Generating Hypergraph Languages by (Context-dependent ... · MINT-Doktorand*innen-Seminar. 2/30 Dissertation I PhD in theoretical computer science I Working title 2018: Hypergraph

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

Splitting/Fusion Grammars

Aaron Lye

University of Bremen, [email protected]

24.01.2020

MINT-Doktorand*innen-Seminar

2/30

Dissertation

I PhD in theoretical computer scienceI Working title 2018:

Hypergraph Transformation

I Working title 2020:

Fusion and Splitting/Fusion Grammars -Generation of Hypergraph Languages andRelation to other Modeling Frameworks

3/30

Hypergraph

I A hypergraph consists of a set of objects (vertices) withrelations (hyperedges) between them (connecting arbitrarymany vertices).

I A hyperedge with attachment vertices looks likevk1

. . .v1

A

wk2

. . .w1

k11

k21

whereA ∈ Σ is a label,v1 · · · vk1 is a sequence of source vertices,w1 · · ·wk2 is a sequence of target vertices.

I labels and arrowheads may be omitted.I The class of all hypergraphs over Σ is denoted by HΣ.

4/30

Hypergraph transformation

I Hypergraphs are static structures.I Dynamics are expressed by hypergraph transformation

combining concepts of hypergraphs and rules.

I (Hyper-)Graph transformation combines:I category theoryI formal language theoryI concurrency theory

I It has a spectrum of applications.

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Hypergraph transformation

I Hypergraphs are static structures.I Dynamics are expressed by hypergraph transformation

combining concepts of hypergraphs and rules.

I (Hyper-)Graph transformation combines:I category theoryI formal language theoryI concurrency theory

I It has a spectrum of applications.

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Social networks

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Social networks

Rule:Let A,B,C be vertices.If (A,B), (B,C), (C ,A) are edges, then add (A,B,C).

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Social networks


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Social networks


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Motivation for fusion grammarsFusion processes in:I DNA

computingI chemistryI tilingI fractal

geometryI visual modelingI etc.

n + 14.1 MeVHe + 3.5 MeV44

H33H22

X

X

getorder A

getorder B

produce A

produce B

send A

send B

8/30

DNA computingAdleman’s experiment (1994): solution of the NP-hardHamiltonian-path problem by a polynomial number of steps

I constructing short DNAdouble strands

I doubling by polymerasechain reaction:n repetitions yield 2n copies

I fusion of complementarysticky endscomplementarity:(A,T ) and (C ,G)

I reading (sequencing):filtering of DNA moleculesof certain lengths and withcertain substrands

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DNA computing ; Fusion grammar





constructing initial hypergraph;connected components acting asmoleculesmultiplication of connectedcomponentsfusion of complementary labeledhyperedgescomplementarity:(A,A) for each fusion label Areading:filtering of connectedcomponents with certainlabeling

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DNA computing ; Fusion grammar





constructing initial hypergraph;connected components acting asmoleculesmultiplication of connectedcomponentsfusion of complementary labeledhyperedgescomplementarity:(A,A) for each fusion label Areading:filtering of connectedcomponents with certainlabeling

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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

k11

k21

k11

k21

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

2. remove the images of the two hyperedges of fr(A)3. identify corresponding source and target vertices of the

removed edges

H

vk1. . .

v1 v ′1. . .

v ′k1

A A

wk2

. . .w1 w ′1

. . .w ′k2

k11

k21

k11

k21

Rule application is denoted by H =⇒fr(A)

H ′.

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Rule application

1. find a matching morphism g of fr(A) in the hypergraph H2. remove the images of the two hyperedges of fr(A)

3. identify corresponding source and target vertices of theremoved edges

I

vk1. . .

v1 v ′1. . .

v ′k1

wk2

. . .w1 w ′1

. . .w ′k2


H ′.

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Rule application

1. find a matching morphism g of fr(A) in the hypergraph H2. 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


H ′.

