Michael L. Tiomkin

Nonmonotonic default modal logics

Conclusions by failure to prove the opposite are frequently used in reasoning about an incompletely specified world. This naturally leads to logics for default reasoning which, in general, are nonmonotonic, i.e., 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. We study nonmonotonic logics based on various sets of defaults and present a necessary and sufficient condition for a nonmonotonic modal theory to be degenerate. In particular, this condition provides several alternative descriptions of degenerate theories. Also we establish some closure properties of sets of defaults defining a nonmonotonic modal logic.

Propositional dynamic logic with local assignment

We propose an extension of Propositional Dynamic Logic which allows a new kind of program terms—local assignments to propositional variables. They are very close to the known array assignments in Dynamic Logic beacuse they allow us to change the truth values of predicate variables. In this logic, many notions, like equivalence of programs, looping and finitely branching, are expressible on a propositional level. In fact, we show that the resulting logic is equivalent in expressive power to first-order logic augmented by a device to express transitive closures. In other words, it is (modulo extra predicate symbols) equivalent to first-order dynamic logic. Not suprisingly, therefore, the validity problem for this extension is Π11-complete.

An environment for logic programming

We describe a programming environment for Prolog, a common logic programming language. The services offered by our system assist a Prolog user in the tasks of composing, editing, and storing logic (rule-based) programs, as well as in the control of their execution for debugging purposes. In order to facilitate effective debugging of Prolog programs, we propose a new model of computation that can handle both pure Prolog and impure (side-effect causing) Prolog operations quite gracefully. This model employs two stacks representing the state of a computation with respect to the two major activities of Prolog execution: recursion and backtracking. This representation, in addition to being quite clear and intuitive, is easy to display on a terminals screen and can be further processed. We also describe a Prolog oriented editor with which the user may easily create and update Prolog source programs and, further, inspect states of computation as generated by the debugger.

A knowledge representation language for university requirements

Abstract Intelligent systems with applications in an academic environment are becoming a reality. Such systems can offer intelligent assistance in tasks ranging from interactive scheduling, academic planning, office automation, database management and computer assisted instruction. The problem of designing a general requirements model for university degrees is investigated. A method for expressing the structure of university requirements is described, and a representation language consisting of a subset of Prolog based on set theoretic primitives is suggested. This work has been motivated by our experience with the Academic Planning Environment (APE) expert system project [9] whose goal has been to study the use of knowledge-based techniques in the advising and assistance of university students in the planning of their studies towards an academic degree.

Decidability of finite probabilistic propositional dynamic logics

Abstract This paper deals with various versions of finite propositional probabilistic dynamic logic. We present probabilistic propositional dynamic logic with simple probabilistic estimations and “almost regular” program language. Besides the logics previously introduced in the literature we present some natural variations and extensions of these logics. We investigate the (un)decidability of these logics and give a complete picture of decidability and undecidability. Some of these logics have the finite model property, and therefore, if they are undecidable, they are exactly Π10. We show that allowing nesting and probabilistic choice often leads to undecidability.

Probabilistic termination versus fair termination (note)

Abstract In this note we show that probabilistic termination of concurrent programs is in many cases much simpler than the “fair” one. For a wide class of definitions of probabilistic termination we may express termination by π 0 2 arithmetic formula, whereas the “fair” termination can be expressed only by Π| second-order arithmetic formula. Proof of “fair” termination usually needs induction on recursive ordinals, but proof of probabilistic termination has the complexity equivalent to that of deterministic program termination.

Finite and Circular Path Models for Branching Time Logics

We define various kinds of branching time logic consisting of path quantifiers and secondorder (<o-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 models 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 circular paths.