Polynomial automata: Zeroness and applications

Benedikt, Michael; Duff, Timothy; Sharad, Aditya; Worrell, James

doi:10.1109/lics.2017.8005101

Cited by 19 publications

(33 citation statements)

References 29 publications

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“…Also our approach to look at the reachability from language-theoretic point of view is in line with a very recent paper [2] on the complexity of the zeroness problem for deterministic polynomial automata over rationals (i.e., a problem whether a given automaton outputs zero on all words). Authors of [2] showed that while this problem is non-primitive recursive in general, there is a subclass of polynomial automata for which the zeroness problem is primitive recursive.…”

Section: Np-h/pspacementioning

confidence: 58%

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Reachability Problems in Nondeterministic Polynomial Maps on the Integers

Niskanen

Potapov

2018

Developments in Language Theory

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We study the reachability problems in various nondeterministic polynomial maps in Z n. We prove that the reachability problem for very simple three-dimensional affine maps (with independent variables) is undecidable and is PSPACE-hard for two-dimensional quadratic maps. Then we show that the complexity of the reachability problem for maps without functions of the form ±x + b is lower. In this case the reachability problem is PSPACE-complete in general, and NP-hard for any fixed dimension. Finally we extend the model by considering maps as language acceptors and prove that the universality problem is undecidable for two-dimensional affine maps.

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Section: Np-h/pspacementioning

confidence: 58%

“…In [2] the authors study the zeroness problem, which is essentially the universality problem we consider in this section. The main difference is that our model is nondeterministic, while in [2] the automaton is deterministic. The authors show that the zeroness problem is Ackermann-complete 5 .…”

Section: Maps Over Aff Z [X] 2 As Language Acceptorsmentioning

confidence: 99%

Reachability Problems in Nondeterministic Polynomial Maps on the Integers

Niskanen

Potapov

2018

Developments in Language Theory

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“…Note that the decidability of SST equivalence also follows from the two recent results [20] and [8].…”

Section: Introductionmentioning

confidence: 69%

Copyful Streaming String Transducers

Filiot

Reynier

2021

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Copyless streaming string transducers (copyless SST) have been introduced by R. Alur and P. Černý in 2010 as a one-way deterministic automata model to define transductions of finite strings. Copyless SST extend deterministic finite state automata with a set of variables in which to store intermediate output strings, and those variables can be combined and updated all along the run, in a linear manner, i.e., no variable content can be copied on transitions. It is known that copyless SST capture exactly the class of MSO-definable string-to-string transductions, and are as expressive as deterministic two-way transducers. They enjoy good algorithmic properties. Most notably, they have decidable equivalence problem (in PSpace). On the other hand, HDT0L systems have been introduced for a while, the most prominent result being the decidability of the equivalence problem. In this paper, we propose a semantics of HDT0L systems in terms of transductions, and use it to study the class of deterministic copyful SST. Our contributions are as follows: (i)HDT0L systems and total deterministic copyful SST have the same expressive power, (ii)the equivalence problem for deterministic copyful SST and the equivalence problem for HDT0L systems are inter-reducible, in quadratic time. As a consequence, equivalence of deterministic SST is decidable, (iii)the functionality of non-deterministic copyful SST is decidable, (iv)determining whether a non-deterministic copyful SST can be transformed into an equivalent non-deterministic copyless SST is decidable in polynomial time.

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“…The discussed models of cost-register automata [4,7,12] are not the only nonlinear extensions of weighted automata that appear in the literature. We are aware of at least two more extensions: weighted context-free grammars [5,8] and weighted MSO logic [9,17].…”

Section: Applicationsmentioning

confidence: 99%

“…polynomial recurrent relations that generalise pushdown automata of order 3 [12,21]; cost-register automata which arose as a variant of streaming transducers [3,4]; polynomial automata, connected to reachability problems for vector addition systems [7]. Surprisingly, these three models, although introduced in different contexts, are all equivalent.…”

Section: Introductionmentioning

confidence: 99%

On Polynomial Recursive Sequences

et al. 2021

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We study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences. These sequences arise naturally in the study of nonlinear extensions of weighted automata, where (non)expressiveness results translate to class separations. A typical example of a polynomial recursive sequence is bn = n!. Our main result is that the sequence un = nn is not polynomial recursive.

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Polynomial automata: Zeroness and applications

Cited by 19 publications

References 29 publications

Reachability Problems in Nondeterministic Polynomial Maps on the Integers

Reachability Problems in Nondeterministic Polynomial Maps on the Integers

Copyful Streaming String Transducers

On Polynomial Recursive Sequences

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