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Rule application


removed edges

H ′

vk1 = v ′k1. . .

v1 = v ′1

wk2 = w ′k2

. . .w1 = w ′1


H ′.

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Rule application


removed edges

H ′

vk1 = v ′k1. . .

v1 = v ′1

wk2 = w ′k2

. . .w1 = w ′1


H ′.

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Fusing social networks

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Fusing social networks

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Fusion grammar FG = (Z , F , T )I Z finite start hypergraph

F ,T ⊆ Σ, fusion, terminal alphabet (finite, disjoint)

I A direct derivation is eitherH =⇒

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.

I Derivations are defined by the reflexive and transitive closure,i.e., sequences of direct derivations.

I The generated language

L(FG) = {Y | Z ∗=⇒H,Y ∈ C(H) ∩HT )}

FG is simplified here: It actually contains also a set of markers Mused for distinguishing components contributing to the generatedlanguage from others.

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F ,T ⊆ Σ, fusion, terminal alphabet (finite, disjoint)I A direct derivation is either

H =⇒fr(A)

H ′ for some A ∈ F or

H =⇒m

m · H =∑





L(FG) = {Y | Z ∗=⇒H,Y ∈ C(H) ∩HT )}


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H =⇒fr(A)


H =⇒m

m · H =∑





L(FG) = {Y | Z ∗=⇒H,Y ∈ C(H) ∩HT )}


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H =⇒fr(A)


H =⇒m

m · H =∑





L(FG) = {Y | Z ∗=⇒H,Y ∈ C(H) ∩HT )}


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SIER

SIER = (Z , {4}, {#}, {∗})where 4 ∈ Σ, 4 = N, k(4) = 3 and

Z =2

•3•

1•

4 #

23

1

+

•2

•3•

2•

3

•1

•1

•2

•3

•1

4 4

4

2

3

1

N +• •

•

23

1

N

A derivation may be the following.

Z =⇒m 2

•3•

1•

4 #

23

1

+

•2

•3•

2•

3

•1

•1

•2

•3

•1

4 4

4

2

3

1

N + 3 ·• •

•

23

1

N

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SIER

SIER = (Z , {4}, {#}, {∗})where 4 ∈ Σ, 4 = N, k(4) = 3 and

Z =2

•3•

1•

4 #

23

1

+

•2

•3•

2•

3

•1

•1

•2

•3

•1

4 4

4

2

3

1

N +• •

•

23

1

N

A derivation may be the following.

Z =⇒m 2

•3•

1•

4 #

23

1

+

•2

•3•

2•

3

•1

•1

•2

•3

•1

4 4

4

2

3

1

N + 3 ·• •

•

23

1

N

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SIER

Z =⇒m 2

•3•

1•

4 #

23

1

+

•2

•3•

2•

3

•1

•1

•2

•3

•1

4 4

4

2

3

1

N + 3 ·• •

•

23

1

N

3=⇒fr(4) 2

•3•

1•

4 #

23

1

+

• •

•

•

• • N

23

1

• •

•

•

• • #

23

1

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SIER

Z =⇒m 2

•3•

1•

4 #

23

1

+

•2

•3•

2•

3

•1

•1

•2

•3

•1

4 4

4

2

3

1

N + 3 ·• •

•

23

1

N

3=⇒fr(4) 2

•3•

1•

4 #

23

1

+

• •

•

•

• • N

23

1

=⇒fr(4)

• •

•

•

• • #

23

1

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SIER

Z =⇒m 2

•3•

1•

4 #

23

1

+

•2

•3•

2•

3

•1

•1

•2

•3

•1

4 4

4

2

3

1

N + 3 ·• •

•

23

1

N

3=⇒fr(4) 2

•3•

1•

4 #

23

1

+

• •

•

•

• • N

23

1

=⇒fr(4)

