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The Features-and-Fluents Semantics for the Fluent Calculus Michael Thielscher and Thomas Witkowski Department of Computer Science Dresden University of Technology, Germany Abstract fluent calculus, is expressive enough to be applicable to the full test suite of example reasoning problems in (Sandewall Based on an elaborate ontological taxonomy, the Features- 1994), let alone to be provably sound and complete wrt. one and-Fluents framework provides an independent action se- of the more expressive ontological classes in the taxonomy. mantics for assessing the range of applicability of action cal- culi. In this paper, we show how the fluent calculus can In this paper, we present a version of the fluent calculus be used to capture the full range of phenomena in K-IA, that is sufficiently expressive to capture K-IA, which is the the broadest ontological class that has been fully formalized broadest class that has been rigorously formalized and in- in (Sandewall 1994). To this end, we develop a significant tensively studied in (Sandewall 1994) and in which correct extension of the fluent calculus for modeling actions with du- knowledge and a fully inertial world is assumed. To this rations and with specific trajectories of changes. We present a end, we develop a significant extension of the basic fluent provably correct translation of scenario descriptions from the calculus by introducing an explicit model for the duration Features-and-Fluents semantics into fluent calculus axioma- of actions and for trajectories of changes. On this basis, we tizations. present a translation function that maps any K-IA scenario into a fluent calculus axiomatization, and we prove that the Introduction intended models of the former coincide with the classical models of the latter. In this way the fluent calculus provides Action formalisms are a core aspect of research in knowl- a sound and complete reasoning method for this ontologi- edge representation and reasoning. The classical approach, cal class of the Features-and-Fluents semantics. As a simple the situation calculus (McCarthy 1963), traces back to the consequence of this result, the fluent calculus is shown to beginning of Artificial Intelligence (McCarthy 1958) and handle the entire test suite of example problems in (Sande- has led to the fundamental frame problem (McCarthy & wall 1994). Hayes 1969). The concept of successor state axioms (Re- The rest of the paper is organized as follows: We begin by iter 1991) has provided a first satisfactory solution to this giving a brief introduction to both the Features-and-Fluents problem in the situation calculus. By adding the explicit semantics and the fluent calculus. Thereafter, we develop an notion of a state, the situation calculus has been developed extension of the fluent calculus for representing and reason- into the fluent calculus (Thielscher 2005b). Both solutions to ing about the trajectories of changes for actions with explicit the frame problem have been extended to capture a variety durations. We then present a mapping from K-IA scenarios of phenomena, for example, nondeterministic actions (Lin into the extended fluent calculus and prove its soundness and 1996; Thielscher 2000a). The two calculi employ pure clas- completeness. A summary and discussion concludes the pa- sical logic and thus are amenable to its standard semantics. per. However, an analysis of their range of applicability based on an independent, equally expressive action semantics has not been carried out. The Ego-World-Semantics The Features-and-Fluents framework of (Sandewall 1994) In the following, we give a condensed introduction to the constitutes such an independent action semantics, which in- Features-and-Fluents framework; we refer to (Sandewall cludes an elaborate ontological taxonomy comprising a vari- 1994) for more details. ety of aspects like conditional effects, nondeterministic out- comes, and actions with durations. This semantics has been Inhabited dynamical systems used to assess the range of applicability of nonmonotonic solutions to the frame problem based on preferential entail- An inhabited dynamical system (IDS) is a collection of pos- ment. Yet it has been an open problem to show that a stan- sibly related objects whose state changes over time influ- dard monotonic approach, like the situation calculus or the enced by an ego. At each point in time, an IDS is in a state, which is an assignment of values to a given set of features. Copyright c 2006, American Association for Artificial Intelli- The domain of features will be represented by the symbol gence (www.aaai.org). All rights reserved. F , the domain of states by R, and the domain of timepoints 362 by T , which throughout