Features and Fluents: The Representation of Knowledge About Dynamical Systems

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The initial situation in the Anomaly and the three actions, with their resulting situations, can be pictured as follows. The natural, expected outcome of these axioms is that the pistol is loaded and Fred is alive after waiting, so that shooting yields a final outcome in which Fred is not alive and the pistol is unloaded. There is no problem in showing that this corresponds to an extension; the problem is the presence of the other, anomalous extension, which looks like this. Here is a narrative version of this extension. At first, Fred is alive and the pistol is unloaded.

After loading, the pistol is loaded and Fred remains alive.

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After waiting, the pistol becomes unloaded and Fred remains alive. Shooting is then vacuous since the pistol is unloaded, so finally, after shooting, Fred remains alive and the pistol remains unloaded. The best way to see clearly that this is an extension is to work through the proof. Less formally, though, you can see that the expected extension violates just one default: the frame default for Alive is violated when Fred changes state in the last step.

But the anomalous extension also violates only one default: the frame default for Loaded is violated when the pistol spontaneously becomes unloaded while waiting. So, if you just go by the number of defaults that are violated, both extensions are equally good.


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The Yale Shooting Anomaly represents a major obstacle in developing a theory of predictive reasoning. A plausible, well-motivated logical solution to the Frame Problem runs afoul of a simple, crisp example in which it clearly delivers the wrong results.

Naturally, the literature concerning the Yale Shooting Problem is extensive. Surveys of some of this work, with bibliographical references, can be found in Shanahan ; Morgenstern Many formalisms have been proposed to deal with the problems surveyed in the previous section.

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Some are more or less neglected today. Several are still advocated and defended by leading experts; some of these are associated with research groups who are not only interested in developments of logical theory, but in applications in planning and cognitive robotics. The leading approaches provide solutions to the main problems mentioned in Section 4.

It is commonly agreed that good solutions need to be generalizable to more complex cases than the early planning formalisms, and that in particular the solutions they offer should be deployable even when continuous time, concurrent actions, and various kinds of ignorance are allowed.


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Also, it is generally agreed that the formalisms should support several kinds of reasoning, and, in particular, not only prediction and plan verification but retrodiction , i. The accounts of the first three in what follows will be fairly brief; fortunately, each approach is well documented in a single reference. The fourth approach is most likely to be interesting to philosophers and to contain elements that will be of lasting importance regardless of future developments in this area.

This approach, described in Sandewall , uses preference semantics as a way to organize nonmonotonic solutions to the problems of reasoning about action and change. Rather than introducing a single logical framework, Sandewall considers a number of temporal logics, including ones that use discrete, continuous, and branching time. The properties of the logics are systematically tested against a large suite of test scenarios. This theory grew out of direct consideration of the problems in temporal reasoning described above in Section 4. The key technical idea of the paper is a rather complicated definition of motivation in an interval-based temporal logic.


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In Morgenstern , Morgenstern presents a summary of the theory, along with reasons for rejecting its causal rivals. The most important of these reasons is that these theories, based on the Situation Calculus, do not appear to generalize to cases allowing for concurrency and ignorance. She also cites the failure of early causal theories to deal with retrodiction.

In Baker , Andrew Baker presented a solution to the version of the Yale Shooting problem in the Situation Calculus, using a circumscriptive inertial axiom. The very brief account of circumscription above in Section 3 indicated that circumscription uses preferred models in which the extensions of certain predicates are minimized.

In the course of this minimization, a set of parameters including, of course, the predicates to be minimized is allowed to vary; the rest are held constant. Which parameters vary and which are held constant is determined by the application. In the earliest circumscriptive solutions to the Frame Problem, the inertial rule CIR is stated using an abnormality predicate. This axiom uses a biconditional, so that it can be used for retrodiction; this is typical of the more recent formulations of common sense inertia.

In circumscribing, the abnormality predicate is minimized while the Holds predicate is allowed to vary and all other parameters are fixed. This formalization succumbs to the Yale Shooting Anomaly in much the same way that default logic does. Circumscription does not involve multiple extensions, so the problem emerges as the nonderivability of the conclusion that Fred is alive after the occurrence of the shooting. It is this feature that eliminates the incorrect model for that scenario; for details, see Baker and Shanahan , Chapter 6.

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Recall that in the anomalous model of the Yale Shooting Scenario the gun becomes unloaded after the performance of the wait action, an action which has no conventional effects—the unloading, then, is uncaused. In the context of a nonmonotonic logic—and without such a logic, the Yale Shooting Anomaly would not arise—it is very natural to formalize this by treating uncaused eventualities as abnormalities to be minimized. This strategy was pursued by Hector Geffner in Geffner , where he formalizes this simple causal solution to the Yale Shooting Anomaly.

But the solution is presented in the context of an ambitious general project in nonmonotonic logic that not only develops properties of the preferred model approach and shows how to apply it to a number of reasoning problems, but that relates nonmonotonic logic to probabilities, using ideas deriving from Adams In Geffner , the causal theory is sketched; it is not developed to show its adequacy in dealing with the battery of problems presented above, and in particular the Ramification Problem is left untouched. The work beginning with Lifschitz has contributed to a sustained line of research in the causal approach—not only by Lifschitz and students of his such as Enrico Giunchiglia and Hudson Turner, but by researchers at other sites.

