A fix-point characterization of Herbrand equivalence of expressions in data flow frameworks

Jasine Babu, K. Murali Krishnan, Vineeth Paleri

The problem of determining Herbrand equivalence of terms at each program point in a data flow framework is a central and well studied question in program analysis. Most of the well-known algorithms for the computation of Herbrand equivalence in data flow frameworks proceed via iterative fix-point computation on some abstract lattice of short expressions relevant to the given flow graph. However the mathematical definition of Herbrand equivalence is based on a meet over all path characterization over the (infinite) set of all possible expressions. The aim of this paper is to develop a lattice theoretic fix-point formulation of Herbrand equivalence on the (infinite) concrete lattice defined over the set of all terms constructible from variables, constants and operators of a program. The present characterization uses an axiomatic formulation of the notion of Herbrand congruence and defines the (infinite) concrete lattice of Herbrand congruences. Transfer functions and non-deterministic assignments are formulated as monotone functions over this concrete lattice. Herbrand equivalence is defined as the maximum fix point of a composite transfer function defined over an appropriate product lattice of the above concrete lattice. A re-formulation of the classical meet-over-all-paths definition of Herbrand equivalence in the above lattice theoretic framework is also presented and is proven to be equivalent to the new lattice theoretic fix-point characterization.

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