5/6/2004J.-H. R. Jiang1 Functional Dependency for Verification Reduction & Logic Minimization EE290N, Spring 2004

5/6/2004J.-H. R. Jiang 2 Outline Motivations Previous work Our formulation Experimental results Conclusions

5/6/2004J.-H. R. Jiang 3 Outline Motivations Previous work Our formulation Experimental results Conclusions

5/6/2004J.-H. R. Jiang 4 Motivations Logic synthesis of state transition systems Remove “redundant” registers using functional dependency Formal verification of state transition systems Reduce state space and compact BDD representations by removing dependent state variables

5/6/2004J.-H. R. Jiang 5 Outline Motivations Previous work Functional dependency Signal correspondence Our formulation Experimental results Conclusions

5/6/2004J.-H. R. Jiang 6 Previous work “Functional” dependency in state transition systems Problem formulation Given a characteristic function F(x 1,x 2, …, x n ), compute a minimal set of irredundant (independent) variables Variable x i is redundant if it can be replaced with a function over other variables Solution – functional deduction Variable x i is redundant if and only if F| x i = 0 Æ F| x i = 1 = false Example F = abc Ç : a : c Minimal independent sets: {a, b}, {b, c} with dependency functions c := a, a := c, respectively

5/6/2004J.-H. R. Jiang 7 Previous work Applications of functional dependency Synthesis Register minimization in hardware synthesis from HDL Verification Minimization of BDDs of reached state sets Embed detection of functional dependency inside reachability analysis as an on-the-fly reduction Weakness Need to perform reachability analysis to derive functional dependency (for applying functional deduction)

5/6/2004J.-H. R. Jiang 8 Unsolved problem How to detect functional dependency without or before computing reached state sets ?

5/6/2004J.-H. R. Jiang 9 Previous work Signal correspondence Problem formulation A signal correspondence C µ s £ s is an equivalence relation (in reachable state subspace) on the set s of state variables ( This definition includes only identical functions, it can be extended to also include complemented functions) An effective solution Compute the equivalence relation by iterative refinement of state variables Valid for an over-approximated reachable space Application of detecting signal correspondence Make sequential equivalence checking more like combinational equivalence checking Detect equivalent state variables

5/6/2004J.-H. R. Jiang 10 Example (219B) s1s s2s2 s3s3 s4s4 s5s5 s 1 =1 s 2 =1 s 3 =1 s 4 =1 s 5 =1 v s 1 = x  v s 4 = x  v v1v1 s 2 =  v s 3 =  v s 5 =  v v2v2 s 1 = x  v 1 s 4 = x  v 1 v1v1 s 2 =  v 1 v 2 ) s 3 =  v 1 v 2 ) s 5 =  v 1 v 2 ) v2v2 Result: {s 1,s 4 } {s 2,s 3,s 5 } Instead of using constraint, use fresh variable for each class

5/6/2004J.-H. R. Jiang 11 Previous work Weakness Signal correspondence is a very limited form of functional dependency

5/6/2004J.-H. R. Jiang 12 Unsolved problem How to characterize a more general form of functional dependency by fixed-point computation?

5/6/2004J.-H. R. Jiang 13 Outline Motivations Previous work Our formulation Observation Combinational dependency Sequential dependency Greatest fixed point Least fixed point Verification Reduction Experimental results Conclusions

5/6/2004J.-H. R. Jiang 14 Our formulation Objective Resolve the unsolved problems (exploiting functional dependency and detecting signal correspondence) in a unified framework Key Conclude functional dependency directly from transition functions of a state transition system. Define combinational dependency Extend to sequential dependency

5/6/2004J.-H. R. Jiang 15 Combinational dependency Given two functions f and g over the same domain C, f functionally depends on g if there exists some function  such that f (·) =  ( g (·) ). A necessary and sufficient condition: f (a)  f (b)  g (a)  g (b), for all a,b  C In such case, we denote g v f Consider multi-valued functions as vectors of Boolean functions

