Archive for the 'Categorical models' Category

A name abstraction functor for named sets

By Vincenzo Ciancia and Ugo Montinari, from CMCS 2008, available from Vincenzo Ciancia’s website:

The problem of dening fully abstract operational models of name passing calculi has been given some elegant solutions, such as coalgebras over presheaf categories or over nominal sets. These formalisms fail to model garbage collection of unused names, hence they do not have nice properties with respects to finite state algorithms. The category of named sets, on the other hand, was designed for the purpose of supporting ecient algorithms to handle the semantics of name passing calculi. However the theory was developed in a rather ad-hoc fashion (e.g. the existence of a final coalgebra was only proved in the finite case). In this work we introduce a name abstraction functor for named sets and show that it provides a simple and effective notion of garbage collection of unused names. Along the way, we survey a number of needed results on the category of permutation algebras, an algebra-theoretic denition of nominal sets. In particular we give a formalization of the adjunction between abstraction and concretion, an example illustrating a nominal syntax alike handling of De Bruijn indexes, and an explicit functor to model the early semantics of the π -calculus in nominal sets.

Monadic presentations of untyped lambda-terms using inductive datatypes

By Thorsten Altenkirch and Bernhard Reus, from CSL 1999, available from Thorsten Altenkirch’s website:

We present a definition of untyped lambda-terms using a heterogeneous datatype, i.e. an inductively defined operator. This operator can be extended a Kleisli triple, which is a concise way to verify the substitution laws for lambda-calculus. We also observe that repetitions in the definition of the monad as well as in the proofs can be avoided by using well-founded recursion and induction instead of structural induction. We extend the construction to the simply typed lambda-calculus using dependent types, and show that this is an instance of a generalisation of Kleisli triples. The proofs for the untyped case have been checked using the LEGO system.

Term rewriting with variable binders: an initial algebra approach

By Makoto Hamana, from PPDP 2003, available from Makoto Haman’s website:

We present an extension of first-order term rewriting systems, which involves variable binding in the term language. We develop the systems called binding term rewriting systems (BTRSs) in a stepwise manner; firstly we present the term language, then formulate equational logic, and finally define rewrite systems by two styles: rewrite logic and pattern matching styles. The novelty of this development is that we follow an initial algebra approach in an extended notion of Σ-algebras in various functor categories. These are based on Fiore-Plotkin Turi’s presheaf semantics of variable binding and Lüth-Ghani’s monadic semantics of term rewriting systems. We characterise the terms, equational logic and rewrite systems for BTRSs are initial algebras in suitable categories. Finally, we discuss our design choice of BTRSs from a semantic perspective.

Abstract syntax and variable binding

By Marcelo P. Fiore, Gordon Plotkin and Daniele Turi, from LICS 1999, available from the Edinburgh Research Archive website:

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.