Unambiguous Constrained Automata

Cadilhac, Michaël; Finkel, Alain; McKenzie, Pierre

doi:10.1142/s0129054113400339

Cited by 9 publications

(4 citation statements)

References 9 publications

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“…All complexity results obtained in this paper are summarized in Figure 1, except for the undecidability of general monogenic classes as it is a family of classes rather than one class. CFM13]. In [FL02], 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: 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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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: 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%

“…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. In [12,13], Cadilhac, Finkel and McKenzie consider an extension of Parikh automata to affine Parikh automata with the finite-monoid restriction like in our paper. These are automata recognizing boolean languages, but the finite-monoid restriction was exploited in a similar way to obtain some decidability results in that context.…”

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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“…Iterating min breaks semilinearity. Deterministic automata equipped with copyless registers with only "+c" updates are quite well-behaved [3,Section 6]; in particular, the set R = {r | r are the values of the registers at the end of an accepting run} is semilinear. Naturally, min{x, y} is expressible in FO[<, +], hence FO[<, +] = FO[<, +, min] (even, and this is not immediate, when the extra value ∞ is added [4]).…”

Section: Introductionmentioning

confidence: 99%

Weak Cost Register Automata Are Still Powerful

Almagor

Cadilhac

Mazowiecki

et al. 2018

Developments in Language Theory

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We consider one of the weakest variants of cost register automata over a tropical semiring, namely copyless cost register automata over N with updates using min and increments. We show that this model can simulate, in some sense, the runs of counter machines with zerotests. We deduce that a number of problems pertaining to that model are undecidable, in particular equivalence, disproving a conjecture of Alur et al. from 2012. To emphasize how weak these machines are, we also show that they can be expressed as a restricted form of linearly-ambiguous weighted automata.

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Unambiguous Constrained Automata

Cited by 9 publications

References 9 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

Weak Cost Register Automata Are Still Powerful

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