Formalisms for Behaviors Specification

1. Formalisms for Behaviors Specification  Dr. Vered Gafni

2. System Behaviors • System - (2E)-- E system signature (finite set of events) • Trace (behavior) - = 0 1 2… where i2E • Assertion – A(2E)(subset of system behaviors) Specification formalism - means to specify assertions (subsets of system behaviors). • Two types of assertions: - state constraints (safety) • e.g., The gate is closed a.l.a. a train is within the crossing • - temporal constraints (liveness). • e.g., Every train that entersthe crossing shall exit eventually  Dr. VeredGafni

3. Specification Formalisms  Dr. Vered Gafni

4. Regular expressions •  a set of symbols, • * is the set of finite words over (including - empty word) • L*– a language over  • the regular expressions over  are: • , , p for p • For r,s regular expressions: rs (concatenation), r+s (selection), r*(0 or more finite repetition ) Notation: r+=rr*, rk=r…r– k times • Semantics: L()=, L()={}, L(p)={p}, L(rs)=L(r)L(s), L(r+s)=L(r)L(s), L(r*)=L(r)* Regular languages (RL) – those represented by regular expressions RL are closed under union, concatenation, Kleene star, intersection, complementation.  Dr. VeredGafni

5. Regular expressions properties • RL are closed under union, concatenation, Kleene star, intersection, complementation. • Notations:  =2E; pE, [p]= +( | p), [~p]=+( | p) • Examples: E={send, ack} • [send][~{ack,send}]*[ack] • ([send][~{ack,send}]*[ack])* • [~send]* + ([~send]*[send][~{ack,send}]*[ack])* • ([~send]* + ([send][~{ack,send}]*[ack]))* • ([~send]*([send][~{ack,send}]*[ack])*)*  A RL may have several representations by different RE  Dr. Vered Gafni

6. -regular expressions - Basic Representation Formalism Noteis not an extension of * To ensure enumerable word • - the set of infinite sequences (words) over  • L– an -language over  • For R*, , • R = { r| rR,  } • R= { r1r2…| iN, riR/{} } •  regular expressions - terms of the form: i=1..nRiSi where Ri, Si are regular expression • Semantics: L(i=1..nRiSi)= i=1..nL(Ri)L(Si) S is  regular (={}S). • Example: ([~send]* + ([send][~{ack,send}]*[ack]))  Dr. Vered Gafni

7. -regular expressions/languages -regular expressionsG1, G2are equivalent ifL(G1)=L(G2)   regular languages are closed under: • Union • Intersection • Complementation  Dr. Vered Gafni

8. Examples of -regular expressions any  that contains p ({p}+{p,q}) Given system signature E=p,q E, = 2E, • p occurs at every time instant [p] • p occurs infinitely often (*[p]) • Every time p occurs q occurs as well ([~p]*[p,q])*[~p] ([~p]*[p,q]) • Every p is (strict) followed by q with no p ([~p]*[p][~q]*[q/p]) ([~p]*[p][~q]*[q/p])*[~p] any word over  any  that does not contain p any  that contain q but not p  Dr. Vered Gafni

9. Railroad crossing assertions Min. delay of 40 seconds between successive trains. ([~Tin]*[Tin][~Tin]40)*[~Tin]  ([~Tin]*[Tin][~Tin]40) No more than one train in XR at a time ([~Tin]*[Tin][~{Tin,Tout}]*[Tout])*[~Tin]  ([~Tin]*[Tin][~{Tin,Tout}]*[Tout])  ([~Tin]*[Tin] [~{Tin,Tout}] Gate closed as long as a train is in XR. ([~Tin]*[{Tin,Closed}][Closed/Tout}]*[{Closed,Tout}])*[~Tin]  ([~Tin]*[{Tin,Closed}][Closed/Tout}]*[{Closed,Tout}])  ([~Tin]*[{Tin,Closed}] [Closed/{Tin,Tout}] No exit