To my parents 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

syntax of the source language ` c : ' f:::; x : ø ; :::g ` x : ø ß ` e : ø !ø ß ` fix e : ø ß [ fx : ø 1 g ` e : ø 2 ß ` x : ø 1 : e : ø 1 !ø 2 ß ` e 0 : ø 1 !ø 2 ß ` e 1 : ø 1 ß ` @ e 0 e 1 : ø 2 ß ` e 1 : ' ß ` e 2 : ø ß ` e 3 : ø ß ` if e 1 then e 2 else e 3 : ø ß ` e 0 : ø 0 ß [ fx : ø 0 g ` e 1 : ø 1 ß ` let x = e 0 in e 1 : ø 1 ß ` e 1 : ø 1 ß ` e 2 : ø 2 ß ` pair e 1 e 2 : ø 1 \Theta ø 2 ß ` e : ø 1 \Theta ø 2 ß ` fst e : ø 1 ß ` e : ø 1 \Theta ø 2 ß ` snd e : ø 2 Fig. 2. Type-checking rules for the source language approach is used by Kesley and Hudak [11] and by Fradet and Le M'etayer [9]. Both include a CPS transformation. Fradet and Le M'etayer compile both CBN and CBV programs by using the CBN and the CBV CPS-transformation. Recently, Burn and Le M'etayer have combined this technique with a global programanalysis [2], which is comparable to our goal here. 1.4 Overview Section 2 presents the syntax of the source language and the strictness-annotated language. We c...

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Abstract. We introduce the notion of effectoid as a way of axiomatising the notion of “computational effect”. Guided by classical algebra, we define several effectoids equationally and explore their relationship with each other. We demonstrate their computational relevance by applying them to global exceptions, partiality, continuations, and global state. 1

We investigate the duality between call-by-name recursion and call-by-value iteration on the -calculi. The duality between call-by-name and call-by-value was first studied by Filinski, and Selinger has studied the category-theoretic duality on the models of the call-by-name -calculus and the call-by-value one. We extend the call-by-name -calculus and the call-by-value one with a fixed-point operator and an iteration operator, respectively. We show that the dual translations constructed by Selinger can be expanded into our extended -calculi, and we also discuss their implications to practical applications.

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 Filinski's fixpoint operator derived from an iterator (infinite loop constructor) in the presence of firstclass 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.

We investigate continuations in the context of idealized call-by-value programming languages. On the semantic side, we analyze the categorical structures that arise from continuation models of call-by-value languages. On the syntactic side, we study the call-by-value continuation-passing transformation as a translation between equational theories. Among the novelties are an unusually simple axiomatization of control operators and a strengthened completeness result with a proof based on a delaying transform.