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ECE 510 OCE
BDDs and Their Applications
Lecture 8. Image Computation and
Reachability Analysis
April 20, 2000
Alan Mishchenko
April 20, 2000 ECE 510 OCE: BDDs and Their Applications 2
Overview
- Relations as the most fundamental(?) representation of discrete phenomena
- Image computation for relations
- Representing FSMs using relations
- Reachability analysis as an exploration of the state space of FSMs using the transition relation
- Applications of reachability analysis
- Computing the transitive closure of relations
- Equivalence checking of FSMs
April 20, 2000 ECE 510 OCE: BDDs and Their Applications 3
Boolean Functions and Relations
• Function is a mapping Bn^ → B, B = {0,1}
• Function with DCs is a mapping B n^ → {0,1,-}
• Relation is a mapping Bn^ → Bm, n > 0, m > 0.
• Example: F( x 1 ,x 2 , y 1 ,y 2 ,y 3 ), n = 2, m = 3
x 1 x 2 y 1 y 2 y 3 B 2 B
00 01 10 11
000 001 010 011 100 101 110 111
April 20, 2000 ECE 510 OCE: BDDs and Their Applications 4
Example
y (^1)
x 1 x 2 x 3 y (^2)
x
x
x 3
y
y
B^3
B^2
00 01 10 11
000 001 010 011 100 101 110 111
April 20, 2000 ECE 510 OCE: BDDs and Their Applications 5
Definition of Image
- Definitions. Given the relation R: B n^ → Bm^ , the set of all vertices in Bn^ is the input domain while the set of all vertices in Bm^ is the output domain. The domain of the relation is the subset of the input domain, for which the relation is defined. The range of the relation is the subset of the output domain which, under some inputs, may be the value of the relation
- Definition. Given subset X of the relation’s domain (X∈Bn^ ), the image of X w.r.t. the relation R is the subset Y of the range (Y∈B m) composed of values the relation can take if its input values belong to X. Y = ImR(X)
April 20, 2000 ECE 510 OCE: BDDs and Their Applications 6
Graphical Interpretation of Image
Bn
X
Bm
Y
R: Bn^ → Bm
April 20, 2000 ECE 510 OCE: BDDs and Their Applications 10
Example (continued)
y (^2)
y (^1)
x 1 x 2 x 3 F
other
x (^1)
x
x
y
y
0 1
April 20, 2000 ECE 510 OCE: BDDs and Their Applications 11
FSM Transition Relation
- FSM is { I, O, S, δ, λ). Suppose r is the number of inputs I, m is the number of states S, and n is the number of outputs O; δk (i,s) and λk(i,s) are vectors of next-state and output functions
- Transition relation is a boolean function T: {0,1}r^ x {0,1}m^ x {0,1} m^ → {0,1} such that T( i, s, n ) = 1 iff state n can be reached in exactly one transition from state s when input i is applied
April 20, 2000 ECE 510 OCE: BDDs and Their Applications 12
FSM Output Relation
- Output relation is a boolean function O: {0,1} r^ x {0,1}m^ x {0,1} n^ → {0,1} such that O( i, x, o) = 1 iff output o can be produced when the FSM is in state x and input i is applied
April 20, 2000 ECE 510 OCE: BDDs and Their Applications 13
Deriving Transition and Output Relations from NS/Output Functions
- If the FSM is given as { I, O, S, δ, λ), where δk (i,s) and λk(i,s) are vectors of next-state and output functions, it is possible to compute the transition and output relations as follows: T( i,s,n ) = Πk ( nk = δk (i,s) ); O( i,s,o ) = Πk ( ok = λk (i,s) )
- Given the transition and output relations, it is possible to derive the next state and output functions
δk(i,s) = ∃∃∃∃nx (T( i,s,n ) & nk), x ≠ k
April 20, 2000 ECE 510 OCE: BDDs and Their Applications 14
Reachability Analysis for FSMs
- Reachability analysis is the use of image computation to derive the set of all reachable states by traversing the STG of the FSM, starting from the set of initial(reset) states
- The set of reachable states can be used:
- to simplify the initial transition and output relations
- to re-encode the FSM
- to check whether a property is true for these states
- in equivalence checking
- in symbolic model checking
April 20, 2000 ECE 510 OCE: BDDs and Their Applications 15
Computing Reachable State Set
Reached = ResetStates; do { ReachedBefore = Reached; Reached += Image(Reached); } while (ReachedBefore != Reached);
(the state sets in this procedure are given by
characteristic functions represented using BDDs)
April 20, 2000 ECE 510 OCE: BDDs and Their Applications 19
Product Machine
M
M
EXOR
a b
a b c
O 1
c O (^2)
- Given FSM { I, O, S, δ, λ}, the product machine is { I, {0,1}, SxS, δ^2 , λ^2 } (r inputs, 2m states, 1 output)
April 20, 2000 ECE 510 OCE: BDDs and Their Applications 20
Equivalence Checking Formulas
- Machines M1 and M2 are equivalent iff ∀∀∀∀i, x, y [ AR(x,y) => O(i,x,y) ] = 1 where AR (x,y) is the characteristic function of the set of reachable states of product machine and O(i,x,y) is the output relation of product machine
- Alternatively, machines M1 and M2 are not equivalent iff ∃∃∃∃i, x, y [ AR(x,y) & O’(i,x,y) ] = 0
