In this paper we analyse the semantical definition of extensions for open default theories. We argue that this definition reflects the domain closure assumption and show how the domain closure assumption for countable and finite domains can be expressed in first-order default logic extended with the Carnap rule of inference. Also we give examples of the domain dependence of extensions for open default theories. In particular, we show that such extensions do not possess the minimality property.

It it shown that the validity questions in prepositional temporal logic of actions can be translated into validity questions of a temporal logic and vice versa. In particular, actions of prepositional temporal logic of actions can be equivalently replaced by prepositional symbols.

Conclusions by failure to prove the opposite are frequently used in reasoning about an incompletely specdied world.This naturally leads to logics for default reasoning that, in general, are nonmonotonic; that is, introducing new facts can invalidate previously made conclusions.Accordingly, a nonmonotonic theory is called (nonmonotonically) degenerate, if adding new axioms does not invalidate already-proved theorems.Nonmonotonic logics are studied on the basis of various sets of defaults and a necessary and sufficient condition is presented for a nonmonotonic modal theory to be degenerate.In particular, this condition provides several alternative descriptions of degenerate theories.Also some closure properties of sets of defaults defining a nonmonotonic modal logic are established.

We define various kinds of branching time logic consisting of path quantifiers and second-order (ω-regular) linear time logic which are extensions of CTL* (computation tree logic). These logics are decidable for the computation tree semantics. However, if we adopt the non-tree semantics reflecting the possible computations of a looping program eventually returning to the same state of computation, the picture is quite different, and one of the logics under consideration becomes highly undecidable. Nevertheless, this semantics allows a simpler model definition, including finite models corresponding to a finite state machine (program), and circular path finite models reflecting programs, which are really deterministic finite state machines, but the model's view does not include all the details of program state. Since circular path finite model may have only countably many possible paths (computations), this semantics is much simpler than the usual one based on sets of all infinite paths in computation tree. We show that for the finite models the circular path semantics is equivalent to the usual one, i.e. that for a finite structure a formula is true in the model of all the possible paths if and only if it is true in the model of all the possible paths if and only if it is true in the model of all the circular paths.

A formal proof system for unprovability in the predicate calculi is developed. This system is shown to be complete with respect to the logic of finite structures. It can be used to extend the 'negation by failure' of Prolog, prevent infinite loops in a deductive data base or Prolog, or prove formulaes in nonmonotonic (default) logic.>