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In this paper we consider the problem of deriving a term assignment system for Girard's Intuitionistic Linear Logic for both the sequent calculus and natural deduction proof systems. Our system differs from previous calculi (e.g. that of Abramsky) and has two important properties which they lack. These are the substitution property (the set of valid deductions is closed under substitution) and subject reduction (reduction on terms is well-typed). We define a simple (but more general than previous proposals) categorical model for Intuitionistic Linear Logic and show how this can be used to derive the term assignment system. We also consider term reduction arising from cut-elimination in the sequent calculus and normalisation in natural deduction. We explore the relationship between these, as well as with the equations which follow from our categorical model. Technical Report 262, University of Cambridge Computer Laboratory. Contents 1 Introduction 3 2 Introduction to Intuitionisti...

Content uploaded by Martin Hyland

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All content in this area was uploaded by Martin Hyland on Jul 24, 2013

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... Section 2 recalls linear λ-calculus and its equational system together with corresponding proofs of soundness, completeness, and the aforementioned equivalence with a category of autonomous categories. The contents of this section are adaptations of results presented in [BBdPH92,MRA93,Cro93,MMdPR05], the main difference being that we forbid the exchange rule to be explicitly part of linear λ-calculus (instead it is only admissible). This choice is important to ensure that judgements in the calculus have unique derivations, which allows to refer to their interpretations unambiguously [Shu21]. ...

... As we will see, the semantic counterpart of moving from equations to V-equations is to move from ordinary categories to categories enriched over V-categories. The latter, often regarded as generalised metric spaces, are central entities in a fruitful area of enriched category theory that aims to treat uniformly different kinds of 'structured sets', such find more details in [MRA93,BBdPH92,MMdPR05]. We then present categories of linear λ-theories and of autonomous categories, and show that they are equivalent. ...

... As detailed in [BBdPH92,MRA93,MMdPR05], linear λ-calculus comes equipped with a class of equations, given in Fig. 3, specifically equations-in-context Γ ▷ v = w : A, that corresponds to the axiomatics of autonomous categories. As usual, we omit the context and typing information of the equations in Fig. 3, which can be reconstructed in the usual way. ...

Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale V, which covers the cases of (in)equations and (ultra)metric equations among others. Our main result is the introduction of a V-equational deductive system for linear {\lambda}-calculus together with a proof that it is sound and complete. In fact we go further than this, by showing that linear {\lambda}-theories based on this V-equational system form a category that is equivalent to a category of autonomous categories enriched over 'generalised metric spaces'. If we instantiate this result to inequations, we get an equivalence with autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an equivalence with autonomous categories enriched over (ultra)metric spaces. We additionally show that this syntax-semantics correspondence extends to the affine setting. We use our results to develop examples of inequational and metric equational systems for higher-order programming in the setting of real-time, probabilistic, and quantum computing.

... We start by presenting our graded λ-calculus. In a nutshell, it is a graded extension of the linear-non-linear λ-calculus in [5,6] and can be seen as a term assignment system for a graded version of intuitionistic linear logic. Aside from the use of grades, the main difference with [5,6] is the use of a shuffling mechanism [52] that allows to refer to a λ-term's denotation unambiguously (more details below). ...

... In a nutshell, it is a graded extension of the linear-non-linear λ-calculus in [5,6] and can be seen as a term assignment system for a graded version of intuitionistic linear logic. Aside from the use of grades, the main difference with [5,6] is the use of a shuffling mechanism [52] that allows to refer to a λ-term's denotation unambiguously (more details below). Types. ...

... The equations concerning the monoidal structure and the closed structure were already discussed elsewhere (e.g. [5,14]). The equations concerning commuting conversions enforce the fact that certain expressions differing in scope such as (ds v. u) ⊗ w and ds v. (u ⊗ w) are intended to have the same meaning. ...

Modern programming frequently requires generalised notions of program equivalence based on a metric or a similar structure. Previous work addressed this challenge by introducing the notion of a V-equation, i.e. an equation labelled by an element of a quantale V, which covers inter alia (ultra-)metric, classical, and fuzzy (in)equations. It also introduced a V-equational system for the linear variant of lambda-calculus where any given resource must be used exactly once. In this paper we drop the (often too strict) linearity constraint by adding graded modal types which allow multiple uses of a resource in a controlled manner. We show that such a control, whilst providing more expressivity to the programmer, also interacts more richly with V-equations than the linear or Cartesian cases. Our main result is the introduction of a sound and complete V-equational system for a lambda-calculus with graded modal types interpreted by what we call a Lipschitz exponential comonad. We also show how to build such comonads canonically via a universal construction, and use our results to derive graded metric equational systems (and corresponding models) for programs with timed and probabilistic behaviour.

