Daniele Turi

Abstract syntax and variable binding

We develop a theory of abstract syntax with variable binding. To every binding signature we associate a category of models consisting of variable sets endowed with compatible algebra and substitution structures. The syntax generated by the signature is the initial model. This gives a notion of initial algebra semantics encompassing the traditional one; besides compositionality, it automatically verifies the semantic substitution lemma.

On the Foundation of Final Semantics: Non-Standard Sets, Metric Spaces, Partial Orders

Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of non-standard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of post-fixed point. They are also used here for giving a new comprehensive presentation of the (still) non-standard theory of nonwell-founded sets (as non-standard sets are usually called). This paper is meant to provide a basis to a more general project aiming at a full exploitation of the finality of the domains in the semantics of programming languages — concurrent ones among them. Such a final semantics enjoys uniformity and generality. For instance, semantic observational equivalences like bisimulation can be derived as instances of a single ‘coalgebraic’ definition (introduced elsewhere), which is parametric of the functor appearing in the domain equation. Some properties of this general form of equivalence are also studied in this paper.

On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces

This paper, a revised version of Rutten and Turi (1993), is part of a programme aiming at formulating a mathematical theory of structural operational semantics to complement the established theory of domains and denotational semantics to form a coherent whole (Turi 1996; Turi and Plotkin 1997). The programme is based on a suitable interplay between the induction principle, which pervades modern mathematics, and a dual, non-standard ‘coinduction principle’, which underlies many of the recursive phenomena occurring in computer science.The aim of the present survey is to show that the elementary categorical notion of a final coalgebra is a suitable foundation for such a coinduction principle. The properties of coalgebraic coinduction are studied both at an abstract categorical level and in some specific categories used in semantics, namely categories of non-well-founded sets, partial orders and metric spaces.

Initial algebra and final coalgebra semantics for concurrency.

The aim of this paper is to relate initial algebra semantics and final coalgebra semantics. It is shown how these two approaches to the semantics of programming languages are each others dual, and some conditions are given under which they coincide. More precisely, it is shown how to derive initial semantics from final semantics, using the initiality and finality to ensure their equality. Moreover, many facts about congruences (on algebras) and (generalized) bisimulations (on coalgebras) are shown to be dual as well.

Categorical Modelling of Structural Operational Rules: Case Studies

This paper aims at substantiating a recently introduced categorical theory of ‘well-behaved’ operational semantics. A variety of concrete examples of structural operational rules is modelled categorically illustrating the versatility and modularity of the theory. Further, a novel functorial notion of guardedness is introduced which allows for a general and formal treatment of guarded recursive programs.

A Two Steps Semantics for Logic Programs with Negation

We analyze programs with negation by transforming them in order to infer constrained atoms which are regarded as basic semantic objects. Two steps are performed. Step I refers to the positive fragment of the program and unfolds all positive literals so that only atoms and conditional atoms remain. Step II refers to the stratified fragment and replaces defined negative literals with inequalities and equalities.