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1 Minimalist proof assistants Interactions of technology and methodology in formal system level verification Ken McMillan Cadence Berkeley Labs

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2 Automated methods curve system size (bits of state) verification probability research systems 110100 10 3 10 4 10 5 10 6 10 7 100%

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3 Automated methods curve system size (bits of state) verification probability unit-level FV decomposition 110100 10 3 10 4 10 5 10 6 10 7 100% bug finding verification

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4 Proof assistants l General purpose proof assistants +Expressive logics +Integration with model checking l Problem: Do not naturally produce finite-state subgoals –Result: Detailed manual case analysis Complex global invariants Long, fragile proof scripts

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5 What's missing... l Proof strategy that produces finite-state lemmas l Infrastructure to support this strategy l Reduced interactivity –more reusable proofs –no global invariants (too design-specific)

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6 Minimalist proof assistant l Approach –Minimal logical expressiveness –Appropriate domain-specific proof strategy –Proof rules designed to support this strategy –All proof obligations reduced to model checking l Result –Short proofs –Proof goals reduced to tractable MC problems –Proofs more easily reused related: Seger 98

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7 Rules built into SMV system... l Circular compositional rule –decomposition by refinement relations –auxiliary state l Temporal case analysis –path splitting l Symmetry reductions –reduce by symmetry to tractable number of cases l Data type reductions –large and infinite types –uninterpreted functions support general proof strategy...

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8 Compositional refinement verification Abstract model System Translations

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9 Localized verification Abstract model System Translations assumeprove

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10 Localized verification Abstract model System Translations assume prove

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11 Circular inference rule SPEC (related: AL 95, AH 96) 1 up to t -1 implies 2 up to t 2 up to t -1 implies 1 up to t always 1 and 2

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12 SPEC Auxiliary variables aux P PA Q Q * A is definitional * Q does not reference A (related: Owicki, Gries)

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13 SPEC P PA Big structures and path splitting i

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14 Temporal case splitting l Prove separately that p holds at all times when v = i. l Path splitting v record register index i

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15 SPEC P PA Case explosion and symmetry i k

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16 Exploiting symmetry l Symmetric types –Semantics invariant under permutations of type. –Enforced by type checking rules. l Symmetry reduction rule * i is of scalarset type * P references only constants 0..k-1 of type (related: Murphi)

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17 Data type reductions l Problem: large data types yield state explosion l Solution: reduce large (or infinite) types where T\i represents all the values in T except i. l Abstract interpretation Note: use in conjunction with case splitting

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18 Data types with large ranges l Words, addresses, sequence numbers, tags, etc... l Example: content addressable memory KEYVALUE = = = = = = = QUERY

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19 Summary of proof strategy l Control logic –Structural decomposition (comp. rule) l Data path –Refinement maps (comp. rule + aux.) –Decompose large structures (path splitting) –Reduce large types (data type reduction) Case reduction (symmetry)

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20 Illustration: Tomasulos algorithm l Execute instructions in data flow order OP,DST opraoprb OP,DST opraoprb OP,DST opraoprb EU OPS TAGGED RESULTS INSTRUCTIONS VAL/TAG REG FILE

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21 Compositional rule l Decompose into two lemmas OP,DST opraoprb OP,DST opraoprb OP,DST opraoprb EU OPS TAGGED RESULTS INSTRUCTIONS VAL/TAG REG FILE Lemma 1: Correct operands Lemma 2: Correct results

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22 Proving the operand lemma OP,DST opraoprb OP,DST opraoprb OP,DST opraoprb EU OPS INSTRUCTIONS REG FILE VAL/TAG TAGGED RESULTS Lemma 1: Correct operands Lemma 2: Correct results "cone of influence" eliminates

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23 Auxiliary variables in Tomasulo l Used to store correct operands and results for each instruction EU RESULTS INSTRUCTIONS OP,DST opraoprb O1O2R if (~stallout & opin=ALU){ next(aux[st].opra) := opra; next(aux[st].oprb) := oprb; next(aux[st].res) := res; } SPEC

