Black Box Black Box CheckingChecking

Book: Chapter 9

Model Checking Finite state description of a system B. LTL formula . Translate into an automaton

P. Check whether L(B) L(P)=. If so, S satisfies . Otherwise, the intersection

includes a counterexample. Repeat for different properties.

Buchi automata (-automata)

S - finite set of states. (B has l n states) S0 S - initial states. (P has m states)

- finite alphabet. (contains p letters) S S - transition relation. F S - accepting states.

Accepting run: passes a state in F infinitely often.

System automata: F=S, deterministic.

Example: check a

a, aa

a<>a

Example: check <>a

a

a

a

a

<>a

Example: check <>a

Use automatic translation algorithms, e.g., [Gerth,Peled,Vardi,Wolper 95]

a

a

a, a<>~a

System

c b

a

Every element in the product is a counter example for the checked property.

ba

a

a

a

s2

c

as1

s3

q2

q1

s1,q1

s1,q2 s3,q2

s2,q1a

b

c

aAcceptance isdetermined byautomaton P.

Testing Unknown deterministic finite state system B. Known: n states and alphabet . An abstract model C of B. C satisfies all the

properties we want from B. Check conformance of B and C. Another version: only a bound n on the number

of states l is known.

Model Checking / Testing Given Finite state

system B. Transition relation of B

known. Property represent by

automaton P. Check if L(B) L(P)=. Graph theory or BDD

techniques. Complexity:

polynomial.

Unknown Finite state system B.

Alphabet and number of states of B or upper bound known.

Specification given as an abstract system C.

Check if B C. Complexity: polynomial

if number states known. Exponential otherwise.

Black box checking

Property represent by automaton P.

Check if L(B) L(P)=.

Graph theory techniques.

Unknown Finite state system B.

Alphabet and Upper bound on Number of states of B known.

Complexity: exponential.

Combination lock automaton

Accepts only words with a specific suffix (cdab in the example).

s1 s2 s3 s4 s5

bdc a

Conformance testing

Cannot distinguish if reduced or not.

aba

a

b

b

ab ab

Conformance testing (cont.)

When the black box is nondeterministic, we might never test some choices.

b a

a

Conformance testing (cont.)

ab b

a

a

a

a b

b

b

a

Need: bound on number of states of B.

a

Need reliable RESET

s1

s3

s2

a

a

a

bb

Vasilevskii algorithm

Known automaton A has l states. Black box automaton has up to n

states. Check each transition. Check that

there are no "combination lock" errors.

Complexity: O(l2 n p n-l+1). When n=l: O(l3p).

Experimentsaa

bb cc

reset

a

a

b

b

c

c

try ba

a

b

b

c

c

try c

fail

Simpler problem: deadlock?

Nondeterministic algorithm:guess a path of length n from the initial state to a deadlock state.Linear time, logarithmic space.

Deterministic algorithm:systematically try paths of length n, one after the other (and use reset), until deadlock is reached.Exponential time, linear space.

Deadlock complexity

Nondeterministic algorithm:Linear time, logarithmic space.

Deterministic algorithm:Exponential (p n-1) time, linear space.

Lower bound: Exponential time (usecombination lock automata).

How does this conform with what we know about complexity theory?

Modeling black box checking

Cannot model using Turing machines: not all the information about B is given. Only certain experiments are allowed.

We learn the model as we make the experiments.

Can use the model of games of incomplete information.

Games of incomplete information

Two players: player, player (here, deterministic).

Finitely many configurations C. Including:Initial Ci , Winning : W+ and W- .

An equivalence relation on C (the player cannot distinguish between equivalent states).

Labels L on moves (try a, reset, success, fail). The player has the moves labeled the same from

configurations that are equivalent. Strategy for the player: will lead to a configuration

in W+ W-. Cannot distinguish equivalent conf. Nondet. strategy: ends with W+. Can distinguish.

Modeling BBC as games Each configuration contains an automaton

and its current state (and more). Moves of the player are labeled with

try a, reset... Moves of the -player withsuccess, fail.

c1 c2 when the automata in c1 and c2 would respond in the same way to the experiments so far.

A naive strategy for BBC Learn first the structure of the black box. Then apply the intersection. Enumerate automata with n states

(without repeating isomorphic automata). For a current automata and new automata,

construct a distinguishing sequence. Only one of them survives.

Complexity: O((n+1)p (n+1)/n!)

On-the-fly strategy Systematically (as in the deadlock case),

find two sequences v1 and v2 of length <=m n.

Applying v1 to P brings us to a state t that is accepting.

Applying v2 to P brings us back to t. Apply v1 (v2 )n+1 to B. If this succeeds,

there is a cycle in the intersection labeled with v2, with t as the P (accepting) component.

Complexity: O(n2p2mnm).

Learning an automaton

Use Angluin’s algorithm for learning an automaton.

The learning algorithm queries whether some strings are in the automaton B.

It can also conjecture an automaton Mi and asks for a counterexample.

It then generates an automaton with more states Mi+1 and so forth.

A strategy based on learning

Start the learning algorithm. Queries are just experiments to B. For a conjectured automaton Mi ,

check if Mi P =

If so, we check conformance of Mi with B (Vasilevskii algorithm).

If nonempty, it contains some v1 (v2) . We test B with v1 (v2)n+1. If this succeeds: error, otherwise, this is a counterexample for Mi .

Black Box Checking Strategy

Incrementallearning

Comparingcounterexample

ModelChecking

Reporterror

No errorfound

black boxtesting

counterexample

no counterexample

false negative

actual error

discrepancy

conformance established

System

PathModel

Complexity l - real size of B. n - an upper bound of size of B. p - size of alphabet. Lower bound: reachability is similar to

deadlock. O(l 3 p l + l 2mn) if there is an error. O(l 3 p l + l 2 n p n-l+1+ l 2mn) if there is no

error.If n is not known, check while time allows.

Some experiments

Basic system written in SML (by Alex Groce, CMU).

Experiment with black box using Unix I/O.

Allows model-free model checking of C code with inter-process communication.

Compiling tested code in SML with BBC program as one process.

Conclusions

Black box checking is a combination of testing and model checking.

If a tight bound on size of B is given: learn B first, then do model checking.

Tight lower bound on complexity, up to polynomial factor.

Use of games of incomplete information to model testing problems.