New PDF release: A first book in logic

Logic

By Henry Bradford Smith

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It is true that this method does not scale well, but it is nevertheless a feature of PCC that the checker does not care whether the code is produced by a trusted compiler and will gladly accept code that was written or optimized by hand. Another possibility is that the annotations can be produced automatically by a certifying compiler. Such a compiler first inserts bounds checks for all array accesses. In the presence of these checks the invariant predicates must include only type declarations for those live registers that are not declared invariant.

First, VCGen initializes the symbolic values of all four registers with fresh new symbolic values to denote unknown initial values in all registers. But the initial values of the registers rd and rl are constrained by the precondition. To account for this, VCGen takes the specified precondition and, without trying to interpret its meaning, substitutes in it the current symbolic values of the registers. The result is the symbolic predicate formula “d0 : array(bool, l0 )” that is added to a stack of assumptions.

Next VCGen adds to the assumption stack the predicate “i1 ≤ l0 ” obtained from the invariant predicate after substitution of the new symbolic values for registers. When VCGen encounters a conditional branch it simulates both possible outcomes. First, the branch is assumed to be taken and a corresponding assumption is placed on the assumption stack. When that symbolic execution of that branch A Scalable Architecture for Proof-Carrying Code 31 1: Generate fresh values rd = d0 , rl = l0 , ri = i0 and rt = t0 1: Assume Precondition d0 : array(bool, l0 ) 2: Set ri = l0 3: Invariant (first hit) l0 ≤ l0 3: Generate fresh values ri = i1 , rt = t1 3: Assume invariant i1 ≤ l0 4: Set rt = ge(i1 , 0) 5: Branch 5 taken not (ge(i1 , 0)) 11: Check postcondition 1 : bool 5: Branch 5 not taken ge(i1 , 0) 6: Set rt = add(d0 , i1 ) saferd(add(d0 , i1 )) 7: Check load 7: Set rt = mem(add(d0 , i1 )) 8: Branch 8 taken not (mem(add(d0 , i1 ))) 10: Check postcondition 0 : bool 8: Branch 8 not-taken mem(add(d0 , i1 )) 9: Set ri = sub(i1 , 1) sub(i1 , 1) ≤ l0 , d0 = d0 , l0 = l0 3: Invariant (second hit) 4: Done Fig.

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A first book in logic by Henry Bradford Smith


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