Generating Hypergraph Languages by (Context-dependent) Fusion - - PowerPoint PPT Presentation

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Generating Hypergraph Languages by (Context-dependent) Fusion Grammars and Splitting/Fusion Grammars

Hans-J¨

• rg 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

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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 2n copies ◮ fusion of complementary sticky ends complementarity: (A, T) and (C, G) ◮ reading (sequencing): filtering of DNA molecules

• f certain lengths and with

certain substrands

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DNA computing ❀ Fusion grammar (ICGT 2017)

◮ constructing short DNA double strands ◮ doubling by polymerase chain reaction: n repetitions yield 2n copies ◮ fusion of complementary sticky ends complementarity: (A, T) and (C, G) ◮ reading (sequencing): filtering of DNA molecules

• f certain lengths and with

certain substrands

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DNA computing ❀ Fusion grammar (ICGT 2017)

◮ constructing short DNA double strands ◮ doubling by polymerase chain reaction: n repetitions yield 2n copies ◮ fusion of complementary sticky ends complementarity: (A, T) and (C, G) ◮ reading (sequencing): filtering of DNA molecules

• f certain lengths and with

certain substrands constructing initial hypergraph; connected components acting as molecules multiplication of connected components fusion of complementary labeled hyperedges complementarity: (A, A) for each fusion label A reading: filtering of connected components with certain labeling

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Hypergraph

We consider hypergraphs over Σ with hyperedges like vk1 . . . v1 A wk2 . . . w1 k1 1 k2 1 where v1 · · · vk1 is a sequence of source nodes w1 · · · wk2 is a sequence of target nodes A ∈ Σ is a label. The class of all hypergraphs over Σ is denoted by HΣ.

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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). fr(A) = vk1 . . . v1 v′

1 . . .

v′

k1

A A wk2 . . . w1 w′

1

. . . w′

k2

k1 1 k2 1 k1 1 k2 1 type(A) = (k1, k2) fr(A) represents a fusion rule corresponding to A

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Rule application

• 1. find a matching morphism g of fr(A) in the hypergraph H

H vk1 . . . v1 v′

1 . . .

v′

k1

A A wk2 . . . w1 w′

1

. . . w′

k2

k1 1 k2 1 k1 1 k2 1

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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)

I vk1 . . . v1 v′

1 . . .

v′

k1

wk2 . . . w1 w′

1

. . . w′

k2

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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 H′ vk1 = v′

k1

. . . v1 = v′

1

wk2 = w′

k2

. . . w1 = w′

1

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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 H′ vk1 = v′

k1

. . . v1 = v′

1

wk2 = w′

k2

. . . w1 = w′

1

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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 H′ vk1 = v′

k1

. . . v1 = v′

1

wk2 = w′

k2

. . . w1 = w′

1

Rule application is denoted by H = ⇒

fr(A) H′.

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Fusion grammar FG = (Z, F, M, T)

◮ Z ∈ HF∪F∪M∪T finite start hypergraph F, M, T ⊆ Σ, fusion, marker, terminal alphabet (all finite) M ∩ (F ∪ F) = ∅, T ∩ (F ∪ F) = ∅ = T ∩ M

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Fusion grammar FG = (Z, F, M, T)

◮ Z ∈ HF∪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 H = ⇒

fr(A) H′

for some A ∈ F or H = ⇒

m m · H = C∈C(H)

m(C) · C for some multiplicity m: C(H) → N. where C(H) denotes the set of connected components of H. ◮ Derivations are defined by the reflexive and transitive closure.

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Fusion grammar FG = (Z, F, M, T)

◮ Z ∈ HF∪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 H = ⇒

fr(A) H′

for some A ∈ F or H = ⇒

m m · H = C∈C(H)

m(C) · C for some multiplicity 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) = {remM(Y ) | Z

∗

= ⇒ H, Y ∈ C(H) ∩ (HT∪M − HT)}, where remM(Y ) removes all marker hyperedges from Y .

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Pseudotori

Let F = {N, W } with k(N) = k(W ) = 1 and N = S, W = E. PSEUDOTORI = (

• µ

W S E N

, F, {µ}, {∗})

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Pseudotori

Let F = {N, W } with k(N) = k(W ) = 1 and N = S, W = E. PSEUDOTORI = (

• µ

W S E N

, F, {µ}, {∗})

• µ

W S E N

= ⇒

m 20·

• µ

W S E N

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Pseudotori

Let F = {N, W } with k(N) = k(W ) = 1 and N = S, W = E. PSEUDOTORI = (

• µ

W S E N

, F, {µ}, {∗})

