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State-of-the-art refutation systems for SAT are largely based on the derivation of clauses meeting some redundancy criteria, ensuring their addition to a formula does not alter its satisfiability. However, there are strong propositional reasoning techniques whose inferences are not easily expressed in such systems. This paper extends the redundancy framework beyond clauses to characterize redundancy for Boolean constraints in general. We show this characterization can be instantiated to develop efficiently checkable refutation systems using redundancy properties for Binary Decision Diagrams (BDDs). Using a form of reverse unit propagation over conjunctions of BDDs, these systems capture, for instance, Gaussian elimination reasoning over XOR constraints encoded in a formula, without the need for clausal translations or extension variables. Notably, these systems generalize those based on the strong Propagation Redundancy (PR) property, without an increase in complexity.

Term rewriting systems have a simple syntax and semantics and facilitate proofs of correctness. However, they are not as popular in industry or academia as imperative languages. We define a term rewriting based abstract programming language with an imperative style and a precise semantics allowing programs to be translatable into efficient imperative languages, to obtain proofs of correctness together with efficient execution. This language is designed to facilitate translations into correct programs in imperative languages with assignment statements, iteration, recursion, arrays, pointers, and side effects. It can also be used in place of a pseudo-programming language to specify algorithms.

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