[PPT] - Equivalence and Independence in Controlled Graph-Rewriting Processes PowerPoint Presentation

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Equivalence and Independence in Controlled Graph-Rewriting Processes

ICGT@STAF 2018, Toulouse G´ eza Kulcs´ ar1, Andrea Corradini2, Malte Lochau1

1Real-Time Systems Lab, TU Darmstadt 2Department of Computer Science, University of Pisa

geza.kulcsar@es.tu-darmstadt.de

June 26, 2018

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Motivation

Graph-rewriting systems are inherently non-deterministic

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Motivation

Graph-rewriting systems are inherently non-deterministic
When designing algorithms, need to add control structures:

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Motivation

Graph-rewriting systems are inherently non-deterministic
When designing algorithms, need to add control structures:
sequentialization

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Motivation

Graph-rewriting systems are inherently non-deterministic
When designing algorithms, need to add control structures:
sequentialization
conditionals

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Motivation

Graph-rewriting systems are inherently non-deterministic
When designing algorithms, need to add control structures:
sequentialization
conditionals
iteration and recursion

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Motivation

Graph-rewriting systems are inherently non-deterministic
When designing algorithms, need to add control structures:
sequentialization
conditionals
iteration and recursion
parallelism

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Motivation

Graph-rewriting systems are inherently non-deterministic
When designing algorithms, need to add control structures:
sequentialization
conditionals
iteration and recursion
parallelism
⇒ Controlled graph rewriting [Bunke, Sch¨

urr, Habel & Plump, Plump & Steinert, Kreowski et al.; PROGRES, Fujaba, eMoflon]

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Motivation

Graph-rewriting systems are inherently non-deterministic
When designing algorithms, need to add control structures:
sequentialization
conditionals
iteration and recursion
parallelism
⇒ Controlled graph rewriting [Bunke, Sch¨

urr, Habel & Plump, Plump & Steinert, Kreowski et al.; PROGRES, Fujaba, eMoflon]

However, mostly I/O semantics so far, not considering:

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Motivation

Graph-rewriting systems are inherently non-deterministic
When designing algorithms, need to add control structures:
sequentialization
conditionals
iteration and recursion
parallelism
⇒ Controlled graph rewriting [Bunke, Sch¨

urr, Habel & Plump, Plump & Steinert, Kreowski et al.; PROGRES, Fujaba, eMoflon]

However, mostly I/O semantics so far, not considering:
Concurrency in graph-rewriting algorithms

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Motivation

Graph-rewriting systems are inherently non-deterministic
When designing algorithms, need to add control structures:
sequentialization
conditionals
iteration and recursion
parallelism
⇒ Controlled graph rewriting [Bunke, Sch¨

urr, Habel & Plump, Plump & Steinert, Kreowski et al.; PROGRES, Fujaba, eMoflon]

However, mostly I/O semantics so far, not considering:
Concurrency in graph-rewriting algorithms
Reactive (non-terminating) specifications

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Introduction Unmarked Processes Marked Processes Abstraction Independence

In this paper...

Process-algebraic (CCS-like) syntax and semantics for

controlled graph rewriting

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Introduction Unmarked Processes Marked Processes Abstraction Independence

In this paper...

Process-algebraic (CCS-like) syntax and semantics for

controlled graph rewriting

Transition systems both for control processes and for their

executions on graphs

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Introduction Unmarked Processes Marked Processes Abstraction Independence

In this paper...

Process-algebraic (CCS-like) syntax and semantics for

controlled graph rewriting

Transition systems both for control processes and for their

executions on graphs

Expressiveness of our control language (encoding graph

programs by Habel & Plump)

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Introduction Unmarked Processes Marked Processes Abstraction Independence

In this paper...

Process-algebraic (CCS-like) syntax and semantics for

controlled graph rewriting

Transition systems both for control processes and for their

executions on graphs

Expressiveness of our control language (encoding graph

programs by Habel & Plump)

Handling of parallel rules and derivations by synchronization

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Introduction Unmarked Processes Marked Processes Abstraction Independence

In this paper...

Process-algebraic (CCS-like) syntax and semantics for

controlled graph rewriting

Transition systems both for control processes and for their

executions on graphs

Expressiveness of our control language (encoding graph

programs by Habel & Plump)

Handling of parallel rules and derivations by synchronization
An abstract semantics, with graphs up to isomorphism

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Introduction Unmarked Processes Marked Processes Abstraction Independence

In this paper...

