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 Felleisen's theory for first-class continuations.

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Abstract. 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.

The structure theorem of Joyal, Street and Verity says that every traced monoidal category C arises as a monoidal full subcategory of the tortile monoidal category Int C. In this paper we focus on a simple observation that a traced monoidal category C is closed if and only if the canonical inclusion from C into Int C 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. IntroductionIn Joyal et al. (1996), Joyal, Street and Verity introduced the notion of traced monoidal categories. They showed that every traced monoidal category C fully faithfully embeds in a tortile monoidal category Int C, and that this Int-construction gives a left biadjoint of the forgetful 2-functor from the 2-category of tortile monoidal categories to that of traced monoidal categories. This remarkable result attracted much attention from theoretical computer scientists, particularly in connection with linear logic (Girard 1987) and the Geometry of Interaction (Girard 1989;Abramsky and Jagadeesan 1994;Abramsky 1996;Haghverdi 2000;Abramsky et al. 2002;Haghverdi and Scott 2006), where the special case of traced symmetric monoidal categories and compact closed categories (Kelly and Laplaza 1980) is of interest.In this paper we shall see that the monoidal closed structure can be tied with the Int-construction in an unexpected manner. Namely, we show the following result (Theorem 4.1). Theorem.A traced monoidal category C is closed if and only if the embedding from C into Int C has a right adjoint.Despite its simplicity, to the best of our knowledge, this fact has not previously been pointed out in the literature. Perhaps this is partly because the Int-construction works too nicely: tortile monoidal categories are closed, therefore every traced monoidal category embeds into a closed one just via the Int-construction. So it seems that for this reason it was not felt that traced monoidal closed categories were themselves interesting (an exception being Coccia et al. (2002), where traced monoidal closed categories with extra structure are used for modelling higher-order cyclic shared structures). However, the tortile structure (or compact closed structure in the symmetric case) just describes a very special http://journals.cambridge.org M. Hasegawa 218 kind of closedness, and not every traced monoidal closed category is tortile. Our result says that the relation between tortile structure and general closed structure is still in harmony: every traced monoidal closed category arises as a monoidal co-reflexive full subcategory of a tortile monoidal category. The embedding from C to Int C may not preserve the closed structure (see Remark 4.2 for a counterexample), but the closed structure on C is determined by the closed struct...

Abstract. We show that the category FinVect k 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 FinVect k which distinguishes them. Therefore two arrows in the free traced category are the same if and only if they agree for all interpretations in FinVect k .