arXiv subjects

Tobias Kappé

Publications and source records attributed to Tobias Kappé.

30 records · Page 2Linked to original sources

arXiv2020

Partially Observable Concurrent Kleene Algebra

We introduce partially observable concurrent Kleene algebra (POCKA), an algebraic framework to reason about concurrent programs with control structures, such as conditionals and loops. POCKA enables reasoning about programs that can access variables and values, which we illustrate through concrete examples. We prove that POCKA is a sound and complete axiomatisation of a model of partial observations, and show the semantics passes an important check for sequential consistency.

cs.LO↗

arXiv2020

Concurrent Kleene Algebra with Observations: from Hypotheses to Completeness

Concurrent Kleene Algebra (CKA) extends basic Kleene algebra with a parallel composition operator, which enables reasoning about concurrent programs. However, CKA fundamentally misses tests, which are needed to model standard programming constructs such as conditionals and $\mathsf{while}$-loops. It turns out that integrating tests in CKA is subtle, due to their interaction with parallelism. In this paper we provide a solution in the form of Concurrent Kleene Algebra with Observations (CKAO). Our main contribution is a completeness theorem for CKAO. Our result resorts on a more general study of CKA "with hypotheses", of which CKAO turns out to be an instance: this analysis is of independent interest, as it can be applied to extensions of CKA other than CKAO.

cs.LO↗

arXiv2019

Guarded Kleene Algebra with Tests: Verification of Uninterpreted Programs in Nearly Linear Time

Guarded Kleene Algebra with Tests (GKAT) is a variation on Kleene Algebra with Tests (KAT) that arises by restricting the union ($+$) and iteration ($*$) operations from KAT to predicate-guarded versions. We develop the (co)algebraic theory of GKAT and show how it can be efficiently used to reason about imperative programs. In contrast to KAT, whose equational theory is PSPACE-complete, we show that the equational theory of GKAT is (almost) linear time. We also provide a full Kleene theorem and prove completeness for an analogue of Salomaa's axiomatization of Kleene Algebra.

cs.LO↗

arXiv2019

Kleene Algebra with Observations

Kleene algebra with tests (KAT) is an algebraic framework for reasoning about the control flow of sequential programs. Generalising KAT to reason about concurrent programs is not straightforward, because axioms native to KAT in conjunction with expected axioms for concurrency lead to an anomalous equation. In this paper, we propose Kleene algebra with observations (KAO), a variant of KAT, as an alternative foundation for extending KAT to a concurrent setting. We characterise the free model of KAO, and establish a decision procedure w.r.t. its equational theory.

cs.LO↗

arXiv2019

Completeness and Incompleteness of Synchronous Kleene Algebra

Synchronous Kleene algebra (SKA), an extension of Kleene algebra (KA), was proposed by Prisacariu as a tool for reasoning about programs that may execute synchronously, i.e., in lock-step. We provide a countermodel witnessing that the axioms of SKA are incomplete w.r.t. its language semantics, by exploiting a lack of interaction between the synchronous product operator and the Kleene star. We then propose an alternative set of axioms for SKA, based on Salomaa's axiomatisation of regular languages, and show that these provide a sound and complete characterisation w.r.t. the original language semantics.

cs.LO↗

arXiv2019

Tree Automata as Algebras: Minimisation and Determinisation

We study a categorical generalisation of tree automata, as $Σ$-algebras for a fixed endofunctor $Σ$ endowed with initial and final states. Under mild assumptions about the base category, we present a general minimisation algorithm for these automata. We then build upon and extend an existing generalisation of the Nerode equivalence to a categorical setting and relate it to the existence of minimal automata. Finally, we show that generalised types of side-effects, such as non-determinism, can be captured by this categorical framework, leading to a general determinisation procedure.

