Transformation of Turing Machines into Context-Dependent Fusion - PowerPoint PPT Presentation

Transformation of Turing Machines into Context-Dependent Fusion Grammars Aaron Lye University of Bremen lye@math.uni-bremen.de 17.07.2019 10th International Workshop on Graph Computation Models (GCM) 1/21

Motivation Fusion grammars were introduced at ICGT 2017 as a generating device for hypergraph languages. Context-dependent fusion grammars were introduced at LATA 2019 to simulate Petri nets. 2/21

Motivation Fusion grammars were introduced at ICGT 2017 as a generating device for hypergraph languages. Context-dependent fusion grammars were introduced at LATA 2019 to simulate Petri nets. How powerful are context-dependent fusion grammars? 2/21

Motivation Fusion grammars were introduced at ICGT 2017 as a generating device for hypergraph languages. Context-dependent fusion grammars were introduced at LATA 2019 to simulate Petri nets. How powerful are context-dependent fusion grammars? ◮ They are at least as powerful as Petri nets. 2/21

Motivation Fusion grammars were introduced at ICGT 2017 as a generating device for hypergraph languages. Context-dependent fusion grammars were introduced at LATA 2019 to simulate Petri nets. How powerful are context-dependent fusion grammars? ◮ They are at least as powerful as Petri nets. ◮ They are powerful enough to simulate Turing machines. 2/21

Motivation Fusion grammars were introduced at ICGT 2017 as a generating device for hypergraph languages. Context-dependent fusion grammars were introduced at LATA 2019 to simulate Petri nets. How powerful are context-dependent fusion grammars? ◮ They are at least as powerful as Petri nets. ◮ They are powerful enough to simulate Turing machines. ◮ They can generate all recursively enumerable string languages (up to representation) and are universal in this respect. 2/21

Hypergraph We consider hypergraphs over Σ with hyperedges like v k 1 v 1 . . . k 1 1 A 1 k 2 . . . w 1 w k 2 where v 1 · · · v k 1 is a sequence of source vertices w 1 · · · w k 2 is a sequence of target vertices A ∈ Σ is a label. The class of all hypergraphs over Σ is denoted by H Σ . 3/21

Fusion rule Let F ⊆ Σ be a fusion alphabet. Let type : F → N × N . Each A ∈ F has a complement A ∈ F where type ( A ) = type ( A ). v ′ v k 1 v ′ v 1 1 . . . k 1 . . . 1 k 1 1 k 1 fr ( A ) = A A k 2 1 1 k 2 . . . . . . w k 2 w ′ w ′ w 1 1 k 2 type ( A ) = ( k 1 , k 2 ) 4/21

Rule application 1. find a matching morphism g of fr ( A ) in the hypergraph H . v ′ v 1 v k 1 v ′ 1 . . . k 1 . . . 1 k 1 1 k 1 H A A 1 k 2 k 2 1 . . . . . . w ′ w k 2 w ′ w 1 1 k 2 5/21

Rule application 1. find a matching morphism g of fr ( A ) in the hypergraph H . 2. remove the images of the two hyperedges of fr ( A ). v ′ v 1 v k 1 v ′ 1 . . . k 1 . . . I . . . . . . w ′ w k 2 w ′ w 1 1 k 2 5/21

Rule application 1. find a matching morphism g of fr ( A ) in the hypergraph H . 2. remove the images of the two hyperedges of fr ( A ). 3. identify corresponding source and target vertices of the removed hyperedges. v k 1 = v ′ v 1 = v ′ 1 k 1 . . . H ′ . . . w k 2 = w ′ w 1 = w ′ 1 k 2 5/21

Rule application 1. find a matching morphism g of fr ( A ) in the hypergraph H . 2. remove the images of the two hyperedges of fr ( A ). 3. identify corresponding source and target vertices of the removed hyperedges. v k 1 = v ′ v 1 = v ′ 1 k 1 . . . H ′ . . . w k 2 = w ′ w 1 = w ′ 1 k 2 5/21

Rule application 1. find a matching morphism g of fr ( A ) in the hypergraph H . 2. remove the images of the two hyperedges of fr ( A ). 3. identify corresponding source and target vertices of the removed hyperedges. v k 1 = v ′ v 1 = v ′ 1 k 1 . . . H ′ . . . w k 2 = w ′ w 1 = w ′ 1 k 2 fr ( A ) H ′ . ⇒ Rule application is denoted by H = 5/21

Context-dependent fusion rule and its application ( fr ( A ) , PC , NC ) where PC , NC are sets of morphisms with domain fr ( A ). It is applicable to hypergraph H via a matching morphism g : fr ( A ) → H if 1. ∀ c ∈ PC 2. ∀ c ∈ NC c c fr ( A ) fr ( A ) C C = = g g ∃ h � ∃ h H H and h is injective on the set of hyperedges. Adaption of the definition in Habel-Heckel-Taentzer:96. 6/21

Multiplication Fusion adds nothing to the hypergraph. For a finite hypergraph the number of possible fusions is finite. One may multiply (copy) connected components. 7/21

Multiplication Fusion adds nothing to the hypergraph. For a finite hypergraph the number of possible fusions is finite. One may multiply (copy) connected components. Let C ( H ) be the set of all connected components of H . Define multiplicity m : C ( H ) → N . � m m · H = ⇒ m ( C ) · C H = C ∈C ( H ) 7/21

