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Incremental formal verification of hardware Hana Chockler Alexander Ivrii Arie Matsliah Shiri Moran Ziv Nevo IBM Research - Haifa Formal verification (hardware) • Effective, but computationally expensive In many scenarios, similar verification tasks are performed repetitively: • Design Spec Verification tool • Regression verification • Update to design • Update to specifications • Coverage verification Can we store and reuse information to reduce amount of redundant computation? • Pass / Fail Incremental formal verification hardware extract relevant part of previously saved information safety propertie s Design Spec ic3 Verification tool DB store reusable information Pass / Fail Outline ● inductive proofs and inductive strengthening ● saving information ● ● – ic3 overview – what is saved? reusing saved information – extracting relevant parts (w.r.t. new design/spec) – checking if verification can be concluded – injecting into ic3 conclusion and experimental results FSMs and safety properties ● x1,x2,…,xn – state variables (latches) ● I – initial states ● T – transition relation ● R – all reachable states ● ● Ri – states reachable within i steps from I P – (safety) property All states ┐P R Rk-1 … R2 R1 I Inductive proofs (for R P) ● Simple induction: – I P, P ^ T P‘ Sufficient but not necessary. Almost never holds in practice.. ● ┐P Solution: find G such that: – IG – G ^ T G‘ – GP G is over-approximation of R I R G All states ! ic3 - basic properties ● Complete – always terminates with correct result ● SAT based, no unrolling ● If P is invariant, produces a CNF formula G, s.t.: – ● IG – G ^ T G’ – GP ┐P I R G All states If not, produces a (generalized) CEX α0, α1,…, αk s.t.: – all α0 states belong to I – all αi states lead to some αi+1 state – αk is in ┐P I a0 a1 … ┐P ak (bounded) inductive invariants in ic3 ● Clause sets/CNF formulas F1,...,Fk ● Initially: k=1, F1 = P ● (assume I P and Img(I) P) Invariants: – I F1 ... Fk P – Img(Fi) Fi+1 – Ri Fi (furthermore, for all i, Fi+1 is a subset of Fi) P Fk-1 Fk F1 F0=I ● Img(F0) … Img(Fk-2) Img(Fk-1) If Fi = Fi+1 for some i<k, then Fi is an inductive strengthening that proves R P ic3 progress and termination ● ● ● Inductive clauses that block “bad state predecessors” are added to the sets Fi, in a way that maintains the containment invariants Once in a while, clauses are “pushed” to higher Fi’s ic3 terminates when either: – Fi=Fi+1 for some i we save the inductive invariant Fi – it finds a CEX: chain of bad state predecessors that starts at I we generalize and save the CEX + we save the absolute invariants * Absolute inductive invariants are those clauses that were “pushed” beyond Fk How to reuse saved invariants? Finding maximal inductive invariant ● Input: I, T, P and C = {c1,…,cm} - candidate invariant clauses ● Output: PASS or maximum subset Q of C such that I Q and Q ^ T Q‘ * Note: if Q1 ^ T Q’1 and Q2 ^ T Q‘2 then (Q1 U Q2) ^ T (Q’1 U Q’2) • Once such Q is found, we can “inject” it into ic3 by conjoining Q with all sets Fi • This saves ic3 the effort of “rediscovering” the invariants from Q Finding Q using a SAT solver * that supports SolveWithAssumptions(a1,...,ak) 1. cnfize T and I, set Q:=C 2. remove from Q all clauses that are not implied by I 3. for every ci in Q, introduce two auxiliary vars: xi and y’i 4. for every i, cnfize xi ci and y’i ┐c’i 5. SolveWithAssumptions(x1, ..., x|Q|, (y’1 v ... v y’|Q|)) 6. if unsat: if sat: Q is invariant remove from Q each ci with assign(y’i)=1 and goto 5 7. if Q P output PASS, ow return Q Overall approach inject maximal inductive-invariant into ic3 Design maximal invariant/ CEX extraction Spec Verification tool DB save inductive invariant / CEX inductive invariants / generalize d CEXes Pass / Fail Experimental results (accumulated runtimes in seconds) 758 designs from HWMCC’10 From scratch After original After mutated Original 30,597 402 10,070 Mutated 50,294 37,348 2,091 17 IBM designs From scratch After original After mutated Original 36,605 1,238 11,710 Mutated 54,160 24,447 883 Concluding remarks ● ● ● ● ● ic3 can be used to save small inductive proofs, and generalized CEXes the technique is robust since ic3 invariants and CEXes involve only state variables makes coverage and regression verification almost immediate parts from inductive proofs can be used even if design/spec has significantly changed saved information is reusable even when verification result changes The End Generalizing assignments ● Input: circuit C and assignment a such that C(a)=y ● Output: partial assignment a’ such that C(b)=y for all extensions b of a’ * a’ is obtained by subst. some of the 0,1 values in a with x (don’t cares) Standard algs: • start from root and propagate “cares” • start from leaves and propagate “don’t cares” Generalizing assigns. with solver ● Input: circuit C and assignment a such that C(a)=y ● Output: partial assignment a’ such that C(b)=y for all extensions b of a’ 1. cnfize C 2. SolveWithAssumptions(┐(C(a)=y), a1, ..., an) * must return unsat (BCP) 3. if ai participates in the conflict set a’i = ai else set a’i = x Generalizing assigns. with solver 1. cnfize C 2. SolveWithAssumptions(┐(C(a)=y), a1, ..., an) * must return unsat (BCP) 3. if ai participates in the conflict set a’i = ai else set a’i = x Advantages: 1. easy to enforce additional constraints (e.g. learnt clauses and invars) 2. can order the variables in the assumptions acc. to some priority 3. can run after standard algs 4. no real solving – just BCP 5. shrinks by additional 30-40% after ternary simulation (like in PDR)