Non-Standard Algorithmic and Dynamic Logic

The present author as well as Andreka's group has experienced, while writing program- verifying programs, the following dilemma. We have to decide which program verification method (i.e. logic of programs) to choose as a framework for our software system. It is not obvious that the strongest method is also the best because it might overload the theorem prover subprogram. So we have to optimise. But for this we need as much information and insight as possible into the natures of available methods. Non-standard algorithmic logic of dynamic logic (NDL from now on) is a complete first order logic with a decidable proof concept for reasoning about programs, developed in 1978 by H. Andreka et al. NDL unifies existing approaches to program verification and turns incompleteness results into completeness results. Still, it is far from being popular and this is because non-standard models of computation are unpopular. It is the aim of the present paper to show that these models play for dynamic logic the same role which, say, complex numbers, play for physics. They are merely a tool for proving very realistic, standard properties of programs (e.g. non-provability by a given standard method) or for reasoning about (existing) program verification methods. The paper also provides two essential proofs, which are missing in their complete or proper form from the publications.