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These include graph-based and proof-theoretic approaches to nonmonotonic logic, results that interrelate the various formalisms, complexity results, tractable special cases of nonmonotonic reasoning, relations between nonmonotonic and abductive reasoning, relations to probability logics, the logical intuitions and apparent patterns of validity underlying nonmonotonic logics, and the techniques used to formalize domains using nonmonotonic logics.

For these and other topics, the reader is referred to the literature. As a start, the chapters in Gabbay et al. Time and temporal reasoning have been associated with logic since the origins of scientific logic with Aristotle.

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Thus, the central logical problems and techniques of tense logic were borrowed from modal logic. For instance, it became a research theme to work out the relations between axiomatic systems and the corresponding model theoretic constraints on temporal orderings. See, for instance, Burgess and van Benthem Priorian tense logic shares with modal logic a technical concentration on issues that arise from using the first-order theory of relations to explain the logical phenomena, an expectation that the important temporal operators will be quantifiers over world-states, and a rather remote and foundational approach to actual specimens of temporal reasoning.

Of course, these temporal logics do yield validities, such as. But at most, these can only play a broadly foundational role in accounting for realistic reasoning about time. It is hard to think of realistic examples in which they play a leading part. This characteristic, of course, is one that modal logic shares with most traditional and modern logical theories; the connection with everyday reasoning is rather weak. Although modern logical techniques do account with some success for the reasoning involved in verifying mathematical proofs and logic puzzles, they do not explain other cases of technical or common sense reasoning with much detail or plausibility.

Even in cases like legal reasoning, where logicians and logically-minded legal theorists have put much effort into formalizing the reasoning, the utility of the results is controversial. Planning problems provide one of the most fruitful showcases for combining logical analysis with AI applications. On the one hand there are many practically important applications of automated planning, and on the other logical formalizations of planning are genuinely helpful in understanding the problems, and in designing algorithms.

In such a problem, an agent in an initial world-state is equipped with a set of actions , which are thought of as partial functions transforming world-states into world-states. Actions are feasible only in world-states that meet certain constraints. A planning problem then becomes a search for a series of feasible actions that successively transform the initial world-state into a desired world-state.

The Situation Calculus , developed by John McCarthy, is the origin of most of the later work in formalizing reasoning about action and change.

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Apparently, Priorian tense logic had no influence on Amarel As in tense logic, these locations are ordered, and change is represented by the variation in truths from one location to another. The crucial difference between the Situation Calculus and tense logic is that change in the situation is dynamic —changes do not merely occur, but occur for a reason.

In general, actions can be successfully performed only under certain limited circumstances. A planning problem starts with a limited repertoire of actions where sets of preconditions and effects are associated with each action , an initial situation, and a goal which can be treated as a formula. A planning problem is a matter of finding a sequence of actions that will achieve the goal, given the initial situation.

That is, given a goal G and initial situation s, the problem will consist of finding a sequence s 1 , The planning problem is in effect a search for a sequence of actions meeting these conditions. The success conditions for the search can be characterized in a formalism like the Situation Calculus, which allows information about the results of actions to be expressed. Nothing has been said up till now about the actual language of the Situation Calculus. The crucial thing is how change is to be expressed.

Nature and varieties of logic

With tense logic in mind, it would be natural to invoke a modality like [ a ] A , with the truth condition. This formalization, in the style of dynamic logic, is in fact a leading candidate; see Section 4. Actions are treated as individuals. And certain propositions whose truth values can change over time propositional fluents are also treated as individuals. Where s is a situation and f is a fluent, Holds f,s says that f is true in s.

Since the pioneering work of the nineteenth and early twentieth century logicians, the process of formalizing mathematical domains has largely become a matter of routine.

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Although as with set theory there may be controversies about what axioms and logical infrastructure best serve to formalize an area of mathematics, the methods of formalization and the criteria for evaluating them are relatively unproblematic. This methodological clarity has not been successfully extended to other domains; even the formalization of the empirical sciences presents difficult problems that have not yet been resolved.

The formalization of temporal reasoning, and in particular of reasoning about actions and plans, is the best-developed successful extension of modern formalization techniques to domains other than mathematical theories. This departure has required the creation of new methodologies. One methodological innovation will emerge in Section 4. Another is the potential usefulness of explicit representations of context; see Guha, Doing this requires predictive reasoning, a type of reasoning that was neglected in the tense-logical literature.

As in mechanics, prediction involves the inference of later states from earlier ones. But in the case of simple planning problems at least the dynamics are determined by actions rather than by differential equations. The investigation of this qualitative form of temporal reasoning, and of related sorts of reasoning e. The essence of prediction is the problem of inferring what holds in the situation that ensues from performing an action, given information about the initial situation.

It is often assumed that the agent has complete knowledge about the initial situation—this assumption is usual in classical formalizations of planning. A large part of the qualitative dynamics that is needed for planning consists in inferring what does not change. The required inference can be thought of as a form of inertia. The Frame Problem is the problem of how to formalize the required inertial reasoning. Both of these volumes document interactions between AI and philosophy.

The quality of these interactions is discouraging. Like any realistic common sense reasoning problem, the Frame Problem is open-ended, and can depend on a wide variety of circumstances. This may account for the temptation that makes some philosophers [ 27 ] want to construe the Frame Problem very broadly, so that very soon it becomes indiscernible from the problem of formalizing general common sense in arbitrary domains. Such a broad construal may serve to introduce speculative discussions concerning the nature of AI, but it loses all contact with the genuine, new logical problems in temporal reasoning that have been discovered by the AI community.

It provides a forum for repeating some familiar philosophical themes, but it brings nothing new to philosophy.

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This way of interpreting the Frame Problem is disappointing, because philosophy can use all the help it can get; the AI community has succeeded in extending and enriching the application of logic to common sense reasoning in dramatic ways that are highly relevant to philosophy. The clearest account of these developments to be found in the volumes edited by Pylyshyn is Morgenstern An extended treatment can be found in Shanahan ; also see Sandewall and Shanahan The purely logical Frame Problem can be solved using monotonic logic, by simply writing explicit axioms stating what does not change when an action is performed.

This technique can be successfully applied to quite complex formalization problems. Some philosophers Fodor , Lormand have felt that contrived propositions will pose special difficulties in connection with the Frame Problem.

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As Shanahan points out Shanahan [p. This is one of the few points about the Frame Problem made by a philosopher that raises a genuine difficulty for the formal solutions.

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But the difficulty is peripheral, since the example is not realistic. Recall that fluents are represented as first-order individuals. Although fluents are situation-dependent functions, an axiom of comprehension is certainly not assumed for fluents. In fact, it is generally supposed that the domain of fluents will be a very limited set of the totality of situation-dependent functions; typically, it will be a relatively small finite set of variables representing features of the domain considered to be important.

In particular cases these will be chosen in much the same way that a set of variables is chosen in statistical modeling. The idea behind nonmonotonic solutions to the Frame Problem is to treat inertia as a default; changes are assumed to occur only if there is some special reason for them to occur. In an action-centered account of change, this means that absence of change is assumed when an action is performed unless a reason for the change can be found in axioms for the action.

To formalize inertia, then, we need to use default rule schemata.

For each fluent f, action a, and situation s, the set of these schemata will include an instance of the following schema:. This way of doing things makes any case in which a fluent changes truth value a prima facie anomaly.