Journal of Articial Intelligence Research Submitted published A Temporal Description Logic for Reasoning ab out Actions and Plans Alessandro Artale artaleirstitcit ITCIRST Cognitive and Communication Technologies Division I Povo TN Italy Enrico Franconi franconicsmanacuk Department of Computer Science University of Manchester Manchester M PL UK Abstract A class of intervalbased temp oral languages for uniformly representing and reasoning ab out actions and plans is presented Actions are represented by describing what is true while the action itself is o ccurring and plans are constructed by temp orally relating actions and world states The temp oral languages are members of the family of Description Logics which are characterized by high expressivity combined with go o d computational prop erties The subsumption problem for a class of temp oral Description Logics is investigated and sound and complete decision pro cedures are given The basic language TLF is considered rst it is the comp osition of a temp oral logic TL able to express interval temp oral networks together with the nontemp oral logic F a Feature Description Logic It is proven that subsumption in this language is an NPcomplete problem Then it is shown how to reason with the more expressive languages T LU FU and TLALC F The former adds disjunction b oth at the temp oral and nontemp oral sides of the language the latter extends the nontemp oral side with setvalued features ie roles and a prop ositionally complete language Introduction The representation of temp oral knowledge has received considerable attention in the Ar ticial Intelligence community in an attempt to extend existing knowledge representation systems to deal with actions and change At the same time many logicbased formalisms were developed and analyzed by logicians and philosophers for the same purp oses In this class of logical formalisms prop erties such as expressive p ower and computability have b een studied as regards typical problems involving events and actions This pap er analyzes from a theoretical p oint of view the logical and computational prop erties of a knowledge representation system that allows us to deal with time actions and plans in a uniform way The most common approaches to mo del actions are based on the notion of state change eg the formal mo dels based on the original situation calculus McCarthy Hayes Sandewall Shoham or the Stripslike planning systems Fikes Nilsson Lifschitz in which actions are generally considered instantaneous and dened as functions from one state to another by means of pre and p ostconditions Here an explicit notion of time is introduced in the mo deling language and actions are dened as occurring over time intervals following the Allen prop osal Allen c AI Access Foundation and Morgan Kaufmann Publishers All rights reserved Artale Franconi In this formalism an action is represented by describing the time course of the world while the action o ccurs Concurrent or overlapping actions are allowed eects of overlapping actions can b e dierent from the sum of their individual eects eects may not directly follow the action but more complex temp oral relations may hold For instance consider the motion of a p ointer on a screen driven by a mouse the p ointer moves b ecause of the movement of the device on the pad there is a causeeect relation but the two events are contemporary in the commonsense notion of the word A class of interval temp oral logics is studied based on Description Logics and inspired by the works of Schmiedel and of Weida and Litman In this class of formalisms a state describ es a collection of prop erties of the world holding at a certain time Actions are represented through temp oral constraints on world states which p ertain to the action itself Plans are built by temp orally relating actions and states To represent the temp oral dimen sion classical Description Logics are extended with temp oral constructors thus a uniform representation for states actions and plans is provided Furthermore the distinction made by Description Logics b etween the terminological and assertional asp ects of the knowledge allows us to describ e actions and plans b oth at an abstract level actionplan types and at an instance level individual actions and plans In this environment the subsumption calculus is the main inference to ol for managing collections of action and plan types Action and plan types can b e organized in a subsumptionbased taxonomy which plays the role of an actionplan library to b e used for the tasks known in the literature as plan retrieval and individual plan recognition Kautz A renement of the plan recognition no tion is prop osed by splitting it into the dierent tasks of plan description classication involving a plan type and specic plan recognition with respect to a plan description involving an individual plan According to the latter reasoning task the system is able to recognize which type of actionplan has taken place at a certain time interval given a set of observations of the world Advantages of using Description Logics are their high expressivity combined with de sirable computational prop erties such as decidability soundness and completeness of de duction pro cedures Buchheit Donini Schaerf Schaerf Donini Lenzerini Nardi Schaerf Donini Lenzerini Nardi Nutt The main purp ose of this work is to investigate a class of decidable temp oral Description Logics and to provide com plete algorithms for computing subsumption To this aim we start with TLF a language b eing the comp osition of a temp oral logic TL able to express interval temp oral networks together with the nontemp oral Description Logic F a Feature Description Logic Smolka It turns out that subsumption for TLF is an NPcomplete problem Then we show how to reason with more expressive languages T LU FU which adds disjunction b oth at the temp oral and nontemp oral sides of the language and TLALC F which extends the nontemp oral side with setvalued features ie roles and a prop ositionally complete De scription Logic Hollunder Nutt In b oth cases we show that reasoning is decidable and we supply sound and complete pro cedures for computing subsumption The pap er is organized as follows After introducing the main features of Description Logics in Section Section organizes the intuitions underlying our prop osal The technical bases are introduced by briey overviewing the temp oral extensions of Description Logics relevant for this approach together with the interrelationships with the interval temp oral mo dal logic sp ecically intended for time and action representation and reasoning The A Temporal Description Logic for Reasoning about Actions and Plans basic feature temp oral language TLF is introduced in Section The language syntax is rst describ ed in Section together with a worked out example illustrating the informal meaning of temp oral expressions Section reveals the mo del theoretic semantics of TLF together with a formal denition of the subsumption and instance recognition problems Section shows that the temp oral language is suitable for action and plan representation and reasoning the well known cooking domain and blocks world domain are taken into consideration The sound and complete calculus for the feature temp oral language TLF is presented in details in Section A pro of that subsumption for TLF is an NPcomplete problem is included The calculus for TLF forms the basic reasoning pro cedure that can b e adapted to deal with logics having an extended prop ositional part An algorithm for checking subsumption in presence of disjunction T LU FU is devised in Section while in Section the nontemp oral part of the language is extended with roles and full prop ositional calculus TLALC F In b oth cases the subsumption problem is still decidable Op erators for homogeneity and p ersistence are presented in Section for an adequate representation of world states In particular a p ossible solution to the frame problem ie the problem to compute what remains unchanged by an action is suggested Section surveys the whole sp ectrum of extensions of Description Logics for representing and reasoning with time and action This Section is concluded by a comparison with State Change based approaches by briey illustrating the eort made in the situation calculus area to temp orally extend this class of formalisms Section concludes the pap er Description Logics Description Logics are formalisms designed for a logical reconstruction of representa tion to ols such as frames semantic networks objectoriented and semantic data mo dels see Calvanese Lenzerini Nardi for a survey Nowadays Description Logics are also considered the most imp ortant unifying formalism for the many ob jectcentered representation languages used in areas other than Knowledge Representation Imp ortant characteristics of Description Logics are high expressivity together with decidability which guarantee the existence of reasoning algorithms that always terminate with the correct answers This Section gives a brief introduction to a basic Description Logic which will serve as the basic representation language for our prop osal As for the formal apparatus the formal ism introduced by SchmidtSchau Smolka and further elab orated by Donini Hollunder Lenzerini Spaccamela Nardi Nutt Donini et al Buchheit et al De Giacomo Lenzerini is followed in this way Description Logics are considered as a structured fragment of predicate logic ALC SchmidtSchau