A note on intersection types

Information & Computation, 2003

Intersection types discipline allows to define a wide variety of models for the type free lambda-calculus, but the Curry-Howard isomorphism breaks down for this kind of type systems. In this paper we show that the correspondence between types and suitable logical formulas can still be recovered appealing to the fact that there is a strict connection between the semantics for lambda-calculus induced by the intersection types and a Kripke style semantics for modal and relevant logics. Indeed, we present a modal logic hinted by the analysis of the sub-typing relation for intersection types, and we show that the deduction relation for such a modal system is a conservative extension of the relation of sub-typing. Then, we define a Kripke-style semantics for the formulas of such a system, present suitable sequential calculi, prove a completeness theorem and give a syntactical proof of the cut elimination property. Finally, we define a decision procedure for theorem-hood and we show that it yields the finite model property and cut-redundancy.

Mathematical Logic Quarterly - MLQ, 1974

In this paper we assume familiarity with, and use the notation of, [ l ] . The system S of [ l ] is the system of type theory with 4-conversion introduced by CHURCH [5], minus axioms of extensionality, descriptions, choice, and infinity. We shall refer to S as elementary type theory, since S simply embodies the logic of propositional connectives, quantifiers, and 4-conversion in the context of type theory. In spite of the fact that S is analogous to first order logic in certain respects, it is a considerably more complex language, and special cases of the decision problem for provability in S seem rather intractable for the most part. We shall use the methods of [ l ] to obtain information about some very special cases of this decision problem.

Logical Methods in Computer Science, 2011

We define a logical framework with singleton types and one universe of small types. We give the semantics using a PER model; it is used for constructing a normalisation-by-evaluation algorithm. We prove completeness and soundness of the algorithm; and get as a corollary the injectivity of type constructors. Then we give the definition of a correct and complete type-checking algorithm for terms in normal form. We extend the results to proof-irrelevant propositions. 1998 ACM Subject Classification: F.4.1. CC Creative Commons 2 A. ABEL, T. COQUAND, AND M. PAGANO

BIT, 1992

To every partial inductive definition D, a natural deduction calculus ND(D) is associated. Not every such system will have the normalization property; specifically, there are definitions D' for which ND(D') permits non-normalizable deductions. A lambda calculus is formulated where the terms are used as objects realizing deductions in ND(D), and is shown to have the Church-Rosser property. Since ND(D) permits non-normalizable deductions, there will be typed terms which are non-normalizable. It will, for example, be possible to obtain a typed fixed-point operator.

Archive for Mathematical Logic, 2002

Inductive characterizations of the sets of terms, the subset of strongly normalizing terms and normal forms are studied in order to reprove weak and strong normalization for the simplytyped λ-calculus and for an extension by sum types with permutative conversions. The analogous treatment of a new system with generalized applications inspired by generalized elimination rules in natural deduction, advocated by von Plato, shows the flexibility of the approach which does not use the strong computability/candidate styleà la Tait and Girard. It is also shown that the extension of the system with permutative conversions by η-rules is still strongly normalizing, and likewise for an extension of the system of generalized applications by a rule of "immediate simplification". By introducing an infinitely branching inductive rule the method even extends to Gödel's T.

This thesis presents a new algorithm for Normalisation by Evaluation for different type theories. This algorithm is later used to define a type-checking algorithm and to prove other meta-theoretical results about predicative type systems.

Archive for Mathematical Logic, 2003

Inductive characterizations of the sets of terms, the subset of strongly normalizing terms and normal forms are studied in order to reprove weak and strong normalization for the simply-typed λ-calculus and for an extension by sum types with permutative conversions. The analogous treatment of a new system with generalized applications inspired by generalized elimination rules in natural deduction, advocated by von Plato, shows the flexibility of the approach which does not use the strong computability/candidate style à la Tait and Girard. It is also shown that the extension of the system with permutative conversions by η-rules is still strongly normalizing, and likewise for an extension of the system of generalized applications by a rule of ``immediate simplification''. By introducing an infinitely branching inductive rule the method even extends to Gödel's T.

