Jelena Ivetic

Characterising Strongly Normalising Intuitionistic Terms

This paper gives a characterisation, via intersection types, of the strongly normalising proof-terms of an intuitionistic sequent calculus where LJ easily embeds. The soundness of the typing system is reduced to that of a well known typing system with intersection types for the ordinary λ-calculus. The completeness of the typing system is obtained from subject expansion at root position. Next we use our result to analyze the characterisation of strong normalisability for three classes of intuitionistic terms: ordinary λ-terms, ΛJ-terms λ-terms with generalised application, and λx-terms λ-terms with explicit substitution. We explain via our system why the type systems in the natural deduction format for ΛJ and λx known from the literature contain extra, exceptional rules for typing generalised application or substitution; and we show a new characterisation of the β-strongly normalising λ-terms, as a corollary to a PSN-result, relating the λ-calculus and the intuitionistic sequent calculus. Finally, we obtain variants of our characterisation by restricting the set of assignable types to sub-classes of intersection types, notably strict types. In addition, the known characterisation of the β-strongly normalising λ-terms in terms of assignment of strict types follows as an easy corollary of our results.

Intersection types for the resource control lambda calculi

We propose intersection type assignment systems for two resource control term calculi: the lambda calculus and the sequent lambda calculus with explicit operators for weakening and contraction. These resource control calculi, λ® and λ®Gtz, respectively, capture the computational content of intuitionistic natural deduction and intuitionistic sequent logic with explicit structural rules. Our main contribution is the characterisation of strong normalisation of reductions in both calculi. We first prove that typability implies strong normalisation in λ® by adapting the reducibility method. Then we prove that typability implies strong normalisation in λ®Gtz by using a combination of well-orders and a suitable embedding of λ®Gtz-terms into λ®-terms which preserves types and enables the simulation of all its reductions by the operational semantics of the λ®-calculus. Finally, we prove that strong normalisation implies typability in both systems using head subject expansion.

Intuitionistic sequent-style calculus with explicit structural rules

In this paper we extend the Curry-Howard correspondence to intuitionistic sequent calculus with explicit structural rules of weakening and contraction. We present a linear term calculus derived from the calculus of Espirito Santo, which captures the computational content of the intuitionistic sequent logic, by adding explicit operators for weakening and contraction. For the proposed calculus we introduce the type assignment system with simple types and prove some operational properties, including the subject reduction and strong normalisation property. We then relate the proposed linear type calculus to the simply typed intuitionistic calculus of Kesner and Lengrand, which handles explicit operators of weakening and contraction in the natural deduction framework.

Probabilistic Reasoning About Simply Typed Lambda Terms

Reasoning with uncertainty has gained an important role in computer science, artificial intelligence and cognitive science. These applications urge for development of formal models which capture reasoning of probabilistic features. We propose a formal model for reasoning about probabilities of simply typed lambda terms. We present its syntax, Kripke-style semantics and axiomatic system. The main results are the corresponding soundness and strong completeness, which rely on two key facts: the completeness of simple type assignment and the existence of a maximal consistent extension of a consistent set.