Tutorial - Philadelphia University

Tutorial

1

Algebraic specifications of Abstract

Data Types Exceptional behavior on the erroneous

arguments

The Stack example

fmod STACK is sorts Stack NeStack Element . subsort NeStack < Stack . op emptystack : -> Stack . op push : Stack Element -> NeStack . op pop : NeStack -> Stack . op top : NeStack -> Element . var S : Stack . var X : Element . eq pop(push(S, X)) = S . eq top(push(S, X)) = X .

endfm

The List example

fmod LIST is sorts List NeList Element . subsort NeList < List . op nil : -> List . op _._ : Element List -> NeList . op head : NeList -> Element . op tail : NeList -> List . var L : List . var E : Element . eq head(E.L) = E . eq tail(E.L) = L .

endfm

The Queue example

fmod QUEUE is

sorts Queue NeQueue Element.

subsort NeQueue < Queue .

op emptyqueue : -> Queue .

op add : Queue Element -> NeQueue .

op remove : NeQueue -> Queue .

op front : NeQueue -> Element .

var X : Element .

var Q : Queue .

eq remove(add(Q, X)) = if Q == emptyqueue

then emptyqueue else add(remove(Q), X)

fi .

eq front(add(Q, X)) = if Q == emptyqueue

then X else front(Q)

fi .

endfm

Petri nets

• Definition

• Graphical structure

Definition Graphical structure

p1 p2

p3

p4

t1

t2

t3 t4

M1 = (1,0,0,1)

M2 = (0,1,0,1)

M3 = (0,0,1,0)

M4 = (0,0,0,1)

Source: The Petri Net Method, by Dr Chris Ling School of Computer Science & Software Engineering, Monash University

Reachability set

Reachability Graph

Matrix analysis

Source: System modelling with Petri nets, Andrea Bobbio, Istituto Elettrotecnico Nazionale Galileo Ferraris, Torino, Italy

II

Specification of data structures

• Algebraic specification

• Z notation

Z notation

• Full notation

– slides 20 – 21

• Short-hand notation (∆ schema)

– Slide 26

Recall State Schema

Operation Schema (full-notation)

Operation Schema (short-hand notation)

Z notation and Object-Z

• Writing Object-Z notation from Z notation and vice-versa

Z notation

Item is the formal generic parameter

Source: The Object-Z Specification Language, Graeme Smith, Software Verification Research Centre University of Queensland, with modification

Object-Z

Source: The Object-Z Specification Language, Graeme Smith, Software Verification Research Centre University of Queensland, with modification

Graph Transformations

• Graph rewrite rule

• Application of graph rewrite rule

– Slides 21-22, 25

Source:Bernhard Westfechtel Lehrstuhl für Informatik III, RWTH Aachen

Type Graph and Instance Graph

Account

balance

Bill

amount

Customer

name

To

Has

:Customer

name=C1

b1:Bill

amount=3

:Account

balance=5

:Account

balance=0

:Customer

name=C2

b2:Bill

amount=2

Pays

Pays Pays

To To

Has

Has

From From From

TG IG

These slides are strongly based on the paper entitled Graph Transformation and Visual Modeling Techniques by Reiko Heckel and Gregor Engels

Petri nets : mutual exclusion

Traffic light

Source: Petri nets, classical Petri nets: The basic model, prof.dr.ir. Will van der Aalst, Technische Universitat Eindoven, University of Technology

Petri nets: producer-consumer

Figure 5: Unbounded buffer

Source: System modelling with Petri nets, Andrea Bobbio, Istituto Elettrotecnico Nazionale Galileo Ferraris, Torino, Italy

Scenario: A token in p1 : P ready to produce Firing t1: P produces Firing t2: P sends produced object to buffer p5, ready to produce again A token in p3 : C ready to consume If an object is in p5, t3 fires and C consumes, C is then ready to consume again

Figure 6: Finite buffer

Source: System modelling with Petri nets, Andrea Bobbio, Istituto Elettrotecnico Nazionale Galileo Ferraris, Torino, Italy

The total number of tokens in p5 and p6 is constant and represents the total available buffer positions If a single token is assigned to p6: strictly sequential ordering of activities

III