Automatic modular abstractions for template numerical constraints

We propose a method for automatically generating abstract transformers for static analysis by abstract interpretation. The method focuses on linear constraints on programs operating on rational, real or floating-point variables and containing linear assignments and tests. In addition to loop-free code, the same method also applies for obtaining least fixed points as functions of the precondition, which permits the analysis of loops and recursive functions. Our algorithms are based on new quantifier elimination and symbolic manipulation techniques. Given the specification of an abstract domain, and a program block, our method automatically outputs an implementation of the corresponding abstract transformer. It is thus a form of program transformation. The motivation of our work is data-flow synchronous programming languages, used for building control-command embedded systems, but it also applies to imperative and functional programming

Step-Indexing: The Good, the Bad and the Ugly

Over the last decade, step-indices have been widely used for the construction of operationally-based logical relations in the presence of various kinds of recursion. We first give an argument that step-indices, or something like them, seem to be required for defining realizability relations between high-level source languages and low-level targets, in the case that the low-level allows egregiously intensional operations such as reflection or comparison of code pointers. We then show how, much to our annoyance, step-indices also seem to prevent us from exploiting such operations as aggressively as we would like in proving program transformations

10351 Executive Summary -- Modelling, Controlling and Reasoning About State

From 29 August 2010 to 3 September 2010, the Dagstuhl Seminar 10351 ``Modelling, Controlling and Reasoning About State \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. 44 researchers, with interests and expertise in many different aspects of modelling and reasoning about mutable state, met to present their current work and discuss ongoing projects and open problems. This executive summary provides a general overview of the goals of the seminar and of the topics discussed

08061 Abstracts Collection -- Types, Logics and Semantics for State

From 3 February to 8 February 2008, the Dagstuhl Seminar 08061 ``Types, Logics and Semantics for State\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

10351 Abstracts Collection -- Modelling, Controlling and Reasoning About State

From 29 August 2010 to 3 September 2010, the Dagstuhl Seminar 10351 ``Modelling, Controlling and Reasoning About State \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. Links to extended abstracts or full papers are provided, if available

08061 Executive Summary -- Types, Logics and Semantics for State

From 3 February to 8 February 2008, the Dagstuhl Seminar 08061 State" Conference and Research Center (IBFI), Schloss Dagstuhl. 45 researchers, with interests and expertise in many different aspects of modelling and reasoning about mutable state, met to present their current work and discuss ongoing projects and open problems

Abstract effects and proofrelevant logical relations. POPL

Abstract We give a denotational semantics for a region-based effect system that supports type abstraction in the sense that only externally visible effects need to be tracked: non-observable internal modifications, such as the reorganisation of a search tree or lazy initialisation, can count as 'pure' or 'read only'. This 'fictional purity' allows clients of a module to validate soundly more effect-based program equivalences than would be possible with previous semantics. Our semantics uses a novel variant of logical relations that maps types not merely to partial equivalence relations on values, as is commonly done, but rather to a proof-relevant generalisation thereof, namely setoids. The objects of a setoid establish that values inhabit semantic types, whilst its morphisms are understood as proofs of semantic equivalence. The transition to proof-relevance solves two awkward problems caused by naïve use of existential quantification in Kripke logical relations, namely failure of admissibility and spurious functional dependencies

Proof-relevant logical relations for name generation

Pitts and Stark's nu-calculus is a paradigmatic total language for studying the problem of contextual equivalence in higher-order languages with name generation. Models for the nu-calculus that validate basic equivalences concerning names may be constructed using functor categories or nominal sets, with a dynamic allocation monad used to model computations that may allocate fresh names. If recursion is added to the language and one attempts to adapt the models from (nominal) sets to(nominal)domains, however, the direct-style construction of the allocation monad no longer works. This issue has previously been addressed by using a monad that combines dynamic allocation with continuations, at some cost to abstraction. This paper presents a direct-style model of a nu-calculus-like language with recursion using the novel framework of proof-relevant logical relations, in which logical relations also contain objects (or proofs) demonstrating the equivalence of (the semantic counterparts of) programs. Apart from providing a fresh solution to an old problem, this work provides an accessible setting in which to introduce the use of proof-relevant logical relations, free of the additional complexities associated with their use for more sophisticated languages

A Model of Cooperative Threads

We develop a model of concurrent imperative programming with threads. We focus on a small imperative language with cooperative threads which execute without interruption until they terminate or explicitly yield control. We define and study a trace-based denotational semantics for this language; this semantics is fully abstract but mathematically elementary. We also give an equational theory for the computational effects that underlie the language, including thread spawning. We then analyze threads in terms of the free algebra monad for this theory.Comment: 39 pages, 5 figure

A Lambda Term Representation Inspired by Linear Ordered Logic

We introduce a new nameless representation of lambda terms inspired by ordered logic. At a lambda abstraction, number and relative position of all occurrences of the bound variable are stored, and application carries the additional information where to cut the variable context into function and argument part. This way, complete information about free variable occurrence is available at each subterm without requiring a traversal, and environments can be kept exact such that they only assign values to variables that actually occur in the associated term. Our approach avoids space leaks in interpreters that build function closures. In this article, we prove correctness of the new representation and present an experimental evaluation of its performance in a proof checker for the Edinburgh Logical Framework. Keywords: representation of binders, explicit substitutions, ordered contexts, space leaks, Logical Framework.Comment: In Proceedings LFMTP 2011, arXiv:1110.668