ing WS1S Systems to Verify Parameterized Networks . . . . . . . . . . . . 188 Kai Baukus, Saddek Bensalem, Yassine Lakhnech and Karsten Stahl FMona: A Tool for Expressing Validation Techniques over Infinite State Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 J.-P. Bodeveix and M. Filali Transitive Closures of Regular Relations for Verifying Infinite-State Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Bengt Jonsson and Marcus Nilsson Diagnostic and Test Generation Using Static Analysis to Improve Automatic Test Generation . . . . . . . . . . . . . 235 Marius Bozga, Jean-Claude Fernandez and Lucian Ghirvu Efficient Diagnostic Generation for Boolean Equation Systems . . . . . . . . . . . . 251 Radu Mateescu Efficient Model-Checking Compositional State Space Generation with Partial Order Reductions for Asynchronous Communicating Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 Jean-Pierre Krimm and Laurent Mounier Checking for CFFD-Preorder with Tester Processes . . . . . . . . . . . . . . . . . . . . . . . 283 Juhana Helovuo and Antti Valmari Fair Bisimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Thomas A. Henzinger and Sriram K. Rajamani Integrating Low Level Symmetries into Reachability Analysis . . . . . . . . . . . . . 315 Karsten Schmidt Model-Checking Tools Model Checking Support for the ASM High-Level Language . . . . . . . . . . . . . . 331 Giuseppe Del Castillo and Kirsten Winter Table of

Tools and Algorithms for the Construction and Analysis of Systems. Proc. TACAS 2009

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Hilbert's tenth problem

More than 30 years after their inception, the decidability proofs for reach ability in vector addition systems (VAS) still retain much of their mystery. These proofs rely crucially on a decomposition of runs successively refined by Mayr, Kosaraju, and Lambert, which appears rather magical, and for which no complexity upper bound is known. We first offer a justification for this decomposition technique, by showing that it computes the ideal decomposition of the set of runs, using the natural embedding relation between runs as well quasi ordering. In a second part, we apply recent results on the complexity of termination thanks to well quasi orders and well orders to obtain a cubic Ackermann upper bound for the decomposition algorithms, thus providing the first known upper bounds for general VAS reach ability.

Demystifying Reachability in Vector Addition Systems

Review: K. A. Mihajlova, (Problema vhozdenia did pramyh proizvedenij grupp):The Occurrence Problem for Direct Products of Groups

1. Preliminaries from Theoretical Computer Science.- 2. Preliminaries from Combinatorial Group Theory.- 3. Algorithms on Compressed Words.- 4. The Compressed Word Problem.- 5. The Compressed Word Problem in Graph Products.- 6. The Compressed Word Problem in HNN-Extensions.- 7.Outlook.- References.- Index.

/pdf/the-compressed-word-problem-for-groups-7gfb07qu30.pdf

The Compressed Word Problem for Groups

It is shown that the knapsack problem (introduced by Myasnikov, Nikolaev, and Ushakov) is undecidable in a direct product of sufficiently many copies of the discrete Heisenberg group (which is nilpotent of class 2). Moreover, for the discrete Heisenberg group itself, the knapsack problem is decidable. Hence, decidability of the knapsack problem is not preserved under direct products. It is also shown that for every co-context-free group, the knapsack problem is decidable. For the subset sum problem (also introduced by Myasnikov, Nikolaev, and Ushakov) we show that it belongs to the class NL (nondeterministic logspace) for every finitely generated virtually nilpotent group and that there exists a polycyclic group with an NP-complete subset sum problem.

Knapsack and subset sum problems in nilpotent, polycyclic, and co-context-free groups

We consider first-order logic over the subword ordering on finite words where each word is available as a constant. Our first result is that the Σ 1 theory is undecidable (already over two letters). We investigate the decidability border by considering fragments where all but a certain number of variables are alternation bounded, meaning that the variable must always be quantified over languages with a bounded number of letter alternations. We prove that when at most two variables are not alternation bounded, the Σ 1 fragment is decidable, and that it becomes undecidable when three variables are not alternation bounded. Regarding higher quantifier alternation depths, we prove that the Σ 2 fragment is undecidable already for one variable without alternation bound and that when all variables are alternation bounded, the entire first-order theory is decidable.

Decidability, complexity, and expressiveness of first-order logic over the subword ordering

The downward closure of a word language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of any language is regular. While the downward closure appears to be a powerful abstraction, algorithms for computing a finite automaton for the downward closure of a given language have been established only for few language classes. 
This work presents a simple general method for computing downward closures. For language classes that are closed under rational transductions, it is shown that the computation of downward closures can be reduced to checking a certain unboundedness property. 
This result is used to prove that downward closures are computable for (i) every language class with effectively semilinear Parikh images that are closed under rational transductions, (ii) matrix languages, and (iii) indexed languages (equivalently, languages accepted by higher-order pushdown automata of order 2).

An approach to computing downward closures

The downward closure of a word language is the set of all not necessarily contiguous subwords of its members It is well-known that the downward closure of any language is regular While the downward closure appears to be a powerful abstraction, algorithms for computing a finite automaton for the downward closure of a given language have been established only for few language classes

This work presents a simple general method for computing downward closures For language classes that are closed under rational transductions, it is shown that the computation of downward closures can be reduced to checking a certain unboundedness property

This result is used to prove that downward closures are computable for i every language class with effectively semilinear Parikh images that are closed under rational transductions, ii matrix languages, and iii indexed languages equivalently, languages accepted by higher-order pushdown automata of orderi¾?2

An Approach to Computing Downward Closures

It is shown that the knapsack problem, which was introduced by Myasnikov et al. for arbitrary finitely generated groups, can be solved in NP for every graph group. This result even holds if the group elements are represented in a compressed form by so called straight-line programs, which generalizes the classical NP-completeness result of the integer knapsack problem. If group elements are represented explicitly by words over the generators, then knapsack for a graph group belongs to the class LogCFL (a subclass of P) if the graph group can be built up from the trivial group using the operations of free product and direct product with $\mathbb {Z}$
 . In all other cases, the knapsack problem is NP-complete.

Georg Zetzsche

Papers

Knapsack and subset sum problems in nilpotent, polycyclic, and co-context-free groups

Decidability, complexity, and expressiveness of first-order logic over the subword ordering

An approach to computing downward closures

An Approach to Computing Downward Closures

Knapsack in Graph Groups