics to a variety of programming languages and logical systems. It has been used to construct the first syntax-independent fully abstract models for a spectrum of programming languages ranging from purely functional languages to languages with non-functional features such as control operators and locally-scoped references [4, 21, 5, 19, 2, 22, 17, 11].
Abstract Particular focus in this paper is on quantum information protocols, which exploit quantu... more Abstract Particular focus in this paper is on quantum information protocols, which exploit quantum-mechanical effects in an essential way. The particular examples we shall use to illustrate our approach will be teleportation (Benett et al., 1993), logic-gate teleportation (Gottesman and Chuang, 1999), and entanglement swapping (Zukowski et al., 1993).
Abstract Particular focus in this paper is on quantum information protocols, which exploit quantu... more Abstract Particular focus in this paper is on quantum information protocols, which exploit quantum-mechanical effects in an essential way. The particular examples we shall use to illustrate our approach will be teleportation (Benett et al., 1993), logic-gate teleportation (Gottesman and Chuang, 1999), and entanglement swapping (Zukowski et al., 1993).
Abstract Some basic topics in the theory of concurrency are studied from the point of view of den... more Abstract Some basic topics in the theory of concurrency are studied from the point of view of denotational semantics, and particularly the “domain theory in logical form” developed by the author. A domain of synchronization trees is defined by means of a recursive domain equation involving the Plotkin powerdomain. The logical counterpart of this domain is described, and shown to be related to it by Stone duality.
The very existence of the CONCUR conference bears witness to the fact that "concurrency theory" h... more The very existence of the CONCUR conference bears witness to the fact that "concurrency theory" has developed into a subject unto itself, with substantiaUy different emphases and techniques to those prominent elsewhere in the semantics of computation. Whatever the past merits of this separate development, it seems timely to look for some convergence and unification. In addressing these issues, I have found it instructive to trace some of the received ideas in concurrency back to their origins in the early 1970's.
Abstract We propose Interaction Categories as a new paradigm for the semantics of computation. Th... more Abstract We propose Interaction Categories as a new paradigm for the semantics of computation. The categories standardly used for denotational semantics have structured sets as objects and functions as morphisms. This limits the really e ective use of denotational methods to the sphere of functional computation. Interaction categories have speci cations as objects, processes as morphisms, and interaction as composition.
Abstract We use the mathematical language of sheaf theory to give a unified treatment of non-loca... more Abstract We use the mathematical language of sheaf theory to give a unified treatment of non-locality and contextuality, in a setting that generalizes the familiar probability tables used in non-locality theory to arbitrary measurement covers; this includes Kochen–Specker configurations and more. We show that contextuality, and non-locality as a special case, correspond exactly to obstructions to the existence of global sections.
For the 1st-generation models, logic could be taken\ as it was"| static and timeless For our 2nd-... more For the 1st-generation models, logic could be taken\ as it was"| static and timeless For our 2nd-generation models, getting an adequate account| a genuine\ logic of interaction"| may require a fundamental reconceptualization of logic itself. This radical revision of our view of logic is happening anyway| prompted partly by the applications, and partly from ideas arising within pure logic.
Genericity is the idea that the same program can work at many different data types. Longo, Milste... more Genericity is the idea that the same program can work at many different data types. Longo, Milsted and Soloviev proposed to capture the inability of generic programs to probe the structure of their instances by the following equational principle: if two generic programs, viewed as terms of type.∀ XA [X], are equal at any given instance A [T], then they are equal at all instances.
Abstract This paper defines the categorical notions of relators and transformations and shows tha... more Abstract This paper defines the categorical notions of relators and transformations and shows that these concepts enable us to give a semantics for polymorphic, higher order functional programs. We demonstrate the pertinence of this semantics to the analysis of polymorphic programs by proving that strictness analysis is a polymorphic invariant.
Abstract: We present a novel coalgebraic formulation of infinite extensive games. We define both ... more Abstract: We present a novel coalgebraic formulation of infinite extensive games. We define both the game trees and the strategy profiles by possibly infinite systems of corecursive equations. Certain strategy profiles are proved to be subgame perfect equilibria using a novel proof principle of predicate coinduction which is shown to be sound by reducing it to Kozen's metric coinduction. We characterize all subgame perfect equilibria for the dollar auction game.
The manipulation of objects with state which changes over time is all-pervasive in computing. Per... more The manipulation of objects with state which changes over time is all-pervasive in computing. Perhaps the simplest example of such objects are the program variables of classical imperative languages. An important strand of work within the study of such languages, pioneered by John Reynolds, focuses on “Idealized Algol”, an elegant synthesis of imperative and functional features. We present a novel semantics for Idealized Algol using games, which is quite unlike traditional denotational models of state.
An intensional model for the programming language PCF is described in which the types of PCF are ... more An intensional model for the programming language PCF is described in which the types of PCF are interpreted by games and the terms by certain history-free strategies. This model is shown to capture definability in PCF. More precisely, every compact strategy in the model is definable in a certain simple extension of PCF. We then introduce an intrinsic preorder on strategies and show that it satisfies some striking properties such that the intrinsic preorder on function types coincides with the pointwise preorder.
Abstract: We define a strongly normalising proof-net calculus corresponding to the logic of stron... more Abstract: We define a strongly normalising proof-net calculus corresponding to the logic of strongly compact closed categories with biproducts. The calculus is a full and faithful representation of the free strongly compact closed category with biproducts on a given category with an involution. This syntax can be used to represent and reason about quantum processes.
Our general motivation is to answer the question:“What is a model of concurrent computation?”. As... more Our general motivation is to answer the question:“What is a model of concurrent computation?”. As a preliminary exercise, we study dataflow networks. We develop a very general notion of model for asynchronous networks. The “Kahn Principle”, which states that a network built from functional nodes is the least fixpoint of a system of equations associated with the network, has become a benchmark for the formal study of dataflow networks.
The aim of these notes is to provide a succinct, accessible introduction to some of the basic ide... more The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain numerous exercises, and hopefully will prove useful for self-study by those seeking a first introduction to the subject, with fairly minimal prerequisites. The coverage is by no means comprehensive, but should provide a good basis for further study; a guide to further reading is included.
Diagonal arguments lie at the root of many fundamental phenomena in the foundations of logic and ... more Diagonal arguments lie at the root of many fundamental phenomena in the foundations of logic and mathematics. Recently, a striking form of diagonal argument has appeared in the foundations of epistemic game theory, in a paper by Adam Brandenburger and H. Jerome Keisler [11]. The core Brandenburger-Keisler result can be seen, as they observe, as a two-person or interactive version of Russell's Paradox.
Abstract: We use a simple relational framework to develop the key notions and results on hidden v... more Abstract: We use a simple relational framework to develop the key notions and results on hidden variables and non-locality. The extensive literature on these topics in the foundations of quantum mechanics is couched in terms of probabilistic models, and properties such as locality and no-signalling are formulated probabilistically. We show that to a remarkable extent, the main structure of the theory, through the major No-Go theorems and beyond, survives intact under the replacement of probability distributions by mere relations.
Game Semantics has emerged as a powerful paradigm for giving semantics to a variety of programmin... more Game Semantics has emerged as a powerful paradigm for giving semantics to a variety of programming languages and logical systems. It has been used to construct the first syntax-independent fully abstract models for a spectrum of programming languages ranging from purely functional languages to languages with non-functional features such as control operators and locally-scoped references [4, 21, 5, 19, 2, 22, 17, 11].
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