Introduction Results Conclusion Weight-reducing Hennie Machines and Their Descriptional Complexity 1 Daniel Pr˚ uˇ sa Czech Technical University in Prague International Conference on Language and Automata Theory and Applications (LATA) 2014 1The author was supported by the Grant Agency of the Czech Republic under the project P103/10/0783. uˇ 1 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Results Conclusion Outline Introduction 1 Descriptional complexity of automata Hennie machine Weight-reducing property Results 2 Comparison with 1DFA Comparison with 1NFA Comparison with 2NFA Comparison with 2DPA Conclusion 3 uˇ 2 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Regular languages and finite automata Basic model: One-way deterministic finite-state automaton (1 DFA ). Extensions: Nondeterminism (1 NFA ), alternation (1 AFA ). Two-way movement (2 DFA , 2 NFA ). Usage of a pebble (2 DPA ). All the models equal in power, however, they differ in succinctness of their descriptions. uˇ 3 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Regular languages and finite automata Basic model: One-way deterministic finite-state automaton (1 DFA ). Extensions: Nondeterminism (1 NFA ), alternation (1 AFA ). Two-way movement (2 DFA , 2 NFA ). Usage of a pebble (2 DPA ). All the models equal in power, however, they differ in succinctness of their descriptions. uˇ 3 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Size of automata description Number of states – frequently studied measure (for example: 1 DFA needs 2 n states to simulate 1 NFA in the worst case). Number of transitions – better suits our purposes. Theorem ([Shannon 1956]) Each Turing machine has an equivalent with only two active states. uˇ 4 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Size of automata description Number of states – frequently studied measure (for example: 1 DFA needs 2 n states to simulate 1 NFA in the worst case). Number of transitions – better suits our purposes. Theorem ([Shannon 1956]) Each Turing machine has an equivalent with only two active states. uˇ 4 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Hennie machine Bounded, single-tape Turing machine. ⊢ a 1 a 2 a 3 a 4 a 5 a 6 ⊣ The number of transitions performed over every tape field limited by a constant k . Theorem ([Hennie 1965]) Each language accepted by a Hennie machine is regular language. The condition can be further relaxed: Linear time [Hennie 1965]. O ( n log n ) time [Hartmanis 1968]. uˇ 5 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Hennie machine Bounded, single-tape Turing machine. ⊢ a 1 a 2 a 3 a 4 a 5 a 6 ⊣ The number of transitions performed over every tape field limited by a constant k . Theorem ([Hennie 1965]) Each language accepted by a Hennie machine is regular language. The condition can be further relaxed: Linear time [Hennie 1965]. O ( n log n ) time [Hartmanis 1968]. uˇ 5 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Hennie machine Bounded, single-tape Turing machine. ⊢ a 1 a 2 a 3 a 4 a 5 a 6 ⊣ The number of transitions performed over every tape field limited by a constant k . Theorem ([Hennie 1965]) Each language accepted by a Hennie machine is regular language. The condition can be further relaxed: Linear time [Hennie 1965]. O ( n log n ) time [Hartmanis 1968]. uˇ 5 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Hennie machine - properties Nonrecursive trade-off with respect to 1 DFA : Let c ( n ) be the cost of the optimal simulation of Hennie machine by 1 DFA . c ( n ) is not bounded by any recursive function. Nonconstructive: Given a Turing machine T , it is undecidable if T is a Hennie machine. uˇ 6 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Hennie machine - properties Nonrecursive trade-off with respect to 1 DFA : Let c ( n ) be the cost of the optimal simulation of Hennie machine by 1 DFA . c ( n ) is not bounded by any recursive function. Nonconstructive: Given a Turing machine T , it is undecidable if T is a Hennie machine. uˇ 6 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Our goal Propose a constructive variant of Hennie machine. 