## Reachability in Succinct and Parametric One-Counter Automata

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@MISC{Haase_reachabilityin,

author = {Christoph Haase and Stephan Kreutzer and Joël Ouaknine and James Worrell},

title = {Reachability in Succinct and Parametric One-Counter Automata},

year = {}

}

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### Abstract

Abstract. One-counter automata are a fundamental and widely-studied class of infinite-state systems. In this paper we consider one-counter automata with counter updates encoded in binary—which we refer to as the succinct encoding. It is easily seen that the reachability problem for this class of machines is in PSpace and is NP-hard. One of the main results of this paper is to show that this problem is in fact in NP, and is thus NP-complete. We also consider parametric one-counter automata, in which counter updates be integer-valued parameters. The reachability problem asks whether there are values for the parameters such that a final state can be reached from an initial state. Our second main result shows decidability of the reachability problem for parametric one-counter automata by reduction to existential Presburger arithmetic with divisibility. 1

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Citation Context ...add(−2 n ) v1 Fig. 1. The first observation is the following. Proposition 1. The reachability problem for one-counter automata is NP-hard. Proof. The proof is by reduction from the subset sum problem =-=[9]-=-. Recall that an instance of the latter consists of a set of positive integers S = {a1, a2, . . . , an} and a goal c, and the question asked is whether there exists a subset T ⊆ S such that ∑ T = c. T... |

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Citation Context ...hability certificate for (v, c) and (v ′ , c ′ ) is in NP. Proof. It can be checked in polynomial time whether f has any positive cycles or any negative cycles, e.g., using the Bellman-Ford algorithm =-=[6]-=-. If f has no positive cycles, then by Proposition 4(i) to show that f is a type-1 reachability certificate we need only guess a decomposition f = f0 + · · · + fn−1 such that ∑ j i=0 weight(fi) ≥ −c, ... |

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Citation Context ...uring machines [19], and there has been considerable interest in subclasses of counter machines for which reachability is decidable, such as Petri nets, one-counter automata and flat counter automata =-=[5, 18]-=-. As originally conceived by Minsky, counters are updated either by incrementation or decrementation instructions. However, for many applications of counter machines, including modelling computer prog... |

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92 | Multiple Counters Automata, Safety Analysis and Presburger Arithmetic
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Citation Context ...uring machines [19], and there has been considerable interest in subclasses of counter machines for which reachability is decidable, such as Petri nets, one-counter automata and flat counter automata =-=[5, 18]-=-. As originally conceived by Minsky, counters are updated either by incrementation or decrementation instructions. However, for many applications of counter machines, including modelling computer prog... |

57 | Programs with Lists are Counter Automata
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Citation Context ...ions. However, for many applications of counter machines, including modelling computer programs, it is natural to consider more general types of updates, such as adding integer constants to a counter =-=[3, 5, 16]-=- or adding integer parameters [4, 12]. Parametric automata are used in various synthesis problems, and to model open programs, whose behaviour depends on values input from the environment [2]. In [20]... |

21 |
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Citation Context ...er constants, where the latter are encoded in binary. We also show decidability of reachability for parametric one-counter automata by reduction to existential Presburger arithmetic with divisibility =-=[17]-=-. We defer consideration of the complexity of the latter problem to the full version of this paper.1.1 Related Work The verification literature contains a large body of work on decidability and compl... |

19 | Flat Parametric Counter Automata
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Citation Context ...counter machines, including modelling computer programs, it is natural to consider more general types of updates, such as adding integer constants to a counter [3, 5, 16] or adding integer parameters =-=[4, 12]-=-. Parametric automata are used in various synthesis problems, and to model open programs, whose behaviour depends on values input from the environment [2]. In [20] parameters are also used to model re... |

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Citation Context ...ions. However, for many applications of counter machines, including modelling computer programs, it is natural to consider more general types of updates, such as adding integer constants to a counter =-=[3, 5, 16]-=- or adding integer parameters [4, 12]. Parametric automata are used in various synthesis problems, and to model open programs, whose behaviour depends on values input from the environment [2]. In [20]... |

