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Model Checking : Making Automatic Formal Verification Scale Shaz Qadeer EECS Department University of California at Berkeley

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Formal Verification Problem –Design or system M - mathematical model –Specification - formal language –Does M satisfy ? Applications –Software (safety-critical) –Hardware (microprocessors) –Protocols (shared memory multiprocessors) –Security

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Specification Intent Implementation Product Property verification Refinement checking Equivalence checking

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Approaches Algorithmic (model checking) –Design - typically finite state machines –Specification - first-order temporal logic – Automatic !! Deductive (interactive theorem proving) –Very expressive –User interacts with the prover Deductive + Algorithmic –Abstraction –Compositional model checking

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Modeling designs Formal model of implementation and specification - finite state machine r1r1 r2r2 g1g1 g2g2 r1r1 r2r2 ~r 1 ~r 2 F1F1 F2F2 G1G1 G2G2 r g r g Semantics of model - trace language on I/O

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Specification - Temporal logic pppppp Finally: F p Globally: G p NeXt: X p Temporal operators: Arbiter specification: G (r 1 F g 1 )

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Specification - FSM r2r2 r1r1 g2g2 g1g1 r2r2 r1r1 F G1G1 G2G2 Nondeterministic finite state machine r1r1 r2r2 g1g1 g2g2 r1r1 r2r2 ~r 1 ~r 2 F1F1 F2F2 G1G1 G2G2 P P’ iff def L(P) is contained in L(P’) iff Every linear property satisfied by P’ is satisfied by P

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Model Checking Explore state space of I exhaustively All possible execution sequences are checked !! FSM I Does I satisfy ? Does I refine S ? Counterexample No

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State space explosion P || Q m n states Designs expressed as composition P 1 || P 2 ||.... || P k - m 1 m 2 .... m k states Model checking exponential in design description a P m states Q n states a bb FSM composition operator ||

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Outline Symbolic model checking in VIS Overview of my contributions Forward symbolic model checking Compositional model checking in assume-guarantee paradigm Future research directions

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VIS Symbolic model checker I 1 || I 2 || I 3 I1I1 I2I2 I3I3 Temporal logic spec Hierarchical description Flat description Compose

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Symbolic Model Checking Manipulation of state sets (not individual states) Implicit representation with boolean expressions –state transition graph –state sets Operations on state sets performed implicitly –boolean ( , , ) –image computation (pre, post)

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Pre Transition relation T S pre(S) T

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Post Transition relation T post(S) S T

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VIS : Limitations 1. Backward Model Checking Invariant checking : init F bad ? initbad Iterate pre until fixpoint Useless exploration of unreachable states

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VIS: Limitations 2. Design structure not utilized Symbolic model checker I 1 || I 2 || I 3 I1I1 I2I2 I3I3 Temporal logic spec Compose

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VIS: Limitations 3. Unsuitable for asynchronous protocols Naïve symbolic exploration explores all orderings of independent events - wasteful ! Independent events a, b a a b b

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Partial-order techniques –explicit state exploration –avoids exploring all orderings of independent events VIS: Limitations 3. Unsuitable for asynchronous protocols “Partial-order reduction in symbolic state exploration,” Alur, Brayton, Henzinger, Qadeer, Rajamani 1997

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VIS: Limitations 4. Limited to finite-state systems Parameterized systems –protocols (communication, multiprocessors) –finite-state methods verify only instantiations “Verifying sequential consistency on shared-memory multiprocessors,” Henzinger, Qadeer, Rajamani 1999

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Outline Symbolic model checking in VIS Overview of my contributions Forward symbolic model checking Compositional model checking in assume-guarantee paradigm Future research directions

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Invariant Verification: Pre init F bad ? initbad

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Invariant Verification: Post init F bad ? initbad

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Response Verification: Pre init F (req G ack) ack G ack

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Response Verification: Pre init F (req G ack) G ackreq init

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Response Verification: Post init F (req G ack) init req R = Reach(init) req req req R

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Response Verification: Post req R ack

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Theorem All linear properties expressible by Buchi automata can be model checked by forward reasoning. “From pre-historic to post-modern symbolic model checking,” Henzinger, Kupferman, Qadeer 1998

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Outline Symbolic model checking in VIS Overview of my contributions Forward symbolic model checking Compositional model checking in assume-guarantee paradigm Future research directions

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Previous approaches Heuristically improve state exploration efficiency –Synchronous hardware: symbolic techniques (BDDs) - VIS –Asynchronous protocols: state reduction with explicit enumeration partial-order reduction - SPIN symmetry reduction - Murphi Algorithms work on flat description of systems State-of-the-art : state variables

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Our approach - M OCHA Complex designs necessarily modular and hierarchical Utilize rather than destroy design structure Partition the verification problem into smaller obligations Use existing algorithms to decide these obligations

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Divide and Conquer P P’ Q Q’ P || Q P’ || Q’ Difficulty : Not applicable in practice

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P P’ Q Q’ PQ P’Q’ P’

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Assume-Guarantee Rule [Stark] [Clarke-Long-McMillan] [Grumberg-Long] [Abadi-Lamport][Alur-Henzinger][McMillan] P || Q’ P’ P’ || Q Q’ P || Q P’ || Q’

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Propositional Validity? p q’ p’ p’ q q’ p q p’ q’ Consider case when p = true q = true p’ = false q’ = false

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Refinement Checking in M OCHA Space Abstraction

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Refinement Checking in M OCHA Space Abstraction Time abstraction

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Refinement Checking in M OCHA

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Refinement Maps Environment signals not present in specification Manually construct abstract constraining modules Design insight is required !

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EQEQ yx a y EPEP Px b a b EQEQ EPEP a a P Q x y b S x y EPEP b S xy Q EQEQ y a b x

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M OCHA - Refinement Examples Asynchronous applications –Sliding window protocol Synchronous applications –Pipeline –Tomasulo’s algorithm “You assume, we guarantee: methodology and case studies,” Henzinger, Qadeer, Rajamani 1998

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VGI architecture 16 clusters with 6 processors in each - 4 compute, 1 memory, 1 I/O ~30K logic gates and ~800 latches per processor 3-stage pipelined compute processors Complex handshake between processors Interconnection between processors configured statically

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Complex handshake FIFO buffer pipeline ISA

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Verification of VGI Level-sensitive latches multiple implementation steps correspond to single specification step Very large design need compositional verification “Assume-guarantee refinement between different time scales,” Henzinger, Qadeer, Rajamani 1999 “Formal specification and verification of a dataflow processor array,” Henzinger, Liu, Qadeer, Rajamani 1999

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MOCHA Verilog Proof Assistant e.g., assume-guarantee,.... Reactive Module Algorithm 1Algorithm 2 Temporal logic specification Abstract Module EsterelJava “Mocha: modularity in model checking,” Alur, Henzinger, Mang, Qadeer, Rajamani, Tasiran 1998

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Formal design 3 subtle bugs in the interaction between datapath and communication control were found and fixed Design insight (through refinement maps) indispensable for model checking Model checking indispensable for producing correct design –Error traces invaluable –Iterative testing of design fixes

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Formal design Symbiotic relationship between design and verification Refinement maps not a big burden for designers Vision –Both activities performed in parallel –Designer uses a model checker interactively during the design phase itself

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P’Q’ PQ TPTP TQTQ PQ TPTP TQTQ P’ P Q’ Q P’

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Sample operator Finite state machine P Predicate on the variables of P A run of P sampled whenever is true is a run of Sample (P) We compare Sample ~clk (VGI) against its specification

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pipeline ISA ~clk

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