Masahito Hasegawa

Finite dimensional vector spaces are complete for traced symmetric monoidal categories

We show that the category FinVectk of finite dimensional vector spaces and linear maps over any field k is (collectively) complete for the traced symmetric monoidal category freely generated from a signature, provided that the field has characteristic 0; this means that for any two different arrows in the free traced category there always exists a strong traced functor into FinVectk which distinguishes them. Therefore two arrows in the free traced category are the same if and only if they agree for all interpretations in FinVectk.

Recursion from Cyclic Sharing: Traced Monoidal Categories and Models of Cyclic Lambda Calculi

Cyclic sharing (cyclic graph rewriting) has been used as a practical technique for implementing recursive computation efficiently. To capture its semantic nature, we introduce categorical models for lambda calculi with cyclic sharing (cyclic lambda graphs), using notions of computation by Moggi/Power and Robinson and traced monoidal categories by Joyal, Street and Verity. The former is used for representing the notion of sharing, whereas the latter for cyclic data structures. Our new models provide a semantic framework for understanding recursion created from cyclic sharing, which includes traditional models for recursion created from fixed points as special cases. Our cyclic lambda calculus serves as a uniform language for this wider range of models of recursive computation.

A sound and complete axiomatization of delimited continuations

The shift and reset operators, proposed by Danvy and Filinski, are powerful control primitives for capturing delimited continuations. Delimited continuation is a similar concept as the standard (unlimited) continuation, but it represents part of the rest of the computation, rather than the whole rest of computation. In the literature, the semantics of shift and reset has been given by a CPS-translation only. This paper gives a direct axiomatization of calculus with shift and reset, namely, we introduce a set of equations, and prove that it is sound and complete with respect to the CPS-translation. We also introduce a calculus with control operators which is as expressive as the calculus with shift and reset, has a sound and complete axiomatization, and is conservative over Sabry and Felleisens theory for first-class continuations.

Models of Sharing Graphs

A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is first-order acyclic sharing graphs represented by let-syntax, and others are extensions with higher-order constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and traced monoidal categories for interpreting higher-order and cyclic features. The models studied here are closely related to structures known as notions of computation, as well as models for intuitionistic linear type theory. As an interesting implication of the latter observation, for the acyclic settings, we show that our calculi conservatively embed into linear type theory. The models for higher-order cyclic sharing are of particular interest as they support a generalized form of recursive computation, and we look at this case in detail, together with the connection with cyclic lambda calculi. We demonstrate that our framework can accommodate Milner’s action calculi, a proposed framework for general interactive computation, by showing that our calculi, enriched with suitable constructs for interpreting parameterized constants called controls, are equivalent to the closed fragments of action calculi and their higher-order/reflexive extensions. The dynamics, the computational counterpart of action calculi, is then understood as rewriting systems on our calculi, and interpreted as local preorders on our models.

On traced monoidal closed categories

The structure theorem of Joyal, Street and Verity says that every traced monoidal category arises as a monoidal full subcategory of the tortile monoidal category Int . In this paper we focus on a simple observation that a traced monoidal category is closed if and only if the canonical inclusion from into Int has a right adjoint. Thus, every traced monoidal closed category arises as a monoidal co-reflexive full subcategory of a tortile monoidal category. From this, we derive a series of facts for traced models of linear logic, and some for models of fixed-point computation. To make the paper more self-contained, we also include various background results for traced monoidal categories.

From Action Calculi to Linear Logic

Milner introduced action calculi as a framework for investigating models of interactive behaviour. We present a type-theoretic account of action calculi using the propositions-as-types paradigm; the type theory has a sound and complete interpretation in Powers categorical models. We go on to give a sound translation of our type theory in the (type theory of) intuitionistic linear logic, corresponding to the relation between Bentons models of linear logic and models of action calculi. The conservativity of the syntactic translation is proved by a model-embedding construction using the Yoneda lemma. Finally, we briefly discuss how these techniques can also be used to give conservative translations between various extensions of action calculi.

Linearly Used Effects: Monadic and CPS Transformations into the Linear Lambda Calculus

We propose a semantic and syntactic framework for modelling linearly used effects, by giving the monadic transforms of the computational lambda calculus (considered as the core calculus of typed call-by-value programming languages) into the linear lambda calculus. As an instance Berdine et al.s work on linearly used continuations can be put in this general picture. As a technical result we show the full completeness of the CPS transform into the linear lambda calculus.

Small-step and big-step semantics for call-by-need

We present natural semantics for acyclic as well as cyclic call-by-need lambda calculi, which are proved equivalent to the reduction semantics given by Ariola and Felleisen (J. Funct. Program., vol. 7, no. 3, 1997). The natural semantics are big-step and use global heaps, where evaluation is suspended and memorized. The reduction semantics are small-step, and evaluation is suspended and memorized locally in let-bindings. Thus two styles of formalization describe the call-by-need strategy from different angles. The natural semantics for the acyclic calculus is revised from the previous presentation by Maraist et al. (J. Funct. Program., vol. 8, no. 3, 1998), and its adequacy is ascribed to its correspondence with the reduction semantics, which has been proved equivalent to call-by-name by Ariola and Felleisen. The natural semantics for the cyclic calculus is inspired by that of Launchbury (1993) and Sestoft (1997), and we state its adequacy using a denotational semantics in the style of Launchbury; adequacy of the reduction semantics for the cyclic calculus is in turn ascribed to its correspondence with the natural semantics.

Classical linear logic of implications

We give a simple term calculus for the multiplicative exponential fragment of Classical Linear Logic, by extending Barber and Plotkins dual-context system for the intuitionistic case. The calculus has the non-linear and linear implications as the basic constructs, and this design choice allows a technically manageable axiomatisation without commuting conversions. Despite this simplicity, the calculus is shown to be sound and complete for category-theoretic models given by *-autonomous categories with linear exponential comonads.

Axioms for Recursion in Call-by-Value

We propose an axiomatization of fixpoint operators in typed call-by-value programming languages, and give its justifications in two ways. First, it is shown to be sound and complete for the notion of uniform T-fixpoint operators of Simpson and Plotkin. Second, the axioms precisely account for Filinskis fixpoint operator derived from an iterator (infinite loop constructor) in the presence of first-class continuations, provided that we define the uniformity principle on such an iterator via a notion of effect-freeness (centrality). We then explain how these two results are related in terms of the underlying categorical structures.