Refusal Graph Construction From a Labelled Transitions System

Introduction • • • Context: Robustness test, systems modelling. Objective: Implementation of a calculating model of the refusal sets. Transformation of a Labelled Transition System (LTS) to a Refusal Graph (RG). Tools: Delphi software using Pascal as programming language.

Labelled Transition System (LTS) • • A basic structure used to represent a communicating system. Represented by a quadruple S=(S, Σ, Δ, s 0) where: S is a finite set of states 1 Σ is a finite set of visible actions Δ is set of transitions a 2 • out(s) returns the output ( a set of actions ) of a state s.

Refusal Graph (RG) ¡ ¡ ¡ A bilabelled graph-structure allowing abstract representation of communicating system. Represented by 5 -tuple (G, Σ, Δ, Ref, g 0) where: Ref defines the sets of events that may be refused after the sequence leading to one state. closing(s) returns a sets of states obtained by removing all intern and empty events.

Difference Between LTS and RG(1) ¡ Determinism: Existence of an isomorphism between a state and a transition. 1 a a 2 3 = 1 a 2

Difference Between LTS and RG(2) ¡ There are not any intern event (τ) neither empty event (ε) into a Refusal Graph. 1 1 ε, τ 2 = a 3 a 2

Algorithm Initialization: Define g 0 the initial state Calculate closing(g 0) , out(g 0) For g Refusal Graph Do Newstate(g) Algorithm: For a out(g) -Look for the successor of the states Newstate(g) by the event a -Calculate their closing -if there is g’ from the Refusal Graph such as Newstate(g)∩Newstate(g’) ≠ Φ Create transition labelled a such as g g’ -else create g’’ such as g g’’ by the transition labelled a Calculate out(g’’)

Implementation ¡ Choice of dynamic tables representing LTS and RG. Source Purpose state Action 1 2 a 1 3 b 2 4 ε Source state Purpose state Action State New. State Out g 1 g 2 a g 1 {1} {a, b} g 2 {2, 4} {Ø} g 3 {3} {Ø} g 1 g 3 b

Demonstration

Test (1) LTS RGraph Algorithm

Test (2) LTS RGraph Algorithm

Test (3) RGraph LTS Algorithm

Conclusion o Aim: Robustness of a protocol § Communication protocol TCP , UDP § Softwares § Automatic system o Limits of the software § Important refusal set § Specific events