Results

**11 - 13**of**13**### 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

### Algebraic Meta-Theories and . . .

"... Fiore and Hur [18] recently introduced a novel methodology—henceforth referred to as Sol—for the Synthesis of equational and rewriting logics from mathematical models. In [18], Sol was successfully applied to rationally reconstruct the traditional equational logic for universal algebra of Birkhoff [ ..."

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Fiore and Hur [18] recently introduced a novel methodology—henceforth referred to as Sol—for the Synthesis of equational and rewriting logics from mathematical models. In [18], Sol was successfully applied to rationally reconstruct the traditional equational logic for universal algebra of Birkhoff [3] and its multi-sorted version [26], and also to synthesise a new version of the Nominal Algebra of Gabbay and Mathijssen [41] and the Nominal Equational Logic of Clouston and Pitts [8] for reasoning about languages with name-binding operators. Based on these case studies and further preliminary investigations, we contend that Sol can make an impact in the problem of engineering logics for modern computational languages. For example, our proposed research on secondorder equational logic will provide foundations for designing a second-order extension of the Maude system [37], a first-order semantic and logical framework used in formal software engineering for specification and programming. Our research strategy can be visualised as follows: (I)

### Monad Combinators, Non-Determinism and Probabilistic Choice [Extended abstract]

"... We test Lüth and Ghani’s proposal [11, 12, 13] to use coproducts of monads as a basis for modularity by applying their ideas to Varacca’s work [20] on combining non-determinism and probabilistic choice. In particular, we discuss i) the coproduct of non-determinism and probabilistic choice; ii) how m ..."

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We test Lüth and Ghani’s proposal [11, 12, 13] to use coproducts of monads as a basis for modularity by applying their ideas to Varacca’s work [20] on combining non-determinism and probabilistic choice. In particular, we discuss i) the coproduct of non-determinism and probabilistic choice; ii) how monad compositions based upon distributivity possess a universal property; and iii) Capretta’s [4] treatment of looping as an effect. We take advantage of the fact that all three effects are usefully modelled by ideal monads. 1