Abstract We consider several novel congruences on the signature of meadows with the aim to survey different notions of fractions. In particular we suggest a notion of “true fraction”.

Let Q_0 denote the rational numbers expanded to a ‘meadow’, that is, after taking its zero-totalized form (0^{−1}=0) as the preferred interpretation. In this paper, we consider ‘cancellation meadows’, i.e. meadows without proper zero divisors, such as Q_0 and prove a generic completeness result. We apply this result to cancellation meadows expanded with differentiation operators, the sign function, and with floor, ceiling and a signed variant of the square root, respectively. We give an equational axiomatization of these operators and thus obtain a finite basis for various expanded cancellation meadows.

Sequential propositional logic deviates from conventional propositional logic by taking into account that during the sequential evaluation of a propositional statement, atomic propositions may yield different Boolean values at repeated occurrences. We introduce “free valuations” to capture this dynamics of a propositional statement's environment. The resulting logic is phrased as an equationally specified algebra rather than in the form of proof rules, and is named “proposition algebra.” It is strictly more general than Boolean algebra to the extent that the classical connectives fail to be expressively complete in the sequential case. The four axioms for free valuation congruence are then combined with other axioms in order define a few more valuation congruences that gradually identify more propositional statements, up to static valuation congruence (which is the setting of conventional propositional logic). Proposition algebra is developed in a fashion similar to the process algebra ACP and the program algebra PGA, via an algebraic specification which has a meaningful initial algebra for which a range of coarser congruences are considered important as well. In addition, infinite objects (i.e., propositional statements, processes and programs respectively) are dealt with by means of an inverse limit construction which allows the transfer of knowledge concerning finite objects to facts about infinite ones while reducing all facts about infinite objects to an infinity of facts about finite ones in return.

Threads as contained in a thread algebra are used for the modeling of sequential program behavior. A thread that may use a counter to control its execution is called a ‘one-counter thread’. In this paper the decidability of risk assessment (a certain form of action forecasting) for one-counter threads is proved. This relates to Cohen’s impossibility result on virus detection (Comput. Secur. 6(1), 22–35, 1984 ). Our decidability result follows from a general property of the traces of one-counter threads: if a state is reachable from some initial state, then it is also reachable along a path in which all counter values stay below a fixed bound that depends only on the initial and final counter value. A further consequence is that the reachability of a state is decidable. These properties are based on a result for ω -one counter machines by Rosier and Yen (SIAM J. Comput. 16(5), 779–807, 1987 ).

Threads as contained in a thread algebra emerge from the behavioral abstraction from programs in an appropriate program algebra. Threads may make use of services such as stacks, and a thread using a single stack is called a pushdown thread. Equivalence of pushdown threads is shown decidable whereas pushdown thread inclusion is undecidable. This is again an example of a borderline crossing where the equivalence problem is decidable, whereas the inclusion problem is not.

We study extensions of the process algebra axiom system ACP with two recursive operations: the binary Kleene star * , which is defined by x * y = x ( x * y + y , and the push-down operation $ , defined by x $ y = x (( x $ y )( x $ y )) + y . In this setting it is easy to represent register machine computation, and an equational theory results that is not decidable. In order to increase the expressive power, abstraction is then added: with rooted branching bisimulation equivalence each computable process can be expressed, and with rooted ô-bisimilarity each semi-computable process that initially is finitely branching can be expressed. Moreover, with abstraction and a finite number of auxiliary actions these results can be obtained without binary Kleene star. Finally, we consider two alternatives for the push-down operation. Each of these gives rise to similar results.

Abstract A process is called computable if it can be modelled by a transition system that has a recursive structure—implying finite branching. The equivalence relation between transition systems considered is strong bisimulation equivalence. The transition systems studied in this paper can be associated to processes specified in common specification languages such as CCS, LOTOS, ACP and PSF. As a means for defining transition systems up to bisimulation equivalence, the specification language μ CRL is used. Two simple fragments of, μ CRL are singled out, yielding universal expressivity with respect to recursive and primitive recursive transition systems. For both these domains the following properties are classified in the arithmetical hierarchy: bisimilarity, perpetuity (both ∏ 1 0 ), regularity (having a bisimilar, finite representation, Σ 2 0 ), acyclic regularity (Σ 1 0 ), and deadlock freedom (distinguishing deadlock from successful termination, ∏ 1 0 ). Finally, it is shown that in the domain of primitive recursive transition systems over a fixed, finite label set, a genuine hierarchy in bisimilarity can be defined by the complexity of the witnessing relations, which extends r.e. bisimilarity. Hence, primitive recursive transition systems already form an interesting class.

We introduce iteration in process algebra by means of (the original, binary version of) Kleene's star operation: x * y is the process that chooses between x and y, and upon termination of x has this choice again. We add this operation to a whole range of process algebra axiom systems, starting from BPA (Basic Process Algebra). In the case of the most complex system under consideration, ACP τ , every regular process can be defined with handshaking (two-party communication) and auxiliary actions. Next we introduce nesting in process algebra: x#y is defined by the equation x#y=x(x#y)x+y

Abstract We extend process algebra with guards, comparable to the guards in guarded commands or conditions in common programming constructs such as ‘if — then — else — fi’ and ‘while — do — od’. The extended language is provided with an operational semantics based on transitions between pairs of a process and a (data-)state. The data-states are given by a data environment that also defines in which data-states guards hold and how atomic actions (non-deterministically) transform these states. The operational semantics is studied modulo strong bisimulation equivalence. For basic process algebra (without operators for parallelism) we present a small axiom system that is complete with respect to a general class of data environments. Given a particular data environmentL we add three axioms to this system, which is then again complete, provided weakest preconditions are expressible andL is sufficiently deterministic. Then we study process algebra with parallelism and guards. A two phase-calculus is provided that makes it possible to prove identities between parallel processes. Also this calculus is complete. In the last section we show that partial correctness formulas can easily be expressed in this setting. We use process algebra with guards to prove the soundness of a Hoare logic for linear processes by translating proofs in Hoare logic into proofs in process algebra.