How Hard is It to Verify Flat Affine Counter Systems with the Finite Monoid Property?

Iosif, Radu

doi:10.1007/978-3-319-46520-3_6

Cited by 7 publications

(7 citation statements)

References 21 publications

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“…We do not know whether an exponential bound on M V holds for any class of afmp-Z-VASS over Z d×d . We are aware that an exponential upper bound holds when M V is generated by a single matrix [IS16]; and when M V is a group then we have an exponential bound but only on |M V | (see [KP02] for an exposition on the group case).…”

Section: Discussionmentioning

confidence: 99%

“…Note that the model of Finkel and Leroux does not allow for control-states and the usual tricks of encoding each control-state by a counter or all control-states into three counters [HP79] do not work over Z since transitions are non-blocking. Iosif and Sangnier [IS16] investigated the complexity of model checking problems for a variant of the model of Finkel and Leroux with guards defined by convex polyhedra and with control-states over a flat structure. Haase and Halfon [HH14] studied the complexity of the reachability, coverability and inclusion problems for Z-VASS and reset Z-VASS, two submodels of the affine Z-VASS that we study in this paper.…”

Section: Introductionmentioning

confidence: 99%

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Affine Extensions of Integer Vector Addition Systems with States

Blondin¹,

Haase²,

Mazowiecki³

et al. 2021

Logical Methods in Computer Science

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We study the reachability problem for affine $\mathbb{Z}$-VASS, which are integer vector addition systems with states in which transitions perform affine transformations on the counters. This problem is easily seen to be undecidable in general, and we therefore restrict ourselves to affine $\mathbb{Z}$-VASS with the finite-monoid property (afmp-$\mathbb{Z}$-VASS). The latter have the property that the monoid generated by the matrices appearing in their affine transformations is finite. The class of afmp-$\mathbb{Z}$-VASS encompasses classical operations of counter machines such as resets, permutations, transfers and copies. We show that reachability in an afmp-$\mathbb{Z}$-VASS reduces to reachability in a $\mathbb{Z}$-VASS whose control-states grow linearly in the size of the matrix monoid. Our construction shows that reachability relations of afmp-$\mathbb{Z}$-VASS are semilinear, and in particular enables us to show that reachability in $\mathbb{Z}$-VASS with transfers and $\mathbb{Z}$-VASS with copies is PSPACE-complete. We then focus on the reachability problem for affine $\mathbb{Z}$-VASS with monogenic monoids: (possibly infinite) matrix monoids generated by a single matrix. We show that, in a particular case, the reachability problem is decidable for this class, disproving a conjecture about affine $\mathbb{Z}$-VASS with infinite matrix monoids we raised in a preliminary version of this paper. We complement this result by presenting an affine $\mathbb{Z}$-VASS with monogenic matrix monoid and undecidable reachability relation.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Affine Extensions of Integer Vector Addition Systems with States

Blondin¹,

Haase²,

Mazowiecki³

et al. 2021

Logical Methods in Computer Science

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“…Our work is primarily related to the work of Finkel and Leroux [23], Iosif and Sangnier [27], Haase and Halfon [24], and Cadilhac, Finkel and McKenzie [12,13]. In [23], Finkel and Leroux consider a model more general than affine Z-VASS in which transitions are additionally equipped with guards which are Presburger formulas defining admissible sets of vectors in which a transition does not block.…”

Section: Related Workmentioning

confidence: 99%

“…Note that the model of Finkel and Leroux does not allow for control-states and the usual tricks of encoding each control-state by a counter or all control-states into three counters [25] do not work over Z since transitions are non-blocking. Iosif and Sangnier [27] investigated the complexity of model checking problems for a variant of the model of Finkel and Leroux with guards defined by convex polyhedra and with control-states over a flat structure. Haase and Halfon [24] studied the complexity of the reachability, coverability and inclusion problems for Z-VASS and reset Z-VASS, two submodels of the affine Z-VASS that we study in this paper.…”

Section: Related Workmentioning

confidence: 99%

Affine Extensions of Integer Vector Addition Systems with States

Blondin,

Haase,

Mazowiecki

et al. 2019

Preprint

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We study the reachability problem for affine Z-VASS, which are integer vector addition systems with states in which transitions perform affine transformations on the counters. This problem is easily seen to be undecidable in general, and we therefore restrict ourselves to affine Z-VASS with the finite-monoid property (afmp-Z-VASS). The latter have the property that the monoid generated by the matrices appearing in their affine transformations is finite. The class of afmp-Z-VASS encompasses classical operations of counter machines such as resets, permutations, transfers and copies. We show that reachability in an afmp-Z-VASS reduces to reachability in a Z-VASS whose control-states grow linearly in the size of the matrix monoid. Our construction shows that reachability relations of afmp-Z-VASS are semilinear, and in particular enables us to show that reachability in Z-VASS with transfers and Z-VASS with copies is PSPACE-complete. We then focus on the reachability problem for affine Z-VASS with monogenic monoids: (possibly infinite) matrix monoids generated by a single matrix. We show that, in a particular case, the reachability problem is decidable for this class, disproving a conjecture about affine Z-VASS with infinite matrix monoids we raised in a preliminary version of this paper. We complement this result by presenting an affine Z-VASS with monogenic matrix monoid and undecidable reachability relation.

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“…Our work is also loosely related to a broader line of research on (variants of) affine VASS dealing with, e.g., modeling power [Val78], accelerability [FL02], formal languages [CFM12], coverability [BFP12], and the complexity of integer reachability for restricted counters [FGH13] and structures [IS16].…”

Section: Introductionmentioning

confidence: 99%

The Complexity of Reachability in Affine Vector Addition Systems with States

Blondin¹,

Raskin²

2021

Logical Methods in Computer Science

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Vector addition systems with states (VASS) are widely used for the formal verification of concurrent systems. Given their tremendous computational complexity, practical approaches have relied on techniques such as reachability relaxations, e.g., allowing for negative intermediate counter values. It is natural to question their feasibility for VASS enriched with primitives that typically translate into undecidability. Spurred by this concern, we pinpoint the complexity of integer relaxations with respect to arbitrary classes of affine operations. More specifically, we provide a trichotomy on the complexity of integer reachability in VASS extended with affine operations (affine VASS). Namely, we show that it is NP-complete for VASS with resets, PSPACE-complete for VASS with (pseudo-)transfers and VASS with (pseudo-)copies, and undecidable for any other class. We further present a dichotomy for standard reachability in affine VASS: it is decidable for VASS with permutations, and undecidable for any other class. This yields a complete and unified complexity landscape of reachability in affine VASS. We also consider the reachability problem parameterized by a fixed affine VASS, rather than a class, and we show that the complexity landscape is arbitrary in this setting.

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How Hard is It to Verify Flat Affine Counter Systems with the Finite Monoid Property?

Cited by 7 publications

References 21 publications

Affine Extensions of Integer Vector Addition Systems with States

Affine Extensions of Integer Vector Addition Systems with States

Affine Extensions of Integer Vector Addition Systems with States

The Complexity of Reachability in Affine Vector Addition Systems with States

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