Second-Order Finite Automata: Expressive Power and Simple Proofs Using Automatic Structures

Kuske, Dietrich

doi:10.1007/978-3-030-81508-0_20

Cited by 3 publications

(7 citation statements)

References 11 publications

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“…is equal to the second language of some second-order finite automaton). This conjecture has recently been confirmed by Kuske in [32]. Similar connections can be established with the framework of uniform classes of automatic structures [39], which are defined with basis on the notion of automatic structures with advice.…”

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

confidence: 53%

“…Second, that the relation R ⊆ ( , w) = {(D, D ) : D, D ∈ B( , w) L(D) ⊆ L(D )} is regular (Proposition 6.6 of [13]). Similar observations have been used in [32] to study second-order finite automata using the framework of the theory of automatic structures [1,2,31]. In particular, some of our decidability and closure results have been rederived in [32] using this framework, and some new applications of second-order finite automata to partial-order theory have been obtained.…”

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

confidence: 60%

“…As pointed out in [32], the fact that automatic structures are closed under first-order interpretations [8,23,31] has also applications in our framework. For instance, one can infer that the relation R ⊆ ( , w) is automatic from the fact that R ∈ ( , w) is automatic, together with the fact that for each two ODDs D, D , the inclusion L(D) ⊆ L(D ) is defined by the first-order formula ∀ s, (D, s) ∈ R ∈ ( , w) ⇒ (D , s) ∈ R ∈ ( , w).…”

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

confidence: 77%

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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: 53%

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

confidence: 60%

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

confidence: 77%

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Second-Order Finite Automata

Melo

Oliveira

2022

Theory Comput Syst

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“…In [17], Kuske generalizes results of de Malo and de Oliveira Oliveira in [7] on secondorder finite automata by using automatic structures. As an application, Kuske shows in [17,Thm.…”

Section: Encoding Of Graphs and Related Workmentioning

confidence: 63%

“…In [17], Kuske generalizes results of de Malo and de Oliveira Oliveira in [7] on secondorder finite automata by using automatic structures. As an application, Kuske shows in [17,Thm. 3.6] how to decide typical properties of languages classes accepted by second-order finite automata.…”

Section: Encoding Of Graphs and Related Workmentioning

confidence: 63%

Properties of graphs specified by a regular language

2022

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Traditionally, graph algorithms get a single graph as input, and then they should decide if this graph satisfies a certain property $$\varPhi $$ Φ . What happens if this question is modified in a way that we get a possibly infinite family of graphs as an input, and the question is if there is a graph satisfying $$\varPhi $$ Φ in the family? We approach this question by using formal languages for specifying families of graphs, in particular by regular sets of words. We show that certain graph properties can be decided by studying the syntactic monoid of the specification language L if a certain torsion condition is satisfied. This condition holds trivially if L is regular. More specifically, we use a natural binary encoding of finite graphs over a binary alphabet $$\varSigma $$ Σ , and we define a regular set $$\mathbb {G}\subseteq \varSigma ^*$$ G ⊆ Σ ∗ such that every nonempty word $$w\in \mathbb {G}$$ w ∈ G defines a finite and nonempty graph. Also, graph properties can then be syntactically defined as languages over $$\varSigma $$ Σ . Then, we ask whether the automaton $$\mathcal {A}$$ A specifies some graph satisfying a certain property $$\varPhi $$ Φ . Our structural results show that we can answer this question for all “typical” graph properties. In order to show our results, we split L into a finite union of subsets and every subset of this union defines in a natural way a single finite graph F where some edges and vertices are marked. The marked graph in turn defines an infinite graph $$F^\infty $$ F ∞ and therefore the family of finite subgraphs of $$F^\infty $$ F ∞ where F appears as an induced subgraph. This yields a geometric description of all graphs specified by L based on splitting L into finitely many pieces; then using the notion of graph retraction, we obtain an easily understandable description of the graphs in each piece.

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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.

show abstract

Second-Order Finite Automata: Expressive Power and Simple Proofs Using Automatic Structures

Cited by 3 publications

References 11 publications

Second-Order Finite Automata

Second-Order Finite Automata

Properties of graphs specified by a regular language

Second-Order Finite Automata

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