Intersection Types: a Proof-Theoretical Approach
نویسندگان
چکیده
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.
منابع مشابه
Some lower bounds for the $L$-intersection number of graphs
For a set of non-negative integers~$L$, the $L$-intersection number of a graph is the smallest number~$l$ for which there is an assignment of subsets $A_v subseteq {1,dots, l}$ to vertices $v$, such that every two vertices $u,v$ are adjacent if and only if $|A_u cap A_v|in L$. The bipartite $L$-intersection number is defined similarly when the conditions are considered only for the ver...
متن کاملTheoretical Pearls: A bargain for intersection types: a simple strong normalization proof
This pearl gives a discount proof of the folklore theorem that every strongly β-normalizing λ-term is typable with an intersection type. (We consider typings that do not use the empty intersection ω which can type any term.) The proof uses the perpetual reduction strategy which finds a longest path. This is a simplification over existing proofs that consider any longest reduction path. The choi...
متن کاملTHEORETICAL PEARL A Bargain for Intersection Types: A Simple Strong Normalization Proof
This pearl gives a discount proof of the folklore theorem that every strongly β-normalizing λ-term is typable with an intersection type. (We consider typings that do not use the empty intersection ω which can type any term.) The proof uses the perpetual reduction strategy which finds a longest path. This is a simplification over existing proofs that consider any longest reduction path. The choi...
متن کاملIntersection Types from a Proof-theoretic Perspective
In this work we present a proof-theoretical justification for the intersection type assignment system (IT) by means of the logical system Intersection Synchronous Logic (ISL). ISL builds classes of equivalent deductions of the implicative and conjunctive fragment of the intuitionistic logic (NJ). ISL results from decomposing intuitionistic conjunction into two connectives: a synchronous conjunc...
متن کاملA General Technique for Analyzing Termination in Symmetric Proof Calculi
Proof-term calculi expressing a computational interpretation of classical logic serve as tools for extracting the constructive content of classical proofs and at the same time can be seen as pure programming languages with explicit representation of control. The inherent symmetry in the underlying logic presents technical challenges to the study of the reduction properties of these systems. We ...
متن کامل