Chapter 2 Effective temporal logics of programs H. Andreka* V. Goranko] S. Mikulas*, I. Nemeti* and I. Sain* * Mathematical Institute Hungarian Academy of Sciences Budapest, Pf. 127, H-1364f Hungary * Department of Mathematics, UniQwa Private Bag X13, Phuthaditjhaba 9866f South Africa Abstract In this chapter we investigate effective proof systems for temporal log- ics both propositional and first-order. The issue of effective proof systems for propositional temporal logic is much easier than for the first-order one. Partly because of this and partly because of applications we dwell on the first-order case much longer than on the propositional case. We prove soundness and completeness theorems for various effective proof systems and compare the program verifying - power of those systems. 2.1 Introduction In this chapter we investigate effective inference systems (i.e. proof systems or calculi) h for temporal logics both propositional (PTL) and first-order (FTL). The issue of effective inference systems for PTL is much easier than for FTL. Partly because of this, and partly because of motivation coming from appli- cations, we will dwell on the FTL case much longer than on the PTL case. (However, the PTL case will be thoroughly presented too.) An inference system h for a logic C is effective if the set of Kproofs is a Tur- ing enumerable (or equivalently, recursively enumerable) set of finite sequences of strings. This restriction is motivated by the applications, but independently *The research of has been supported by Hungarian NSF grant no. 1810. tThe work was partly supported by research grant GUN 2019536 of the Foundation for Research Development of South Africa