Conference Paper

A Dependent Type Theory with Names and Binding

DOI: 10.1007/978-3-540-30124-0_20 Conference: Computer Science Logic, 18th International Workshop, CSL 2004, 13th Annual Conference of the EACSL, Karpacz, Poland, September 20-24, 2004, Proceedings
Source: DBLP


We consider the problem of providing formal support for working with abstract syntax involving variable binders. Gabbay and Pitts have shown in their work on Fraenkel-Mostowski (FM) set theory how to address this through first-class names: in this paper we present a dependent type theory for programming and reasoning with such names. Our development is based on a categorical axiomatisation of names, with freshness as its central notion. An associated adjunction captures constructions known from FM theory: the freshness quantifier И, name-binding, and unique choice of fresh names. The Schanuel topos — the category underlying FM set theory — is an instance of this axiomatisation. Working from the categorical structure, we define a dependent type theory which it models. This uses bunches to integrate the monoidal structure corresponding to freshness, from which we define novel multiplicative dependent products ∏ * and sums ∑ *, as well as a propositions-as-types generalisation H of the freshness quantifier.

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    • "Typing for restriction and freshness quantification have been introduced, either in the context of nominal logic [14], of nominal abstraction [15], or of more expressive spatial logics [5]. A different notion of intensional binding is available in bunched logic [24], an affine variant of which has been used in [26] — however, the idea there is to have quantification ranging over linear terms, a quite different approach from that adopted in [7] and followed here. "
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    ABSTRACT: We introduce a system of linear dependent types, extended with quantifiers that ensure separation between distinct bound variables. Such variables may be interpreted as resources that can be accessed only locally. The main motivation for this system, is to make more manageable the logic encoding of specification formalisms based on graphs and state-transition models. The proof system is based on a sequent calculus presentation of quantified intuitionistic linear logic, relying on double-entry sequents. We prove the admissibility of cut, and show that this result can be used to prove subject reduction.
    Full-text · Article · Jun 2014 · Journal of Logic and Computation
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    • "These extensions were motivated by a denotational interpretation of SNTT using nominal sets (following [28]). We will not develop a denotational semantics of λ Π N here; however, the topos of nominal sets provides all of the necessary structure to interpret dependent types, and it seems clear that the extensions we consider can be justified using Schöpp and Stark's semantics for a more general nominal type theory [37] [35] or using Pitts' approach to recursion in a slightly different nominal type theory [30] [31]. "
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    ABSTRACT: Nominal abstract syntax is an approach to representing names and binding pioneered by Gabbay and Pitts. So far nominal techniques have mostly been studied using classical logic or model theory, not type theory. Nominal extensions to simple, dependent and ML-like polymorphic languages have been studied, but decidability and normalization results have only been established for simple nominal type theories. We present a LF-style dependent type theory extended with name-abstraction types, prove soundness and decidability of beta-eta-equivalence checking, discuss adequacy and canonical forms via an example, and discuss extensions such as dependently-typed recursion and induction principles.
    Full-text · Article · Jan 2012 · Logical Methods in Computer Science
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    • "Unfortunately, name-bindings are well-known to be cumbersome to encode and reason about in logic and type theory, making it tedious to prove properties about programming languages and other formalisms that use namebindings . This has led to much research into techniques for encoding name-bindings that mitigate this burden [16] [10] [8] [20] [7] [19] [13] [24] [21] [18] [4]. "
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    ABSTRACT: Although name-bindings are ubiquitous in computer science, they are well-known to be cumbersome to encode and reason about in logic and type theory. There are many proposed solutions to this problem in the literature, but most of these proposals, however, have been extensional, meaning they are defined in terms of other concepts in the theory. This makes it difficult to apply these pro-posals in intensional theories like the Calculus of Inductive Con-structions, or CIC. In this paper, we introduce an approach to encoding name-bindings that is intensional, as it attempts to capture the meaning of a name-binding in itself. This approach combines in a straightfor-ward manner with CIC to form the Calculus of Nominal Inductive Constructions, or CNIC. CNIC supports induction over data con-taining bindings, comparing of names for equality, and associating meta-language types with names in a fashion similar to HOAS, fea-tures which have been shown difficult to support in practice.
    Preview · Article · Aug 2009
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