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

Demonic fixed points

We deal with a relational model for the demonic semantics of programs. The demonic semantics of a while loop is given as a fixed point of a function involving the demonic operators. This motivates us to investigate the fixed points of these functions. We give the expression of the greatest fixed point with respect to the demonic ordering (demonic inclusion) of the semantic function. We prove that this greatest fixed coincides with the least fixed point with respect to the usual ordering (angelic inclusion) of the same function. This is followed by an example of application

Axioms for signatures with domain and demonic composition

Demonic composition ∗ is an associative operation on binary relations, and demonic refinement ⊑ is a partial order on binary relations. Other operations on binary relations considered here include the unary domain operation D and the left restrictive multiplication operation ∘ given by s∘t=D(s)∗t. We show that the class of relation algebras of signature {⊑,D,∗}, or equivalently {⊆,∘,∗}, has no finite axiomatisation. A large number of other non-finite axiomatisability consequences of this result are also given, along with some further negative results for related signatures. On the positive side, a finite set of axioms is obtained for relation algebras with signature {⊑,∘,∗}, hence also for {⊆,∘,∗}

Nondeterministic Relational Semantics of a while Program

A relational semantics is a mapping of programs to relations. We consider that the input-output semantics of a program is given by a relation on its set of states; in a nondeterministic context, this relation is calculated by considering the worst behavior of the program (demonic relational semantics). In this paper, we concentrate on while loops. Calculating the relational abstraction (semantics) of a loop is difficult, but showing the correctness of any candidate abstraction is much easier. For functional programs, Mills has described a checking method known as the while statement verification rule. A programming theorem for iterative constructs is proposed, proved, demonstrated and applied for an example. This theorem can be considered as a generalization of the while statement verification to nondeterministic loops.&nbsp

Domain and Antidomain Semigroups

Abstract. We axiomatise and study operations for relational domain and antidomain on semigroups and monoids. We relate this approach with previous axiomatisations for semirings, partial transformation semi-groups and dynamic predicate logic.

Automatic Probabilistic Program Verification through Random Variable Abstraction

The weakest pre-expectation calculus has been proved to be a mature theory to analyze quantitative properties of probabilistic and nondeterministic programs. We present an automatic method for proving quantitative linear properties on any denumerable state space using iterative backwards fixed point calculation in the general framework of abstract interpretation. In order to accomplish this task we present the technique of random variable abstraction (RVA) and we also postulate a sufficient condition to achieve exact fixed point computation in the abstract domain. The feasibility of our approach is shown with two examples, one obtaining the expected running time of a probabilistic program, and the other the expected gain of a gambling strategy. Our method works on general guarded probabilistic and nondeterministic transition systems instead of plain pGCL programs, allowing us to easily model a wide range of systems including distributed ones and unstructured programs. We present the operational and weakest precondition semantics for this programs and prove its equivalence

Entailment systems for stably locally compact locales

The category SCFrU of stably continuous frames and preframe ho-momorphisms (preserving ¯nite meets and directed joins) is dual to the Karoubi envelope of a category Ent whose objects are sets and whose morphisms X ! Y are upper closed relations between the ¯nite powersets FX and FY . Composition of these morphisms is the \cut composition" of Jung et al. that interfaces disjunction in the codomains with conjunctions in the domains, and thereby relates to their multi-lingual sequent calculus. Thus stably locally compact locales are represented by \entailment systems" (X; `) in which `, a generalization of entailment relations,is idempotent for cut composition. Some constructions on stably locally compact locales are represented in terms of entailment systems: products, duality and powerlocales. Relational converse provides Ent with an involution, and this gives a simple treatment of the duality of stably locally compact locales. If A and B are stably continuous frames, then the internal preframe hom A t B is isomorphic to e A ­ B where e A is the Hofmann-Lawson dual. For a stably locally compact locale X, the lower powerlocale of X is shown to be the dual of the upper powerlocale of the dual of X

Modal Kleene algebra and applications - a survey

Modal Kleene algebras are Kleene algebras with forward and backward modal operators defined via domain and codomain operations. They provide a concise and convenient algebraic framework that subsumes various other calculi and allows treating quite a variety of areas. We survey the basic theory and some prominent applications. These include, on the system semantics side, Hoare logic and PDL (Propositional Dynamic Logic), wp calculus and predicate transformer semantics, temporal logics and termination analysis of rewrite and state transition systems. On the derivation side we apply the framework to game analysis and greedy-like algorithms