Runtime verification for biochemical programs

The biochemical paradigm is well-suited for modelling autonomous systems and new programming languages are emerging from this approach. However, in order to validate such programs, we need to define precisely their semantics and to provide verification techniques. In this paper, we consider a higher-order biochemical calculus that models the structure of system states and its dynamics thanks to rewriting abstractions, namely rules and strategies. We extend this calculus with a runtime verification technique in order to perform automatic discovery of property satisfaction failure. The property specification language is a subclass of LTL safety and liveness properties

Deciding the First Levels of the Modal mu Alternation Hierarchy by Formula Construction

We construct, for any sentence of the modal mu calculus Psi, derived sentences in the modal fragment and the fragment without least fixpoints of the modal mu calculus such that Psi is equivalent to a formula in these fragments if and only if it is equivalent to these formulas. The formula without greatest fixpoints that Psi is equivalent to if and only if it is equivalent to any formula without greatest fixpoint is obtained by duality. This yields a constructive proof of decidability of the first levels of the modal mu alternation hierarchy. The blow-up incurred by turning Psi into the modal formula is shown to be necessary: there are modal formulas that can be expressed sub-exponentially more efficiently with the use of fixpoints. For the fragments with only greatest or least fixpoints however, as long as formulas are in disjunctive form, the transformation into a formula syntactically in these fragments does not increase the size of the formula

Coinduction in Flow: The Later Modality in Fibrations

This paper provides a construction on fibrations that gives access to the so-called later modality, which allows for a controlled form of recursion in coinductive proofs and programs. The construction is essentially a generalisation of the topos of trees from the codomain fibration over sets to arbitrary fibrations. As a result, we obtain a framework that allows the addition of a recursion principle for coinduction to rather arbitrary logics and programming languages. The main interest of using recursion is that it allows one to write proofs and programs in a goal-oriented fashion. This enables easily understandable coinductive proofs and programs, and fosters automatic proof search. Part of the framework are also various results that enable a wide range of applications: transportation of (co)limits, exponentials, fibred adjunctions and first-order connectives from the initial fibration to the one constructed through the framework. This means that the framework extends any first-order logic with the later modality. Moreover, we obtain soundness and completeness results, and can use up-to techniques as proof rules. Since the construction works for a wide variety of fibrations, we will be able to use the recursion offered by the later modality in various context. For instance, we will show how recursive proofs can be obtained for arbitrary (syntactic) first-order logics, for coinductive set-predicates, and for the probabilistic modal mu-calculus. Finally, we use the same construction to obtain a novel language for probabilistic productive coinductive programming. These examples demonstrate the flexibility of the framework and its accompanying results

Disjunctive form and the modal μ\mu alternation hierarchy

This paper studies the relationship between disjunctive form, a syntactic normal form for the modal mu calculus, and the alternation hierarchy. First it shows that all disjunctive formulas which have equivalent tableau have the same syntactic alternation depth. However, tableau equivalence only preserves alternation depth for the disjunctive fragment: there are disjunctive formulas with arbitrarily high alternation depth that are tableau equivalent to alternation-free non-disjunctive formulas. Conversely, there are non-disjunctive formulas of arbitrarily high alternation depth that are tableau equivalent to disjunctive formulas without alternations. This answers negatively the so far open question of whether disjunctive form preserves alternation depth. The classes of formulas studied here illustrate a previously undocumented type of avoidable syntactic complexity which may contribute to our understanding of why deciding the alternation hierarchy is still an open problem.Comment: In Proceedings FICS 2015, arXiv:1509.0282