Rewriting is a framework for reasoning about functional programming. The dependency pair criterion is a well-known mechanism to analyze termination of term rewriting systems. Functional specifications with an operational semantics based on evaluation are related, in the rewriting framework, to the innermost reduction relation. This paper presents a PVS formalization of the dependency pair criterion for the innermost reduction relation: a term rewriting system is innermost terminating if and only if it is terminating by the dependency pair criterion. The paper also discusses the application of this criterion to check termination of functional specifications.
An approach is introduced to formally verify the logical correctness of reconfigurable hardware implementations of algebraic operators. Since Hardware Description Languages describe circuits/systems in an imperative style and formalization tools use recursive specification languages, the kernel of our approach is based on a conservative translation from imperative into recursive implementations. The main challenge of this approach is that proofs follow an inductive schema that is based on guaranteeing pre and postconditions and preservation of invariants during all steps of the recursive execution such as in the Floyd-Hoare's logical approach for verification of imperative procedures. The applicability of the methodology is illustrated in the Prototype Verification System (PVS) by proving the logical correctness of an FPGA implementation of the Gauss-Jordan matrix inversion algorithm (GJ). Correctness of this FPGA implementation is based on proving its functional equivalence (FEq) with an algebraic imperative definition of GJ. The approach allows formal verification of fragments of the implementations either simultaneously or afterwards the design process has been finished, avoiding in this way hardware development delays.
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