# Bit-State Space Exploration It’s a variation on reachability analysis The reachability analysis: –Keeps track of the already explored states –Performs.

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Bit-State Space Exploration It’s a variation on reachability analysis The reachability analysis: –Keeps track of the already explored states –Performs full state space search –Most implementations use Hashing to quickly access an element on the table of already explored states

Bit-State Space Exploration Hashing function –Given a hash space table of H slots –and a Hashing function h(s) for A states –h(s) points a slot in the hash table without the need to search for the states in the whole table h(s)

In case of collision –the use of a linked list is a common option To minimize collision, two hashing functions are used. Bit-State Space Exploration h(s) S2S3S1

Bit-State Space Exploration Say we have two states s1 and s2. –With only one hash function, these are the possibilities: h(s1) = x and h(s2) = x or h(s1) = x and h(s2) = y

Bit-State Space Exploration Say now we use two hashing functions h1 and h2. The four possibilities are: h1(s1)=x, h1(s2)=x, h2(s1)=v, h2(s2)=w h1(s1)=x, h1(s2)=x, h2(s1)=v, h2(s2)=v h1(s1)=x, h1(s2)=y, h2(s1)=v, h2(s2)=w h1(s1)=x, h1(s2)=y, h2(s1)=v, h2(s2)=v Only the green shaded row causes collision. We have thus reduced collision risk.

Bit-State Space Exploration Memory space analysis –Hash table size (H) –Pointer size (B) –Hash table will occupy HxB bytes –State data will use (S+B)xA bytes –Total memory: HxB + (S+B)xA Example –H = 1,000,000, B=4  Table size = 4Mb –S and A depend on the specification under test next pointer state data size

Bit-State Space Exploration Workaround: Bit-state space exploration –By using a depth search algorithm, there is no need any more for storing the visited states, so: –M = HxB + (S+B)xA  M = H, or H/8, –where M is the total amount of memory used for the hash table. –Constraints Collision avoidance is a matter of probability of occurrence has to use depth search algorithm only goes until maximum depth is reached Because one state now can be represented by only one bit: reached (1), or not (0)

Bit-State Space Exploration Example 2: (Even better than previous) –if M = 1,000,000, H=8,000,000 –previous example: 4Mb = 10 6 states only for the hashing table –this example: 4Mb = 32x 10 6 states, and no need for extra storage for the states data

Bit-state Space exploration How it works –Storing the state data in a stack as it goes… –Go until any of the following conditions... G.S. 1 G.S. = Global State G.S. 1.1 G.S. 1.2 G.S. 1.3 G.S. 1.1.1 G.S. 1.1.2 G.S. 1.1.3 Depth-first

Bit-state Space exploration a) b) G.S. i No new state or possible action Simply backtrack in the stack G.S. i G.S. j (already visited) G.S. k No new state or possible action backtrack and go the next on the right

Bit-state Space exploration c) d) Problem encountered: –e.g. Unspecified reception, or deadlock –reads the whole stack, creates an MSC –adds a report to the list of reports with the MSC –backtrack and go again. Rem: For each new visited state (hash table bit = 0), sets the hash table to 1. G.S. i Maximum depth Simply backtrack

Bit-state Space exploration In the TAU Validator Bit-State space exploration tool, the results are like: –No of reports: No of reports generated –Generated states: No of global states generated –Truncated paths: No of states cut by the maximum depth constraint –Unique system states: No of unique global states from the generated ones –Size of hash table: The size of the hash table (H) –No of bits set in hash table: No of bits in the hash table set to 1 (visited) –Collision risk: the risk of having two states colliding in the same slot –Max Depth: the maximum depth set for this bit-state exploration –Current depth: the depth after the execution (should be -1 if went ok) –Min state size and Max state size: (limit sizes of states, used by h(s)) –Symbol coverage: The percentage of the SDL symbols covered

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