Process-algebraic (CCS-like) syntax and semantics for

controlled graph rewriting

Transition systems both for control processes and for their

executions on graphs

Expressiveness of our control language (encoding graph

programs by Habel & Plump)

Handling of parallel rules and derivations by synchronization
An abstract semantics, with graphs up to isomorphism
Equivalence and congruence notions of CCS are reflected in
ur setting

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Graph-Rewriting Actions

DPO rules:

p : (L ← K → R)

. . . Parallel composition of DPO rules:

p1|p2 : ( L1 + L2 ← K1 + K2 → R1 + R2 )

R is the set of rule names, R∗ the set of parallel rule names

ranged over by ρ

Actions are pairs (ρ, N) ∈ Act where ρ ∈ R∗ and N ⊆ R is a

set of non-applicability conditions

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Unmarked Processes

Unmarked processes specify control over rule applications,

having a CCS-like syntax

Definition (Unmarked Process Term Syntax)

P, Q ::= 0 | γ.P | A | P + Q | P || Q where γ ranges over Act and A := P.

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Unmarked Transition System (UTS)

The semantics of unmarked processes is an LTS where states

are processes and transitions are labeled with actions

PRE

γ.P

γ

− → P

STOP

− → 0

CHOICE

P

γ

− → P′ P + Q

γ

− → P′

PAR

P

γ

− → P′ P || Q

γ

− → P′ || Q

REC

A := P P

α

− → P′ A α − → P′

SYNC

P

(ρ1,N1)

− − − − → P′ Q

(ρ2,N2)

− − − − → Q′ P || Q

(ρ1|ρ2,N1∪N2)

− − − − − − − − → P′ || Q′

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Rule Applications

G

ρ@m

= = ⇒ H

L K

l

G

m

D (PO)

f k

R H

r n g

(PO)

ρ

A linear derivation from G0 is a sequence of rule applications

G0

p1@m1

= = = ⇒ . . .

pn@mn

= = = = ⇒ Gn with no parallel rules

A parallel derivation from G0 is a sequence of rule applications

G0

ρ1@m1

= = = ⇒ . . .

ρn@mn

= = = = ⇒ Gn with (potentially) parallel rules ρi = pi1| . . . |pik

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Marked Transition System (MTS)

MARK

P

(ρ,N)

− − − → P′ G

ρ@m

= = ⇒ H ∀p ∈ N : G

p

= ⇒ (P, G)

(ρ,δ,N)

− − − − →M (P′, H)

L K

δ :=

l

G

m

D (PO)

f k

R H

r n g

(PO)

ρ

STOP

P

−

→ P′ (P, G) − →M (P′, G)

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Equivalence of Graph-Rewriting Processes

A trace of a state P is a sequence of transition labels starting

from it, e.g., (ρ1, δ1, N1)(ρ2, δ2, N2) . . . (ρn, δn, Nn)

P and Q are trace equivalent (P ≃T Q) if they have the same

set of traces

P and Q are bisimilar (P ≃BS Q) if for each P

α

− →M P′, there is Q

α

− →M Q′ such that P′ ≃BS Q′ (and vice versa)

Proposition

For any unmarked processes P, Q and graph G,

P ≃BS Q implies (P, G) ≃BS

M (Q, G)

P ≃T Q implies (P, G) ≃T

M (Q, G)

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Correspondence between Traces and Derivations

Each successful trace of a marked process (P, G) uniquely

identifies an underlying parallel derivation of R from G. (ρ1, δ1, N1)(ρ2, δ2, N2) . . . (ρn, δn, Nn)

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Correspondence between Traces and Derivations

Each successful trace of a marked process (P, G) uniquely

identifies an underlying parallel derivation of R from G. (ρ1, δ1, N1)(ρ2, δ2, N2) . . . (ρn, δn, Nn)

We can construct a process

PR = 0 +

p∈R

p.PR such that (PR, G) has as successful trace each linear derivation starting from a graph G...

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Correspondence between Traces and Derivations

Each successful trace of a marked process (P, G) uniquely

identifies an underlying parallel derivation of R from G. (ρ1, δ1, N1)(ρ2, δ2, N2) . . . (ρn, δn, Nn)

We can construct a process

PR = 0 +

p∈R

p.PR such that (PR, G) has as successful trace each linear derivation starting from a graph G...

...and we can do the same for parallel derivations:

QR = 0 +

(
p∈R

p.0 + ε.0) || QR

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Expressiveness: Encoding Graph Programs

Reference: a minimal control language, graph programs, being

computationally complete [Habel & Plump 2001]

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Expressiveness: Encoding Graph Programs

Reference: a minimal control language, graph programs, being

computationally complete [Habel & Plump 2001]

choice, sequential composition, as-long-as-possible iteration

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Expressiveness: Encoding Graph Programs

Reference: a minimal control language, graph programs, being

computationally complete [Habel & Plump 2001]

choice, sequential composition, as-long-as-possible iteration
Graph programs can be encoded into unmarked processes,

even without using parallel composition:

If GP = {p1, . . . , pn} is an elementary graph program, then

[ [GP] ] := n

i=1 pi.0.

