On Equivalence and Canonical Forms in the LF Type Theory

Decidability of definitional equality and conversion of terms into canonical form play a central role in the meta-theory of a type-theoretic logical framework. Most studies of definitional equality are based on a confluent, strongly-normalizing notion of reduction. Coquand has considered a different approach, directly proving the correctness of a practical equivalance algorithm based on the shape of terms. Neither approach appears to scale well to richer languages with unit types or subtyping, and neither directly addresses the problem of conversion to canonical. In this paper we present a new, type-directed equivalence algorithm for the LF type theory that overcomes the weaknesses of previous approaches. The algorithm is practical, scales to richer languages, and yields a new notion of canonical form sufficient for adequate encodings of logical systems. The algorithm is proved complete by a Kripke-style logical relations argument similar to that suggested by Coquand. Crucially, both the algorithm itself and the logical relations rely only on the shapes of types, ignoring dependencies on terms.Comment: 41 page

Kripke Semantics for Martin-L\"of's Extensional Type Theory

It is well-known that simple type theory is complete with respect to non-standard set-valued models. Completeness for standard models only holds with respect to certain extended classes of models, e.g., the class of cartesian closed categories. Similarly, dependent type theory is complete for locally cartesian closed categories. However, it is usually difficult to establish the coherence of interpretations of dependent type theory, i.e., to show that the interpretations of equal expressions are indeed equal. Several classes of models have been used to remedy this problem. We contribute to this investigation by giving a semantics that is standard, coherent, and sufficiently general for completeness while remaining relatively easy to compute with. Our models interpret types of Martin-L\"of's extensional dependent type theory as sets indexed over posets or, equivalently, as fibrations over posets. This semantics can be seen as a generalization to dependent type theory of the interpretation of intuitionistic first-order logic in Kripke models. This yields a simple coherent model theory, with respect to which simple and dependent type theory are sound and complete

A dependent nominal type theory

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

Soft Contract Verification

Behavioral software contracts are a widely used mechanism for governing the flow of values between components. However, run-time monitoring and enforcement of contracts imposes significant overhead and delays discovery of faulty components to run-time. To overcome these issues, we present soft contract verification, which aims to statically prove either complete or partial contract correctness of components, written in an untyped, higher-order language with first-class contracts. Our approach uses higher-order symbolic execution, leveraging contracts as a source of symbolic values including unknown behavioral values, and employs an updatable heap of contract invariants to reason about flow-sensitive facts. We prove the symbolic execution soundly approximates the dynamic semantics and that verified programs can't be blamed. The approach is able to analyze first-class contracts, recursive data structures, unknown functions, and control-flow-sensitive refinements of values, which are all idiomatic in dynamic languages. It makes effective use of an off-the-shelf solver to decide problems without heavy encodings. The approach is competitive with a wide range of existing tools---including type systems, flow analyzers, and model checkers---on their own benchmarks.Comment: ICFP '14, September 1-6, 2014, Gothenburg, Swede