• µ

W S E N

= ⇒

m 20·

• µ

W S E N

fr(N), fr(W ) 22

W W E E N S N S S N N S N S N S N S W E W E W E N N W S E W E S W S E N

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Pseudotori

Let F = {N, W } with k(N) = k(W ) = 1 and N = S, W = E. PSEUDOTORI = (

• µ

W S E N

, F, {µ}, {∗})

• µ

W S E N

= ⇒

m 12·

• µ

W S E N

fr(N), fr(W ) 17

N S N S N S N S W E W E W E

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Pseudotori

Let F = {N, W } with k(N) = k(W ) = 1 and N = S, W = E. PSEUDOTORI = (

• µ

W S E N

, F, {µ}, {∗})

• µ

W S E N

= ⇒

m 12·

• µ

W S E N

fr(N), fr(W ) 17

N S N S N S N S W E W E W E

∗ ∗

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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 ∈ HΣ, ext sequence of k(A) vertices of R.

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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 ∈ HΣ, ext sequence of k(A) vertices of R. Application of r:

• A

2 1 k(A)

= ⇒

r

• R

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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 ∈ HΣ, ext sequence of k(A) vertices of R. Application of r:

• A

2 1 k(A)

= ⇒

r

• R

L(HRG) = {H | S

∗

= ⇒ H, H ∈ HT}

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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 ∈ HΣ, ext sequence of k(A) vertices of R. Application of r:

• A

2 1 k(A)

= ⇒

r

• R

L(HRG) = {H | S

∗

= ⇒ H, H ∈ HT} Idea of the transformation: F = N

R A

• 2

1 k(A)

fusion component

• f r in the fusion

grammar’s start hypergraph S with marker

Theorem

L(HRG) = L(FG(HRG))

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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 S N S N S N S W E W E W E

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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)

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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 H = ⇒

cdfr H′ for some cdfr ∈ P,

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 H = ⇒

m m · H = C∈C(H)

m(C) · C for some multiplicity m: C(H) → N. ◮ derivations and generated languages as before.

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Generative power of context-dependent fusion grammars (GCM 2019)

Transformation of Turing machines into corresponding context-dependent fusion grammars.

Theorem

Let TM be a Turing machine. Let CDFG(TM) be the corresponding context-dependent fusion grammar. L(CDFG(TM)) = {sg(w) | w ∈ L(TM)} generated language recognized language sg(w) graph representation of a string w.

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Transformation of Turing Machines into Context-Dependent Fusion Grammars

Main construction steps:

• 1. Representation of the TM by a hypergraph (using the usual

state graph representation)

• 2. Generation of arbitrary inputs on the tape (using the string

graph representation of strings)

• 3. Simulation of a transition step of the TM

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Transformation of Turing Machines into Context-Dependent Fusion Grammars

Main construction steps:

• 1. Representation of the TM by a hypergraph (using the usual

state graph representation)

• 2. Generation of arbitrary inputs on the tape (using the string

graph representation of strings)

• 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.

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Splitting rule with fixed disjoint context (ICGT 2018)

splitting is the inverse of fusion srfdc(A, a) consists of a splitting rule sr(A) and a morphism a: [k(A)] → X for some context X.

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Splitting rule with fixed disjoint context (ICGT 2018)

splitting is the inverse of fusion srfdc(A, a) consists of a splitting rule sr(A) and a morphism a: [k(A)] → X for some context X. It is applicable to H if H can be split into H′ and X (with an additional A-hyperedge)

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Splitting rule with fixed disjoint context (ICGT 2018)

splitting is the inverse of fusion srfdc(A, a) consists of a splitting rule sr(A) and a morphism a: [k(A)] → X for some context X. It is applicable to H if H can be split into H′ and X (with an additional A-hyperedge) Example cut = (A, 2• 1•

• ⊇ [2])
• =

⇒

cut

• A +

A •

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Splitting/fusion grammar SFG = (Z, F, M, T, SR)

◮ (Z, F, M, T) fusion grammar SR finite set of splitting rules with fixed disjoint context.

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Splitting/fusion grammar SFG = (Z, F, M, T, SR)

◮ (Z, F, M, T) fusion grammar SR finite set of splitting rules with fixed disjoint context. ◮ A direct derivation is either H = ⇒

fr(A) H′

for some A ∈ F or H = ⇒

m m · H =

• C∈C(H)

m(C) · C for some m: C(H) → N or H = ⇒

srfdc(A,a) H′

for some A ∈ F and a: K → X.

