Peano Arithmetic as Axiomatization of the Time Frame in Logics of Programs and in Dynamic Logics

Biro, B. and I. Sain, Peano arithmetic as axiomatization of the time frame in logics of programs and in dynamic logics, Annals of Pure and Applied Logic 63 (1993) 201-225. We show that one can prove the partial correctness of more programs using Peano’s axioms for the time frames of three-sorted time models than using only Presburger’s axioms, that is it is useful to allow multiplication of time points at program verification and in dynamic and temporal logics. We organized the paper as follows: 1. Preliminaries, 2. The main result, 3. Peano arithmetic with bounded multiplication, 4. Connections with temporal logics and dynamic logics, Acknowledgements, References. In the present paper we solve a problem raised in e.g. [18] and [3] common in the fields of comperative study of program verification methods, lattice of logics of programs, dynamic logics, nonstandard dynamic logic (NDL), temporal logics of programs etc. Roughly speaking, the problem is as follows. When reasoning about programs, is it useful to postulate the full set Tpa of axioms of Peano Arithmetic on the time scale (with addition and multiplication of the time points)? In other words, does there exist a partial correctness assertion not provable from the set Tpres of Presburger’s axioms plus Ind, the set of so called computational induction axioms, but provable from Tpa plus Ind? (In the Correspondence to: I. Sain, Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Pf. 127, H-1364, Hungary. Email: [email protected]. Research supported by Hungarian National Foundation for Scientific Research grant No 1911. The final version of this paper was prepared in Prague, November l-5 1992, at the Institute of Computer Science of the Czechoslovak Academy of Sciences. 0168-0072/93/$06.00