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**11 - 16**of**16**### Languages, Theory

"... Recently there has been a great deal of interest in higherorder syntax which seeks to extend standard initial algebra semantics to cover languages with variable binding by using functor categories. The canonical example studied in the literature is that of the untyped λ-calculus which is handled as ..."

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Recently there has been a great deal of interest in higherorder syntax which seeks to extend standard initial algebra semantics to cover languages with variable binding by using functor categories. The canonical example studied in the literature is that of the untyped λ-calculus which is handled as an instance of the general theory of binding algebras, cf. Fiore, Plotkin, Turi [8]. Another important syntactic construction is that of explicit substitutions. The syntax of a language with explicit substitutions does not form a binding algebra as an explicit substitution may bind an arbitrary number of variables. Nevertheless we show that the language given by a standard signature Σ and explicit substitutions is naturally modelled as the initial algebra of the endofunctor Id + FΣ ◦ + ◦ on a functor category. We also comment on the apparent lack of modularity in syntax with variable binding as compared to first-order languages. Categories and Subject Descriptors

### Substitution in Structural Operational Semantics and value-passing process calculi

"... Consider a process calculus that allows agents to communicate values. The structural operational semantics involves substitution of values for variables. Existing rule formats, such as the GSOS format, do not allow this kind of explicit substitution in the semantic rules. We investigate how to deriv ..."

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Consider a process calculus that allows agents to communicate values. The structural operational semantics involves substitution of values for variables. Existing rule formats, such as the GSOS format, do not allow this kind of explicit substitution in the semantic rules. We investigate how to derive rule formats for languages with substitution, by using categorical logic to interpret the framework of the GSOS format in different categories. The categories in question are categories of ‘substitution actions’. 1 A simple language for value-passing To set the scene, fix a set of channel names, and consider a set V of value-expressions, that includes the channel names. A simple untyped value-passing process language, V-CCS, is given in Figure 1 (c.f. [8]). The precise value expressions of V are not important, but note that since V includes the (static) channel names, V-CCS is a very primitive applied π-calculus without restriction or name generation; c.f. [1]. For the sake of illustration, consider the set Vex of value expressions determined by the following grammar: v:: = n | v + v | (v, v) | π1(v) | π2(v) | c (n is a number, c is a channel name). We will always work with value expressions up-to the evident equations (2 + 3 = 5; π1(v, w) = v; etc.), rather than explicitly evaluating or normalizing them; this is to simplify the presentation. The following transitions are derivable in Vex-CCS. (¯c〈3〉.0) | (c(v).¯c〈2 + v〉.0) τ

### GFOL: A Term-Generic Logic for Defining λ-Calculi

, 2006

"... Generic first-order logic (GFOL) is a first-order logic parameterized with terms de ned axiomatically (rather than constructively), by requiring them to only provide generic notions of free variable and substitution satisfying reasonable properties. GFOL has a complete Gentzen system generalizing th ..."

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Generic first-order logic (GFOL) is a first-order logic parameterized with terms de ned axiomatically (rather than constructively), by requiring them to only provide generic notions of free variable and substitution satisfying reasonable properties. GFOL has a complete Gentzen system generalizing that of FOL. An important fragment of GFOL, called HORN 2, possesses a much simpler Gentzen system, similar to traditional context-based derivation systems of λ-calculi. HORN 2 appears to be sufficient for defining virtually any λ-calculi (including polymorphic and type-recursive ones) as theories inside the logic. GFOL endows its theories with a default loose semantics, complete for the specified calculi.

### Standardization for the Coinductive Lambda-Calculus

, 2002

"... In the calculus of possibly non-wellfounded -terms, standardization is proved for a parallel notion of reduction. For this system confluence has recently been established by means of a bounding argument for the number of reductions provoked by the joining function which witnesses the conflue ..."

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In the calculus of possibly non-wellfounded -terms, standardization is proved for a parallel notion of reduction. For this system confluence has recently been established by means of a bounding argument for the number of reductions provoked by the joining function which witnesses the confluence statement. Similarly,

### Substitution in non-wellfounded . . .

- ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE 82 NO. 1 (2003)
, 2003

"... Inspired from the recent developments in theories of non-wellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and non-wellfounded terms is studied for a very general notion of signature permitting both simple variable bin ..."

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Inspired from the recent developments in theories of non-wellfounded syntax (coinductively defined languages) and of syntax with binding operators, the structure of algebras of wellfounded and non-wellfounded terms is studied for a very general notion of signature permitting both simple variable binding operators as well as operators of explicit substitution. This is done in an extensional mathematical setting of initial algebras and final coalgebras of endofunctors on a functor category. In the non-wellfounded case, the fundamental operation of substitution is more beneficially defined in terms of primitive corecursion than coiteration.

### MSc in Logic at the Universiteit van Amsterdam.

, 2012

"... Interaction, observation and denotation A study of dialgebras for program semantics ..."

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Interaction, observation and denotation A study of dialgebras for program semantics