Context-dependent fusion grammar CDFG = ( Z , F , M , T , P ) Z : finite start hypergraph F , M , T ⊆ Σ: fusion, marker, terminal alphabet (all finite and pairwise disjoint) P : finite set of context-dependent fusion rules. 8/21

Context-dependent fusion grammar CDFG = ( Z , F , M , T , P ) Z : finite start hypergraph F , M , T ⊆ Σ: fusion, marker, terminal alphabet (all finite and pairwise disjoint) P : finite set of context-dependent fusion rules. A direct derivation is either cdfr H ′ H = ⇒ for some cdfr ∈ P or H = m m · H = � ⇒ m ( C ) · C for some multiplicity m : C ( H ) → N . C ∈C ( H ) 8/21

Context-dependent fusion grammar CDFG = ( Z , F , M , T , P ) Z : finite start hypergraph F , M , T ⊆ Σ: fusion, marker, terminal alphabet (all finite and pairwise disjoint) P : finite set of context-dependent fusion rules. A direct derivation is either cdfr H ′ H = ⇒ for some cdfr ∈ P or H = m m · H = � ⇒ m ( C ) · C for some multiplicity m : C ( H ) → N . C ∈C ( H ) Derivations are defined by the reflexive and transitive closure. 8/21

Context-dependent fusion grammar CDFG = ( Z , F , M , T , P ) Z : finite start hypergraph F , M , T ⊆ Σ: fusion, marker, terminal alphabet (all finite and pairwise disjoint) P : finite set of context-dependent fusion rules. A direct derivation is either cdfr H ′ H = ⇒ for some cdfr ∈ P or H = m m · H = � ⇒ m ( C ) · C for some multiplicity m : C ( H ) → N . C ∈C ( H ) Derivations are defined by the reflexive and transitive closure. The generated language ∗ L ( CDFG ) = { rem M ( Y ) | Z = ⇒ H , Y ∈ C ( H ) ∩ ( H T ∪ M − H T ) } , where rem M ( Y ) removes all marker hyperedges from Y . 8/21

Transformation of Turing Machines into Context-Dependent Fusion Grammars 9/21

Turing machine TM = ( Q , Ω , Γ , ∆) Q : finite set of states, q start , q accept ∈ Q , q start � = q accept Ω: input alphabet Γ: tape alphabet with Ω ⊆ Γ and � ∈ Γ \ Ω ∆: transition relation ∆ ⊆ ( Q \ { q accept } ) × Q × Γ × Γ × { l , n , r } state transition, symbol replacement, head movement 10/21

Turing machine TM = ( Q , Ω , Γ , ∆) Q : finite set of states, q start , q accept ∈ Q , q start � = q accept Ω: input alphabet Γ: tape alphabet with Ω ⊆ Γ and � ∈ Γ \ Ω ∆: transition relation ∆ ⊆ ( Q \ { q accept } ) × Q × Γ × Γ × { l , n , r } state transition, symbol replacement, head movement It has one two-sided infinite tape. 10/21

Turing machine TM = ( Q , Ω , Γ , ∆) Q : finite set of states, q start , q accept ∈ Q , q start � = q accept Ω: input alphabet Γ: tape alphabet with Ω ⊆ Γ and � ∈ Γ \ Ω ∆: transition relation ∆ ⊆ ( Q \ { q accept } ) × Q × Γ × Γ × { l , n , r } state transition, symbol replacement, head movement It has one two-sided infinite tape. Q × � ∞ Γ ∗ × Γ ∗ � ∞ : configurations with the usual transition step c ⊢ TM c ′ based on ∆. 10/21

Turing machine TM = ( Q , Ω , Γ , ∆) Q : finite set of states, q start , q accept ∈ Q , q start � = q accept Ω: input alphabet Γ: tape alphabet with Ω ⊆ Γ and � ∈ Γ \ Ω ∆: transition relation ∆ ⊆ ( Q \ { q accept } ) × Q × Γ × Γ × { l , n , r } state transition, symbol replacement, head movement It has one two-sided infinite tape. Q × � ∞ Γ ∗ × Γ ∗ � ∞ : configurations with the usual transition step c ⊢ TM c ′ based on ∆. The recognized language: w ∈ Ω ∗ contributes to L ( TM ) if and only if q accept is reachable from start configuration ( q start , � ∞ , w � ∞ ). 10/21

Transformation of Turing Machines into Context-Dependent Fusion Grammars Main construction steps: 1. Representation of the TM by a hypergraph. 2. Generation of arbitrary inputs on the tape. 3. Simulation of a transition step of the TM. 11/21

Transformation of Turing Machines into Context-Dependent Fusion Grammars Main construction steps: 1. Representation of the TM by a hypergraph. 2. Generation of arbitrary inputs on the tape. 3. Simulation of a transition step of the TM. (Context-dependent) fusion rules can only consume two complementary labeled hyperedges by a rule application. All modifications must be expressed in this way. 11/21

Representation of the Turing machine by a hypergraph TM is represented by its usual state graph. b / � / r q accept q start q aux a / c / r b / b / n 12/21