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ECE 510 OCE
BDDs and Their Applications
Lecture 11. FSM Equivalence Checking
and FSM State Minimization
May 2, 2000 Alan Mishchenko
May 2, 2000 ECE 510 OCE: BDDs and Their Applications 2
Overview
- Equivalence for FSMs and FSM states
- Product machine (PM)
- Solving the problem of FSM equivalence
- Derive transition and output relation of the PM
- Perform reachability on the PM and verify property Output = 1
- Generate an error trace if the equivalence check has failed
- Solving the problem of FSM state minimization
- Derive transition and output relation of the PM
- Perform reachability on the PM and derive the state equivalence relation
- Transform the initial FSM’s transition and output relations
- Compatible projection operator
May 2, 2000 ECE 510 OCE: BDDs and Their Applications 3
FSM Equivalence
- Definition. Two state machines are equivalent, if starting from their reset states, for any sequence of input vectors, they produce identical sequences of output vectors
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FSM State Equivalence
- Definition. Two states s 1 and s 2 of an FSM are equivalent, if for any sequence of input vectors, the FSM starting from state s 1 produces the same sequence of output vectors as the FSM starting from state s 2
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FSM Equivalence Checking
- Find the transition relations and output functions of M 1 and M 2. Find the transition relation and output function of the PM
- Perform reachability for the PM, while checking its output
- If the output of the product machine is 1 for all reachable states, M 1 and M 2 are equivalent; otherwise, generate an error trace (It is possible to define equivalence relative to anysubset of inputs and outputs of the FSM)
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Equivalence Checking Formulas
- Property expresses equivalence of M 1 and M 2 in states s 1 and s 2 which constitute state s of PM
P(s) = ∀∀∀∀i [ λPM (i,s) ]
- Machines M 1 and M 2 are equivalent iff
∀∀∀∀s [ AR(s) => P(s) ] = 1
where AR(s) is the set of reachable states of the PM and P(s) is the property that expresses equivalence of M 1 and M 2 in the product state s
- Alternatively, M 1 and M 2 arenot equivalent iff
∃∃^ ∃∃s [ AR(s) & ( P(s) )’ ] = 0
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Reachability Analysis Procedure
bool VerifyPropertyUsingReachabilityAnalysis( FSM* pM, bdd Property ) { bdd InitState = FindBddCube( 0, pM->NBits, CSVars, 0 ); bdd Reached = InitState, From = InitState, New[MAXITERNUM]; int NIter = 0; do { bdd To= bdd_appex(pM->TransRel,From,bddop_and,AllCSVars); To = bdd_replace( To, pNS4CS ); New[ NIter ] = To - Reached; bdd Check = ( New[ NIter ] >> Property ); if ( Check != bddtrue ) return false; From = New[ NIter ]; Reached = Reached | New[ NIter ]; } while ( New[ NIter++ ] != bddfalse ); return true; }
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FSM State Minimization
- Find the transition relations and the output functions of M. Find the transition relation and output function of the PM created by two identical instances of M
- Compute the state equivalence relation, describing the sets of all equivalent state pairs
- Compute the equivalence class characterization relation, by selecting a representative state from each class of equivalence states
- Compute the transformed transition relation and the transformed output relation
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Properties of Equivalence Relation
- Equivalence relation is reflexive, symmetric, and transitive
- Suppose the equivalence classes are {(00,01),(11),(10)}
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Computing Equivalence Relation
- E(s 1 , s 2 ) can be computed using the
following procedure (iterated until
E j(s) = E j+1 (s) )
E 0 (s) = ∀∀∀∀i [ λPM(i,s) ]
E j+1 (s) = E j(s) & ∀∀∀∀i∃∃∃∃n [T( i,s,n ) & E j(n)]
where E j (n) = R(s→n)[E j(s)] and R(s→n)
is the variable replacement operator
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Computing Distinquishability Relation
- D(s 1 , s 2 ) can be computed using the
following procedure (iterated until
D j(s) = Dj+1 (s) )
D 0 (s) = ∃i [ λPM(i,s)’ ]
D j+1 (s) = Dj (s) + ∃∃∃∃i∃∃∃∃n [ T( i,s,n ) & D j(n) ]
where D j(n) = R(s→n)[D j (s)] and R(s→n)
is the variable replacement operator
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Deriving E(s 1 , s 2 ) from D(s 1 , s 2 )
- AR(s 1 ) is the set of reachable states
AR(s 1 ) = ∃∃∃∃s2 [E(s 1 ,s 2 )] or
AR(s 1 ) = ∃∃∃∃s2 [D(s 1 ,s 2 )]
- The equivalence relation is derived as follows E(s 1 ,s 2 ) = [D(s 1 ,s 2 )]’ & AR(s 1 ) & AR(s 2 )
- Similarly, for the distinquishability relation D(s 1 ,s 2 ) = [E(s 1 ,s 2 )]’ & AR(s 1 ) & AR(s 2 )
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Compatible Projection Operator
- Given an equivalence relation E(x 1 , x 2 ) : {0,1} m^ x {0,1} m^ → {0,1}, the compatible projection is a boolean function F(x 1 , x 2 ) = { (x 1 ,x 2 ) | (x 1 , x 2 )∈E, x 2 = SEL(x 1 ) }, where SEL(x 1 ) is a selection function that uniquely selects one representative from each equivalence class
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Pseudocode of Compatible Projection
function CProjection( E, α ) if ( α = 1 ) return E; if ( E = 0 ) return 0; if ( E = 1 ) return α; y 1 is the top variable in α; if ( αy1 = 0 ) α 1 = x 1 ’; else /if ( αy1’= 0 )/ α 1 = x 1 ;
γ = ∃∃∃∃x1 Eα 1 ;
return α 1 & CProjection( Eα 1 , αα 1 ) + γ’α 1 ’ & CProjection( Eα1’ , α (^) α 1 );
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Complete Source Code for CProjection
bdd CProjection( bdd F, bdd Ref ) { assert( Ref != bddfalse ); if ( Ref == bddtrue ) return F; if ( F == bddfalse ) return bddfalse; if ( F == bddtrue ) return Ref; // check cache for ready-made results bdd NextRef, Literal; int CurVar = bdd_var( Ref ); if ( bdd_low( Ref ) == bddfalse ) { // the top var is positive NextRef = bdd_high( Ref ); Literal = bdd_ithvar(CurVar); } else if ( bdd_high( Ref ) == bddfalse ) { // the top var is negative NextRef = bdd_low( Ref ); Literal =!bdd_ithvar(CurVar); } else // Ref is not a cube! assert( 0 );
// cofactors of F with respect to this literal bdd PosCofF = bdd_restrict( F, Literal ); bdd NegCofF = bdd_restrict( F, !Literal ); // the domain where projection does exist bdd Domain = bdd_exist( PosCofF, AllAVars0 ); bdd PosPart = CProjection( PosCofF, NextRef ); bdd NegPart = !Domain & CProjection( NegCofF, NextRef ); bdd Result = bdd_ite( Literal, PosPart, NegPart ); // insert the result into cache return Result; }
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Homework: A Study of Random FSMs
- Generate transition relations of random FSMs with N states and K transitions in each state
- Perform reachability analysis using the generated transition relations and determine the number of reachable states and the number of iterations in the FSM traversal
- Assume, N = 10000, K = {1,2,…,10}. Draw a graph visualizing the number of reachable states, the number of iterations, and the time needed to complete the reachability analysis as a function of K.