Here, we briefly describe some of theories developed by the Texas Action Group, leading up to the causal solution presented in Turner Turner returns to the ideas of Geffner , but places them in a simpler logical setting and applies them to the formalization of more complex scenarios that illustrate the interactions of causal inertia with other considerations, especially the Ramification Problem.

Ramification is induced by the presence of static laws which relate the direct consequences of actions to other changes. There is a fluent Ig tracking whether the ignition is on, a fluent Dead tracking whether the battery is dead, and a fluent Run tracking whether the engine is running. But contraposition of laws makes it difficult to devise a principled solution. This law not only is true in our scenario, but would be used to explain a failed attempt to start the car. The battery is dead in this outcome because of causal inertia. This paper presents an increasingly powerful and sophisticated series of action languages.

Their language incorporates an ad hoc or at least purely syntactic solution to the Ramification Problem. Gelfond and Lifschitz impose a weak closure condition on static laws: where s is a set of literals, s is restricted-closed with respect to a B theory T , R B Cl T s if and only if every literal that would be added by starting with s and forward-chaining through the static laws of B is already in s.

In other words:. This has some somewhat counterintuitive effects. With the addition of this law, there is a model in which preserving the fact that the car is not running makes the battery become dead when the ignition is turned on. This makes it very plausible to suppose that the source of the problem is a representation of underlying causal information in action language B that is somehow inadequate.

Features and Fluents: The Representation of Knowledge about Dynamical Systems Volume 1

Gelfond and Lifschitz go on to describe another action language, C , which invokes an explicit notion of causality—motivated, in all likelihood, in part by the need to provide a more principled solution to the problem. Instead of describing that language, we now discuss the similar theory of Turner In the preferred models of this logic, the caused propositions coincide with the propositions that are true, and this must be the only possibility consistent with the extensional part of the model.

To make this more explicit, recall that in the possible worlds interpretation of S5 , it is possible to identify possible worlds with state descriptions , which we can represent as sets I of literals atomic formulas and their negations. Consult Turner for details. The axioms that specify the effects of actions treat these effects as caused; for instance, the axiom schema for loading would read as follows:. Ramifications of the immediate effects of actions are also treated as caused. And the nonmonotonic inertial axiom schemata take the form. Thus, a true proposition can be caused either because it is the direct or indirect effect of an action, or because it involves the persistence of a caused proposition.

Initial conditions are also considered to be caused, by stipulation. As in the Yale Shooting Problem, there are no axioms for wait ; this action can always be performed and has no associated effects. M 1 is the intended model, in which nothing changes. M 2 is an anomalous model, in which the fluent ceases spontaneously. So, while M 1 is a preferred model, M 2 is not. The task of clarifying the foundations of causal theories of action and change may not yet be complete. And the causal theory, as initiated by Geffner and developed by Turner, has many interesting detailed features.

For instance, while philosophical work on causality has concentrated on the causal relation, this work in logical AI shows that a great deal can be done by using only a nonrelational causal predicate. The relation between causality and conditionals can be explored and exploited in various ways. Lewis undertakes to account for causality in terms of conditionals. The motivation for this idea is than an explicit solution to the frame problem automatically provides a semantics for such conditionals.

As work on the approach continues, progress is being made in these areas. But the constraints that a successful logic of action and change must meet are so complex that it seems to be a reasonable research methodology to concentrate initially on a restricted logical setting. Although for many AI logicists, the goal of action formalisms is to illuminate an important aspect of common sense reasoning, most of their research is uninformed by an important source of insights into the common sense view of time—namely, natural language. Linguists concerned with the semantics of temporal constructions in natural language, like the AI community, have begun with ideas from philosophical logic but have discovered that these ideas need to be modified in order to deal with the phenomena.

The Features-and-Fluents Semantics for the Fluent Calculus

A chief discovery of the AI logicists has been the importance of actions and their relation to change. The goal of articulating a logical framework tailored to a representational system that is motivated by systematic evidence about meanings in natural languages is not acknowledged by all linguistic semanticists. Nevertheless, it is a significant theme in the linguistic literature. This goal is remarkably similar to those of the common sense logicists, but the research methodology is entirely different. Can the insights of these separate traditions be reconciled and unified?

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Is it possible to constrain theories of temporal representations and reasoning with the insights and research methodologies of both traditions? In Steedman and , listed in the Other Internet Resources Section , these important questions are addressed, and a theory is developed that extends action formalisms like the Situation Calculus, and that incorporates many of the insights from linguistic semantics. The project reported in Steedman is still incomplete, but the results reported there make a convincing case that the event-based ideas from linguistics can be fruitfully combined with the action-centered formalisms in the AI literature.

The possibility of this unification is one of the most exciting logical developments in this area, bringing together as it does two independent descendants of the earlier work in the logic of time.