joint work with: José Meseguer (UIUC) Ugo Montanari (UNIPI) Vladimiro Sassone (COGS)

1

Algebraic Theories for Contextual Pre-NetsRoberto Bruni - Dipartimento di Informatica, Università di Pisa

joint work with:

José Meseguer (UIUC)Ugo Montanari (UNIPI)Vladimiro Sassone (COGS)

ICTCS 2003, Bertinoro (FC), 13-15 Ottobre 2003

Algebraic Theories for Contextual Pre-

Nets 2

Plan of the Talk Introduction P/T Petri nets and pre-nets Read arcs, molecules and match-

share Partial membership equational logic Algebraic theories Final remarks and conclusions


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Petri Nets and Read Arcs P/T Petri nets [Petri’62]

basic model of concurrency supported by a large community and by industrial-quality tools

a media for conveying ideas to non-expert thanks to a suggestive graphical presentation

Nets with read arcs aka nets with test arcs, aka nets with activator

arcs, aka contextual nets model a non-destructive read operation

multiple concurrent readings of resources are possible naïve encoding in P/T nets would serialize these accesses,

with a dramatic loss of concurrency and efficiency


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Semantic Approaches Petri nets

processes causality and

concurrency within a single run

unfolding causality, concurrency

and conflict between all possible events

algebraic algebra of computations

logic deduction theories

Contextual nets processes unfolding

(asymmetric conflict)

algebraic molecules

tokens are not atomic entities

match-share too many fictitious

behaviours logic


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Aim Neat algebraic / logic characterization of the

(monoidal) category of concurrent computation of contextual nets

1. Functorial construction net morphisms are preserved

2. Universal construction compositionality

3. Exploit PMEqtl theories proofs simplified by PMEqtl techniques

4. Follow the individual token philosophy more informative semantics

5. 1 + 4 = Pre-nets must be used


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Terminology and Notation Places a,b,c,… classes of resources Transitions t,t’,… basic activities Tokens instances of a resource class Markings u,v,w,… multisets of resources

Multiset union uw Empty marking Places are unary markings a

Pre-sets t resources necessary to execute t multiset fetched by the execution of t

Post-sets t resources produced by t We write t:uv for t=u and t=v


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Formal Definition A P/T Petri net is a graph N=(S,T,pre,post,u0)

S is the set of markings (is the free monoid over the set of places S)

(Nodes of the graph) T is the set of transitions

(Arcs of the graph) pre:T S assigns pre-sets to transitions

pre(t) = t (Source map of the graph)

post:T S assigns post-sets to transitions post(t) = t

(Target map of the graph) u0 is the initial marking


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Circles and Boxes

2

a1

t1

a2

t2

a3

t3

a4

2

Circles are places Boxes are transitions Weighted arcs model

pre- and post-sets Bullets are tokens

uo=a1a2

t1:a12a3

t2:a2 2a3

t3:2a3a4

2


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CTPh vs. ITPh Collective Token Philosophy (CTPh)

Any two tokens in the same place are indistinguishable one from the other computationally equivalent

Individual Token Philosophy (ITPh) Any token carries its own history

tokens have unique origins fetching one token makes an activity causally

dependent from the activities that produced it such analysis can be important for recovery

purposes, detecting intrusions, increase parallelism, …


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CTPh vs. ITPh: Graphically

2

a1

t1

a2

t2

a3

t3

a4

2

2

2

a1

t1

a2

t2

a3

t3

a4

2

2


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2

a1

t1

a2

t2

a3

t3

a4

2

2

2

a1

t1

a2

t2

a3

t3

a4

2

2e1 e2


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2

a1

t1

a2

t2

a3

t3

a4

2

2

2

a1

t1

a2

t2

a3

t3

a4

2

2e1 e2

e’3 e’’3


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2

a1

t1

a2

t2

a3

t3

a4

2

2

2

a1

t1

a2

t2

a3

t3

a4

2

2e1 e2

e’3 e’’3


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Petri Nets are Monoids Proof terms denote CTPh computations:

commutative processes

uS

u:uNu[idle]

:uNv :u’Nv’

:uu’Nvv’[parallel composition]

commutative associativeunit :N

:uNv :vNw

;:uNw[sequential composition]monoid homomorphism

identities u:uNu

t:uvT

t:uNv[firing]


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ITPh and Pre-Nets T(.) : Petri CMonCat

CTPh, functorial, universal, expressed in PMEqtl P(.), DP(.), Q(.) : Petri SMonCat

