Marc Bezem

Strongly Majorizable Functionals of Finite Type: A Model for Barrecursion Containing Discontinuous Functionals

In this paper a model for barrecursion is presented. It has as a novelty that it contains discontinuous functionals. The model is based on a concept called strong majorizability. This concept is a modification of Howards majorizability notion; see [T, p. 456].

Invariants in Process Algebra with Data

We provide rules for calculating with invariants in process algebra with data, and illustrate these with examples. The new rules turn out to be equivalent to the well known Recursive Specification Principle which states that guarded recursive equations have at most one solution. In the setting with data this is reformulated as ‘every convergent linear process operator has at most one fixed point’ (CL-RSP). As a consequence, one can carry out verifications in well-known process algebras satisfying CL-RSP using invariants.

Strong termination of logic programs

We study a powerful class of logic programs which terminate for a large class of goals. Both classes are characterized in a natural way in terms of mappings from variable-free atoms to natural numbers. Based on this idea we present a technique which improves the termination behaviour and allows a more multidirectional use of Prolog programs. The class of logic programs is shown to be strong enough to compute every total recursive function. The class of goals considerably extends the variable-free ones.

A correctness proof of a one-bit sliding window protocol in µCRL

We model a one-bit sliding window protocol and prove that its external behaviour is a bi-directional buffer of capacity 2. The proof is given in μCRL, which is a process algebra extended with data. Due to the abundant parallelism in this protocol, the behaviour is quite complicated. The complexity has been mastered by explicitly identifying invariants and foci of cones in the protocol. Both concepts seem promising as tools for the verification of larger and more complex protocols.

On the computational content of the axiom of choice

We present a possible computational content of the negative translation of classical analysis with the Axiom of (countable) Choice. Interestingly, this interpretation uses a refinement of the realizability semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis of ∃-statements and how to extract algorithms from proofs of ∀∃-statements. Our interpretation seems computationally more direct than the one based on Godels Dialectica interpretation.

Automating coherent logic

First-order coherent logic (CL) extends resolution logic in that coherent formulas allow certain existential quantifications. A substantial number of reasoning problems (e.g., in confluence theory, lattice theory and projective geometry) can be formulated directly in CL without any clausification or Skolemization. CL has a natural proof theory, reasoning is constructive and proof objects can easily be obtained. We prove completeness of the proof theory and give a linear translation from FOL to CL that preserves logical equivalence. These properties make CL well-suited for providing automated reasoning support to logical frameworks. The proof theory has been implemented in Prolog, generating proof objects that can be verified directly in the proof assistant Coq. The prototype has been tested on the proof of Hessenberg’s Theorem, which could be automated to a considerable extent. Finally, we compare the prototype to some automated theorem provers on selected problems.

Hard problems in max-algebra, control theory, hypergraphs and other areas

We introduce the max-atom problem (MAP): solving (in Z) systems of inequations of the form max(x, y)+k z, where x, y, z are variables and k ∈ Z. Our initial motivation for MAP was reasoning on delays in circuits using SAT Modulo Theories [10], viewing MAP as a natural extension of Difference Logic, i.e., inequations of the form x+ k y. Here we show that MAP is PTIME-equivalent to several rather different well-known problems for which no PTIME algorithm has been found so far, in spite of decades of independent efforts. One is on solving two-sided linear max-plus systems (Section 3 of this paper) that arise in Control Theory when modeling Discrete Event Systems, and another one on shortest paths in directed weighted hypergraphs (Section 4).

The Max-Atom Problem and Its Relevance

Let F be a conjunction of atoms of the form max (x ,y ) + k ≥ z , where x , y , z are variables and k is a constant value. Here we consider the satisfiability problem of such formulas (e.g., over the integers or rationals). This problem, which appears in unexpected forms in many applications, is easily shown to be in NP. However, decades of efforts (in several research communities, see below) have not produced any polynomial decision procedure nor an NP-hardness result for this --- apparently so simple --- problem. Here we develop several ingredients (small-model property and lattice structure of the model class, a polynomially tractable subclass and an inference system) which altogether allow us to prove the existence of small unsatisfiability certificates, and hence membership in NP intersection co-NP. As a by-product, we also obtain a weakly polynomial decision procedure. We show that the Max-atom problem is PTIME-equivalent to several other well-known --- and at first sight unrelated --- problems on hypergraphs and on Discrete Event Systems, problems for which the existence of PTIME algorithms is also open. Since there are few interesting problems in NP intersection co-NP that are not known to be polynomial, the Max-atom problem appears to be relevant.

Formalizing process algebraic verifications in the calculus of constructions

This paper reports on the first steps towards the formal verification of correctness proofs of real-life protocols in process algebra. We show that such proofs can be verified, and partly constructed, by a general purpose proof checker. The process algebra we use isμCRL, ACPτ augmented with data, which is expressive enough for the specification of real-life protocols. The proof checker we use is Coq, which is based on the Calculus of Constructions, an extension of simply typed lambda calculus. The focus is on the translation of the proof theory ofμCRL andμCRL-specifications to Coq. As a case study, we verified the Alternating Bit Protocol.

On the Mechanization of the Proof of Hessenberg's Theorem in Coherent Logic

We propose to combine interactive proof construction with proof automation for a fragment of first-order logic called Coherent Logic (CL). CL allows enough existential quantification to make Skolemization unnecessary. Moreover, CL has a constructive proof system based on forward reasoning, which is easy to automate and where standardized proof objects can easily be obtained. We have implemented in Prolog a CL prover which generates Coq proof scripts. We test our approach with a case study: Hessenberg’s Theorem, which states that in elementary projective plane geometry Pappus’ Axiom implies Desargues’ Axiom. Our CL prover makes it possible to automate large parts of the proof, in particular taking care of the large number of degenerate cases.