# 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:

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

**Sender-Receiver Model**

### 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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