5/6/2004J.-H. R. Jiang 16 Combinational dependency

5/6/2004J.-H. R. Jiang 17 Combinational dependency

5/6/2004J.-H. R. Jiang 18 Sequential dependency Extend combinational dependency for state transition systems Find invariant  such that s dep =  (s ind ) and  dep =  (  ind ) where s represents the set of state variable and  represents the set of transition functions. Two approaches of computing fixed points Greatest fixed-point (gfp); least fixed-point (lfp)

5/6/2004J.-H. R. Jiang 19 Sequential dependency Greatest fixed-point (gfp) computation Initially, all state variables are distinct. In each iteration, compute the combinational dependency among independent state variables from the previous iteration.

5/6/2004J.-H. R. Jiang 20 Sequential dependency (gfp)

5/6/2004J.-H. R. Jiang 21 Sequential dependency Least fixed-point (lfp) computation Initially, select one state var as the representative.  (0) is determined by initial state information. In each iteration of computing functional dependency, try to reuse  ’s from the previous iteration. If restrict  ’s to be identity functions, the computation reduces to detecting signal correspondences.

5/6/2004J.-H. R. Jiang 22 Sequential dependency (lfp)

5/6/2004J.-H. R. Jiang 23 Legitimacy for logic synthesis Dependency may not hold for initial states which have no predecessors Localize conflicting state variables and declare them as independent state variables

5/6/2004J.-H. R. Jiang 24 Verification reduction Reachability analysis on reduced state space Static verification reduction Before a reachability analysis, derive sequential dependency (using lfp or gfp computation). Dynamic (on-the-fly) verification reduction In each iteration of a reachability analysis, derive a minimal set of independent state variables before the image computation. (No need to try to reuse  ’s.) Thus, the image computation is over the reduced state space. Prior work on exploiting functional dependency is not effective because the detection of functional dependency is done after the image computation.

5/6/2004J.-H. R. Jiang 25 Verification reduction

5/6/2004J.-H. R. Jiang 26 Outline Motivations Previous work Our formulation Experimental results Conclusions

5/6/2004J.-H. R. Jiang 27 Experimental results Dependency in original FSM CircuitRegSignal CorrespondenceSequential Dependency GfpSequential Dependency Lfp Indp.Iter.MbsecIndp.Iter.MbsecIndp.Iter.Mbsec s298-rt s526n-rt s838-rt s991-rt mult16a-rt tbk-rt s s s s s

5/6/2004J.-H. R. Jiang 28 Experimental results Dependency in product FSM CircuitRegSignal CorrespondenceSequential Dependency GfpSequential Dependency Lfp Indp.Iter.MbsecIndp.Iter.MbsecIndp.Iter.Mbsec s s s s s s s526n s s s mult16a tbk

5/6/2004J.-H. R. Jiang 29 Experimental results On-the-fly reduction CircuitIter.Reach. Analysis w/o Dep. ReductionReach. Analysis w Dep. Reduction Peak (bdd nodes) Reached (bdd nodes) MbsecPeak (bdd nodes) Reached (bdd nodes) Mbsec s ,819,30116,158, ,843,83710,746, s ,527,781248, ,0068, s53782N/A >2GN/A1,151,439113, s ,842,8899,961, ,667,0766,356, ,663,7491,701, ,830,6021,338,

5/6/2004J.-H. R. Jiang 30 Outline Motivations Previous work Our formulation Experimental results Conclusions

5/6/2004J.-H. R. Jiang 31 Conclusions Proposed a computation of functional dependency w/o reachability analysis. Unified two previously independent studies on detecting signal correspondence and exploiting functional dependency. Detecting signal correspondence is a special case of lfp computation of sequential dependency. Previous approach on exploiting functional dependency can be improved with our dynamic reduction. In addition to verification reduction, our results can be used to minimize state transition systems.