Guarded Kleene Algebra with Tests (GKAT) is a variant of Kleene algebra which allows for reasoning about simple imperative programs, and which features a decision procedure for program equivalence in nearly linear time. In the current paper, we address the challenge of reasoning under assumptions about these programs. In particular, we develop a form of Hoare hypotheses, which allow modelling basic domain knowledge on pre- and post-conditions of uninterpreted basic programs, and which are well-developed for classical Kleene algebra but not yet for GKAT. We show that the resulting axiomatisation is sound and complete. We then extend Hoare hypotheses to the more general form of word hypotheses. Based on an automata-theoretic approach, we show that equivalence of GKAT under word hypotheses is as efficiently decidable as for plain GKAT.

Fixed points are a recurring theme in computer science and are often constructed as limits of suitably seeded fixed point iterations. We present the algebra of iterative constructions (AIC) - a purely algebraic approach to reasoning about fixed point iterations of continuous endomaps on complete lattices. AIC allows derivations of constructive fixed point theorems via equational logic and avoids explicit computations with indices. For example, F ◇ F^* ⊥ = ◇ F^* ⊥ states in AIC that sup_n Fⁿ (⊥) - a construction known from the Kleene fixed point theorem - is a fixed point of F. We demonstrate the applicability of AIC by providing algebraic proofs of several well- and less-well-known fixed point theorems: Among others, we prove the Tarski-Kantorovich principle - a generalization of the Kleene fixed point theorem - as well as a fixed point-theoretic generalization of k-induction - a technique used in software verification. We moreover present a novel fixed point theorem. It improves a recent generalization of the Tarski-Kantorovich principle due to Olszewski for obtaining pre- and postfixed points from lattice-theoretic limit inferiors and limit superiors through iterating an endomap on an arbitrary seed element: We identify sufficient continuity conditions on the endomaps so that these limits become proper fixed points. We have mechanized our algebra in Isabelle/HOL. Isabelle’s sledgehammer tool is able to find proofs of the above fixed point theorems fully automatically. Finally, we investigate the completeness of our axiomatization of AIC. We prove that our finite set of finitary axioms is (a) sound but incomplete for standard models of AIC (sequences of elements from a complete lattice) and that (b) a different finite set of infinitary axioms is complete. We also prove that infinitary axioms are unavoidable: there exists no complete axiomatization of standard models given by finitely many finitary axioms.

We present the first sound and complete axiomatization of infinite trace semantics for generative probabilistic transition systems. Our approach is categorical, and we build on recent results on proper functors over convex sets. At the core of our proof is a characterization of infinite traces as the final coalgebra of a functor over convex algebras. Somewhat surprisingly, our axiomatization of infinite trace semantics coincides with that of finite trace semantics, even though the techniques used in the completeness proof are significantly different.

Abstract We consider process algebras with branching parametrized by an equational theory $$\textsf{T}$$ T , and show that it is possible to axiomatize bisimilarity under certain conditions on $$\textsf{T}$$ T . Our proof abstracts an earlier argument due to Grabmayer and Fokkink (LICS’20), and yields new completeness theorems for skip-free process algebras with probabilistic (guarded) branching, while also covering existing completeness results.

We forge connections between the theory of fractal sets obtained as attractors of iterated function systems and process calculi. To this end, we reinterpret Milner's expressions for processes as contraction operators on a complete metric space. When the space is, for example, the plane, the denotations of fixed point terms correspond to familiar fractal sets. We give a sound and complete axiomatization of fractal equivalence, the congruence on terms consisting of pairs that construct identical self-similar sets in all interpretations. We further make connections to labelled Markov chains and to invariant measures. In all of this work, we use important results from process calculi. For example, we use Rabinovich's completeness theorem for trace equivalence in our own completeness theorem. In addition to our results, we also raise many questions related to both fractals and process calculi.

Abstract Guarded Kleene Algebra with Tests (GKAT) is a fragment of Kleene Algebra with Tests (KAT) that was recently introduced to reason efficiently about imperative programs. In contrast to KAT, GKAT does not have an algebraic axiomatization, but relies on an analogue of Salomaa’s axiomatization of Kleene Algebra. In this paper, we present an algebraic axiomatization and prove two completeness results for a large fragment of GKAT consisting of skip-free programs.

We introduce Probabilistic Guarded Kleene Algebra with Tests (ProbGKAT), an extension of GKAT that allows reasoning about uninterpreted imperative programs with probabilistic branching. We give its operational semantics in terms of special class of probabilistic automata. We give a sound and complete Salomaa-style axiomatisation of bisimilarity of ProbGKAT expressions. Finally, we show that bisimilarity of ProbGKAT expressions can be decided in $O(n^3 \log n)$ time via a generic partition refinement algorithm.

We develop a (co)algebraic framework to study a family of process calculi with monadic branching structures and recursion operators. Our framework features a uniform semantics of process terms and a complete axiomatisation of semantic equivalence. We show that there are uniformly defined fragments of our calculi that capture well-known examples from the literature like regular expressions modulo bisimilarity and guarded Kleene algebra with tests. We also derive new calculi for probabilistic and convex processes with an analogue of Kleene star.

There is a large amount of literature on the topic of covarieties, coequations and coequational specifications, dating back to the early seventies. Nevertheless, coequations have not (yet) emerged as an everyday practical specification formalism for computer scientists. In this review paper, we argue that this is partly due to the multitude of syntaxes for writing down coequations, which seems to have led to some confusion about what coequations are and what they are for. By surveying the literature, we identify four types of syntaxes: coequations-as-corelations, coequations-as-predicates, coequations-as-equations, and coequations-as-modal-formulas. We present each of these in a tutorial fashion, relate them to each other, and discuss their respective uses.

Guarded Kleene Algebra with Tests (GKAT) is an efficient fragment of KAT, as it allows for almost linear decidability of equivalence. In this paper, we study the (co)algebraic properties of GKAT. Our initial focus is on the fragment that can distinguish between unsuccessful programs performing different actions, by omitting the so-called early termination axiom. We develop an operational (coalgebraic) and denotational (algebraic) semantics and show that they coincide. We then characterize the behaviors of GKAT expressions in this semantics, leading to a coequation that captures the covariety of automata corresponding to these behaviors. Finally, we prove that the axioms of the reduced fragment are sound and complete w.r.t. the semantics, and then build on this result to recover a semantics that is sound and complete w.r.t. the full set of axioms.