Proving Termination of Graph Transformation Systems using Weighted Type Graphs over Semirings

We introduce techniques for proving uniform termination of graph transformation systems, based on matrix interpretations for string rewriting. We generalize this technique by adapting it to graph rewriting instead of string rewriting and by generalizing to ordered semirings. In this way we obtain a framework which includes the tropical and arctic type graphs introduced in a previous paper and a new variant of arithmetic type graphs. These type graphs can be used to assign weights to graphs and to show that these weights decrease in every rewriting step in order to prove termination. We present an example involving counters and discuss the implementation in the tool Grez

Kruskal's Tree Theorem for Acyclic Term Graphs

In this paper we study termination of term graph rewriting, where we restrict our attention to acyclic term graphs. Motivated by earlier work by Plump we aim at a definition of the notion of simplification order for acyclic term graphs. For this we adapt the homeomorphic embedding relation to term graphs. In contrast to earlier extensions, our notion is inspired by morphisms. Based on this, we establish a variant of Kruskal's Tree Theorem formulated for acyclic term graphs. In proof, we rely on the new notion of embedding and follow Nash-Williams' minimal bad sequence argument. Finally, we propose a variant of the lexicographic path order for acyclic term graphs.Comment: In Proceedings TERMGRAPH 2016, arXiv:1609.0301

Coalgebraic Trace Semantics for Continuous Probabilistic Transition Systems

Coalgebras in a Kleisli category yield a generic definition of trace semantics for various types of labelled transition systems. In this paper we apply this generic theory to generative probabilistic transition systems, short PTS, with arbitrary (possibly uncountable) state spaces. We consider the sub-probability monad and the probability monad (Giry monad) on the category of measurable spaces and measurable functions. Our main contribution is that the existence of a final coalgebra in the Kleisli category of these monads is closely connected to the measure-theoretic extension theorem for sigma-finite pre-measures. In fact, we obtain a practical definition of the trace measure for both finite and infinite traces of PTS that subsumes a well-known result for discrete probabilistic transition systems. Finally we consider two example systems with uncountable state spaces and apply our theory to calculate their trace measures

Complexity of Acyclic Term Graph Rewriting

Term rewriting has been used as a formal model to reason about the complexity of logic, functional, and imperative programs. In contrast to term rewriting, term graph rewriting permits sharing of common sub-expressions, and consequently is able to capture more closely reasonable implementations of rule based languages. However, the automated complexity analysis of term graph rewriting has received little to no attention. With this work, we provide first steps towards overcoming this situation. We present adaptions of two prominent complexity techniques from term rewriting, viz, the interpretation method and dependency tuples. Our adaptions are non-trivial, in the sense that they can observe not only term but also graph structures, i.e. take sharing into account. In turn, the developed methods allow us to more precisely estimate the runtime complexity of programs where sharing of sub-expressions is essential

Stable states of perturbed Markov chains

Given an infinitesimal perturbation of a discrete-time finite Markov chain, we seek the states that are stable despite the perturbation, \textit{i.e.} the states whose weights in the stationary distributions can be bounded away from $0$ as the noise fades away. Chemists, economists, and computer scientists have been studying irreducible perturbations built with exponential maps. Under these assumptions, Young proved the existence of and computed the stable states in cubic time. We fully drop these assumptions, generalize Young's technique, and show that stability is decidable as long as $f\in O(g)$ is. Furthermore, if the perturbation maps (and their multiplications) satisfy $f\in O(g)$ or $g\in O(f)$, we prove the existence of and compute the stable states and the metastable dynamics at all time scales where some states vanish. Conversely, if the big-$O$ assumption does not hold, we build a perturbation with these maps and no stable state. Our algorithm also runs in cubic time despite the general assumptions and the additional work. Proving the correctness of the algorithm relies on new or rephrased results in Markov chain theory, and on algebraic abstractions thereof

Tools and Algorithms for the Construction and Analysis of Systems

This open access two-volume set constitutes the proceedings of the 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2021, which was held during March 27 – April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The total of 41 full papers presented in the proceedings was carefully reviewed and selected from 141 submissions. The volume also contains 7 tool papers; 6 Tool Demo papers, 9 SV-Comp Competition Papers. The papers are organized in topical sections as follows: Part I: Game Theory; SMT Verification; Probabilities; Timed Systems; Neural Networks; Analysis of Network Communication. Part II: Verification Techniques (not SMT); Case Studies; Proof Generation/Validation; Tool Papers; Tool Demo Papers; SV-Comp Tool Competition Papers