Generating Hypergraph Languages by (Context-dependent) Fusion Grammars and Splitting/Fusion Grammars Hans-J¨ org Kreowski, Sabine Kuske and Aaron Lye University of Bremen, Germany { kreo , kuske , lye } @informatik.uni-bremen.de 26.10.2019 29. GI-Theorietag: Automaten und Formale Sprachen 1/20
DNA computing Adleman’s experiment (1994): solution of the NP-hard Hamiltonian-path problem by a polynomial number of steps ◮ constructing short DNA double strands ◮ doubling by polymerase chain reaction: n repetitions yield 2 n copies ◮ fusion of complementary sticky ends complementarity: ( A , T ) and ( C , G ) ◮ reading (sequencing): filtering of DNA molecules of certain lengths and with certain substrands 2/20
DNA computing ❀ Fusion grammar (ICGT 2017) ◮ constructing short DNA double strands ◮ doubling by polymerase chain reaction: n repetitions yield 2 n copies ◮ fusion of complementary sticky ends complementarity: ( A , T ) and ( C , G ) ◮ reading (sequencing): filtering of DNA molecules of certain lengths and with certain substrands 3/20
DNA computing ❀ Fusion grammar (ICGT 2017) ◮ constructing short DNA constructing initial hypergraph; double strands connected components acting as ◮ doubling by polymerase molecules chain reaction: multiplication of connected n repetitions yield 2 n copies components ◮ fusion of complementary fusion of complementary labeled sticky ends hyperedges complementarity: complementarity: ( A , T ) and ( C , G ) ( A , A ) for each fusion label A ◮ reading (sequencing): reading: filtering of DNA molecules filtering of connected of certain lengths and with components with certain certain substrands labeling 3/20
Hypergraph We consider hypergraphs over Σ with hyperedges like v 1 v k 1 . . . k 1 1 A k 2 1 . . . w k 2 w 1 where v 1 · · · v k 1 is a sequence of source nodes w 1 · · · w k 2 is a sequence of target nodes A ∈ Σ is a label. The class of all hypergraphs over Σ is denoted by H Σ . 4/20
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 k 1 v ′ v ′ v 1 1 . . . k 1 . . . k 1 k 1 1 1 fr ( A ) = type ( A ) = ( k 1 , k 2 ) A A 1 k 2 k 2 1 . . . . . . w ′ w 1 w k 2 w ′ k 2 1 fr ( A ) represents a fusion rule corresponding to A 5/20
Rule application 1. find a matching morphism g of fr ( A ) in the hypergraph H v ′ v k 1 v ′ v 1 1 . . . k 1 . . . k 1 1 k 1 1 H A A k 2 1 k 2 1 . . . . . . w ′ w 1 w k 2 w ′ 1 k 2 6/20
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 k 1 v ′ v 1 1 . . . k 1 . . . I . . . . . . w ′ w 1 w k 2 w ′ 1 k 2 6/20
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 edges v k 1 = v ′ v 1 = v ′ 1 k 1 . . . H ′ . . . w k 2 = w ′ w 1 = w ′ 1 k 2 6/20
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 edges v k 1 = v ′ v 1 = v ′ 1 k 1 . . . H ′ . . . w k 2 = w ′ w 1 = w ′ 1 k 2 6/20
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 edges 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 = 6/20
Fusion grammar FG = ( Z , F , M , T ) ◮ Z ∈ H F ∪ F ∪ M ∪ T finite start hypergraph F , M , T ⊆ Σ, fusion, marker, terminal alphabet (all finite) M ∩ ( F ∪ F ) = ∅ , T ∩ ( F ∪ F ) = ∅ = T ∩ M 7/20
Fusion grammar FG = ( Z , F , M , T ) ◮ Z ∈ H F ∪ F ∪ M ∪ T finite start hypergraph F , M , T ⊆ Σ, fusion, marker, terminal alphabet (all finite) M ∩ ( F ∪ F ) = ∅ , T ∩ ( F ∪ F ) = ∅ = T ∩ M ◮ A direct derivation is either fr ( A ) H ′ ⇒ for some A ∈ F or H = m m · H = � ⇒ m ( C ) · C H = for some multiplicity C ∈C ( H ) m : C ( H ) → N . where C ( H ) denotes the set of connected components of H . ◮ Derivations are defined by the reflexive and transitive closure. 7/20
Fusion grammar FG = ( Z , F , M , T ) ◮ Z ∈ H F ∪ F ∪ M ∪ T finite start hypergraph F , M , T ⊆ Σ, fusion, marker, terminal alphabet (all finite) M ∩ ( F ∪ F ) = ∅ , T ∩ ( F ∪ F ) = ∅ = T ∩ M ◮ A direct derivation is either fr ( A ) H ′ ⇒ for some A ∈ F or H = m m · H = � ⇒ m ( C ) · C H = for some multiplicity C ∈C ( H ) m : C ( H ) → N . where C ( H ) denotes the set of connected components of H . ◮ Derivations are defined by the reflexive and transitive closure. ◮ The generated language ∗ L ( FG ) = { rem M ( Y ) | Z = ⇒ H , Y ∈ C ( H ) ∩ ( H T ∪ M − H T ) } , where rem M ( Y ) removes all marker hyperedges from Y . 7/20
