Ralph Matthes

Verification of the Schorr-Waite algorithm - from trees to graphs

This article proposes a method for proving the correctness of graph algorithms by manipulating their spanning trees enriched with additional references. We illustrate this concept with a proof of the correctness of a (pseudo-)imperative version of the Schorr-Waite algorithm by refinement of a functional one working on trees. It is composed of two orthogonal steps of refinement - functional to imperative and tree to graph - finally merged to obtain the result. Our imperative specifications use monadic constructs and syntax sugar, making them close to common imperative languages. This work has been realized within the Isabelle/HOL proof assistant.

Coinductive Graph Representation: the Problem of Embedded Lists

When trying to obtain formally certified model transformations, one may want to represent models as graphs and graphs as greatest fixed points. To do so, one is rather naturally led to define co-inductive types that use lists (to represent a finite but unbounded number of children of internal nodes). These concepts are rather well supported in the proof assistant Coq. However, their use in our intended applications may cause problems since the co-recursive call in the type definition occurs in the list parameter. When defining co-recursive functions on such structures, one will face guardedness issues, and in fact, the syntactic criterion applied by the Coq system is too rigid here. We offer a solution using dependent types to overcome the guardedness issues that arise in our graph transformations. We also give examples of further properties and results, among which finiteness of represented graphs. All of this has been fully formalized in Coq.

Permutations in Coinductive Graph Representation

In the proof assistant Coq, one can model certain classes of graphs by coinductive types. The coinductive aspects account for infinite navigability already in finite but cyclic graphs, as in rational trees. Coq’s static checks exclude simple-minded definitions with lists of successors of a node. In previous work, we have shown how to mimic lists by a type of functions and built a Coq theory for such graphs. Naturally, these coinductive structures have to be compared by a bisimulation relation, and we defined it in a generic way.

Monadic translation of classical sequent calculus

We study monadic translations of the call-by-name (cbn) and call-by-value (cbv) fragments of the classical sequent calculus λμ˜ due to Curien and Herbelin, and give modular and syntactic proofs of strong normalisation. The target of the translations is a new meta-language for classical logic, named monadic λμ. This language is a monadic reworking of Parigot’s λμ-calculus, where the monadic binding is confined to commands, thus integrating the monad with the classical features. Also, its μ-reduction rule is replaced by a rule expressing the interaction between monadic binding and μ-abstraction. Our monadic translations produce very tight simulations of the respective fragments of λμ˜ within monadic λμ, with reduction steps of λμ˜ being translated in a 1–1 fashion, except for β steps, which require two steps. The monad of monadic λμ can be instantiated to the continuations monad so as to ensure strict simulation of monadic λμ within simply typed λ-calculus with β -a ndη-reduction. Through strict simulation, the strong normalisation of simply typed λ -calculus is inherited by monadicλμ, and then by cbn and cbv λμ˜, thus reproving strong normalisation in an elementary syntactical way for these fragments of λμ˜, and establishing it for our new calculus. These results extend to second-order logic, with polymorphic λ-calculus as the target, giving new strong normalisation results for classical second-order logic in sequent calculus style. CPS translations of cbn and cbv λμ˜ with the strict simulation property are obtained by composing our monadic translations with the continuations-monad instantiation. In an appendix to the paper, we investigate several refinements of the continuations-monad instantiation in order to obtain in a modular way improvements of the CPS translations enjoying extra properties like simulation by cbv β-reduction or reduction of administrative redexes at compile time.

From Signatures to Monads in UniMath

The term UniMath refers both to a formal system for mathematics, as well as a computer-checked library of mathematics formalized in that system. The UniMath system is a core dependent type theory, augmented by the univalence axiom. The system is kept as small as possible in order to ease verification of it—in particular, general inductive types are not part of the system. In this work, we partially remedy the lack of inductive types by constructing some set-level datatypes and their associated induction principles from other type constructors. This involves a formalization of a category-theoretic result on the construction of initial algebras, as well as a mechanism to conveniently use the datatypes obtained. We also connect this construction to a previous formalization of substitution for languages with variable binding. Altogether, we construct a framework that allows us to concisely specify, via a simple notion of binding signature, a language with variable binding. From such a specification we obtain the datatype of terms of that language, equipped with a certified monadic substitution operation and a suitable recursion scheme. Using this we formalize the untyped lambda calculus and the raw syntax of Martin-Löf type theory.

Proceedings Tenth International Workshop on Fixed Points in Computer Science

This volume contains the proceedings of the Tenth International Workshop on Fixed Points in Computer Science (FICS 2015) which took place on September 11th and 12th, 2015 in Berlin, Germany, as a satellite event of the conference Computer Science Logic (CSL 2015). Fixed points play a fundamental role in several areas of computer science. They are used to justify (co)recursive definitions and associated reasoning techniques. The construction and properties of fixed points have been investigated in many different settings such as: design and implementation of programming languages, logics, verification, databases. The aim of this workshop is to provide a forum for researchers to present their results to those members of the computer science and logic communities who study or apply the theory of fixed points. Each of the 11 contributed papers of this volume were evaluated by three or four reviewers. Some of the papers were re-reviewed after revision. Additionally, this volume contains the abstracts of the FICS 2015 invited talks given by Bartek Klin and James Worrell.