... We start by presenting our graded λ-calculus. In a nutshell, it is a graded extension of the linear-non-linear λ-calculus in [2,3] and can be seen as a term assignment system for a graded version of intuitionistic linear logic. Aside from the use of grades, the main difference with [2,3] is the use of a shuffling mechanism [49] that allows to refer to a λ-term's denotation unambiguously (more details below). ...

... In a nutshell, it is a graded extension of the linear-non-linear λ-calculus in [2,3] and can be seen as a term assignment system for a graded version of intuitionistic linear logic. Aside from the use of grades, the main difference with [2,3] is the use of a shuffling mechanism [49] that allows to refer to a λ-term's denotation unambiguously (more details below). Types. ...

... The equations concerning the monoidal structure and the closed structure were already discussed elsewhere (e.g. [2,11]). The equations concerning commuting conversions enforce the fact that certain expressions differing in scope such as (ds v. u) ⊗ w and ds v. (u ⊗ w) are intended to have the same meaning. ...

Modern programming frequently requires generalised notions of program equivalence based on a metric or a similar structure. Previous work addressed this challenge by introducing the notion of a V-equation, i.e. an equation labelled by an element of a quantale V, which covers inter alia (ultra-)metric, classical, and fuzzy (in)equations. It also introduced a V-equational system for the linear variant of lambda-calculus where any given resource must be used exactly once. In this paper we drop the (often too strict) linearity constraint by adding graded modal types which allow multiple uses of a resource in a controlled manner. We show that such a control, whilst providing more expressivity to the programmer, also interacts more richly with V-equations than the linear or Cartesian cases. Our main result is the introduction of a sound and complete V-equational system for a lambda-calculus with graded modal types interpreted by what we call a Lipschitz exponential comonad. We also show how to build such comonads canonically via a universal construction, and use our results to derive graded metric equational systems (and corresponding models) for programs with timed and probabilistic behaviour.

... §2 recalls linear λ-calculus and its equational system together with corresponding proofs of soundness, completeness, and the aforementioned equivalence with a category of autonomous categories (in fact a 'quasi-category', see § 2). The contents of this section are slight adaptations of results presented in [BBdPH92,MRA93,Cro93,MMDPR05], the main difference being that we forbid the exchange rule to be explicitly part of linear λcalculus (instead it is only admissible). This choice is important to ensure that judgements in the calculus have unique derivations, which allows us to refer to their interpretations unambiguously [Shu19]. ...

... Then we recall that it is sound and complete w.r.t. autonomous categories -we mention only what is needed to present our results, the interested reader will find a more detailed exposition in [MRA93,BBdPH92,MMDPR05]. Subsequently we present a quasi-category [AHS09] of linear λ-theories, a quasi-category of autonomous categories, and show that they are equivalent. ...

... As detailed in [BBdPH92,MRA93,MMDPR05], linear λ-calculus comes equipped with a class of equations, given in Fig. 3, specifically equations-in-context Γ ⊲ v = w : A, that corresponds to the axiomatics of autonomous categories. As usual, we omit the context and typing information of the equations in Fig. 3, which can be reconstructed in the usual way. ...

Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale V, which covers the cases of (in)equations and (ultra)metric equations among others. Our main result is the introduction of a V-equational deductive system for linear {\lambda}-calculus together with a proof that it is sound and complete. In fact we go further than this, by showing that linear {\lambda}-theories based on this V-equational system form a category that is equivalent to a category of autonomous categories enriched over 'generalised metric spaces'. In other words, we prove the existence of a so-called syntax-semantics duality between both structures which are parametrised by V: if we choose linear {\lambda}-calculus based on inequations, we obtain a correspondence with autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get a correspondence with autonomous categories enriched over (ultra)metric spaces. We also show that this correspondence extends to the affine setting. We use our results to develop examples of inequational and metric equational systems for higher-order programming in the setting of real-time, probabilistic, and quantum computing.

