Ahj Aad Mathijssen

Nominal (Universal) Algebra

In informal mathematical discourse (such as the text of a paper on theoretical computer science), we often reason about equalities involving binding of object-variables. We find ourselves writing assertions involving meta-variables and captureavoidance constraints on where object-variables can and cannot occur free. Formalizing such assertions is problematic because the standard logical frameworks cannot express capture-avoidance constraints directly. In this article, we make the case for extending the logic of equality with meta-variables and capture-avoidance constraints, to obtain ‘nominal algebra’. We use nominal techniques that allow for a direct formalization of meta-level assertions, while remaining close to informal practice. We investigate proof-theoretical properties, we provide a sound and complete semantics in nominal sets and we compare and contrast our design decisions with other possibilities leading to similar systems.

Capture-avoiding substitution as a nominal algebra

Substitution is fundamental to the theory of logic and computation. Is substitution something that we define on syntax on a case-by-case basis, or can we turn the idea of substitution into a mathematical object? We give axioms for substitution and prove them sound and complete with respect to a canonical model. As corollaries we obtain a useful conservativity result, and prove that equality-up-to-substitution is a decidable relation on terms. These results involve subtle use of techniques both from rewriting and algebra. A special feature of our method is the use of nominal techniques. These give us access to a stronger assertion language, which includes so-called ‘freshness’ or ‘capture-avoidance’ conditions. This means that the sense in which we axiomatise substitution (and prove soundness and completeness) is particularly strong, while remaining quite general.

One-and-a-halfth-order Logic

The practice of first-order logic is replete with meta-level concepts. Most notably there are meta-variables ranging over formulae, variables, and terms, and properties of syntax such as alpha-equivalence, capture-avoiding substitution and assumptions about freshness of variables with respect to meta-variables. We present one-and-a-halfth-order logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of one-and-a-halfth-order logic derivability, show them equivalent, show that the derivations satisfy cut-elimination, and prove correctness of an interpretation of first-order logic within it. We discuss the technicalities in a wider context as a case-study for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation for future implementation.

A formal calculus for informal equality with binding

In informal mathematical usage we often reason using languages with binding.We usually find ourselves placing capture-avoidance constraints on where variables can and cannot occur free. We describe a logical derivation system which allows a direct formalisation of such assertions, along with a direct formalisation of their constraints. We base our logic on equality, probably the simplest available judgement form. In spite of this, we can axiomatise systems of logic and computation such as first-order logic or the lambda-calculus in a very direct and natural way. We investigate the theory of derivations, prove a suitable semantics sound and complete, and discuss existing and future research.

From µCRL to mCRL2 : motivation and outline

We sketch the language mCRL2, the successor of μCRL, which is a process algebra with data, devised in 1990 to model and study the behaviour of interacting programs and systems. The language is improved in several respects guided by the experience obtained from numerous applications where realistic systems have been modelled and analysed. Just as with μCRL, the leading principle is to provide a minimal set of primitives that allow effective specifications, that conform to standard mathematics and that allow standard mathematical manipulations and proof methodologies. In the first place the equational abstract datatypes have been enhanced with higher-order constructs and standard data types, ranging from booleans, numbers and lists to sets, bags and higher-order function types. In the second place multi-actions have been introduced to allow a seamless integration with Petri nets. In the last place communication is made local to enable compositionality.

A Nominal Axiomatization of the Lambda Calculus

The lambda calculus is fundamental in computer science. It resists an algebraic treatment because of capture-avoidance sideconditions. Nominal algebra is a logic of equality designed for specifications involving binding. We axiomatize the lambda calculus using nominal algebra, demonstrate how proofs with these axioms reflect the informal arguments on syntax and we prove the axioms to be sound and complete. We consider both non-extensional and extensional versions (alpha-beta and alpha-beta-eta equivalence). This connects the nominal approach to names and binding with the view of variables as a syntactic convenience for describing functions. The axiomatization is finite, close to informal practice and it fits into a context of other research such as nominal rewriting and nominal sets.

One-and-a-halfth-order logic

The practice of first-order logic is replete with meta-level concepts. Most notably there are the meta-variables themselves (ranging over predicates, variables, and terms), assumptions about freshness of variables with respect to these meta-variables, alpha-equivalence and capture-avoiding substitution. We present one-and-a-halfth-order logic, in which these concepts are made explicit. We exhibit both algebraic and sequent specifications of one-and-a-halfth-order logic derivability, show them equivalent, show that the derivations satisfy cut-elimination, and prove correctness of an interpretation of first-order logic within itWe discuss the technicalities in a wider context as a case-study for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation for future implementation.

Behavioural Analysis of an I2C Linux Driver

Introduction . Formal methods for the analysis of system behaviour offer solutions to problems with concurrency, such as race conditions and deadlocks.We employ two such methods that are presently most applied in industry: model checking and static analysis on a common case study to analyse the behaviour of a Linux driver for I2C (Inter-Integrated Circuit). An industrial client provided us with the source code of the driver for which it was known that it contained defects. Based on the code, some documentation, and feedback by the developers we extracted a model of the device driver. The model was checked using the mCRL2 toolset [3] and some potential defects were revealed which were later confirmed by the developers. The errors were caused by inconsistent use of routines for interrupt enabling and disabling, resulting in unprotected references to shared memory and calls to lower-level functions. In addition, we performed checks with UNO [4], a static analysis tool that works directly with the source code. We employed UNO to statically detect the errors that were found by the dynamic analysis in the model checking phase. Based on our findings, we modified the source code to avoid the discovered potential defects. Although some errors remained unsolved, an improvement was observed in the standard tests that were carried out with our fixed version.