Efficiently Simulating Higher-Order Arithmetic by a First-Order Theory Modulo

Efficiently Simulating Higher-Order Arithmetic by a First-Order Theory Modulo

Deduction modulo is a paradigm which consists in applying the inference rules of a deductive system—such as for instance natural deduction—modulo a rewrite system over terms and propositions. It has been shown that higher-order logic can be simulated into the first-order natural deduction modulo. However, a theorem stated by Gödel and proved by Parikh expresses that proofs in second-order arith...

First-Order Proof Theory of Arithmetic

First-order reasoning for higher-order concurrency

We present a practical first-order theory of a higher-order ⇡-calculus which is both sound and complete with respect to a standard semantic equivalence. The theory is a product of combining and simplifying two of the most prominent theories for HO⇡ of Sangiorgi et al. and Je↵rey and Rathke [10, 21], and a novel approach to scope extrusion. In this way we obtain an elementary labelled transition...

From Higher-Order to First-Order Rewriting

We show how higher-order rewriting may be encoded into rst-order rewriting modulo an equational theory E. We obtain a characterization of the class of higher-order rewriting systems which can be encoded by rst-order rewriting modulo an empty theory (that is, E = ;). This class includes of course the-calculus. Our technique does not rely on a particular substitution calculus but on a set of abst...

First-Order Proofs for Higher-Order Languages

We study the use of the-calculus for semantical descriptions of higher-order concurrent languages with state. As an example, we choose Concurrent Idealized ALGOL (CIA). CIA is particularly interesting as, yet being a core language, it combines imperative and parallel features with a procedural mechanism of full higher order. It can thus be used as a formal model for concurrent programs as, e.g....