the paper will be the set of nonneg- scd1 [s1; t1]Load ative integers. An IDS history is a function R(t; f) which, scd2 [s2; t2]Spin for a given timepoint t and a feature f , assigns an appro- scd3 [s3; t3]Fire priate value to that feature. scd4 t1 < s2 ^ t2 < s3 In the IDS reality, an event occurrence happens over an obs1 [0]alive=T ^ [t3]alive=F interval of time and is denoted by hs; E; ti 2 T × E × b b T , where s < t and E denotes the domain of occurrence Figure 1: A chronicle for a variant of the Russian Turkey designators (also called actions). A development of an IDS example. can be understood as a tuple which contains all information about occurrences, feature-values, and timepoints during the run of the system. Chronicles Definition 1. A 5-tuple hB; M; R; A; Ci is a development In the following, we formally define scenario descriptions, of an IDS if called chronicles, in K-IA. This requires some preparatory • B ⊆ T is a finite set of timepoints, whose members are definitions. called breakpoints; the largest member of B is called A feature expression is of the form F (!1;:::;!n), where n now and is written nB ; F 2 F is an n-ary feature symbol and each !i is either • M is a valuation which maps every object constant to an an object constant or an object variable (n ≥ 0). Using element of a given object domain O (see also Definition 2 the standard logical connectives, fluent formulas are built below) and every timepoint constant to a nonnegative in- from atomic formulas of the form f=X , meaning that fea- teger; ture expression f has one of the valuesb X = fx1; : : : ; xng (n ≥ 1). If X = fxg is a singleton, then we simply write • R is an IDS history defined up to timepoint nB ; f=x. Using the standard logical connectives, logic formulas •A, the past action set, is a set of occurrences hs; E; ti, areb built from elementary formulas, which are where s; t 2 B; • relational formulas among object constants or timepoints, •C, the current action set, is a set of pairs hs; Ei, where T s 2 B. which are either timepoint constants or members of (i.e., nonnegative integers); or The interaction between an ego and a world is under- stood as a game, where both the ego and the world take • expressions of the form [τ]φ, where τ is a timepoint ex- pression (i.e., an arithmetic term over timepoints) and φ turns and where, starting from an initial state r0 of the world at time zero, they construct a development. For a is a fluent formula. given valuation M , the initial development is defined as An example of a logic formula is obs1 in Figure 1, where hf0g; M; f0 7! r0g; ;; ;i. The ego, when it makes a move, alive is a nullary feature symbol with domain fF; Tg. An either starts an action by adding hs; Ei to C (with nB < s) action statement [&; τ]" consists of variable-free timepoint and s to B, or the ego terminates an action by removing expressions & and τ and a variable-free occurrence expres- hs; Ei from C (with s < nB ) and a corresponding addi- sion " = A(!1;:::;!n), where A is an n-ary action sym- tion of hs; E; nBi to A. Afterwards, the world extrapolates bol and each !i is an object constant. An example of an what happens as a result of the current state of the world action statement is scd1 in Figure 1. and the actions that the ego has initiated. It can either add A 4-tuple hO; hInfl; Trajsi; SCD; OBSi exactly one further member n to B and construct a new de- Definition 2. is a chronicle if velopment as a revision up to n, or it can leave the current state unchanged and extend the history R so that it becomes •O is an object domain; complete. • the world description hInfl; Trajsi consists of Ontological taxonomy – a mapping Infl(E; r) from actions E 2 E and par- tial states1 r to finite subsets of F (defining the set of The definition of an ontological taxonomy over the set features that are potentially affected by E if executed of worlds and scenario descriptions allows one to classify in a state of the world that satisfies r), and worlds and problems which fulfill certain epistemological E; r assumptions. While the Features-and-Fluents framework is – a finite, non-empty set Trajs( ) whose elements a very general semantics, the broadest class that has been are finite and non-empty sequences of partial states fully formalized and intensively studied in (Sandewall 1994) (the trajectories) assigning values to the features in Infl(E; r); is denoted by K-IA, where K means that the ego has cor- rect knowledge of the world, I means that the world is in- • the schedule set SCD consists of action statements along ertial, and A stands for alternative effects of actions (e.g., with timing statements, which are formulas that use only conditional or nondeterministic).

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