Regular languages closed under word operations Szil´ ard Zsolt Fazekas Preliminaries Regular languages closed under word operations Subsequence / supersequence Duplication Timeline Szil´ ard Zsolt Fazekas Duplication closure of languages Hairpin Akita University completion Timeline Pseudopalindromic Workshop “Topology and Computer 2016” completion Power of a language Timeline Decidability Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
Regular languages closed under word Σ - finite non-empty set, alphabet operations Szil´ ard Zsolt Fazekas Preliminaries Subsequence / supersequence Duplication Timeline Duplication closure of languages Hairpin completion Timeline Pseudopalindromic completion Power of a language Timeline Decidability Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
Regular languages closed under word Σ - finite non-empty set, alphabet operations Σ ∗ - the free monoid generated by Σ Szil´ ard Zsolt Fazekas Preliminaries Subsequence / supersequence Duplication Timeline Duplication closure of languages Hairpin completion Timeline Pseudopalindromic completion Power of a language Timeline Decidability Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
Regular languages closed under word Σ - finite non-empty set, alphabet operations Σ ∗ - the free monoid generated by Σ Szil´ ard Zsolt Fazekas Preliminaries Example Subsequence / supersequence Σ = { a , b } Σ ∗ = { λ, a , b , aa , ab , ba , bb , . . . } Duplication Timeline Duplication closure of languages Hairpin completion Timeline Pseudopalindromic completion Power of a language Timeline Decidability Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
Regular languages closed under word Σ - finite non-empty set, alphabet operations Σ ∗ - the free monoid generated by Σ Szil´ ard Zsolt Fazekas Preliminaries Example Subsequence / supersequence Σ = { a , b } Σ ∗ = { λ, a , b , aa , ab , ba , bb , . . . } Duplication Timeline Duplication closure of languages Hairpin L ⊆ Σ ∗ : language completion Timeline Pseudopalindromic completion Power of a language Timeline Decidability Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
Regular languages closed under word Σ - finite non-empty set, alphabet operations Σ ∗ - the free monoid generated by Σ Szil´ ard Zsolt Fazekas Preliminaries Example Subsequence / supersequence Σ = { a , b } Σ ∗ = { λ, a , b , aa , ab , ba , bb , . . . } Duplication Timeline Duplication closure of languages Hairpin L ⊆ Σ ∗ : language completion Timeline Pseudopalindromic completion w 0 = λ and w n +1 = w n w , ∀ n ≥ 0: powers of a word Power of a language w ∈ Σ ∗ Timeline Decidability Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
Regular languages closed under word Σ - finite non-empty set, alphabet operations Σ ∗ - the free monoid generated by Σ Szil´ ard Zsolt Fazekas Preliminaries Example Subsequence / supersequence Σ = { a , b } Σ ∗ = { λ, a , b , aa , ab , ba , bb , . . . } Duplication Timeline Duplication closure of languages Hairpin L ⊆ Σ ∗ : language completion Timeline Pseudopalindromic completion w 0 = λ and w n +1 = w n w , ∀ n ≥ 0: powers of a word Power of a language w ∈ Σ ∗ Timeline Decidability w ∗ = { w 0 , w 1 , w 2 , . . . } . Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
Regular languages closed under word operations Szil´ ard Zsolt A finite automaton is a quintuple A = � Σ , Q , q 0 , F , σ }� where Fazekas Σ is the input alphabet , Preliminaries Q is a finite set called the set of states , Subsequence / supersequence q 0 ∈ Q is the initial state , Duplication Timeline F ⊆ Q is the set of final states and Duplication closure of languages σ : Q × Σ → 2 Q is the transition function . Hairpin completion Timeline Pseudopalindromic completion Power of a language Timeline Decidability Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
Regular languages closed under word operations Szil´ ard Zsolt A finite automaton is a quintuple A = � Σ , Q , q 0 , F , σ }� where Fazekas Σ is the input alphabet , Preliminaries Q is a finite set called the set of states , Subsequence / supersequence q 0 ∈ Q is the initial state , Duplication Timeline F ⊆ Q is the set of final states and Duplication closure of languages σ : Q × Σ → 2 Q is the transition function . Hairpin completion Timeline If ∀ q ∈ Q , a ∈ Σ : | σ ( q , a ) | ≤ 1 then A is deterministic, Pseudopalindromic completion otherwise nondeterministic. Power of a language Timeline Decidability Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
Regular languages closed under word operations Szil´ ard Zsolt L ( A ), the language accepted by the finite automaton A is the Fazekas set of all words a 1 a 2 . . . a n ( a i ∈ Σ), such that there exist states Preliminaries p 0 , . . . , p n such that Subsequence / supersequence ∀ i ∈ { 1 , . . . , n } : p i ∈ σ ( p i − 1 , a i ) , Duplication Timeline Duplication closure of p 0 = q 0 , p n ∈ F languages Hairpin completion . Timeline Pseudopalindromic completion Power of a language Timeline Decidability Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
Regular languages closed under word operations Szil´ ard Zsolt L ( A ), the language accepted by the finite automaton A is the Fazekas set of all words a 1 a 2 . . . a n ( a i ∈ Σ), such that there exist states Preliminaries p 0 , . . . , p n such that Subsequence / supersequence ∀ i ∈ { 1 , . . . , n } : p i ∈ σ ( p i − 1 , a i ) , Duplication Timeline Duplication closure of p 0 = q 0 , p n ∈ F languages Hairpin completion . Timeline Pseudopalindromic completion A language is regular iff it is accepted by a finite automaton. Power of a language Timeline Decidability Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
Example Regular languages closed under A = �{ 0 , 1 } , { q 0 , q 1 } , q 0 , { q 1 } , σ � , where the transition word operations function σ is: Szil´ ard Zsolt Fazekas σ 0 1 Preliminaries q 0 q 0 q 1 Subsequence / q 1 q 1 q 0 supersequence L ( A ) = { w ∈ Σ ∗ |∃ k ≥ 0 : | w | 1 = 2 k + 1 } , Duplication Timeline Duplication closure of that is all binary words having an odd number of 1’s. languages Hairpin completion Timeline Pseudopalindromic completion Power of a language Timeline Decidability Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
Regular languages closed under word operations Szil´ ard Zsolt Fazekas Quasi order (preorder): reflexive and transitive binary relation Preliminaries Subsequence / supersequence Duplication Timeline Duplication closure of languages Hairpin completion Timeline Pseudopalindromic completion Power of a language Timeline Decidability Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
Regular languages closed under word operations Szil´ ard Zsolt Fazekas Quasi order (preorder): reflexive and transitive binary relation Preliminaries Subsequence / Well quasi order: any infinite sequence of elements x 0 , x 1 , . . . supersequence contains an increasing pair x i ≤ x j with i < j . So: Duplication Timeline Duplication no infinite decreasing series closure of languages no antichain (infinite series of pairwise incomparable Hairpin completion elements) Timeline Pseudopalindromic completion Power of a language Timeline Decidability Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
Regular languages closed under word operations Definition Szil´ ard Zsolt Fazekas For u , v ∈ Σ ∗ : u ≤ v : u is a subsequence (subword, scattered subword) of v if Preliminaries u = x 1 · · · x n and v = y 0 x 1 y 1 x 2 y 2 · · · y n for some x i , y j ∈ Σ ∗ . Subsequence / supersequence v is a supersequence of u . Duplication Timeline Duplication closure of languages Hairpin completion Timeline Pseudopalindromic completion Power of a language Timeline Decidability Workshop “Topology and Computer 2016” Szil´ ard Zsolt Fazekas (Akita University) Regular languages closed under word operations / 30
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