research-article

Logical relations for a logical framework

Authors:

Kristina SojakovaAuthors Info & Claims

ACM Transactions on Computational Logic (TOCL), Volume 14, Issue 4

Article No.: 32, Pages 1 - 34

https://doi.org/10.1145/2536740.2536741

Published: 28 November 2013 Publication History

Abstract

Logical relations are a central concept used to study various higher-order type theories and occur frequently in the proofs of a wide variety of meta-theorems. Besides extending the logical relation principle to more general languages, an important research question has been how to represent and thus verify logical relation arguments in logical frameworks.

We formulate a theory of logical relations for Dependent Type Theory (DTT) with β η-equality which guarantees that any valid logical relation satisfies the Basic Lemma. Our definition is syntactic and reflective in the sense that a relation at a type is represented as a DTT type family but also permits expressing certain semantic definitions. We use the Edinburgh Logical Framework (LF) incarnation of DTT and implement our notion of logical relations in the type-checker Twelf. This enables us to formalize and mechanically decide the validity of logical relation arguments. Furthermore, our implementation includes a module system so that logical relations can be built modularly. We validate our approach by formalizing and verifying several syntactic and semantic meta-theorems in Twelf. Moreover, we show how object languages encoded in DTT can inherit a notion of logical relation from the logical framework.

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Cited By

CAVE APIENTKA B(2018)Mechanizing proofs with logical relations – Kripke-styleMathematical Structures in Computer Science10.1017/S096012951800015428:09(1606-1638)Online publication date: 2-Aug-2018
https://doi.org/10.1017/S0960129518000154
Rabe F(2015)Lax Theory MorphismsACM Transactions on Computational Logic10.1145/281864417:1(1-36)Online publication date: 28-Oct-2015
https://dl.acm.org/doi/10.1145/2818644
Rabe F(2014)How to identify, translate and combine logics?Journal of Logic and Computation10.1093/logcom/exu07927:6(1753-1798)Online publication date: 14-Dec-2014
https://doi.org/10.1093/logcom/exu079
Show More Cited By

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Logical relations for a logical framework

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cover image ACM Transactions on Computational Logic

ACM Transactions on Computational Logic Volume 14, Issue 4

November 2013

282 pages

ISSN:1529-3785

EISSN:1557-945X

DOI:10.1145/2555591

Issue’s Table of Contents

Copyright © 2013 ACM.

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Publication History

Published: 28 November 2013

Accepted: 01 March 2013

Revised: 01 February 2013

Received: 01 September 2012

Published in TOCL Volume 14, Issue 4

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Cited By

CAVE APIENTKA B(2018)Mechanizing proofs with logical relations – Kripke-styleMathematical Structures in Computer Science10.1017/S096012951800015428:09(1606-1638)Online publication date: 2-Aug-2018
https://doi.org/10.1017/S0960129518000154
Rabe F(2015)Lax Theory MorphismsACM Transactions on Computational Logic10.1145/281864417:1(1-36)Online publication date: 28-Oct-2015
https://dl.acm.org/doi/10.1145/2818644
Rabe F(2014)How to identify, translate and combine logics?Journal of Logic and Computation10.1093/logcom/exu07927:6(1753-1798)Online publication date: 14-Dec-2014
https://doi.org/10.1093/logcom/exu079
Rasmussen UFilinski AMomigliano APientka BPollack R(2013)Structural logical relations with case analysis and equality reasoningProceedings of the Eighth ACM SIGPLAN international workshop on Logical frameworks & meta-languages: theory & practice10.1145/2503887.2503891(43-54)Online publication date: 23-Sep-2013
https://dl.acm.org/doi/10.1145/2503887.2503891

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