A Proof Theoretic View of Constraint Programming

We provide here a proof theoretic account of constraint programming that attempts to capture the essential ingredients of this programming style. We exemplify it by presenting proof rules for linear constraints over interval domains, and illustrate their use by analyzing the constraint propagation process for the {\tt SEND + MORE = MONEY} puzzle. We also show how this approach allows one to build new constraint solvers.Comment: 25 page

Operational Semantics of Resolution and Productivity in Horn Clause Logic

This paper presents a study of operational and type-theoretic properties of different resolution strategies in Horn clause logic. We distinguish four different kinds of resolution: resolution by unification (SLD-resolution), resolution by term-matching, the recently introduced structural resolution, and partial (or lazy) resolution. We express them all uniformly as abstract reduction systems, which allows us to undertake a thorough comparative analysis of their properties. To match this small-step semantics, we propose to take Howard's System H as a type-theoretic semantic counterpart. Using System H, we interpret Horn formulas as types, and a derivation for a given formula as the proof term inhabiting the type given by the formula. We prove soundness of these abstract reduction systems relative to System H, and we show completeness of SLD-resolution and structural resolution relative to System H. We identify conditions under which structural resolution is operationally equivalent to SLD-resolution. We show correspondence between term-matching resolution for Horn clause programs without existential variables and term rewriting.Comment: Journal Formal Aspect of Computing, 201

A Type-Theoretic Approach to Structural Resolution

Structural resolution (or S-resolution) is a newly proposed alternative to SLD-resolution that allows a systematic separation of derivations into term-matching and unification steps. Productive logic programs are those for which term-matching reduction on any query must terminate. For productive programs with coinductive meaning, finite term-rewriting reductions can be seen as measures of observation in an infinite derivation. Ability of handling corecursion in a productive way is an attractive computational feature of S-resolution. In this paper, we make first steps towards a better conceptual understanding of operational properties of S-resolution as compared to SLD-resolution. To this aim, we propose a type system for the analysis of both SLD-resolution and S-resolution. We formulate S-resolution and SLD-resolution as reduction systems, and show their soundness relative to the type system. One of the central methods of this paper is realizability transformation, which makes logic programs productive and non-overlapping. We show that S-resolution and SLD-resolution are only equivalent for programs with these two properties.Comment: LOPSTR 201

Applying Formal Methods to Networking: Theory, Techniques and Applications

Despite its great importance, modern network infrastructure is remarkable for the lack of rigor in its engineering. The Internet which began as a research experiment was never designed to handle the users and applications it hosts today. The lack of formalization of the Internet architecture meant limited abstractions and modularity, especially for the control and management planes, thus requiring for every new need a new protocol built from scratch. This led to an unwieldy ossified Internet architecture resistant to any attempts at formal verification, and an Internet culture where expediency and pragmatism are favored over formal correctness. Fortunately, recent work in the space of clean slate Internet design---especially, the software defined networking (SDN) paradigm---offers the Internet community another chance to develop the right kind of architecture and abstractions. This has also led to a great resurgence in interest of applying formal methods to specification, verification, and synthesis of networking protocols and applications. In this paper, we present a self-contained tutorial of the formidable amount of work that has been done in formal methods, and present a survey of its applications to networking.Comment: 30 pages, submitted to IEEE Communications Surveys and Tutorial

Recursive Definitions of Monadic Functions

Using standard domain-theoretic fixed-points, we present an approach for defining recursive functions that are formulated in monadic style. The method works both in the simple option monad and the state-exception monad of Isabelle/HOL's imperative programming extension, which results in a convenient definition principle for imperative programs, which were previously hard to define. For such monadic functions, the recursion equation can always be derived without preconditions, even if the function is partial. The construction is easy to automate, and convenient induction principles can be derived automatically.Comment: In Proceedings PAR 2010, arXiv:1012.455