... Section 2 recalls linear λ-calculus, its equational system, and the well-known correspondence to autonomous categories, via soundness, completeness, and internal language theorems. The results of this section are slight adaptations of those presented in [24,4], the main difference being that we forbid the exchange rule to be explicitly part of linear λ-calculus (instead it is only admissible). This choice is important to ensure that judgements in the calculus have unique derivations, which allows us to refer to their interpretations unambiguously [33]. ...

... Our paper falls in this line of research. Specifically, our aim is to integrate notions of approximation and refinement into the equational system of linear λ-calculus [4,24]. The core idea that we explore in this paper is to have equations t = q s labelled by elements q of a quantale V. ...

... , A n → A in Σ we postulate an interpretation f : A 1 ⊗ · · · ⊗ A n → A as a C-morphism. The interpretation of judgements is defined by induction over the structure of judgement derivation according to the rules in Fig. 2. Linear λ-calculus comes equipped with a class of equations ( Fig. 3), specifically equationsin-context Γ ▷ v = w : A, that corresponds to the axiomatics of autonomous categories [4,24]. For simplicity we omit the context and typing information of the equations in Fig. 3, which can be reconstructed in the usual way. ...

Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale V, which covers e.g. the cases of inequations and (ultra)metric equations. Our main result is the development of a V-equational deductive system for linear lambda-calculus together with a proof that it is sound and complete (in fact, an internal language) for a class of enriched autonomous categories. In the case of inequations, we get an internal language for autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an internal language for autonomous categories enriched over (ultra)metric spaces. We use our results to obtain examples of inequational and metric equational systems for higher-order programs that contain real-time and probabilistic behaviour.

... Our categorical account is based on the established categorical semantics of ILL [35], [36], [37], [28] and of IL [26], [27], as well as the study of the relation between monad and comonad [29] and its application in game semantics [38]. ...

... 16 concisely. Recall that NSCs give a (equationally sound and complete) semantics of ILL without ⊥ or ⊕ (w.r.t. the term calculus given in [36], [37]): ...

... Note that !D, !W and !C may be handled just as in the interpretation of ILL in NSCs [36], [37]; ?D, ?W and ?C are just symmetric. Also, XL and XR are interpreted by symmetries w.r.t. ...

The present work aims to give a unity of logic via standard sequential, unpolarized games. Specifically, our vision is that there must be mathematically precise concepts of linear refinement and intuitionistic restriction of logic such that the linear refinement of classical logic (CL) coincides with (classical) linear logic (LL), and its intuitionistic restriction with the linear refinement of intuitionistic logic (IL) into intuitionistic LL (ILL). However, LL is, in contradiction to the name, cannot be the linear refinement of CL at least from the game-semantic point of view due to its concurrency and polarization. In fact, existing game semantics of LL employs concurrency, which is rather exotic to game semantics of ILL, IL or CL. Also, linear negation in LL is never true in (game semantics of) ILL, IL or CL. In search for the truly linear refinement of CL, we carve out (a sequent calculus of) linear logic negative ($LL^-$) from (the two-sided sequent calculus of) LL, and introducing a new distribution axiom $! ? A \vdash ? ! A$ (for a translation of sequents $\Delta \vdash \Gamma$ for CL into the sequents $! \Delta \vdash ? \Gamma$ for $LL^-$). We then give a categorical semantics of $LL^-$, for which we introduce why not monad ?, dual to the well-known of course comonad !, giving a categorical translation $\Delta \rightarrow \Gamma = ? (\Delta \multimap \Gamma) \cong ! \Delta \multimap ? \Gamma$ of CL into $LL^-$, which is the Kleisli extension of the standard translation $\Delta \rightarrow \Gamma = ! \Delta \multimap \Gamma$ of IL into ILL. Moreover, we instantiate the categorical semantics by fully complete (sequential, unpolarized) game semantics of $LL^-$ (without atoms), for which we introduce linearity of strategies.

... obtained from σ using correspondence (6) and symmetry transformations. We take the set of literals ...

... In this section we assume that the reader is familiar with basic notions of λ-calculus, see [4] for a reference. We use [6] as a reference for syntax and semantics of linear λ-calculus and intuitionistic linear logic. We note though that we consider only the simplest, implicational fragment, while definitions and results in [6] are formulated for the full system. ...

