Contextual modal type theory

Nanevski, Aleksandar; Pfenning, Frank; Pientka, Brigitte

doi:10.1145/1352582.1352591

Cited by 204 publications

(219 citation statements)

References 61 publications

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“…This proof follows essentially the proof by Nanveski et al [11] and we generalize the property to types, kinds, and contexts. Since we prove the substitution lemma modulo A = K B, we use Lemma 1 when we for example consider the variable case.…”

Section: Typing Modulomentioning

confidence: 78%

Higher-Order Dynamic Pattern Unification for Dependent Types and Records

Abel

Pientka

2011

Lecture Notes in Computer Science

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Abstract. While higher-order pattern unification for the λ Π -calculus is decidable and unique unifiers exists, we face several challenges in practice: 1) the pattern fragment itself is too restrictive for many applications; this is typically addressed by solving sub-problems which satisfy the pattern restriction eagerly but delay solving sub-problems which are non-patterns until we have accumulated more information. This leads to a dynamic pattern unification algorithm. 2) Many systems implement λ ΠΣ calculus and hence the known pattern unification algorithms for λ Π are too restrictive. In this paper, we present a constraint-based unification algorithm for λ ΠΣ -calculus which solves a richer class of patterns than currently possible; in particular it takes into account type isomorphisms to translate unification problems containing Σ-types into problems only involving Π-types. We prove correctness of our algorithm and discuss its application.

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Section: Typing Modulomentioning

confidence: 78%

Higher-Order Dynamic Pattern Unification for Dependent Types and Records

Abel

Pientka

2011

Lecture Notes in Computer Science

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“…In beluga, we pair a term M together with the context Ψ in which it is meaningful, written as [Ψ M]. These are called contextual LF objects [13]. We can then embed contextual objects and types into the reasoning level; in particular, we can state inductive properties about contexts, contextual objects and contextual types.…”

Section: Representing Reducibility Using Indexed Typesmentioning

confidence: 99%

Inductive Beluga: Programming Proofs

Pientka

Cave

2015

Automated Deduction - CADE-25

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Abstract. beluga is a proof environment which provides a sophisticated infrastructure for implementing formal systems based on the logical framework LF together with a first-order reasoning language for implementing inductive proofs about them following the Curry-Howard isomorphism. In this paper we describe four significant extensions to beluga: 1) we enrich our infrastructure for modelling formal systems with first-class simultaneous substitutions, a key and common concept when reasoning about formal systems 2) we support inductive definitions in our reasoning language which significantly increases beluga's expressive power 3) we provide a totality checker which guarantees that recursive programs are well-founded and correspond to inductive proofs 4) we describe an interactive program development environment. Taken together these extensions enable direct and compact mechanizations. To demonstrate beluga's strength and illustrate these new features we develop a weak normalization proof using logical relations.

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“…18 See [Primiero, 2012] for the full formal language: it is a variant interpretation of the basic system of constructive type-theory that links hypotheses and refutable contents. It extends to a modal type-theory, variating on a theme first proposed in [Pfenning, Dvies, 2001] and later expanded in [Nanevski et al, 2008]. We shall here only focus on the appropriate introduction rules for justified and assumed contents and expand on the use of modalities in the next section.…”

Section: Data For Functional Informationmentioning

confidence: 99%

“…The latter is the form of judgements used for the modal extension of Martin-Löf's Type Theory in [Pfenning, Dvies, 2001] and [Nanevski et al, 2008], whereas we use the judgemental modalities both in [Primiero, 2012] and [Primiero, 2010] in order to express the contextual nature of our proof-terms. 26 See also [Bellin et al, 2001] for a formal language that requires the same kind of rules.…”

Section: Epistemic Modalitiesmentioning

confidence: 99%

Offline and online data: on upgrading functional information to knowledge

Primiero

2012

Philos Stud

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This paper addresses the problem of upgrading functional information to knowledge. Functional information is defined as syntactically wellformed, meaningful and collectively opaque data. Its use in the formal epistemology of information theories is crucial to solve the debate on the veridical nature of information, and it represents the companion notion to standard strongly semantic information, defined as well-formed, meaningful and true data. The formal framework, on which the definitions are based, uses a contextual version of the verificationist principle of truth in order to connect functional to semantic information, avoiding Gettierization and decoupling from true informational contents. The upgrade operation from functional information uses the machinery of epistemic modalities in order to add data localization and accessibility as its main properties. We show in this way the conceptual worthiness of this notion for issues in contemporary epistemology debates, such as the explanation of knowledge process acquisition from information retrieval systems, and open data repositories.

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Contextual modal type theory

Cited by 204 publications

References 61 publications

Higher-Order Dynamic Pattern Unification for Dependent Types and Records

Higher-Order Dynamic Pattern Unification for Dependent Types and Records

Inductive Beluga: Programming Proofs

Offline and online data: on upgrading functional information to knowledge

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