Automata, Logic and Algebra for (Finite) Word Transductions - PowerPoint PPT Presentation

Introduction Transducers Logic Algebra New logic Summary Automata, Logic and Algebra for (Finite) Word Transductions Emmanuel Filiot Universit´ e libre de Bruxelles & FNRS ACTS 2017, Chennai 1 / 31

Introduction Transducers Logic Algebra New logic Summary Trinity for Regular Languages Algebra Regular languages L ⊆ Σ ∗ Logic Automata

Introduction Transducers Logic Algebra New logic Summary Trinity for Regular Languages Finite monoids Algebra Regular languages L ⊆ Σ ∗ Logic Automata MSO[S] DFA=NFA=2DFA=2NFA 2 / 31

Introduction Transducers Logic Algebra New logic Summary Objective of the talk ? Algebra Transductions f : Σ ∗ → Σ ∗ Logic Automata ? ? 3 / 31

Introduction Transducers Logic Algebra New logic Summary Automata models for transductions 4 / 31

Introduction Transducers Logic Algebra New logic Summary Automata for transductions: transducers b : ǫ b : ǫ a : a f del : a : a 5 / 31

Introduction Transducers Logic Algebra New logic Summary Automata for transductions: transducers b : ǫ b : ǫ a : a f del : a : a aabaa �→ aaaa 5 / 31

Introduction Transducers Logic Algebra New logic Summary Automata for transductions: transducers b : ǫ b : ǫ a : a f del : a : a aabaa �→ aaaa aaba �→ undefined 5 / 31

Introduction Transducers Logic Algebra New logic Summary Automata for transductions: transducers b : ǫ b : ǫ a : a f del : a : a aabaa �→ aaaa aaba �→ undefined dom ( f del ) = ’even number of a ’ 5 / 31

Introduction Transducers Logic Algebra New logic Summary Non-determinism In general, transducers define binary relations in Σ ∗ × Σ ∗ σ : ǫ realizes { ( u, v ) | v is a subword of u } σ : σ 6 / 31

Introduction Transducers Logic Algebra New logic Summary Sequential vs Non-deterministic functional Non-deterministic transducers may define functions: σ : σ q b b : ǫ σ : bσ f sw : for all σ ∈ Σ σ : aσ a : ǫ q a σ : σ 7 / 31

Introduction Transducers Logic Algebra New logic Summary Sequential vs Non-deterministic functional Non-deterministic transducers may define functions: σ : σ q b b : ǫ σ : bσ f sw : for all σ ∈ Σ σ : aσ a : ǫ q a σ : σ babaa �→ ababa 7 / 31

Introduction Transducers Logic Algebra New logic Summary Sequential vs Non-deterministic functional Non-deterministic transducers may define functions: σ : σ q b b : ǫ σ : bσ f sw : for all σ ∈ Σ σ : aσ a : ǫ q a σ : σ babaa �→ ababa uσ �→ σu | u | ≥ 1 input-determinism (aka sequential) < non-determinism ∩ functions 7 / 31

Introduction Transducers Logic Algebra New logic Summary Determinizability = white space : ǫ a : a : ǫ : : ǫ 2 0 1 a : a 8 / 31

Introduction Transducers Logic Algebra New logic Summary Determinizability = white space : ǫ a : a : ǫ : : ǫ 2 0 1 a : a aa a �→ aa a 8 / 31

Introduction Transducers Logic Algebra New logic Summary Determinizability = white space : ǫ a : a : ǫ : : ǫ 2 0 1 a : a aa a �→ aa a Is non-determinism needed ? 8 / 31

Introduction Transducers Logic Algebra New logic Summary Determinizability = white space : ǫ a : a : ǫ : : ǫ 2 0 1 a : a aa a �→ aa a a : a : ǫ : ǫ 3 4 a : a Is non-determinism needed ? No. 8 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 output 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 output 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 output 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 output 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 output 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 output 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 output 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 output 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 output 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 output 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 output d 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 e output d 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 e s output d 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 e s s output d 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 e s s e output d 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 e s s e r output d 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 e s s e r t output d 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 e s s e r s t output d 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e ⊢ t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 3 e s s e r s t output d 9 / 31

Introduction Transducers Logic Algebra New logic Summary Two-way transducers input s r e s s e t d ⊣ σ : ǫ, → σ : σ, ← ⊣ : ǫ, ← ⊢ : ǫ 1 2 e s s e r s t output d one-way < two-way � decidable equivalence problem (Culik, Karhumaki, 87). � closed under composition ◦ (Chytil, Jakl, 77) 9 / 31

Introduction Transducers Logic Algebra New logic Summary Landscape of Transducer Classes ⊂ ⊂ SFTs FT 2DFT=2FT expressiveness sequential rational regular transductions transductions transductions 10 / 31

Introduction Transducers Logic Algebra New logic Summary Landscape of Transducer Classes ⊂ ⊂ SFTs FT 2DFT=2FT PTime expressiveness Chof77 sequential rational regular WK95 transductions transductions transductions BealCartonPS03 10 / 31

Introduction Transducers Logic Algebra New logic Summary Landscape of Transducer Classes ⊂ ⊂ SFTs FT 2DFT=2FT PTime expressiveness decidable Chof77 FGRS13 sequential rational regular WK95 transductions transductions transductions BealCartonPS03 10 / 31

Introduction Transducers Logic Algebra New logic Summary Landscape of Transducer Classes valuedness NFT 2NFT ⊂ ⊂ SFTs FT 2DFT=2FT PTime expressiveness decidable Chof77 FGRS13 sequential rational regular WK95 transductions transductions transductions BealCartonPS03 10 / 31

Introduction Transducers Logic Algebra New logic Summary Landscape of Transducer Classes valuedness ⊂ NFT 2NFT ⊂ ⊂ ⊂ ⊂ SFTs FT 2DFT=2FT PTime expressiveness decidable Chof77 FGRS13 sequential rational regular WK95 transductions transductions transductions BealCartonPS03 10 / 31