cs.FL↗

arXiv2018

On Series-Parallel Pomset Languages: Rationality, Context-Freeness and Automata

Concurrent Kleene Algebra (CKA) is a formalism to study concurrent programs. Like previous Kleene Algebra extensions, developing a correspondence between denotational and operational perspectives is important, for both foundations and applications. This paper takes an important step towards such a correspondence, by precisely relating bi-Kleene Algebra (BKA), a fragment of CKA, to a novel type of automata, pomset automata (PAs). We show that PAs can implement the BKA semantics of series-parallel rational expressions, and that a class of PAs can be translated back to these expressions. We also characterise the behavior of general PAs in terms of context-free pomset grammars; consequently, universality, equivalence and series-parallel rationality of general PAs are undecidable.

cs.FL↗

arXiv2018

Decision problems for Clark-congruential languages

A common question when studying a class of context-free grammars is whether equivalence is decidable within this class. We answer this question positively for the class of Clark-congruential grammars, which are of interest to grammatical inference. We also consider the problem of checking whether a given CFG is Clark-congruential, and show that it is decidable given that the CFG is a DCFG.

cs.CL↗

arXiv2018

Concurrent Kleene Algebra: Free Model and Completeness

Concurrent Kleene Algebra (CKA) was introduced by Hoare, Moeller, Struth and Wehrman in 2009 as a framework to reason about concurrent programs. We prove that the axioms for CKA with bounded parallelism are complete for the semantics proposed in the original paper; consequently, these semantics are the free model for this fragment. This result settles a conjecture of Hoare and collaborators. Moreover, the techniques developed along the way are reusable; in particular, they allow us to establish pomset automata as an operational model for CKA.

cs.FL↗

arXiv2017

Brzozowski Goes Concurrent - A Kleene Theorem for Pomset Languages

Concurrent Kleene Algebra (CKA) is a mathematical formalism to study programs that exhibit concurrent behaviour. As with previous extensions of Kleene Algebra, characterizing the free model is crucial in order to develop the foundations of the theory and potential applications. For CKA, this has been an open question for a few years and this paper makes an important step towards an answer. We present a new automaton model and a Kleene-like theorem that relates a relaxed version of CKA to series-parallel pomset languages, which are a natural candidate for the free model. There are two substantial differences with previous work: from expressions to automata, we use Brzozowski derivatives, which enable a direct construction of the automaton; from automata to expressions, we provide a syntactic characterization of the automata that denote valid CKA behaviours.

cs.FL↗

arXiv2017

A Component-oriented Framework for Autonomous Agents

The design of a complex system warrants a compositional methodology, i.e., composing simple components to obtain a larger system that exhibits their collective behavior in a meaningful way. We propose an automaton-based paradigm for compositional design of such systems where an action is accompanied by one or more preferences. At run-time, these preferences provide a natural fallback mechanism for the component, while at design-time they can be used to reason about the behavior of the component in an uncertain physical world. Using structures that tell us how to compose preferences and actions, we can compose formal representations of individual components or agents to obtain a representation of the composed system. We extend Linear Temporal Logic with two unary connectives that reflect the compositional structure of the actions, and show how it can be used to diagnose undesired behavior by tracing the falsification of a specification back to one or more culpable components.

cs.LO↗

arXiv2016

A Compositional Framework for Preference-Aware Agents

A formal description of a Cyber-Physical system should include a rigorous specification of the computational and physical components involved, as well as their interaction. Such a description, thus, lends itself to a compositional model where every module in the model specifies the behavior of a (computational or physical) component or the interaction between different components. We propose a framework based on Soft Constraint Automata that facilitates the component-wise description of such systems and includes the tools necessary to compose subsystems in a meaningful way, to yield a description of the entire system. Most importantly, Soft Constraint Automata allow the description and composition of components' preferences as well as environmental constraints in a uniform fashion. We illustrate the utility of our framework using a detailed description of a patrolling robot, while highlighting methods of composition as well as possible techniques to employ them.

cs.LO↗