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Register a new userSimulation Problems Over One-Counter Nets
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Added by
Patrick Totzke
Publication date
2016
fields
Informatics Engineering
and research's language is
English
Authors
Piotr Hofman
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No Arabic abstract
One-counter nets (OCN) are finite automata equipped with a counter that can store non-negative integer values, and that cannot be tested for zero. Equivalently, these are exactly 1-dimensional vector addition systems with states. We show that both strong and weak simulation preorder on OCN are PSPACE-complete.
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One-counter nets (OCN) are Petri nets with exactly one unbounded place. They are equivalent to a subclass of one-counter automata with just a weak test for zero. Unlike many other semantic equivalences, strong and weak simulation preorder are decidable for OCN, but the computational complexity was an open problem. We show that both strong and weak simulation preorder on OCN are PSPACE-complete.
We study complexity of the model-checking problems for LTL with registers (also known as freeze LTL) and for first-order logic with data equality tests over one-counter automata. We consider several classes of one-counter automata (mainly deterministic vs. nondeterministic) and several logical fragments (restriction on the number of registers or variables and on the use of propositional variables for control locations). The logics have the ability to store a counter value and to test it later against the current counter value. We show that model checking over deterministic one-counter automata is PSPACE-complete with infinite and finite accepting runs. By constrast, we prove that model checking freeze LTL in which the until operator is restricted to the eventually operator over nondeterministic one-counter automata is undecidable even if only one register is used and with no propositional variable. As a corollary of our proof, this also holds for first-order logic with data equality tests restricted to two variables. This makes a difference with the facts that several verification problems for one-counter automata are known to be decidable with relatively low complexity, and that finitary satisfiability for the two logics are decidable. Our results pave the way for model-checking memoryful (linear-time) logics over other classes of operational models, such as reversal-bounded counter machines.
We study the (parameter) synthesis problem for one-counter automata with parameters. One-counter automata are obtained by extending classical finite-state automata with a counter whose value can range over non-negative integers and be tested for zero. The updates and tests applicable to the counter can further be made parametric by introducing a set of integer-valued variables called parameters. The synthesis problem for such automata asks whether there exists a valuation of the parameters such that all infinite runs of the automaton satisfy some omega-regular property. Lechner showed that (the complement of) the problem can be encoded in a restricted one-alternation fragment of Presburger arithmetic with divisibility. In this work (i) we argue that said fragment, called AERPADPLUS, is unfortunately undecidable. Nevertheless, by a careful re-encoding of the problem into a decidable restriction of AERPADPLUS, (ii) we prove that the synthesis problem is decidable in general and in N2EXP for several fixed omega-regular properties. Finally, (iii) we give a polynomial-space algorithm for the special case of the problem where parameters can only be used in tests, and not updates, of the counter.
We interpret Linear Logic Proof Nets in a term language based on Solos calculus. The system includes a synchronisation mechanism, obtained by a conservative extension of the logic, that enables to define non-deterministic behaviours and multiparty sessions.
We consider approaches for causal semantics of Petri nets, explicitly representing dependencies between transition occurrences. For one-safe nets or condition/event-systems, the notion of process as defined by Carl Adam Petri provides a notion of a run of a system where causal dependencies are reflected in terms of a partial order. A well-known problem is how to generalise this notion for nets where places may carry several tokens. Goltz and Reisig have defined such a generalisation by distinguishing tokens according to their causal history. However, this so-called individual token interpretation is often considered too detailed. A number of approaches have tackled the problem of defining a more abstract notion of process, thereby obtaining a so-called collective token interpretation. Here we give a short overview on these attempts and then identify a subclass of Petri nets, called structural conflict nets, where the interplay between conflict and concurrency due to token multiplicity does not occur. For this subclass, we define abstract processes as equivalence classes of Goltz-Reisig processes. We justify this approach by showing that we obtain exactly one maximal abstract process if and only if the underlying net is conflict-free with respect to a canonical notion of conflict.
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