1 Temporal Logic u Classical logic:  Good for describing static conditions u Temporal logic:  Adds temporal operators  Describe how static conditions change over time u Two main ways to represent temporal logic:  Linear time: describes single possible time line o LTL (Linear Temporal Logic): Spin  Branching time: describes all possible time lines o CTL (Computation Tree Logic): SMV

2 What can you do with TL? u Use Automaton to describe system behavior  Describes actions of system  In terms of state sequences u Use temporal logic to describe property of state sequences u Use model checker to verify TL property  Checks TL property against automaton  Exhaustively checks the automaton  Automatic checking

3 Basic Idea of Temporal Logic u Truth changes over time u Must say when things are true,  Not just what is true u Most model checking tools do not allow quantification  Use propositional TLs (PTLs)

4 Semantics of Temporal Logic u 4 basic operators: : always : sometime or eventually : next time or next step : until u Conceptual representation: Let S be a sequence of states  S[ 0 ]: first state; S[ j ]: jth state  S[ j.. ] sequence starting from jth state  S[ j..k ] sequence from j to k U

5 More Formal Semantics u Classical Logic: S f: sequence S satisfies formula f S true: for any S S (f and g): S f and S g u Temporal Logic: S f : if for any j, S[j..] f S f : if for some j, S[j..] f Sf : if S[1..] f S f g : if for some k, S[k..] g,  and for any j < k, S[ j.. ] f U

6 Example u Automaton/machine produces state sequence: abcabcabcabc… u Sequence satisfies property:  (a  b) and It’s always the case that a implies that the next step will be b  (a  a)) It’s always the case that a implies that the next step will eventually be an a. ab c

7 More Semantics of PTL u Next f: S[j] f iff S[j+1] f u Always f: S[j] f iff (  k: k  j  S[k] f ) u Sometime f: S[j] f iff (  k: k  j  S[k] f ) u f U g: S[j] f U g iff (  k: k  j  S[k] g )  (   l: j  l  k  S[l] f )

8 Sample Specifications u Mutual Exclusion: (  inCsA  inCsB) u Response: (wantsInA  inCsA)

9 Computation Tree Logic u Most distributed, reactive systems are nondeterministic Cannot be represented by sequence of possible states or transitions Has a tree of possible computations u Can use CTL to represent these cases Computation Tree Logic

10 Kripke Structure ab bc ac s0 s2

11 CTL operator combinations

12 CTL Syntax u Basic logic, then add in… u Temporal expressions:  Temporal operators are defined in pairs  Path part: A: means “all paths” (inevitably) E: means “on some path” (possibly)  Property part: F : same as % For some G : same as % Globally holds X : same as % neXt U : same as U

13 CTL Syntax Sample expression forms: AGp : On all paths, property p always holds EGp : On some paths, property p always holds AFp : On all paths, property p eventually holds EFp : On some path, property p eventually holds

14 Specification Patterns u Safety: bad thing never happens: AG (  bad-thing ) u Liveness: good thing eventually happens AF ( good-thing ) u Bad thing could happen: EF ( bad-thing )

15 System satisfying EFp

16 System satisfying EGp

17 System satisfying AGp

18 System satisfying AFp

19 Sample Specification Patterns u Possible to get to started state, but ready does not hold EF(started)  (  ready)) u If a request occurs, then it will eventually be acknowledged AG (requested  AF acknowledged)  Process is enabled  infinitely often on every path AG (AF enabled) u Whatever happens, a certain process will eventually be permanently deadlocked: AF (AG deadlocked) u From any state, it is possible to get to a restart state AG (AF restart)

20 Example Elevator u An elevator at the 2 nd floor traveling upward towards 5 th floor destination does not change its direction AG (floor = 2  direction = up  ButtonPressed5   A [direction = up  U floor 

21 Tool Support u Model checkers: check that system satisfies property u Reachability analysis: describe paths of behavior of system u Each tool uses different algorithms for optimization purposes  SPIN (Holzmann et al)  SMV (Clarket et al)  Nitpick (D. Jackson)  COSPAN (Kurshan, mostly for HW)  Verisoft  FDR  Etc.