[

[GP1; GP2] ] := [ [GP1] ] [ [GP2] ].

[

[GP ↓] ] := AGP↓ ∈ K where AGP↓ := [ [GP] ] AGP↓ + [ [GP] ].

where P Q := P[Q/0] (syntactic substitution)

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Expressiveness: Encoding Graph Programs

Reference: a minimal control language, graph programs, being

computationally complete [Habel & Plump 2001]

choice, sequential composition, as-long-as-possible iteration
Graph programs can be encoded into unmarked processes,

even without using parallel composition:

If GP = {p1, . . . , pn} is an elementary graph program, then

[ [GP] ] := n

i=1 pi.0.

[

[GP1; GP2] ] := [ [GP1] ] [ [GP2] ].

[

[GP ↓] ] := AGP↓ ∈ K where AGP↓ := [ [GP] ] AGP↓ + [ [GP] ].

where P Q := P[Q/0] (syntactic substitution)

[ [GP] ] is a process representing the termination criterion for GP iteration

defined inductively using non-applicability conditions
successful exactly if [

[GP] ] fails

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Abstract MTS

This “concrete” MTS definition is infinitely branching

Definition (Abstract Marked Transition System)

[G] denotes the isomorphism class of graph G and [δ] the isomorphism class of DPO diagram δ.

MARK

P

(ρ,N)

− − − → P′ G

ρ@m

= = ⇒ H ∀p ∈ N : G

p

= ⇒ (P, [G])

(ρ,[δ],N)

− − − − − →A (P′, [H])

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Abstraction Preserves Bisimilarity and Trace Equivalence

Proposition

For any unmarked processes P, Q and graphs G, H,

(P, G) ≃BS

M (Q, H) implies (P, [G]) ≃BS A

(Q, [H])

(P, G) ≃T

M (Q, H) implies (P, [G]) ≃T A (Q, [H])

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Abstract Bisimilarity is a Congruence

Given P, Q, R ∈ P with P ≃BS Q. (P + R, [G]) ≃BS

A

(Q + R, [G]), (P || R, [G]) ≃BS

A

(Q || R, [G]), and (γ.P, [G]) ≃BS

A

(γ.Q, [G]) for any γ ∈ Act

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Parallel Independence and Bisimulation

Let P0 := ρ1.(ρ2 || P) + ρ2.(ρ1 || P) and Q0 := ρ1.P || ρ2.0. There exist parallel independent applications ρ1@m1, ρ2@m2 on G, if and

nly if (P0, [G]) ≃BS

D (Q0, [G]).

P0 ρ1 || P

ρ2

ρ2 || P

ρ1

P

ρ2 ρ1

Q0 ρ1.P

ρ2

ρ2 || P

ρ1

P

ρ2 ρ1 ρ1 | ρ2

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Related Work

Concurrent Semantics
Formal notion of (concurrent) graph processes (Corradini et al.

1996, Baldan et al. 1999)

Semantics of control:
A denotational input/output semantics for controlled

graph-rewriting processes (Sch¨ urr 1996)

Composable graph transformation units (Kreowski et al. 2008)
Graph Programs, a graph programming language with an
perational semantics and results regarding computational

completeness (Plump and Habel 2001, Plump and Steinert 2009)

Tool support: Henshin, PROGRES, eMoflon, ...

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Conclusion and Ongoing Work

We proposed to extend graph rewriting by a process-algebraic

control layer and obtained (preserved) results from CCS theory

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Conclusion and Ongoing Work

We proposed to extend graph rewriting by a process-algebraic

control layer and obtained (preserved) results from CCS theory

The concrete MTS has too much information (and is too

strict)

⇒ an abstract interpretation semantics to obtain equivalences

weaker then isomorphism

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Conclusion and Ongoing Work

We proposed to extend graph rewriting by a process-algebraic

control layer and obtained (preserved) results from CCS theory

The concrete MTS has too much information (and is too

strict)

⇒ an abstract interpretation semantics to obtain equivalences

weaker then isomorphism

The abstract MTS cannot capture the truly concurrent

semantics of graph rewriting

⇒ capture independence in an asynchronous transition system

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Introduction Unmarked Processes Marked Processes Abstraction Independence

Backup: Encoding Termination Criterion

[

[GP] ] is an inductive termination criterion:

[ [GP] ] := (ε, {p1, . . . , pn}).0 if GP = {p1, . . . , pn} is an elementary program;

[

[GP1; GP2] ] := [ [GP1] ] + [ [GP1] ] [ [GP2] ];

[

[GP ↓] ] := (p, {p}).0, where p is any rule.

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