Building Models of Linear Logic - Semantic Scholar
Building Models of Linear Logic (Extended Abstract)
Valeria de Paiva1 and Andrea Schalk2 School of Computer Science, University of Birmingham Dept of Pure Maths and Math Stats, Cambridge University 1
2
Abstract. A generic method for constructing categorical models of Lin-
ear Logic is provided and instantiated to yield traditional models such as coherence spaces, hypercoherences, phase spaces, relations, etc. The generic construction is modular, as expected. Hence we discuss multiplicative connectives, modalities and additive connectives in turn. Modelling the multiplicative connectives of Linear Logic is a generalisation of previous work, requiring a few non-standard concepts. More challenging is the modelling of the modalities `!' (and, respectively `?'), which is achieved in the surprisingly general setting of this construction by considering !-candidates and showing that they exist and constitute a modality, under appropriate conditions.
1 Introduction This paper recasts some well-known models of Linear Logic into a more general framework, that allows us to explicate some of their similarities and di erences. It is pleasing (and surprising) to nd that coherence spaces (Girard's original domain-theoretic model of Linear Logic), hypercoherences (Ehrhard's categorical explanation of sequentiality), phase spaces and even (the category of) sets and relations Rel can all be seen as speci c instances of our generic construction. This generic construction can cope both with the intuitionistic and the classical
avours of Linear Logic and it allows us to model fragments of Linear Logic, in a modular fashion, as one would expect to be able to. The motivation for this generic construction arose from comparing Chu's construction [Bar79] with dialectica categories [dP89]. We wanted to discover how far could we get when modelling Linear Logic, simply by mapping into some (linear) algebraic structure, forgetting all about any built-in duality. However, this work can be understood independently from the original motivation. It gives an account of several well-known models of Linear Logic in a uni ed framework, making them all instances of our particular categorical construction. This paper is organised as follows. We rst review coherence spaces and hypercoherences, our motivating examples. Then we describe our generic construction and show that the categories we obtain have a multiplicative structure. In the next section we describe the modality or exponential \!", which is the hard part when modelling Linear Logic. Then we describe the additives and discuss some other examples of models that can be seen as instances of our generic construction.
2 Motivating Examples We rst present a di erent perspective on two well-known models of linear Logic, namely coherence spaces and hypercoherences. Recall that a coherence space X is given by a set jX j (its `web'), and a re exive binary relation _ ^ on jX j. We use _ to denote the relation resulting from removing the diagonal from _ ^. Viewing this model from a di erent angle, we encode this structure via a function X from jX j jX j to the three element ordered set 3 := f^< 1
Valeria de Paiva1 and Andrea Schalk2 School of Computer Science, University of Birmingham Dept of Pure Maths and Math Stats, Cambridge University 1
2
Abstract. A generic method for constructing categorical models of Lin-
ear Logic is provided and instantiated to yield traditional models such as coherence spaces, hypercoherences, phase spaces, relations, etc. The generic construction is modular, as expected. Hence we discuss multiplicative connectives, modalities and additive connectives in turn. Modelling the multiplicative connectives of Linear Logic is a generalisation of previous work, requiring a few non-standard concepts. More challenging is the modelling of the modalities `!' (and, respectively `?'), which is achieved in the surprisingly general setting of this construction by considering !-candidates and showing that they exist and constitute a modality, under appropriate conditions.
1 Introduction This paper recasts some well-known models of Linear Logic into a more general framework, that allows us to explicate some of their similarities and di erences. It is pleasing (and surprising) to nd that coherence spaces (Girard's original domain-theoretic model of Linear Logic), hypercoherences (Ehrhard's categorical explanation of sequentiality), phase spaces and even (the category of) sets and relations Rel can all be seen as speci c instances of our generic construction. This generic construction can cope both with the intuitionistic and the classical
avours of Linear Logic and it allows us to model fragments of Linear Logic, in a modular fashion, as one would expect to be able to. The motivation for this generic construction arose from comparing Chu's construction [Bar79] with dialectica categories [dP89]. We wanted to discover how far could we get when modelling Linear Logic, simply by mapping into some (linear) algebraic structure, forgetting all about any built-in duality. However, this work can be understood independently from the original motivation. It gives an account of several well-known models of Linear Logic in a uni ed framework, making them all instances of our particular categorical construction. This paper is organised as follows. We rst review coherence spaces and hypercoherences, our motivating examples. Then we describe our generic construction and show that the categories we obtain have a multiplicative structure. In the next section we describe the modality or exponential \!", which is the hard part when modelling Linear Logic. Then we describe the additives and discuss some other examples of models that can be seen as instances of our generic construction.
2 Motivating Examples We rst present a di erent perspective on two well-known models of linear Logic, namely coherence spaces and hypercoherences. Recall that a coherence space X is given by a set jX j (its `web'), and a re exive binary relation _ ^ on jX j. We use _ to denote the relation resulting from removing the diagonal from _ ^. Viewing this model from a di erent angle, we encode this structure via a function X from jX j jX j to the three element ordered set 3 := f^< 1