• •

•

•

• • #

23

1

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SIER

Z =⇒m 2

•3•

1•

4 #

23

1

+

•2

•3•

2•

3

•1

•1

•2

•3

•1

4 4

4

2

3

1

N + 3 ·• •

•

23

1

N

3=⇒fr(4) 2

•3•

1•

4 #

23

1

+

• •

•

•

• • N

23

1

=⇒fr(4)

• •

•

•

• • #

23

1

∈ L(SIER)

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

PSEUDOTORI = ( •

•

•

•

W

S

E

N

,F , {µ}, {∗})

•

•

•

•

W

S

E

N

=⇒m

20· •

•

•

•

W

S

E

N

fr(N), fr(W )22

W

W

E

E

N

S

N

S

SN

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

19/30


PSEUDOTORI = ( •

•

•

•

W

S

E

N

,F , {µ}, {∗})

•

•

•

•

W

S

E

N

=⇒m

20· •

•

•

•

W

S

E

N

fr(N), fr(W )22

W

W

E

E

N

S

N

S

SN

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 = ( •

•

•

•

W

S

E

N

,F , {µ}, {∗})

•

•

•

•

W

S

E

N

=⇒m

20· •

•

•

•

W

S

E

N

fr(N), fr(W )22

W

W

E

E

N

S

N

S

SN

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

20/30


PSEUDOTORI = ( •

•

•

•

W

S

E

N

,F , {µ}, {∗})

•

•

•

•

W

S

E

N

=⇒m

12· •

•

•

•

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 = ( •

•

•

•

W

S

E

N

,F , {µ}, {∗})

•

•

•

•

W

S

E

N

=⇒m

12· •

•

•

•

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 intofusion grammars

HRG = (N,T ,P,S)N,T , non-terminal, terminal alphabet, P set of rules (all finite), S ∈ NRules of the form r = (A,R, ext)A ∈ N,R ∈ HΣ, ext sequence of k(A) vertices of R.

Application of r :

••

•

A2 1

k(A) =⇒r

••

•

R

L(HRG) = {H | S ∗=⇒H,H ∈ HT}

Idea of the transformation:F = N

RA

••

•

21

k(A)

fusion componentof r in the fusiongrammar’s starthypergraph

S with marker

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

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Application of r :

••

•

A2 1

k(A) =⇒r

••

•

R

L(HRG) = {H | S ∗=⇒H,H ∈ HT}


RA

••

•

21

k(A)


S with marker


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Application of r :

••

•

A2 1

k(A) =⇒r

••

•

R

L(HRG) = {H | S ∗=⇒H,H ∈ HT}


RA

••

•

21

k(A)


S with marker


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Application of r :

••

•

A2 1

k(A) =⇒r

••

•

R

L(HRG) = {H | S ∗=⇒H,H ∈ HT}


RA

••

•

21

k(A)


S with marker


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The converse is not possible

TheoremFusion grammars are more powerful than hyperedge replacementgrammars.

Proof: L(pseudotori) contain tori of arbitrary size with underlyingrectangular grids. Therefore, the language has unboundedtreewidth whereas hyperedge replacement languages have boundedtreewidth (Courcelle/Engelfriet).

N

S

N

S

N

S

N

S

W E

W E

W E

23/30

Context-dependent fusion grammarCDFG = (Z , F , M, T , P)

I (Z ,F ,M,T ) fusion grammarP finite set of context-dependent fusion ruleswith rules of the form (fr(A),PC ,NC)where PC ,NC : sets of hypergraph morphisms with domain fr(A)


cdfrH ′ for some cdfr ∈ P,

i.e.. application of fr(A) provided thatthe PC -contexts are present, and the NC -contexts not present(in the usual way of context conditions), orH =⇒

mm · H =

∑C∈C(H)

m(C) · C for some multiplicity m : C(H)→ N.

I derivations and generated languages as before.

