Uli Fahrenberg

Logic and languages of higher-dimensional automata

Abstract

In this paper we study finite higher-dimensional automata (HDAs) from the logical point of view. Languages of HDAs are sets of finite bounded-width interval pomsets with interfaces (iiPoms$\le k$) closed under order extension. We prove that languages of HDAs are MSO-definable. For the converse, we show that the order extensions of MSO-definable sets of iiPoms$\le k$ are languages of HDAs. Furthermore, both constructions are effective. As a consequence, unlike the case of all pomsets, the order extension of any MSO-definable set of iiPoms$\le k$ is MSO-definable.

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Presenting interval pomsets with interfaces

Abstract

Interval-order partially ordered multisets with interfaces (ipomsets) have shown to be a versatile model for executions of concur- rent systems in which both precedence and concurrency need to be taken into account. In this paper, we develop a presentation of ipomsets as generated by a graph of certain discrete ipomsets (starters and terminators) under the relation which composes subsequent starters and subsequent ter- minators. Using this presentation, we show that also subsumptions are generated by elementary relations. We develop a similar correspondence on the automata side, relating higher-dimensional automata, which gen- erate ipomsets, and ST-automata, which generate step sequences, and their respective languages.

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Myhill-Nerode theorem for higher-dimensional automata

Abstract

We establish a Myhill-Nerode type theorem for higher-dimensional automata (HDAs), stating that a language is regular if and only if it has finite prefix quotient. HDAs extend standard automata with additional structure, making it possible to distinguish between interleavings and concurrency. We also introduce deterministic HDAs and show that not all HDAs are determinizable, that is, there exist regular languages that cannot be recognised by a deterministic HDA. Using our theorem, we develop an internal characterisation of deterministic languages. Lastly, we develop analogues of the Myhill-Nerode construction and of determinacy for HDAs with interfaces.

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Closure and decision properties for higher-dimensional automata

Abstract

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Languages of higher-dimensional timed automata

Abstract

We present a new language semantics for real-time concurrency. Its operational models are higher-dimensional timed automata (HDTAs), a generalization of both higher-dimensional automata and timed automata. We define languages of HDTAs as sets of interval-timed pomsets with interfaces. As an application, we show that language inclusion of HDTAs is undecidable. On the other hand, using a region construction we can show that untimings of HDTA languages have enough regularity so that untimed language inclusion is decidable.

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A Myhill-Nerode theorem for higher-dimensional automata

Abstract

We establish a Myhill-Nerode type theorem for higher-dimensional automata (HDAs), stating that a language is regular precisely if it has finite prefix quotient. HDAs extend standard automata with additional structure, making it possible to distinguish between interleavings and concurrency. We also introduce deterministic HDAs and show that not all HDAs are determinizable, that is, there exist regular languages that cannot be recognised by a deterministic HDA. Using our theorem, we develop an internal characterisation of deterministic languages.

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Catoids and modal convolution algebras

Abstract

We show how modal quantales arise as convolution algebras $Q^X$ of functions from catoids $X$, that is, multisemigroups with a source map $\ell$ and a target map $r$, into modal quantales $Q$, which can be seen as weight or value algebras. In the tradition of boolean algebras with operators we study modal correspondences between algebraic laws in $X$, $Q$ and $Q^X$. The class of catoids we introduce generalises Schweizer and Sklar’s function systems and object-free categories to a setting isomorphic to algebras of ternary relations, as they are used for boolean algebras with operators and substructural logics. Our results provide a generic construction of weighted modal quantales from such multisemigroups. It is illustrated by many examples. We also discuss how these results generalise to a setting that supports reasoning with stochastic matrices or probabilistic predicate transformers.

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Energy problems in finite and timed automata with Büchi conditions

Abstract

We show how to efficiently solve energy Büchi problems in finite weighted automata and in one-clock weighted timed automata. Solving the former problem is our main contribution and is handled by a modified version of Bellman-Ford interleaved with Couvreur’s algorithm. The latter problem is handled via a reduction to the former relying on the corner-point abstraction. All our algorithms are freely available and implemented in a tool based on the open-source tools TChecker and Spot.

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Higher-dimensional timed and hybrid automata

Abstract

We introduce a new formalism of higher-dimensional timed automata, based on Pratt and van Glabbeek’s higher-dimensional automata and Alur and Dill’s timed automata. We prove that their reachability is PSPACE-complete and can be decided using zone-based algorithms. We also extend the setting to higher-dimensional hybrid automata. The interest of our formalism is in modeling systems which exhibit both real-time behavior and concurrency. Other existing formalisms for real-time modeling identify concurrency and interleaving, which, as we shall argue, is problematic.

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Introduction to the special issue on distributed hybrid systems

Abstract

This special issue contains seven papers within the broad subject of Distributed Hybrid Systems, that is, systems combining hybrid discrete-continuous state spaces with elements of concurrency and logical or spatial distribution. It follows up on several workshops on the same theme which were held between 2017 and 2019 and organized by the editors of this volume. The first of these workshops was held in Aalborg, Denmark, in August 2017 and associated with the MFCS conference. It featured invited talks by Alessandro Abate, Martin Fränzle, Kim G. Larsen, Martin Raussen, and Rafael Wisniewski. The second workshop was held in Palaiseau, France, in July 2018, with invited talks by Luc Jaulin, Thao Dang, Lisbeth Fajstrup, Emmanuel Ledinot, and André Platzer. The third workshop was held in Amsterdam, The Netherlands, in August 2019, associated with the CONCUR conference. It featured a special theme on distributed robotics and had invited talks by Majid Zamani, Hervé de Forges, and Xavier Urbain. The vision and purpose of the DHS workshops was to connect researchers working in real-time systems, hybrid systems, control theory, formal verification, distributed computing, and concurrency theory, in order to advance the subject of distributed hybrid systems. Such systems are abundant and often safety-critical, but ensuring their correct functioning can in general be challenging. The investigation of their dynamics by analysis tools from the aforementioned domains remains fragmentary, providing the rationale behind the workshops: it was conceived that convergence and interaction of theories, methods, and tools from these different areas was needed in order to advance the subject.

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