Structures without Scattered-Automatic Presentation

Kartzow, Alexander; Schlicht, Philipp

doi:10.1007/978-3-642-39053-1_32

Cited by 5 publications

(3 citation statements)

References 18 publications

(44 reference statements)

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“…the ODD-width w necessary to represent languages/structures in the class). Since the concepts in [29,39] have applications in the fields of learning theory [7,24,27,28] and algebra [9,30,38,39], an interesting line of research would be the investigation of potential applications of our fixed-parameter tractable algorithms to problems in these fields.…”

Section: Connections With the Theory Of Automatic Structures Finite A...mentioning

confidence: 99%

Second-Order Finite Automata

Melo

Oliveira

2022

Theory Comput Syst

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Traditionally, finite automata theory has been used as a framework for the representation of possibly infinite sets of strings. In this work, we introduce the notion of second-order finite automata, a formalism that combines finite automata with ordered decision diagrams, with the aim of representing possibly infinite sets of sets of strings. Our main result states that second-order finite automata can be canonized with respect to the second-order languages they represent. Using this canonization result, we show that sets of sets of strings represented by second-order finite automata are closed under the usual Boolean operations, such as union, intersection, difference and even under a suitable notion of complementation. Additionally, emptiness of intersection and inclusion are decidable. We provide two algorithmic applications for second-order automata. First, we show that several width/size minimization problems for deterministic and nondeterministic ODDs are solvable in fixed-parameter tractable time when parameterized by the width of the input ODD. In particular, our results imply FPT algorithms for corresponding width/size minimization problems for ordered binary decision diagrams (OBDDs) with a fixed variable ordering. Previously, only algorithms that take exponential time in the size of the input OBDD were known for width minimization, even for OBDDs of constant width. Second, we show that for each k and w one can count the number of distinct functions computable by ODDs of width at most w and length k in time h(|Σ|,w) ⋅ kO(1), for a suitable $h:\mathbb {N}\times \mathbb {N}\rightarrow \mathbb {N}$ h : ℕ × ℕ → ℕ . This improves exponentially on the time necessary to explicitly enumerate all such functions, which is exponential in both the width parameter w and in the length k of the ODDs.

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Section: Connections With the Theory Of Automatic Structures Finite A...mentioning

confidence: 99%

Second-Order Finite Automata

Melo

Oliveira

2022

Theory Comput Syst

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“…• The sum-of-box-augmentation technique of Delhommé [3] for tree-automatic structures has an analogue for ordinal-automatic structures which allows to classify all (α)-automatic ordinals [14] and give sharp bounds on the ranks of (α)-automatic scattered linear orderings [14] and well-founded order trees [8]. • The word-automatic Boolean algebras [11] and the (ω n )-automatic Boolean algebras [6] have been classified.…”

Section: Introductionmentioning

confidence: 99%

The field of the Reals and the Random Graph are not Finite-Word Ordinal-Automatic

Kartzow

2014

Preprint

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Recently, Schlicht and Stephan lifted the notion of automatic-structures to the notion of (finite-word) ordinal-automatic structures. These are structures whose domain and relations can be represented by automata reading finite words whose shape is some fixed ordinal α. We lift Delhommé's relative-growth-technique from the automatic and tree-automatic setting to the ordinal-automatic setting. This result implies that the random graph is not ordinal-automatic and infinite integral domains are not ordinal-automatic with respect to ordinals below ω1 + ω ω where ω1 is the first uncountable ordinal.

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“…the ODD-width necessary to represent languages/structures in the class). Since the concepts in [29,39] have applications the fields of learning theory [7,24,27,28] and algebra [9,30,38,39], an interesting line of research would be the investigation of potential applications of our fixed-parameter tractable algorithms to problems in these fields.…”

mentioning

confidence: 99%

Second-Order Finite Automata

Melo

Oliveira

2020

Computer Science – Theory and Applications

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Traditionally, finite automata theory has been used as a framework for the representation of possibly infinite sets of strings. In this work, we introduce the notion of second-order finite automata, a formalism that combines finite automata with ordered decision diagrams, with the aim of representing possibly infinite sets of sets of strings. Our main result states that second-order finite automata can be canonized with respect to the second-order languages they represent. Using this canonization result, we show that sets of sets of strings represented by second-order finite automata are closed under the usual Boolean operations, such as union, intersection, difference and even under a suitable notion of complementation. Additionally, emptiness of intersection and inclusion are decidable. We provide two algorithmic applications for second-order automata. First, we show that several width/size minimization problems for deterministic and nondeterministic ODDs are solvable in fixed-parameter tractable time when parameterized by the width of the input ODD. In particular, our results imply FPT algorithms for corresponding width/size minimization problems for ordered binary decision diagrams (OBDDs) with a fixed variable ordering. Previously, only algorithms that take exponential time in the size of the input OBDD were known for width minimization, even for OBDDs of constant width. Second, we show that for each and one can count the number of distinct functions computable by ODDs of width at most and length in time 1 , for a suitable . This improves exponentially on the time necessary to explicitly enumerate all such functions, which is exponential in both the width parameter and in the length of the ODDs.

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Structures without Scattered-Automatic Presentation

Cited by 5 publications

References 18 publications

Second-Order Finite Automata

Second-Order Finite Automata

The field of the Reals and the Random Graph are not Finite-Word Ordinal-Automatic

Second-Order Finite Automata

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