... We use [6] as a reference for syntax and semantics of linear λ-calculus and intuitionistic linear logic. We note though that we consider only the simplest, implicational fragment, while definitions and results in [6] are formulated for the full system. In fact, theorems of [6] that we cite are rather straightforward in the purely implicational case. ...

(This is an improved version of the previously posted paper "Classical linear logic, cobordisms and categorical semantics of categorial grammars", with reduced amount of category theory, much simplified definitions and a number of examples. Also, treatment of tree languages is added.)
We propose a categorial grammar based on classical multiplicative linear logic. This can be seen as an extension of abstract categorial grammars (ACG) and is at least as expressive. However, constituents of linear logic grammars (LLG) are not abstract λ-terms, but simply tuples of words with labeled endpoints and supplied with specific plugging instructions: the sets of endpoints are subdivided into the incoming and the outgoing parts. We call such objects word cobordisms. A key observation is that word cobordisms can be organized in a category , very similar to the familiar category of topological cobordisms. This category is symmetric monoidal closed and compact closed and thus is a model of linear λ-calculus and classical, as well as intuitionistic linear logic. This allows us using linear logic as a typing system for word cobor-disms. At least, this gives a concrete and intuitive representation of ACG. We think, however, that the category of word cobordisms, which has a rich structure and is independent of any grammar, might be interesting on its own right.

... If P is an lnl multicategory with ⊗, 1, F, U, the symmetric monoidal category P L admits a linear exponential comonad[BBdPH92,HS03], i.e. it is a linear category in the sense of[Ben95].Proof. Let ! ...

We define and study LNL polycategories, which abstract the judgmental structure of classical linear logic with exponentials. Many existing structures can be represented as LNL polycategories, including LNL adjunctions, linear exponential comonads, LNL multicategories, IL-indexed categories, linearly distributive categories with storage, commutative and strong monads, CBPV-structures, models of polarized calculi, Freyd-categories, and skew multicategories, as well as ordinary cartesian, symmetric, and planar multicategories and monoidal categories, symmetric polycategories, and linearly distributive and *-autonomous categories. To study such classes of structures uniformly, we define a notion of LNL doctrine, such that each of these classes of structures can be identified with the algebras for some such doctrine. We show that free algebras for LNL doctrines can be presented by a sequent calculus, and that every morphism of doctrines induces an adjunction between their 2-categories of algebras.

... Proposition 3.3. If P is an lnl multicategory with ⊗, 1, F, U, the symmetric monoidal category P L admits a linear exponential comonad [BBdPH92,HS03], i.e. it is a linear category in the sense of [Ben95]. ...

We define and study LNL polycategories, which abstract the judgmental structure of classical linear logic with exponentials. Many existing structures can be represented as LNL polycategories, including LNL adjunctions, linear exponential comonads, LNL multicategories, IL-indexed categories, linearly distributive categories with storage, commutative and strong monads, CBPV-structures, models of polarized calculi, skew multicategories, as well as ordinary cartesian and symmetric multicategories and monoidal categories, polycategories, and linearly distributive and *-autonomous categories. To study such classes of structures uniformly, we define a notion of LNL doctrine, such that each of these classes of structures can be identified with the algebras for some such doctrine. We show that free algebras for LNL doctrines can be presented by a sequent calculus, and that every morphism of doctrines induces an adjunction between their 2-categories of algebras.

The Sweedler semantics of intuitionistic differential linear logic takes values in the category of vector spaces, using the cofree cocommutative coalgebra to interpret the exponential and primitive elements to interpret the differential structure. In this paper, we explicitly compute the denotations under this semantics of an interesting class of proofs in linear logic, introduced by Girard: the encodings of step functions of Turing machines. Along the way we prove some useful technical results about linear independence of denotations of Church numerals and binary integers.

A typed, modular paradigm for polynomial time computation is proposed.

Logical Systems and Semantics.- Introducing HPC.- The Kripke, Beth and Topological Interpretations for HPC.- Heyting's Propositional Calculus and Extensions.- Three Intermediate Logics.- Formulas in One Variable.- Propositional Connectives.- The Interpolation Theorem.- Second Order Propositional Calculus.- Modified Kripke Interpretation.- Theories in HPC 1.- Theories in HPC 2.- Completeness of HPC with Respect to RE and Post Structures.- Undecidability Results.- Decidability Results.