1 Study the descriptional complexity of the model. 2 uˇ 7 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Weight-reducing property Weight function defined on working symbols: µ : Γ → N Each transition has to decrease the weight of the scanned symbol ⇒ the bound k is incorporated in Γ ( k ≤ | Γ | ). a a a µ ( b ) < µ ( a ) Sgraffito automaton [Prusa and Mraz 2012] - 2D automaton for recognition of picture languages, the same principle applied. uˇ 8 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Weight-reducing property Weight function defined on working symbols: µ : Γ → N Each transition has to decrease the weight of the scanned symbol ⇒ the bound k is incorporated in Γ ( k ≤ | Γ | ). a b a µ ( b ) < µ ( a ) Sgraffito automaton [Prusa and Mraz 2012] - 2D automaton for recognition of picture languages, the same principle applied. uˇ 8 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Weight-reducing property Weight function defined on working symbols: µ : Γ → N Each transition has to decrease the weight of the scanned symbol ⇒ the bound k is incorporated in Γ ( k ≤ | Γ | ). a b a µ ( b ) < µ ( a ) Sgraffito automaton [Prusa and Mraz 2012] - 2D automaton for recognition of picture languages, the same principle applied. uˇ 8 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Formal definition of weight-reducing Hennie machine M = ( Q , Σ , Γ , δ, q 0 , Q F , µ ) , where Σ is an input alphabet Γ is a working alphabet, Γ ⊇ Σ , Γ ∩ {⊢ , ⊣} = ∅ Q is a finite, non-empty set of states q 0 is the initial state, q 0 ∈ Q Q F is the set of final states, Q F ⊆ Q µ a weight function, µ : Γ → N δ a transition relation, δ : ( Q \ Q F ) × (Γ ∪ {⊢ , ⊣} ) → 2 Q × (Γ ∪{⊢ , ⊣} ) ×{← , 0 , →} each transition over the input is weight-reducing ( q ′ , a ′ , d ) ∈ δ ( q , a ) ⇒ µ ( a ′ ) <µ ( a ) for all q , q ′ ∈ Q , d ∈ {← , 0 , →} , a , a ′ ∈ Γ automaton is bounded uˇ 9 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Formal definition of weight-reducing Hennie machine M = ( Q , Σ , Γ , δ, q 0 , Q F , µ ) , where Σ is an input alphabet Γ is a working alphabet, Γ ⊇ Σ , Γ ∩ {⊢ , ⊣} = ∅ Q is a finite, non-empty set of states q 0 is the initial state, q 0 ∈ Q Q F is the set of final states, Q F ⊆ Q µ a weight function, µ : Γ → N δ a transition relation, δ : ( Q \ Q F ) × (Γ ∪ {⊢ , ⊣} ) → 2 Q × (Γ ∪{⊢ , ⊣} ) ×{← , 0 , →} each transition over the input is weight-reducing ( q ′ , a ′ , d ) ∈ δ ( q , a ) ⇒ µ ( a ′ ) <µ ( a ) for all q , q ′ ∈ Q , d ∈ {← , 0 , →} , a , a ′ ∈ Γ automaton is bounded uˇ 9 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
Introduction Descriptional complexity of automata Results Hennie machine Conclusion Weight-reducing property Formal definition of weight-reducing Hennie machine M = ( Q , Σ , Γ , δ, q 0 , Q F , µ ) , where Σ is an input alphabet Γ is a working alphabet, Γ ⊇ Σ , Γ ∩ {⊢ , ⊣} = ∅ Q is a finite, non-empty set of states q 0 is the initial state, q 0 ∈ Q Q F is the set of final states, Q F ⊆ Q µ a weight function, µ : Γ → N δ a transition relation, δ : ( Q \ Q F ) × (Γ ∪ {⊢ , ⊣} ) → 2 Q × (Γ ∪{⊢ , ⊣} ) ×{← , 0 , →} each transition over the input is weight-reducing ( q ′ , a ′ , d ) ∈ δ ( q , a ) ⇒ µ ( a ′ ) <µ ( a ) for all q , q ′ ∈ Q , d ∈ {← , 0 , →} , a , a ′ ∈ Γ automaton is bounded uˇ 9 / 21 Daniel Pr˚ sa Weight-reducing Hennie Machines
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