# Algebraic Nets

## Algebraic Net Schemes

An algebraic net scheme is a pair consisting of an algebraic specification and a net with sorted places, with transtitions which have predicate (a set of equations) and with arcs marked with terms over the specification.

An algebraic net scheme consists in a basic coloured net where colours and firing rules are represented by interpretations of terms over a given algebraic specification.

### Definition

(according to [dhp91])

An algebraic net scheme is a tuple N=(SPEC,X,P,T,Pre,Post,eqns,sort) where:

1. SPEC=(SO,OP,EQ) is an algebraic specification of abstract data types
2. X is a family of SO-sorted variables
3. P and T are the sets of places and transitions
4. Pre and Post are functions which assign a set of terms with variables to every pair place/transition
5. eqns is a function which associates with each transition a set of equations
6. sort is a function which assigns a sort SO to every place

## Algebraic High-Level Nets

Algebraic high-level nets are couples (algebraic net scheme, algebra).

### Definition

An algebraic high-level net (AHL net) is a tuple (N,A) where N is an algebraic net scheme and A is a SPEC-algebra.

### Transition Rule

The semantics of an algebraic high-level net is that of the associated coloured net where the terms and equations are interpreted in the given algebra.

### Tools

PAPETRI, an integrated tool for editing and analysing Petri Nets.
SANDS-COOPN, a development system to construct Concurrent Object-Oriented Petri Net specifications based upon algebraic nets.

### References

• J. Vautherin: Un modèle algébrique, basé sur les rèseaux de Petri, pour l'ètude des systèmes parallèles [vau85]
• J. Vautherin: Paralles Systems Specifications with Colored Petri Nets and Algebraic Specifications [vau87]
• U. Hummert: Algebraische Theorie von High-Level-Netzen [hum89]
• C. Dimitrovici, U. Hummert, L. Petrucci: Semantics, Composition and Net Properties of Algebraic High-level Nets [dhp91]

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