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Splitting/fusion grammar SFG = (Z, F, M, T, SR)

◮ (Z, F, M, T) fusion grammar SR finite set of splitting rules with fixed disjoint context. ◮ A direct derivation is either H = ⇒

fr(A) H′

for some A ∈ F or H = ⇒

m m · H =

• C∈C(H)

m(C) · C for some m: C(H) → N or H = ⇒

srfdc(A,a) H′

for some A ∈ F and a: K → X. ◮ derivation and generated language as before

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Transformation of Chomsky grammars into splitting/fusion grammars

Let (N, T, P, S) be a Chomsky grammar. Let p = (u1 . . . uk, v1 . . . vl) ∈ P. Let x1 . . . xn = x1 . . . xi−1u1 . . . ukxi+k . . . xn Then x1 . . . xi−1u1 . . . ukxi+k . . . xn = ⇒

p x1 . . . xi−1v1 . . . vlxi+k . . . xn

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Transformation of Chomsky grammars into splitting/fusion grammars

Let (N, T, P, S) be a Chomsky grammar. Let p = (u1 . . . uk, v1 . . . vl) ∈ P. Let x1 . . . xn = x1 . . . xi−1u1 . . . ukxi+k . . . xn Then x1 . . . xi−1u1 . . . ukxi+k . . . xn = ⇒

p x1 . . . xi−1v1 . . . vlxi+k . . . xn

Adapting a transformation of Chomsky grammars into iterated splicing systems (cf. [P˘ aun,Rozenberg,Salomaa:1998]). cyc(x1 . . . xn) = u1 uk xn x1 xi−1 xi+k be

∗

= ⇒ v1 vl xn x1 xi−1 xi+k be

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Simulating a direct derivation

p = (u1 . . . uk, v1 . . . vl) ∈ P cyc(x1 . . . xn)act,i = xn x1 x2 be xi−1 xi xi+1 xn x1 x2 act = u1 uk xn x1 xi−1 xi+k be act

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Simulating a direct derivation

p = (u1 . . . uk, v1 . . . vl) ∈ P cyc(x1 . . . xn)act,i = xn x1 x2 be xi−1 xi xi+1 xn x1 x2 act = u1 uk xn x1 xi−1 xi+k be act

• 1. splitting, i.e.,

cyc(x1 . . . xn)act,i = ⇒

sr(Ap,u1...,uk)

xi−1 xi+k xn x1 Ap be +

• . . .
• u1

uk

• Ap

act u1 . . . uk =

• 1
• . . .•
• 2

act u1 uk ⊇

• 1
• 2

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Simulating a direct derivation

p = (u1 . . . uk, v1 . . . vl) ∈ P cyc(x1 . . . xn)act,i = xn x1 x2 be xi−1 xi xi+1 xn x1 x2 act = u1 uk xn x1 xi−1 xi+k be act

• 1. splitting, i.e.,

cyc(x1 . . . xn)act,i = ⇒

sr(Ap,u1...,uk)

xi−1 xi+k xn x1 Ap be +

• . . .
• u1

uk

• Ap

act u1 . . . uk =

• 1
• . . .•
• 2

act u1 uk ⊇

• 1
• 2
• 2. fusion, i.e.,
• . . . •
• v1

vl Ap act = ⇒

fr(Ap)

+ cyc(x1 . . . xi−1Apxi+k . . . xn) v1 vl xn x1 xi−1 xi+k be act

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Generative power of splitting/fusion grammars

Theorem

Let CG = (N, T, P, S) be a Chomsky grammar and SFG(CG) the corresponding splitting/fusion grammar. Then cyc(L(CG)) = L(SFG(CG)). CG SFG(CG) L(CG) cyc(L(CG)) = L(SFG(CG)) transform generate generate cyc

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Conclusion

• 1. Fusion grammars can simulate hyperedge replacement

grammars; but are more powerful.

• 2. Context-dependent fusion grammars and splitting/fusion

grammars can generate all recursively enumerable string languages (up to representation) and are universal in this respect.

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Conclusion

• 1. Fusion grammars can simulate hyperedge replacement

grammars; but are more powerful.

• 2. Context-dependent fusion grammars and splitting/fusion

grammars can generate all recursively enumerable string languages (up to representation) and are universal in this respect. Future work

• 1. Is it true (as we conjecture) that fusion grammars are not

universal?

• 2. Is it true (as we conjectiure) that fusion grammars with only

positive context conditions (or only negative ones) are universal?

• 3. How does a natural transformation of context-dependent

fusion grammars into splitting/fusion grammars or the other way round look like?

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Conclusion

• 1. Fusion grammars can simulate hyperedge replacement

grammars; but are more powerful.

• 2. Context-dependent fusion grammars and splitting/fusion

grammars can generate all recursively enumerable string languages (up to representation) and are universal in this respect. Future work

• 1. Is it true (as we conjecture) that fusion grammars are not

universal?

• 2. Is it true (as we conjectiure) that fusion grammars with only

positive context conditions (or only negative ones) are universal?

• 3. How does a natural transformation of context-dependent

fusion grammars into splitting/fusion grammars or the other way round look like? Thank you! Questions?