PARCEL and MIPRAC: Parallelizers for Symbolic and Numeric Programs
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Semantics for MIL The abstract interpretation that is MIL's interprocedural analysis algorithm is derived in several steps. We begin with a standard semantics over the following domains: S = Id ! Z ! B stores B = (Z \Theta Y ) ! V objects V = L + C + Z values L = Id \Theta Z \Theta Z locations C = Proc \Theta E closures E = Id ! Z environments Z integers Y ae Z sizes T booleans Proc procedures Expr expressions Id identifiers The domains Z, Y , T , Expr , Proc, and Id are flat: each has a least element (?Z , ?T , etc.), and its other elements are incomparable. The structure of the non-bottom elements of Expr and Proc is given by the grammar of MIL above. None of the above domains is reflexive, since each is defined in terms of those strictly below it in the table. Said another way, the equation for each can be written (finitely) as products (\Theta), sums (+) and function spaces (!) of the flat domains Z, T , Expr , Proc, and Id . For example, B is equal to (Z \Theta Y ) ...