Pseudotori Let F = { N , W } with k ( N ) = k ( W ) = 1 and N = S , W = E . • N , F , { µ } , {∗} ) PSEUDOTORI = ( • • W E µ • S 8/20
Pseudotori Let F = { N , W } with k ( N ) = k ( W ) = 1 and N = S , W = E . • N , F , { µ } , {∗} ) PSEUDOTORI = ( • • W E µ • S • • N N = m 20 · ⇒ • • • • W E W E µ µ • • S S 8/20
Pseudotori Let F = { N , W } with k ( N ) = k ( W ) = 1 and N = S , W = E . • N , F , { µ } , {∗} ) PSEUDOTORI = ( • • W E µ • S • • N N = m 20 · ⇒ • • • • W E W E µ µ • • S S fr ( N ) , fr ( W ) 22 N N N N N N N W E W E N E S W W E W E S N N S W W E W E E S 8/20 S S S S S S
Pseudotori Let F = { N , W } with k ( N ) = k ( W ) = 1 and N = S , W = E . • N , F , { µ } , {∗} ) PSEUDOTORI = ( • • W E µ • S • • N N = m 12 · ⇒ • • • • W E W E µ µ • • S S fr ( N ) , fr ( W ) 17 N N N N W E W E W E S S S S 9/20
Pseudotori Let F = { N , W } with k ( N ) = k ( W ) = 1 and N = S , W = E . • N , F , { µ } , {∗} ) PSEUDOTORI = ( • • W E µ • S • • N N = m 12 · ⇒ • • • • W E W E µ µ • • S S fr ( N ) , fr ( W ) 17 N N N N W E ∗ ∗ W E W E S S S S 9/20
Transformation of hyperedge replacement grammars into fusion grammars HRG = ( N , T , P , S ) N , T , non-terminal, terminal alphabet, P set of rules (all finite), S ∈ N Rules of the form r = ( A , R , ext ) A ∈ N , R ∈ H Σ , ext sequence of k ( A ) vertices of R . 10/20
Transformation of hyperedge replacement grammars into fusion grammars HRG = ( N , T , P , S ) N , T , non-terminal, terminal alphabet, P set of rules (all finite), S ∈ N Rules of the form r = ( A , R , ext ) A ∈ N , R ∈ H Σ , ext sequence of k ( A ) vertices of R . Application of r : • • • • 2 1 A R = ⇒ k ( A ) r • • 10/20
Transformation of hyperedge replacement grammars into fusion grammars HRG = ( N , T , P , S ) N , T , non-terminal, terminal alphabet, P set of rules (all finite), S ∈ N Rules of the form r = ( A , R , ext ) A ∈ N , R ∈ H Σ , ext sequence of k ( A ) vertices of R . Application of r : • • • • 2 1 A R = ⇒ k ( A ) r • • ∗ L ( HRG ) = { H | S = ⇒ H , H ∈ H T } 10/20
Transformation of hyperedge replacement grammars into fusion grammars HRG = ( N , T , P , S ) N , T , non-terminal, terminal alphabet, P set of rules (all finite), S ∈ N Rules of the form r = ( A , R , ext ) A ∈ N , R ∈ H Σ , ext sequence of k ( A ) vertices of R . Application of r : Idea of the transformation: F = N 1 • • • • • • fusion component 2 2 1 of r in the fusion R A R A = ⇒ k ( A ) grammar’s start r • • • k ( A ) hypergraph S with marker ∗ L ( HRG ) = { H | S = ⇒ H , H ∈ H T } Theorem L ( HRG ) = L ( FG ( HRG )) 10/20
The converse is not possible Theorem Fusion grammars are more powerful than hyperedge replacement grammars. Proof: L(pseudotori) contain tori of arbitrary size with underlying rectangular grids. Therefore, the language has unbounded treewidth whereas hyperedge replacement languages have bounded treewidth (Courcelle/Engelfriet). N N N N W E W E W E S S S S 11/20
Context-dependent fusion grammar CDFG = ( Z , F , M , T , P ) (LATA 2019) ◮ ( Z , F , M , T ) fusion grammar P finite set of context-dependent fusion rules with rules of the form ( fr ( A ) , PC , NC ) where PC , NC : sets of hypergraph morphisms with domain fr ( A ) 12/20
Context-dependent fusion grammar CDFG = ( Z , F , M , T , P ) (LATA 2019) ◮ ( Z , F , M , T ) fusion grammar P finite set of context-dependent fusion rules with rules of the form ( fr ( A ) , PC , NC ) where PC , NC : sets of hypergraph morphisms with domain fr ( A ) ◮ A direct derivation is either cdfr H ′ for some cdfr ∈ P , H = ⇒ i.e.. application of fr ( A ) provided that the PC -contexts are present, and the NC -contexts not present (in the usual way of context conditions), or m m · H = � H = ⇒ m ( C ) · C for some multiplicity m : C ( H ) → N . C ∈C ( H ) ◮ derivations and generated languages as before. 12/20
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