Partial Correctness Assertions Provable in Dynamic Logics

Abstract

We consider a formalism DL for first order Dynamic Logic, based on Segerberg’s axioms for modalities, and observe that DL is not conservative over Hoare Logic (HL) when the background theory is empty, but is conservative if the background theory is the complete theory of an expressive structure (in the sense of Cook). We identify Peano Arithmetic (PA) as the transition point between these two states of affairs: DL is conservative over HL in the presence of a number theory that contains PA, and is not conservative for the sub-theories of PA with a bound on the complexity of induction formulas.

We proceed to delineate a natural sub-formalism of DL, with Segerberg’s induction restricted to first order formulas, and prove that the resulting calculus proves exactly the same partial correctness assertions as HL, regardless of the background first order theory.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.