2015

In my dissertation I address some questions concerning the proof theory of second order logic and its constructive counterpart, System F (Girard 1971). These investigations follow two distinct (though historically related) viewpoints in proof theory, which are compared throughout the text: on the one side, the proof theoretic semantics tradition inaugurated by Dummett and Prawitz (Prawitz 1971, Dummett 1991), focusing on the analysis of the inferential content of proofs; on the other side, the interactionist tradition arising from Kleene's realizability (Kleene 1945) and the Tait/Girard reducibility technique (Tait 1967, Girard 1971), which interprets proofs as untyped programs and focuses, rather, on the behavioral content of proofs, i.e. the way in which they interact through the cut-elimination algorithm. A distinction is made between the issues of justifying and understanding ("explaining why" and "explaining how", as in Girard 2000) impredicative reasoning, i.e. between non elementary results like the Hauptsatz and the combinatorial analysis of proofs, seen as programs, i.e. recursive objects. As for justifi cation, an epistemological analysis of the circularity involved in the second order Hauptsatz is developed; it is shown that the usual normalization arguments for second order logic do not run into the vicious circularity claimed by Poincaré and Russell, but involve a diff erent, epistemic, form of circularity. Still, this weaker circularity makes justifi cation, in a sense, pointless; in particular, some examples of inconsistent higher order theories admitting epistemically circular normalization arguments are discussed. As for the explanation issue, a constructive and combinatorial (i.e. independent from normalization) analysis of higher order order quantifi cation is developed along two directions, with some related technical results. The fi rst direction arises from the parametric and dinatural interpretations of polymorphism (Reynolds 1983, Girard-Scott-Scedrov 1992), which provide a clear mathematical meaning to Carnap's defense of impredicative quanti fication (Carnap 1983). In particular, the violation of the parametric condition leads to paradoxes which are often ignored in the philosophical literature (with the exception of Longo-Fruchart 1997). The analysis of the combinatorial content of these interpretations leads to a 1-completeness theorem (every normal closed -term in the universal closure of a simple type is typable in simple type theory), which connects the interactionist and the inferential conceptions of proof. The second direction follows the analysis of the typing conditions of the -terms associated with intuitionistic second order proofs. To the \vicious circles" in the proofs there correspond recursive (i.e. circular) speci fications for the types of the -terms. The geometrical structure of these vicious circles is investigated (following Lechenadec 1989, Malecki 1990, Giannini - Ronchi Della Rocca 1991), leading to a combinatorial characterization of typability in some inconsistent extension of System F: since, as Girard's paradox shows, a typable term need not be normalizing, one is indeed naturally led to consider not normalizing theories. Such investigations go in the direction both of a mathematical understanding of the structure generated by the vicious circles of impredicative theories and of the development of a proof-theoretic analysis of potentially incorrect or uncertain proofs.

The goal of this work is to present a proof-theoretical justification for IT. In particular, we discuss the relationship between the intersection connective and the intuitionistic conjunction. For this purpose, we define a new logical system called Intersection Synchronous Logic (ISL), that proves properties of sets of deductions of the implicationconjunction fragment of NJ. The main idea behind ISL is the decomposition of the intuitionistic conjunction into two connectives, one with synchronous and the other with asynchronous behavior. Then we show how proofs of ISL can be decorated with terms in a way that it matches the standard IT assignment system when only the synchronous conjunction is taken into account, and the simple types assignment with pairs and projections when the asynchronous conjunction is considered. Finally, we prove that ISL enjoys both the strong normalization and subformula properties.

2015

Abstract. A proof theoretical analysis suggests that the process of cut elimination in a sequent calculus corresponds to the normalisation of the proofs in natural deduction. If one moves to proofs decorated with lambda terms according to the Curry-Howard’s isomorphism, the same observation leads to realising that the process of normalisation of the lambda-terms decorating the proofs in natural deduction is deeply connected with the property of closure under substitution of the lambda terms decorating the proofs in a sequent calculus. In this paper we show that this observation becomes a criterion to recognise the strongly normalising Pure Type Systems.