Shortest Paths in One-Counter Systems

Chistikov, Dmitry; Czerwiński, Wojciech; Hofman, Piotr; Pilipczuk, Michał; Wehar, Michael

doi:10.1007/978-3-662-49630-5_27

Cited by 12 publications

(7 citation statements)

References 18 publications

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“…Consider d ∈ 2 O(|A| 4 ) due to Lemma 3.5. A standard argument in OCA with n states is that, for reachability, it is actually sufficient to consider runs up to some counter value in O(n 3 ) [8,26]. We can apply the same argument here to deduce, together with Lemma 3.5, that parameter and counter values can be bounded by d + O(|Q| 3 ).…”

Section: Lemma 33 There Is An A2amentioning

confidence: 86%

The Complexity of Flat Freeze LTL

Bollig,

Quaas,

Sangnier

2016

Preprint

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We consider the model-checking problem for freeze LTL on one-counter automata (OCA). Freeze LTL extends LTL with the freeze quantifier, which allows one to store different counter values of a run in registers so that they can be compared with one another. As the model-checking problem is undecidable in general, we focus on the flat fragment of freeze LTL, in which the usage of the freeze quantifier is restricted. In a previous work, Lechner et al. showed that model checking for flat freeze LTL on OCA with binary encoding of counter updates is decidable and in 2NEXPTIME. In this paper, we prove that the problem is, in fact, NEXPTIME-complete no matter whether counter updates are encoded in unary or binary. Like Lechner et al., we rely on a reduction to the reachability problem in OCA with parameterized tests (OCA(P)). The new aspect is that we simulate OCA(P) by alternating two-way automata over words. This implies an exponential upper bound on the parameter values that we exploit towards an NP algorithm for reachability in OCA(P) with unary updates. We obtain our main result as a corollary. As another application, relying on a reduction by Bundala and Ouaknine, we show that reachability in closed parametric timed automata with one parametric clock is in NEXPTIME (matching the known lower bound).

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Section: Lemma 33 There Is An A2amentioning

confidence: 86%

The Complexity of Flat Freeze LTL

Bollig,

Quaas,

Sangnier

2016

Preprint

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“…We can assume ∥Λ∥, k > 0 where k is the number of cycles in Λ. Consider any shortest admissible π ∈ Λ from 0 to 0, and let M def = 16∥Λ∥ 7 . First, we show that at all points visited by π where one coordinate is less than M, the other coordinate must be less than M ′ def = 969k ∥Λ∥ 13 = 19(3k)(17∥Λ∥ 7 )∥Λ∥ 6 ≥ 19(k + 2)(M + 1)∥Λ∥ 6 .…”

Section: Reachability Witnesses In Simple Linear Path Schemesmentioning

confidence: 99%

“…Motivated both by the unsettled status of the complexity of the general problem, and by the wide interest in classes of one-counter and two-counter automata (see e.g. [4,7,14]), the reachability problem for VASS of small fixed dimensions has attracted considerable attention. Deciding reachability of 1-VASS assuming unary encoding of numbers is NL-complete: the lower bound is inherited from directed graph reachability [36,Theorem 16.2] and the upper bound follows from a straightforward depumping argument [42].…”

Section: Introductionmentioning

confidence: 99%

The Reachability Problem for Two-Dimensional Vector Addition Systems with States

Blondin

EnglertMatthias

FinkelAlain

et al. 2021

J. ACM

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We prove that the reachability problem for two-dimensional vector addition systems with states is NL-complete or PSPACE-complete, depending on whether the numbers in the input are encoded in unary or binary. As a key underlying technical result, we show that, if a configuration is reachable, then there exists a witnessing path whose sequence of transitions is contained in a bounded language defined by a regular expression of pseudo-polynomially bounded length. This, in turn, enables us to prove that the lengths of minimal reachability witnesses are pseudo-polynomially bounded.

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“…In this situation, a valence automaton over H is equivalent to a one-counter automaton (OCA). It is folklore that an n-state OCA accepts a word of length m if and only if it does so with counter values at most O((mn) 2 ) [22]. We can thus compute in logspace a finite automaton for the language R = L (B) ∩ Σ ≤m .…”

Section: Intersection Under Bounded Context Switching (Bcsint(g H))mentioning

confidence: 99%

Bounded Context Switching for Valence Systems

Meyer,

Muskalla,

Zetzsche

2018

Preprint

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We study valence systems, finite-control programs over infinite-state memories modeled in terms of graph monoids. Our contribution is a notion of bounded context switching (BCS). Valence systems generalize pushdowns, concurrent pushdowns, and Petri nets. In these settings, our definition conservatively generalizes existing notions. The main finding is that reachability within a bounded number of context switches is in NP, independent of the memory (the graph monoid). Our proof is genuinely algebraic, and therefore contributes a new way to think about BCS. In addition, we exhibit a class of storage mechanisms for which BCS reachability belongs to P.

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Shortest Paths in One-Counter Systems

Cited by 12 publications

References 18 publications

The Complexity of Flat Freeze LTL

The Complexity of Flat Freeze LTL

The Reachability Problem for Two-Dimensional Vector Addition Systems with States

Bounded Context Switching for Valence Systems

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