Demonic Lattices and Semilattices in Relational Semigroups with Ordinary Composition

Relation algebra and its reducts provide us with a strong tool for reasoning about nondeterministic programs and their partial correctness. Demonic calculus, introduced to model the behaviour of a machine where the demon is in control of nondeterminism, has also provided us with an extension of that reasoning to total correctness.We formalise the framework for relational reasoning about total correctness in nondeterministic programs using semigroups with ordinary composition and demonic lattice operations. We show that the class of representable demonic join semigroups is not finitely axiomatisable and that the representation class of demonic meet semigroups does not have the finite representation property for its finite members.For lattice semigroups (with composition, demonic join and demonic meet) we show that the representation problem for finite algebras is undecidable, moreover the finite representation problem is also undecidable. It follows that the representation class is not finitely axiomatisable, furthermore the finite representation property fails

Constructing programs or processes

We define interacting sequential programs, motivated originally by constructivist considerations. We use them to investigate notions of implementation and determinism. Process algebras do not define what can be implemented and what cannot. As we demonstrate it is problematic to do so on the set of all processes. Guided by constructivist notions we have constructed interacting sequential programs which we claim can be readily implemented and are a subset of processes

Classes of Terminating Logic Programs

Termination of logic programs depends critically on the selection rule, i.e. the rule that determines which atom is selected in each resolution step. In this article, we classify programs (and queries) according to the selection rules for which they terminate. This is a survey and unified view on different approaches in the literature. For each class, we present a sufficient, for most classes even necessary, criterion for determining that a program is in that class. We study six classes: a program strongly terminates if it terminates for all selection rules; a program input terminates if it terminates for selection rules which only select atoms that are sufficiently instantiated in their input positions, so that these arguments do not get instantiated any further by the unification; a program local delay terminates if it terminates for local selection rules which only select atoms that are bounded w.r.t. an appropriate level mapping; a program left-terminates if it terminates for the usual left-to-right selection rule; a program exists-terminates if there exists a selection rule for which it terminates; finally, a program has bounded nondeterminism if it only has finitely many refutations. We propose a semantics-preserving transformation from programs with bounded nondeterminism into strongly terminating programs. Moreover, by unifying different formalisms and making appropriate assumptions, we are able to establish a formal hierarchy between the different classes.Comment: 50 pages. The following mistake was corrected: In figure 5, the first clause for insert was insert([],X,[X]

Stochastic Invariants for Probabilistic Termination

Termination is one of the basic liveness properties, and we study the termination problem for probabilistic programs with real-valued variables. Previous works focused on the qualitative problem that asks whether an input program terminates with probability~1 (almost-sure termination). A powerful approach for this qualitative problem is the notion of ranking supermartingales with respect to a given set of invariants. The quantitative problem (probabilistic termination) asks for bounds on the termination probability. A fundamental and conceptual drawback of the existing approaches to address probabilistic termination is that even though the supermartingales consider the probabilistic behavior of the programs, the invariants are obtained completely ignoring the probabilistic aspect. In this work we address the probabilistic termination problem for linear-arithmetic probabilistic programs with nondeterminism. We define the notion of {\em stochastic invariants}, which are constraints along with a probability bound that the constraints hold. We introduce a concept of {\em repulsing supermartingales}. First, we show that repulsing supermartingales can be used to obtain bounds on the probability of the stochastic invariants. Second, we show the effectiveness of repulsing supermartingales in the following three ways: (1)~With a combination of ranking and repulsing supermartingales we can compute lower bounds on the probability of termination; (2)~repulsing supermartingales provide witnesses for refutation of almost-sure termination; and (3)~with a combination of ranking and repulsing supermartingales we can establish persistence properties of probabilistic programs. We also present results on related computational problems and an experimental evaluation of our approach on academic examples.Comment: Full version of a paper published at POPL 2017. 20 page