A Type Checker for a Logical Framework with Union and Intersection Types
We present the syntax, semantics, and typing rules of Bull, a prototype theorem prover based on the Delta-Framework, i.e. a fully-typed lambda-calculus decorated with union and intersection types, as described in previous papers by the authors. Bull also implements a subtyping algorithm for the Type...
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ftdatacite:10.48550/arxiv.2002.10803 2023-05-15T15:41:56+02:00 A Type Checker for a Logical Framework with Union and Intersection Types Liquori, Luigi Stolze, Claude 2020 https://dx.doi.org/10.48550/arxiv.2002.10803 https://arxiv.org/abs/2002.10803 unknown arXiv arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Logic in Computer Science cs.LO Programming Languages cs.PL FOS Computer and information sciences F.3.1; F.4.1 Article CreativeWork article Preprint 2020 ftdatacite https://doi.org/10.48550/arxiv.2002.10803 2022-03-10T16:01:48Z We present the syntax, semantics, and typing rules of Bull, a prototype theorem prover based on the Delta-Framework, i.e. a fully-typed lambda-calculus decorated with union and intersection types, as described in previous papers by the authors. Bull also implements a subtyping algorithm for the Type Theory Xi of Barbanera-Dezani-de'Liguoro. Bull has a command-line interface where the user can declare axioms, terms, and perform computations and some basic terminal-style features like error pretty-printing, subexpressions highlighting, and file loading. Moreover, it can typecheck a proof or normalize it. These terms can be incomplete, therefore the typechecking algorithm uses unification to try to construct the missing subterms. Bull uses the syntax of Berardi's Pure Type Systems to improve the compactness and the modularity of the kernel. Abstract and concrete syntax are mostly aligned and similar to the concrete syntax of Coq. Bull uses a higher-order unification algorithm for terms, while typechecking and partial type inference are done by a bidirectional refinement algorithm, similar to the one found in Matita and Beluga. The refinement can be split into two parts: the essence refinement and the typing refinement. Binders are implemented using commonly-used de Bruijn indices. We have defined a concrete language syntax that will allow the user to write Delta-terms. We have defined the reduction rules and an evaluator. We have implemented from scratch a refiner which does partial typechecking and type reconstruction. We have experimented Bull with classical examples of the intersection and union literature, such as the ones formalized by Pfenning with his Refinement Types in LF. We hope that this research vein could be useful to experiment, in a proof theoretical setting, forms of polymorphism alternatives to Girard's parametric one. Article in Journal/Newspaper Beluga Beluga* DataCite Metadata Store (German National Library of Science and Technology) Lambda ENVELOPE(-62.983,-62.983,-64.300,-64.300) |
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DataCite Metadata Store (German National Library of Science and Technology) |
op_collection_id |
ftdatacite |
language |
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topic |
Logic in Computer Science cs.LO Programming Languages cs.PL FOS Computer and information sciences F.3.1; F.4.1 |
spellingShingle |
Logic in Computer Science cs.LO Programming Languages cs.PL FOS Computer and information sciences F.3.1; F.4.1 Liquori, Luigi Stolze, Claude A Type Checker for a Logical Framework with Union and Intersection Types |
topic_facet |
Logic in Computer Science cs.LO Programming Languages cs.PL FOS Computer and information sciences F.3.1; F.4.1 |
description |
We present the syntax, semantics, and typing rules of Bull, a prototype theorem prover based on the Delta-Framework, i.e. a fully-typed lambda-calculus decorated with union and intersection types, as described in previous papers by the authors. Bull also implements a subtyping algorithm for the Type Theory Xi of Barbanera-Dezani-de'Liguoro. Bull has a command-line interface where the user can declare axioms, terms, and perform computations and some basic terminal-style features like error pretty-printing, subexpressions highlighting, and file loading. Moreover, it can typecheck a proof or normalize it. These terms can be incomplete, therefore the typechecking algorithm uses unification to try to construct the missing subterms. Bull uses the syntax of Berardi's Pure Type Systems to improve the compactness and the modularity of the kernel. Abstract and concrete syntax are mostly aligned and similar to the concrete syntax of Coq. Bull uses a higher-order unification algorithm for terms, while typechecking and partial type inference are done by a bidirectional refinement algorithm, similar to the one found in Matita and Beluga. The refinement can be split into two parts: the essence refinement and the typing refinement. Binders are implemented using commonly-used de Bruijn indices. We have defined a concrete language syntax that will allow the user to write Delta-terms. We have defined the reduction rules and an evaluator. We have implemented from scratch a refiner which does partial typechecking and type reconstruction. We have experimented Bull with classical examples of the intersection and union literature, such as the ones formalized by Pfenning with his Refinement Types in LF. We hope that this research vein could be useful to experiment, in a proof theoretical setting, forms of polymorphism alternatives to Girard's parametric one. |
format |
Article in Journal/Newspaper |
author |
Liquori, Luigi Stolze, Claude |
author_facet |
Liquori, Luigi Stolze, Claude |
author_sort |
Liquori, Luigi |
title |
A Type Checker for a Logical Framework with Union and Intersection Types |
title_short |
A Type Checker for a Logical Framework with Union and Intersection Types |
title_full |
A Type Checker for a Logical Framework with Union and Intersection Types |
title_fullStr |
A Type Checker for a Logical Framework with Union and Intersection Types |
title_full_unstemmed |
A Type Checker for a Logical Framework with Union and Intersection Types |
title_sort |
type checker for a logical framework with union and intersection types |
publisher |
arXiv |
publishDate |
2020 |
url |
https://dx.doi.org/10.48550/arxiv.2002.10803 https://arxiv.org/abs/2002.10803 |
long_lat |
ENVELOPE(-62.983,-62.983,-64.300,-64.300) |
geographic |
Lambda |
geographic_facet |
Lambda |
genre |
Beluga Beluga* |
genre_facet |
Beluga Beluga* |
op_rights |
arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
op_doi |
https://doi.org/10.48550/arxiv.2002.10803 |
_version_ |
1766374807372300288 |