ITPh, non functorial

Pre-Net RN as implementation of N [BMMS99] pre- and post-sets are strings, not multisets for each t:uv, one implementation tp,q:pq, with

p,qS (p)=u and (q)=v Z(.) : PreNets SMonCat

ITPh, functorial, universal, expressed in PMEqtl all pre-nets implementations of N have the same

semantics


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Pre-Nets are Monoids Proof terms denote ITPh computations:

concatenable processes

pS

p:pRp[idle]

t:pqT

t:pRq[firing]

:pRq :p’Rq’

:pp’Rqq’[parallel composition]

:pRq :qRr

;:pRr[sequential composition]

commutativeassociativeunit :R

monoid homomorphismidentities p:pRp

p,qS

p,q:pqRqp[symmetry]


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Examplea1 a2

t1 t2

a3 a3 a3 a3

t3t3

p0=a1a2

(t1t2) ; (t3t3) =(t1;t3) (t2;t3)


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Examplea1 a2

t1 t2

a3 a3 a3 a3

t3t3

p0=a1a2

(t1t2) ; (a3a3,a3a3) ; (t3t3)


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Nets with Read Arcs Graphically rendered as undirected arcs Equivalent to self-loops from the point of view of

reachability Allow more parallelism

Ex. readL and readR can fire concurrently!

readL readR

aL aR

b

cL cR

u0=aLbaR

(aLreadR) ; (readLcR) (readLaR) ; (cLreadR)

we need syntax to writereadL(?)readR


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Molecules, Atoms, Electrons

Non free monoid of places [Mes96] a = {a|0} and {a|n} = {a|n+1}[a]

Molecules for CTPh and ITPh [BM01] Two monoid homomorphisms: (.)+ and (.)-

(uv)+ = (u)+ (v)+ and ()+ = (uv)- = (u)- (v)- and ()- =

Suitable laws u = (u)+ (u)- ((u)+)- = ((u)-)+ = (u)- ((u)-)- = (u)k = ((u)+ …)+

(u)0 = u and (u)1 = (u)+ and (u)k+1 = ((u)k)+

atom a nucleus (a)k

electron (a)-

token a context [a]


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Example

readL readR

aL aR

b-

cL cR

u0=aLbaR

readL b2 readR =(readLb+aR) ; (cLb+readR) =(aLb+readR) ; (readLb+cR)

b+

Drawback:Tokens are not atomic entities


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Match-Share Categories Match-Share categories [GM98]

symmetric monoidal categories, plus

with suitable axioms p,p;p = p and p;p,p = p (pp);p = (pp);p and p;(pp) = p;(pp) = and = pq = (pq,pq);(pq) and pq = (p q);

(pp,qq) p;p = (pp);(pp) p;p = p

pS

p:pp Rp[match]

pS

p:pRpp[share]

commutativityassociativity

monoidality {

MAIN AXIOMS {


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Match-Share Graphically

a

a a a

aa a

a

a ba,b

b a

a aa,a

a a

a

a

=a a

a

a


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a

a a a

aa a

a

a ba,b

b a

a a

a

a

=

a aa

a

a a

a

a

a aa

a


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a

a a a

aa a

a

a ba,b

b a

a a

a

a

=a

a

a a


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a

a a a

aa a

a

a ba,b

b a

a

a

=

a aa

a

a

a

a

a a

a

a

a a

a

= a

a

a

a a

a

a

a a


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Transitions as Arrows

aL

readL

b

cL b

b

b

aR

readR

b

cRb

b

b


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ExampleaL

readL

b

cL b aR

readR

cRb

aR

cL


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ExampleaL

readL

b

cL b aR

readR

cRb

aR

cL


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ExampleaL

readL

b

cL aR

readR

cRb

aR

cL


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ExampleaL

readL

b

cL aR

readR

cRb

aR

cL


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ExampleaL

readL

b

cL aR

readR

cRb

aR

cL


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ExampleaL

readL

b

readR

cRb

aR

cL


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ExampleaL

readL

b

readR

cRb

aR

cL


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ExampleaL

readL

b

readR

cRb

aR

cL

S(R) free SMonCat distinguishes too much

MS(R) free MSCat too many arrows

E(R) : S(R) MS(R) E(a) = a E(t:uwv) = (uw) ; (tw) ; (vw)

Drawback: is MS(R) meaningful?