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24 Lemmas in SMV l Operand correctness forall (k in TAG) layer lemma1 : if (rs[k].valid & rs[k].opra.valid) rs[k].opra.val := aux[k].opra; l Result correctness forall (i in TAG) layer lemma2[i] : if (rb.tag = i & rb.valid) rb.val := aux[i].res;

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25 Path splitting in Tomasulo OP,DST opraoprb OP,DST opraoprb OP,DST opraoprb EU OPS INSTRUCTIONS REG FILE VAL/TAG TAGGED RESULTS "cone of influence" eliminates

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26 SMV implementation l Split cases of operand correctness on –producer reservation station –holding register l SMV implementation subcase lemma1[i][j] of rs[k]//lemma1 for rs[k].opra.tag = i & aux[k].srca = j;

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27 Case explosion problem l Number of cases in operand correctness property: TAGS REGS TAGS = O(n 3 ) l Symmetric data type declarations scalarset REG 0..31; scalarset TAG 0..31; l SMV verifies types used in symmetric way

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28 Symmetry reduction in operands lemma OP,DST opraoprb OP,DST opraoprb OP,DST opraoprb EU OPS INSTRUCTIONS REG FILE VAL/TAG TAGGED RESULTS i j k l Reduces by symmetry to two cases: (i = 0, j = 0, k = 0) (i = 0, j = 0, k = 1)

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29 Type reduction: infinite-state Tomasulo l Scalarsets with undefined range l Data type reduction TAG->{i,k}, REG ->{j} l Only include values we care about in reduced type l Reduces variable encodings to –1 bit per variable of type REG –2 bits per variable of type TAG (related: Kurshan)

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30 OP,DST Uninterpreted functions l Verify Tomasulo for arbitrary EU function f(a,b). f(a,b) RESULTS INSTRUCTIONS SPEC OP,DST opraoprb opraoprb OP,DST opraoprb f(a,b) OPS INSTRUCTIONS REG FILE VAL/TAG TAGGED RESULTS f(a,b) (related: Burch, Dill, Jones, etc...)

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31 Case splitting l Prove result correctness only for specific cases, e.g. opra = 0, oprb = 1, f[0][1] = 2 OP,DST 01 opraoprb OP,DST opraoprb f(a,b) OPS INSTRUCTIONS REG FILE VAL/TAG 2 3! = 6 cases verified (related: Hojati, Singhal, Bryant, Clarke)

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32 Result l Verification problem reduced to tractable MC problems –Max 25 state bits –11 cases of lemmas to verify after symmetry –Verification time less than 4 seconds l Tomasulo implementation proved for –Arbitrary number of registers, reservation stations –Arbitrary data word size and EU function l The proof is concise

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33 Summary of approach l Auxiliary variables and circular rule –operand and result lemmas l Temporal case splitting –data path splitting l Symmetry reductions –reduce to tractable number of cases l Data type reductions –reduce large or infinite data types to small finite –uninterpreted functions for data operations

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34 More examples l Applications of the same general strategy: –Infopad packet multiplexer –SGI cache coherence

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35 InfoPad example (Truman 98) Bus Bridge ARM processor subsystem RX ASIC FPGA (FEC, timing,…) Video Pen Speech TX Audio/ Speech VGA control Wireless Modems 32-bit word 8-bit word Serial, FEC encoded Packet Buffer Memory

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36 Packet streams Decomposition -- data integrity tag data Packet mux P P path splitting induction

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37 Cache coherence (Eiriksson 98) S/F network protocol host protocol host protocol host Distributed cache coherence INTF PP MIO to net l Nondeterministic abstract model l Atomic actions l Single address abstraction l Verified coherence, etc...

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38 Mapping protocol to RTL S/F network protocol host other hosts Abstract model CAM TABLES TAGS ~30K lines of verilog

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39 Conclusions l Goal –System-level verification by model checking l Approach –Appropriate domain-specific proof strategy –Proof rules designed to support this strategy l Result –Proof goals reduced to tractable MC problems –Short proofs -- no global invariants

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