17 |
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Citation Context ...counter machines, including modelling computer programs, it is natural to consider more general types of updates, such as adding integer constants to a counter [3, 5, 16] or adding integer parameters =-=[4, 12]-=-. Parametric automata are used in various synthesis problems, and to model open programs, whose behaviour depends on values input from the environment [2]. In [20] parameters are also used to model re... |

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10 |
DP lower bounds for equivalence-checking and model-checking of one-counter automata
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(Show Context)
Citation Context ...w nondeterminism. Thus our Theorem 4 is incomparable with this open problem. Aside from reachability, similarity and bisimilarity for one-counter automata and one-counter nets have been considered in =-=[1, 13, 14]-=-, among others. For automata with more than one counter, other restrictions are required to recover decidability of the reachability problem: for example, flatness [5, 16] and reversal boundedness [11... |

7 | A solvable class of quadratic Diophantine equations with applications to verification of infinite state systems
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Citation Context ... 16] or adding integer parameters [4, 12]. Parametric automata are used in various synthesis problems, and to model open programs, whose behaviour depends on values input from the environment [2]. In =-=[20]-=- parameters are also used to model resources (e.g., time, memory, dollars) consumed by transitions. The reachability problem for parametric counter automata asks whether there exist values of the para... |

6 | Efficient Verification Algorithms for One-Counter Processes
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(Show Context)
Citation Context ...w nondeterminism. Thus our Theorem 4 is incomparable with this open problem. Aside from reachability, similarity and bisimilarity for one-counter automata and one-counter nets have been considered in =-=[1, 13, 14]-=-, among others. For automata with more than one counter, other restrictions are required to recover decidability of the reachability problem: for example, flatness [5, 16] and reversal boundedness [11... |

5 | K.Cerans. Simulation is decidable for one-counter nets (extended abstract
- Abdulla
- 1998
(Show Context)
Citation Context ...w nondeterminism. Thus our Theorem 4 is incomparable with this open problem. Aside from reachability, similarity and bisimilarity for one-counter automata and one-counter nets have been considered in =-=[1, 13, 14]-=-, among others. For automata with more than one counter, other restrictions are required to recover decidability of the reachability problem: for example, flatness [5, 16] and reversal boundedness [11... |

4 |
R.: The effects of bounding syntactic resources on Presburger LTL
- Demri, Gascon
- 2009
(Show Context)
Citation Context ...omplexity for various problems on restricted classes of counter automata. The work that is closest to our own is that of Demri and Gascon on model checking extensions of LTL over one-counter automata =-=[8]-=-. They consider automata with one integer-valued counter, with updates encoded in unary, and with sign tests on the counter. They show that reachability in this model is NL-complete. Determining the c... |

4 |
On the solvability of a class of diophantine equations and applications
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Citation Context ...14], among others. For automata with more than one counter, other restrictions are required to recover decidability of the reachability problem: for example, flatness [5, 16] and reversal boundedness =-=[11]-=-. Bozga, Iosif and Lakhnech [4] show decidability of the reachability problem for flat parametric counter automata with a single loop, by reduction to a decidable problem concerning quadratic diophant... |

2 |
Logiques pour la spécification et vérification, Mémoire d’habilitation, Université Paris 7
- DEMRI
(Show Context)
Citation Context ...n tests on the counter. They show that reachability in this model is NL-complete. Determining the complexity of reachability when updates are encoded in binary is posed as an open problem by Demri in =-=[7]-=-, Page 61, Problem 13. Since this last problem assumes an integer-valued counter with sign tests, it is more general than the one considered in our Theorem 1, and it remains open. Another work closely... |

1 |
On two-way finite automata with monotonic counters and quadratic diophantine equations
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- 2004
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Citation Context ... the existential fragment of Presburger arithmetic with divisibility. This last result has been used to show the decidability of certain problems concerning systems of quadratic Diophantine equations =-=[10, 11]-=-. We give a simple application of this kind below. Let {y1, . . . , yk} and {x1, . . . , xn} be disjoint sets of integer variables. For 1 ≤ i ≤ k let Ri denote the quadratic polynomial yiAi + Bi, wher... |