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PMEqtl Partial Membership Equational Logic

convenient definition of theories for nets and their computations

subsort for regarding objects as idle steps partiality of sequential composition membership predicates for selecting admissible

behaviours tensor product of theories for MonCat efficiently supported by Maude 2.0

proof of freeness follows by PMEqtl results adjunctions proved by existence of theory morphisms


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MON

fth MON is sort Monoid . op e : Monoid . op __ : Monoid Monoid -> Monoid [assoc id:e] .endfth


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CATfth CAT is sorts Object Arrow . subsort Object < Arrow . ops d(_) c(_) : Arrow -> Object . op _;_ . var a : Object . vars f g h : Arrow . eq d(a) = a . eq c(a) = a . ceq a;f = f if d(f)=a . ceq f;a = f if c(f)=a . cmb f;g : Arrow if c(f)=d(g) . ceq d(f;g) = d(f) if c(f)=d(g) . ceq c(f;g) = c(g) if c(f)=d(g) . ceq f;(g;h) = (f;g);h if c(f)=d(g) and c(g)=d(h) .endfth


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SMONCAT

fth SMONCAT is including MONCAT . op (_,_) : Object Object -> Arrow . vars a a’ b b’ c : Object . vars f f’ : Arrow . eq d((a,b)) = ab . eq c((a,b)) = ba . eq (a,e) = a . eq (e,a) = a . eq (ab,c) = (a(b,c)) ; ((a,c)b) . eq (a,b);(b,a) = ab . ceq (ff’);(b,b’) = (a,a’);(f’f) if d(f)=a and d(f’)=a’ and c(f)=b and c(f’)=b’ .endfth


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MSCATfth MSCAT is including SMONCAT . ops (_) (_) : Object -> Arrow . vars a b : Object . eq d((a)) = aa . eq d((a)) = a . eq c((a)) = a . eq c((a)) = aa . eq (e) = e . eq (e) = e . eq (a,a);(a) = (a) . eq (a);(a,a) = (a) . eq (ab) = (a(b,a)b);((a)(b)) . eq (ab) = ((a)(b));(a(a,b)b) . eq ((a)a);(a) = (a(a));(a) . eq (a);((a)a) = (a);(a(a)) .

eq (a);(a) = a . eq (a);(a) = (a(a));((a)a).endfth


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RAUT (Read-Automata)

fth RAUT is including MON . sort Rtrans . subsort Monoid < Rtrans . ops pre(_) post(_) ctx(_) : Rtrans -> Monoid . var u : Monoid . eq pre(u) = e . eq post(u) = e . eq ctx(u) = u .endfth


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RCOMP (Read-Computation)

fth RCOMP is including MSCAT . sort Rtrans Rarrow . subsort Object < Rtrans < Rarrow < Arrow . ops pre(_) post(_) ctx(_) : Rtrans -> Object . op mk(_) : Rtrans -> Arrow vars h k : Rarrow . var t : Rtrans . vars u v : Object . mb hk : Rarrow . mb (u,v) : Rarrow . cmb h;k : Rarrow if c(h)=d(k) . eq d(t) = pre(t)ctx(t) . eq c(t) = post(t)ctx(t) . eq pre(u) = e . eq post(u) = e . eq ctx(u) = u . eq d(mk(t)) = d(t) . eq c(mk(t)) = c(t) . eq mk(u) = u . eq t = (pre(t)(ctx(t)));(mk(t)ctx(t));(post(t)(ctx(t))) .endfth

self-loop version

relation between t and its self-loop version

markings idle+transitions

E(MS(R)) MS(R)

Rarrow is symmetric monoidal


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t and mk(t)

a

t

b

c b

b

b

b

bb

b


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t and mk(t)

a

mk(t)

b

c b

b

b

b

b

b

bb

b


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t and mk(t)

a

mk(t)

b

c b

b

b

b

b

b

bb

b


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t and mk(t)

a

mk(t)

b

c b

b

b

b

bb

b


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t and mk(t)

a

t

c

b

b


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RV (Theory Morphism)

view RV from RAUT to RCOMP is sort Monoid to Object .endview

The view RV induces a forgetful functor URV from the category of RCOMP-algebras (computational models) to the category of RAUT-algebras (contextual pre-nets)with a left-adjoint FRV


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Results

FRV(R) is a match-share category isomorphic to MS(R) via a match-share functor

FRV(R)|Rarrow is a symmetric monoidal category isomorphic to E(S(R)) via a symmetric monoidal functor

A computation is meaningful iff RCOMP |- : Rarrow


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Conclusions Typing discipline via membership

predicates Functorial and universal construction

contextual pre-nets seen as implementations of contextual nets

proof via PMEqtl techniques Results exploited in WADT 2002 [BBM02]

to functorially reconcile all net semantics Maude can be used as proof engine


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The End

Algebraic Theories for Contextual Pre-Nets

a paper by Roberto Bruni

José Meseguer

Ugo Montanari

Vladimiro Sassone

ICTCS 2003 presentation by Roberto Bruni

Research supported by FET-GC Project IST-2001-32747 AGILEMIUR Project COFIN 2001013518 COMETAItalia CNR Fellowship and CS Department at UIUC (first author)

Front cover: electronic watercolor by Roberto Bruni

Documents

joint work with: José Meseguer (UIUC) Ugo Montanari (UNIPI) Vladimiro Sassone (COGS)