Nondeterminism and Uncertainty in the Situation Calculus

A novel approach is presented to modeling action and change within the Situation Calculus in the presence of both non-determinism and probabilistic behavior. Two examples are given to illustrate the approach: the Russian roulette and the alternating bit protocol. Introduction and Motivation When agents act in the world they encounter situations where the actions that they perform seem to have uncertain effects. For example, throwing a dart at a dartboard might result in the dart being in several possible places, depending on the accuracy and intentions of the thrower. For example, picture a novice player of darts in an Irish pub after drinking a couple of pints of Guinness. It is quite likely that this dart player hits any point on the dartboard, or on the wall, without much control on his part. On the other hand, picture a professional darts player with the ability to direct her throw with the utmost accuracy. The novice darts player is performing an indeterminate action, whose results cannot be accurately predicted; whereas, the pro performs a nearly determinate action. In the light of this example, we might ask “what is the source of indeterminacy?” The point of the example is to suggest that primitive actions are not indeterminate. One can easily imagine a darts player so good that he can always throw the dart wherever he wants. Thus, a dart throw is not in and of itself an indeterminate action. What happens is that the agent might not have the ability to execute the action that she or he wants to perform. In a previous article (Pinto 1998), a proposal to view indeterminacy as arising from an indeterminate choice between primitive completely determinate actions was advanced. I.e., actions do not have indeterminate effects. Indeterminate actions were modeled as sets of primitive, determinate actions. There is one crucial advantage of this view of indeterminacy: it is not necessary to deal with the specification of the effects of indeterminate primitive actions. The effects of indeterminate actions are derived from the effects of the Copyright c © 1999, American Association for Artificial Intelligence (www.aaai.org). All rights reserved. individual primitive actions. Therefore, the frame problem, the ramification problem, the qualification problem, etc. are orthogonal to the issue of indeterminacy in theories of actions in this style. Therefore, solutions to these problems in completely determinate worlds would carry over to worlds in which agents might perform indeterminate actions (viewed as choice among a set of primitive, determinate actions). In this article, we depart from the proposal mentioned before, and follow another route. Instead of appealing to sets in order to model indetermination, we model indeterminate actions by decomposing them into a non-deterministic part and its possible outcomes. For example, flipping a coin is modeled as the nondeterministic flipping, along with the possible outcomes heads and tails. Therefore, one action is flip-heads, and the other is flip-tails. Also, we associate a probability distribution to the outcomes, given the nondeterministic action part. Notice, however, that the action flip-heads remains deterministic. Its effects are described with standard effect axioms. Therefore, as in (Pinto 1998), the solutions to the frame and related problems still apply. Our view of indeterminacy is independent of the formalism that one chooses to write theories of action and change. Thus, we believe, one should be able to formalize indeterminacy with this view by extending any of the logical languages that have been proposed to write theories of (determinate) action and change. However, our technical proposal has been developed within the framework of the Situation Calculus (McCarthy & Hayes 1969), and relying on Reiter’s approach to deal with the frame problem (Reiter 1991). Another approach to this problem using fuzzy logics in (Pereira et al. 1997; Saffiotti, Konolige, & Ruspini 1995) The article is organized as follows. First we present the logical framework that constitutes the basis for our technical proposal. Then, we extend the language presented in order to accommodate indeterminate actions (as pairs), and to give probabilities to the outcomes. Afterwards, we deal with ordered sequences of indeterminate actions. Later, we model the alternating bit protocol, which is used in order to ensure that a set of messages are correctly communicated (in order) from a From: Proceedings of the Twelfth International FLAIRS Conference. Copyright © 1999, AAAI (www.aaai.org). All rights reserved. sender to a receiver. The non-determinism that arises is due to losses in the communication channels. Finally we present our conclusions and discuss our future work on this subject. Standard Situation Calculus A many sorted second order language. The sorts are A for primitive actions, S for situations, and F for fluents (i.e., we chose to reify fluents). We also include a sort D for domain objects. We also have the following distinguished symbols: The constant S0 for the initial situation; the function do : A × S → S; the predicate holds ⊆ F × S; the predicate Poss ⊆ A× S. Also, we make use of R0 , the positive reals, as an interpreted sort. The following are foundational axioms for the situation calculus: (∀φ)[φ(S0) ∧ (∀ s, a) (φ(s) ⊃ φ(do(a, s)))] ⊃ (∀ s) φ(s), (1) (∀ a1, a2, s1, s2) do(a1, s1) = do(a2, s2) ⊃ a1 = a2, (2)