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

Algebraic Theories for Contextual Pre-

Nets 3

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

Algebraic Theories for Contextual Pre-

Nets 4

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

Algebraic Theories for Contextual Pre-

Nets 5

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

Algebraic Theories for Contextual Pre-

Nets 6

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

Algebraic Theories for Contextual Pre-

Nets 7

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

Algebraic Theories for Contextual Pre-

Nets 8

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

Algebraic Theories for Contextual Pre-

Nets 9

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

Algebraic Theories for Contextual Pre-

Nets 10

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

Algebraic Theories for Contextual Pre-

Nets 11

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, …

Algebraic Theories for Contextual Pre-

Nets 12

CTPh vs. ITPh: Graphically

2

a1

t1

a2

t2

a3

t3

a4

2

2

2

a1

t1

a2

t2

a3

t3

a4

2

2

Algebraic Theories for Contextual Pre-

Nets 13

CTPh vs. ITPh: Graphically

2

a1

t1

a2

t2

a3

t3

a4

2

2

2

a1

t1

a2

t2

a3

t3

a4

2

2e1 e2

Algebraic Theories for Contextual Pre-

Nets 14

CTPh vs. ITPh: Graphically

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

Algebraic Theories for Contextual Pre-

Nets 15

CTPh vs. ITPh: Graphically

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

Algebraic Theories for Contextual Pre-

Nets 16

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]

Algebraic Theories for Contextual Pre-

Nets 17

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

Algebraic Theories for Contextual Pre-

Nets 18

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]

Algebraic Theories for Contextual Pre-

Nets 19

Examplea1 a2

t1 t2

a3 a3 a3 a3

t3t3

p0=a1a2

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

Algebraic Theories for Contextual Pre-

Nets 20

Examplea1 a2

t1 t2

a3 a3 a3 a3

t3t3

p0=a1a2

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

Algebraic Theories for Contextual Pre-

Nets 21

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

Algebraic Theories for Contextual Pre-

Nets 22

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

Algebraic Theories for Contextual Pre-

Nets 23

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]

Algebraic Theories for Contextual Pre-

Nets 24

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

Algebraic Theories for Contextual Pre-

Nets 25

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 {

Algebraic Theories for Contextual Pre-

Nets 26

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

Algebraic Theories for Contextual Pre-

Nets 27

Match-Share Graphically

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

Algebraic Theories for Contextual Pre-

Nets 28

Match-Share Graphically

a

a a a

aa a

a

a ba,b

b a

a a

a

a

=a

a

a a

Algebraic Theories for Contextual Pre-

Nets 29

Match-Share Graphically

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

Algebraic Theories for Contextual Pre-

Nets 30

Transitions as Arrows

aL

readL

b

cL b

b

b

aR

readR

b

cRb

b

b

Algebraic Theories for Contextual Pre-

Nets 31

ExampleaL

readL

b

cL b aR

readR

cRb

aR

cL

Algebraic Theories for Contextual Pre-

Nets 32

ExampleaL

readL

b

cL b aR

readR

cRb

aR

cL

Algebraic Theories for Contextual Pre-

Nets 33

ExampleaL

readL

b

cL aR

readR

cRb

aR

cL

Algebraic Theories for Contextual Pre-

Nets 34

ExampleaL

readL

b

cL aR

readR

cRb

aR

cL

Algebraic Theories for Contextual Pre-

Nets 35

ExampleaL

readL

b

cL aR

readR

cRb

aR

cL

Algebraic Theories for Contextual Pre-

Nets 36

ExampleaL

readL

b

readR

cRb

aR

cL

Algebraic Theories for Contextual Pre-

Nets 37

ExampleaL

readL

b

readR

cRb

aR

cL

Algebraic Theories for Contextual Pre-

Nets 38

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?

Algebraic Theories for Contextual Pre-

Nets 39

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

Algebraic Theories for Contextual Pre-

Nets 40

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

Algebraic Theories for Contextual Pre-

Nets 41

MON

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

Algebraic Theories for Contextual Pre-

Nets 42

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

Algebraic Theories for Contextual Pre-

Nets 43

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

Algebraic Theories for Contextual Pre-

Nets 44

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

Algebraic Theories for Contextual Pre-

Nets 45

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

Algebraic Theories for Contextual Pre-

Nets 46

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

Algebraic Theories for Contextual Pre-

Nets 47

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

Algebraic Theories for Contextual Pre-

Nets 48

t and mk(t)

a

t

b

c b

b

b

b

bb

b

Algebraic Theories for Contextual Pre-

Nets 49

t and mk(t)

a

mk(t)

b

c b

b

b

b

b

b

bb

b

Algebraic Theories for Contextual Pre-

Nets 50

t and mk(t)

a

mk(t)

b

c b

b

b

b

b

b

bb

b

Algebraic Theories for Contextual Pre-

Nets 51

t and mk(t)

a

mk(t)

b

c b

b

b

b

bb

b

Algebraic Theories for Contextual Pre-

Nets 52

t and mk(t)

a

t

c

b

b

Algebraic Theories for Contextual Pre-

Nets 53

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

Algebraic Theories for Contextual Pre-

Nets 54

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

Algebraic Theories for Contextual Pre-

Nets 55

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

Algebraic Theories for Contextual Pre-

Nets 56

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

Algebraic Theories for Contextual Pre-

Nets 57

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