23/30

Context-dependent fusion grammarCDFG = (Z , F , M, T , P)

I (Z ,F ,M,T ) fusion grammarP finite set of context-dependent fusion ruleswith rules of the form (fr(A),PC ,NC)where PC ,NC : sets of hypergraph morphisms with domain fr(A)


cdfrH ′ for some cdfr ∈ P,

i.e.. application of fr(A) provided thatthe PC -contexts are present, and the NC -contexts not present(in the usual way of context conditions), orH =⇒

mm · H =

∑C∈C(H)

m(C) · C for some multiplicity m : C(H)→ N.

I derivations and generated languages as before.

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Generative power of context-dependent fusion grammars

Transformation of Turing machines into correspondingcontext-dependent fusion grammars.

TheoremLet TM be a Turing machine. Let CDFG(TM) be thecorresponding 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-dependentfusion grammars

Main construction steps:

1. Representation of the TM by a hypergraph (using the usualstate graph representation)

2. Generation of arbitrary inputs on the tape (using the stringgraph representation of strings)

3. Simulation of a transition step of the TM

(Context-dependent) fusion rules can only consume twocomplementary labeled hyperedges by a rule application.All modifications must be expressed in this way.

25/30

Transformation of Turing machines into context-dependentfusion grammars

Main construction steps:

1. Representation of the TM by a hypergraph (using the usualstate graph representation)

2. Generation of arbitrary inputs on the tape (using the stringgraph representation of strings)

3. Simulation of a transition step of the TM

(Context-dependent) fusion rules can only consume twocomplementary labeled hyperedges by a rule application.All modifications must be expressed in this way.

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DNA computing

fusion(ligation)

splitting(triggered by enzymes)

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DNA computing

fusion(ligation)

splitting(triggered by enzymes)

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Splitting rule with fixed disjoint contextsplitting is the inverse of fusionsrfdc(A, a) consists of a splitting rule sr(A) and a morphisma : [k(A)]→ X for some context X .

It is applicable to H if H can be split into H ′ and X (with anadditional A-hyperedge)

Example

cut = (A,2•1•

•••⊇ [2])

••••••

••

•••

=⇒cut •

•••••

••

A + A ••

•••

27/30

Splitting rule with fixed disjoint contextsplitting is the inverse of fusionsrfdc(A, a) consists of a splitting rule sr(A) and a morphisma : [k(A)]→ X for some context X .It is applicable to H if H can be split into H ′ and X (with anadditional A-hyperedge)

Example

cut = (A,2•1•

•••⊇ [2])

••••••

••

•••

=⇒cut •

•••••

••

A + A ••

•••

27/30

Splitting rule with fixed disjoint contextsplitting is the inverse of fusionsrfdc(A, a) consists of a splitting rule sr(A) and a morphisma : [k(A)]→ X for some context X .It is applicable to H if H can be split into H ′ and X (with anadditional A-hyperedge)

Example

cut = (A,2•1•

•••⊇ [2])

••••••

••

•••

=⇒cut •

•••••

••

A + A ••

•••

28/30

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

I (Z ,F ,M,T ) fusion grammarSR finite set of splitting rules with fixed disjoint context.

I A direct derivation is either

H =⇒fr(A)


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 .

I derivation and generated language as before

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H =⇒fr(A)


H =⇒m

m · H =∑


H =⇒srfdc(A,a)



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H =⇒fr(A)


H =⇒m

m · H =∑


H =⇒srfdc(A,a)



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Generative power of splitting/fusion grammars

TheoremLet CG = (N,T ,P,S) be a Chomsky grammarand 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 generatecyc

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Conclusion

1. Fusion grammars can simulate hyperedge replacementgrammars; but are more powerful.

2. Context-dependent fusion grammars and splitting/fusiongrammars can generate all recursively enumerable stringlanguages (up to representation) and are universal in thisrespect.

Future work: How powerful are fusion grammars?

Thank you! Questions?

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Conclusion





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Conclusion





Documents

Generating Hypergraph Languages by (Context-dependent ... · MINT-Doktorand*innen-Seminar. 2/30 Dissertation I PhD in theoretical computer science I Working title 2018: Hypergraph