The Relevance of Semantic Subtyping

Dezani‐Ciancaglini, Mariangiola; Frisch, Alain; Giovannetti, Elio; Motohama, Yoko

doi:10.1016/s1571-0661(04)80492-4

Cited by 11 publications

(13 citation statements)

References 9 publications

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“…What we devised is the first approach for a higher order λ-calculus in which union, intersection, and negation types have a set-theoretic interpretation. The logical relevance of the approach was independently confirmed by Dezani-Ciancaglini et al [2002] who showed that the subtyping relation induced by the universal model of Section 6.8 restricted to its positive part (i.e., arrows, unions, intersections but no negations) coincides with the relevant entailment of the B + logic (defined 30 years before we started our work). This same approach can be applied to paradigms other than λ-calculi: Castagna et al [ , 2007 use our technique to define the π-calculus, a π-calculus where Boolean combinators are added to the type constructors ch + (t) and ch − (t) which classify all the channels on which it is possible to read or, respectively, to write a value of type t. The technique using the extensional interpretation is still needed for cardinality reasons, however bootstrapping in π has a different flavor, since it generates a model that is much closer to the model of values.…”

Section: Resultsmentioning

confidence: 74%

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Semantic subtyping

Frisch¹,

Castagna

Benzaken

2008

J. ACM

Self Cite

112

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Subtyping relations are usually defined either syntactically by a formal system or semantically by an interpretation of types into an untyped denotational model. This work shows how to define a subtyping relation semantically in the presence of Boolean connectives, functional types and dynamic dispatch on types, without the complexity of denotational models, and how to derive a complete subtyping algorithm.

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Section: Resultsmentioning

confidence: 74%

“…Instead, our direct treatment gives a new and nontrivial subtyping rule for arrow types, which turned out to be useful in other contexts. In particular, a connection has been established between this rule and the minimal relevant logic B + [Dezani-Ciancaglini et al 2002].…”

Section: Related Workmentioning

confidence: 98%

Semantic subtyping

Frisch¹,

Castagna

Benzaken

2008

J. ACM

Self Cite

112

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“…In [19] a relevant logic is proposed for modelling "subtype" relationships between types. Under a certain interpretation of types and type constructors (cf.…”

Section: Logics Based On Dlos In Applicationsmentioning

confidence: 99%

“…Under a certain interpretation of types and type constructors (cf. [19]), subtype checking can be expressed, in algebraic terms, as a uniform word problem with respect to a class of distributive lattices with an additional binary operator, →: L × L → L, which is a joinhemimorphism in the second argument and maps joins to meets in the first argument. -Shape analysis.…”

Section: Logics Based On Dlos In Applicationsmentioning

confidence: 99%

Automated theorem proving by resolution in non-classical logics

Sofronie-Stokkermans

2007

Ann Math Artif Intell

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This paper is an overview of a variety of results, all centered around a common theme, namely embedding of non-classical logics into first order logic and resolution theorem proving. We present several classes of non-classical logics, many of which are of great practical relevance in knowledge representation, which can be translated into tractable and relatively simple fragments of classical logic. In this context, we show that refinements of resolution can often be used successfully for automated theorem proving, and in many interesting cases yield optimal decision procedures

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“…But Dezani and her colleagues, for their part, are not to be denied. For it was (what we call) the Better Bubbling Lemma (henceforth, ) of Dezani et al in [6] that led to our new verification of the primitive combinatory equations in prime B+T theories.…”

mentioning

confidence: 98%

Basic Relevant Theories for Combinators at Levels I and II

Pal

Meyer

2005

AJL

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The system B+ is the minimal positive relevant logic. B+ is trivially extended to B+T on adding a greatest truth (Church constant) T. If we leave ∨ out of the formation apparatus, we get the fragment B∧T. It is known that the set of ALL B∧T theories provides a good model for the combinators CL at Level-I, which is the theory level. Restoring ∨ to get back B+T was not previously fruitful at Level-I, because the set of all B+T theories is NOT a model of CL. It was to be expected from semantic completeness arguments for relevant logics that basic combinator laws would hold when restricted to PRIME B+T theories. Overcoming some previous difficulties, we show that this is the case, at Level I. But this does not form a model for CL. This paper also looks for corresponding results at Level-II, where we deal with sets of theories that we call propositions. We adapt work by Ghilezan to note that at Level-II also there is a model of CL in B∧T propositions. However, the corresponding result for B+T propositions extends smoothly to Level-II only in part. Specifically, only some of the basic combinator laws are proved here. We accordingly leave some work for the reader.

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The Relevance of Semantic Subtyping

Cited by 11 publications

References 9 publications

Semantic subtyping

Semantic subtyping

Automated theorem proving by resolution in non-classical logics

Basic Relevant Theories for Combinators at Levels I and II

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