Which simple types have a unique inhabitant?

International audienceWe study the question of whether a given type has a unique inhabitant modulo program equivalence. In the setting of simply-typed lambda-calculus with sums, equipped with the strong βη-equivalence, we show that uniqueness is decidable. We present a saturating focused logic that introduces irreducible cuts on positive types "as soon as possible". Backward search in this logic gives an effective algorithm that returns either zero, one or two distinct inhabitants for any given type. Preliminary application studies show that such a feature can be useful in strongly-typed programs, inferring the code of highly-polymorphic library functions, or "glue code" inside more complex terms

Notions of Anonymous Existence in Martin-L\"of Type Theory

As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda.Comment: 36 pages, to appear in the special issue of TLCA'13 (LMCS

Notions of anonymous existence in Martin-Löf type theory

As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda

Notions of anonymous existence in Martin-Löf type theory

As the groupoid model of Hofmann and Streicher proves, identity proofs in intensional Martin-L\"of type theory cannot generally be shown to be unique. Inspired by a theorem by Hedberg, we give some simple characterizations of types that do have unique identity proofs. A key ingredient in these constructions are weakly constant endofunctions on identity types. We study such endofunctions on arbitrary types and show that they always factor through a propositional type, the truncated or squashed domain. Such a factorization is impossible for weakly constant functions in general (a result by Shulman), but we present several non-trivial cases in which it can be done. Based on these results, we define a new notion of anonymous existence in type theory and compare different forms of existence carefully. In addition, we show possibly surprising consequences of the judgmental computation rule of the truncation, in particular in the context of homotopy type theory. All the results have been formalized and verified in the dependently typed programming language Agda

Implementing and reasoning about hash-consed data structures in Coq

We report on four different approaches to implementing hash-consing in Coq programs. The use cases include execution inside Coq, or execution of the extracted OCaml code. We explore the different trade-offs between faithful use of pristine extracted code, and code that is fine-tuned to make use of OCaml programming constructs not available in Coq. We discuss the possible consequences in terms of performances and guarantees. We use the running example of binary decision diagrams and then demonstrate the generality of our solutions by applying them to other examples of hash-consed data structures

Experience Implementing a Performant Category-Theory Library in Coq

We describe our experience implementing a broad category-theory library in Coq. Category theory and computational performance are not usually mentioned in the same breath, but we have needed substantial engineering effort to teach Coq to cope with large categorical constructions without slowing proof script processing unacceptably. In this paper, we share the lessons we have learned about how to represent very abstract mathematical objects and arguments in Coq and how future proof assistants might be designed to better support such reasoning. One particular encoding trick to which we draw attention allows category-theoretic arguments involving duality to be internalized in Coq's logic with definitional equality. Ours may be the largest Coq development to date that uses the relatively new Coq version developed by homotopy type theorists, and we reflect on which new features were especially helpful.Comment: The final publication will be available at link.springer.com. This version includes a full bibliography which does not fit in the Springer version; other than the more complete references, this is the version submitted as a final copy to ITP 201

Relational Parametricity and Control

We study the equational theory of Parigot's second-order λμ-calculus in connection with a call-by-name continuation-passing style (CPS) translation into a fragment of the second-order λ-calculus. It is observed that the relational parametricity on the target calculus induces a natural notion of equivalence on the λμ-terms. On the other hand, the unconstrained relational parametricity on the λμ-calculus turns out to be inconsistent with this CPS semantics. Following these facts, we propose to formulate the relational parametricity on the λμ-calculus in a constrained way, which might be called ``focal parametricity''.Comment: 22 pages, for Logical Methods in Computer Scienc

Reading Wikipedia to Answer Open-Domain Questions

This paper proposes to tackle open- domain question answering using Wikipedia as the unique knowledge source: the answer to any factoid question is a text span in a Wikipedia article. This task of machine reading at scale combines the challenges of document retrieval (finding the relevant articles) with that of machine comprehension of text (identifying the answer spans from those articles). Our approach combines a search component based on bigram hashing and TF-IDF matching with a multi-layer recurrent neural network model trained to detect answers in Wikipedia paragraphs. Our experiments on multiple existing QA datasets indicate that (1) both modules are highly competitive with respect to existing counterparts and (2) multitask learning using distant supervision on their combination is an effective complete system on this challenging task.Comment: ACL2017, 10 page

On Berry's conjectures about the stable order in PCF

PCF is a sequential simply typed lambda calculus language. There is a unique order-extensional fully abstract cpo model of PCF, built up from equivalence classes of terms. In 1979, G\'erard Berry defined the stable order in this model and proved that the extensional and the stable order together form a bicpo. He made the following two conjectures: 1) "Extensional and stable order form not only a bicpo, but a bidomain." We refute this conjecture by showing that the stable order is not bounded complete, already for finitary PCF of second-order types. 2) "The stable order of the model has the syntactic order as its image: If a is less than b in the stable order of the model, for finite a and b, then there are normal form terms A and B with the semantics a, resp. b, such that A is less than B in the syntactic order." We give counter-examples to this conjecture, again in finitary PCF of second-order types, and also refute an improved conjecture: There seems to be no simple syntactic characterization of the stable order. But we show that Berry's conjecture is true for unary PCF. For the preliminaries, we explain the basic fully abstract semantics of PCF in the general setting of (not-necessarily complete) partial order models (f-models.) And we restrict the syntax to "game terms", with a graphical representation.Comment: submitted to LMCS, 39 pages, 23 pstricks/pst-tree figures, main changes for this version: 4.1: proof of game term theorem corrected, 7.: the improved chain conjecture is made precise, more references adde