New Notions of Reduction and Non-Semantic Proofs of β-Strong Normalization in Typed λ-Calculi

Two new notions of reduction for terms of the λ-calculus are introduced and the question of whether a λ-term is beta-strongly normalizing is reduced to the question of whether a λ-term is merely normalizing under one of the new notions of reduction. This leads to a new way to prove beta-strong normalization for typed λ-calculi. Instead of the usual semantic proof style based on Girard's "candidats de réductibilité'', termination can be proved using a decreasing metric over a well-founded ordering in a style more common in the field of term rewriting. This new proof method is applied to the simply-typed λ-calculus and the system of intersection types.

Addendum to "New Notions of Reduction and Non-Semantic Proofs of β-Strong Normalization in Typed λ-Calculi"

This is an addendum to our technical report BUCS TR-94-014 of December 19, 1994. It clarifies some statements, adds information on some related research, includes a comparison with research be de Groote, and fixes two minor mistakes in a proof.

A New Version of Toom's Proof

There are several proofs now for the stability of Toom's example of a two-dimensional stable cellular automaton and its application to fault-tolerant computation. Simon and Berman simplified and strengthened Toom's original proof: the present report is simplified exposition of their proof.

An Infinite Pebble Game and Applications

We generalize the well-known pebble game to infinite dag's, and we use this generalization to give new and shorter proofs of results in different areas of computer science (as diverse as "logic of programs" and "formal language theory"). Our applications here include a proof of a theorem due to Salomaa, asserting the existence of a context-free language with infinite index, and a proof of a theorem due to Tiuryn and Erimbetov, asserting that unbounded memory increases the power of logics of programs. The original proofs by Salomaa, Tiuryn, and Erimbetov, are fairly technical. The proofs by Tiuryn and Erimbetov also involve advanced techniques of model theory, namely, back-and-forth constructions based on a variant of Ehrenfeucht-Fraisse games. By contrast, our proofs are not only shorter, but also elementary. All we need is essentially finite induction and, in the case of the Tiuryn-Erimbetov result, the compactness and completeness of first-order logic.

System F with Constraint Types

System F is a type system that can be seen as both a proof system for second-order propositional logic and as a polymorphic programming language. In this work we explore several extensions of System F by types which express subtyping constraints. These systems include terms which represent proofs of subtyping relationships between types. Given a proof that one type is a subtype of another, one may use a coercion term constructor to coerce terms from the first type to the second. The ability to manipulate type constraints as first-class entities gives these systems a lot of expressive power, including the ability to encode generalized algebraic data types and intensional type analysis. The main contributions of this work are in the formulation of constraint types and a proof of strong normalization for an extension of System F with constraint types.

Safe Compositional Specification of Networking Systems: TRAFFIC The Language and Its Type Checking

This paper formally defines the operational semantic for TRAFFIC, a specification language for flow composition applications proposed in BUCS-TR-2005-014, and presents a type system based on desired safety assurance. We provide proofs on reduction (weak-confluence, strong-normalization and unique normal form), on soundness and completeness of type system with respect to reduction, and on equivalence classes of flow specifications. Finally, we provide a pseudo-code listing of a syntax-directed type checking algorithm implementing rules of the type system capable of inferring the type of a closed flow specification.