Principal Typings in a Restricted Intersection Type System for Beta Normal Forms with De Bruijn Indices

The lambda-calculus with de Bruijn indices assembles each alpha-class of lambda-terms in a unique term, using indices instead of variable names. Intersection types provide finitary type polymorphism and can characterise normalisable lambda-terms through the property that a term is normalisable if and only if it is typeable. To be closer to computations and to simplify the formalisation of the atomic operations involved in beta-contractions, several calculi of explicit substitution were developed mostly with de Bruijn indices. Versions of explicit substitutions calculi without types and with simple type systems are well investigated in contrast to versions with more elaborate type systems such as intersection types. In previous work, we introduced a de Bruijn version of the lambda-calculus with an intersection type system and proved that it preserves subject reduction, a basic property of type systems. In this paper a version with de Bruijn indices of an intersection type system originally introduced to characterise principal typings for beta-normal forms is presented. We present the characterisation in this new system and the corresponding versions for the type inference and the reconstruction of normal forms from principal typings algorithms. We briefly discuss the failure of the subject reduction property and some possible solutions for it

Investigations in intersection types : confluence, and semantics of expansion in the -calculus, and a type error slicing method

Type systems were invented in the early 1900s to provide foundations for Mathematics where types were used to avoid paradoxes. Type systems have then been developed and extended throughout the years to serve different purposes such as efficiency or expressiveness. The λ-calculus is used in programming languages, logic, mathematics, and linguistics. Intersection types are a kind of types used for building semantic models of the λ-calculus and for static analysis of computer programs. The confluence property was used to prove the λ-calculus’ consistency and the uniqueness of normal forms. Confluence is useful to show that logics are sensibly designed, and to make equality decision procedures for use in theorem provers. Some proofs of the λ-calculus’ confluence are based on syntactic concepts (reduction relations and λ-term sets) and some on semantic concepts (type interpretations). Part I of this thesis presents an original syntactic proof that is a simplification of a semantic proof based on a sound type interpretation w.r.t. an intersection type system. Our proof can be seen as bridging some semantic and syntactic proofs. Expansion is an operation on typings (pairs of type environments and result types) in type systems for the λ-calculus. It was introduced to prove that the principal typing property (i.e., that every typable term has a strongest typing) holds in intersection type systems. Expansion variables were introduced to simplify the expansion mechanism. Part II of this thesis presents a complete realisability semantics w.r.t. an intersection type system with infinitely many expansion variables. This represents the first study on semantics of expansion. Providing sound (and complete) realisability semantics allows one to study the algorithmic behaviour of typed λ-terms through their types w.r.t. a type system. We believe such semantics will cast some light on the not yet well understood expansion operation. Intersection types were used in a type error slicer for the SML programming language. Existing compilers for many languages have confusing type error messages. Type error slicing (TES) helps the programmer by isolating the part of a program contributing to a type error (a slice). TES was initially done for a tiny toy language (the λ-calculus with polymorphic let-expressions). Extending TES to a full language is extremely challenging, and for SML we needed a number of innovations. Some issues would be faced for any language, and some are SML-specific but representative of the complexity of language-specific issues likely to be faced for other languages. Part III of this thesis solves both kinds of issues and presents an original, simple, and general constraint system for providing type error slices for ill-typed programs. We believe TES helps demystify language features known to confuse users

Programming Languages and Systems

This open access book constitutes the proceedings of the 28th European Symposium on Programming, ESOP 2019, which took place in Prague, Czech Republic, in April 2019, held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2019

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.

Proceedings of the Workshop on the lambda-Prolog Programming Language

The expressiveness of logic programs can be greatly increased over first-order Horn clauses through a stronger emphasis on logical connectives and by admitting various forms of higher-order quantification. The logic of hereditary Harrop formulas and the notion of uniform proof have been developed to provide a foundation for more expressive logic programming languages. The λ-Prolog language is actively being developed on top of these foundational considerations. The rich logical foundations of λ-Prolog provides it with declarative approaches to modular programming, hypothetical reasoning, higher-order programming, polymorphic typing, and meta-programming. These aspects of λ-Prolog have made it valuable as a higher-level language for the specification and implementation of programs in numerous areas, including natural language, automated reasoning, program transformation, and databases