Introduction to the Fluent Calculus

Linkoping Electronic Articles in Computer and Information Science Vol nr Intro duction To The Fluent Calculus Michael Thielscher Department of Computer Science Dresden UniversityofTechnology Dresden Germany Linkoping University Electronic Press Linkoping Sweden httpwwwepliuseeacis Revised version Revised version publishedonFebruary by Linkoping University Electronic Press Linkoping Sweden Original version was published on October Linkoping Electronic Articles in Computer and Information Science ISSN Series editor Erik Sandewal l c Michael Thielscher A Typeset by the author using L T X E Formattedusingetendu style Recommended citation Author Title Linkoping Electronic Articles in Computer and Information Science Vol nr http wwwepliuseeacisOctober The URL wil l also contain links to both the original version and the present revised version as wel l as to the authors home page The publishers wil l keep this article online on the Internet or its possible replacement network in the future 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expiration of the publication period For additional information about the Linkoping University Electronic Press and its procedures for publication and for assuranceofdocument integrity ple ase refer to its WWW home page http wwwepliuse or by conventional mail to the address statedabove Motivation The purp ose of the Fluent Calculus is to solve the inferential Frame Prob lem While the representational asp ect of the Frame Problem means the problem of sp ecifying all noneects of actions the inferential asp ect means the problem of actually computing these noneects The latter concerns each uentvalue which when proving a theorem is needed in a situation other than the one for whichitisgiven or in which it arises as an eect of an action or event Apparently onebyone and using separate instances of the relevant axioms every such uentvalue needs to b e carried from the p oint of its app earance past eachintermediate situation to the p oint of its use This is done for instance in the Situation Calculus if successor state axioms are used no matter whether reasoning is p erformed forward in time or via regression and in the Event Calculus where survival needs to b e proven indep endently for eachuentvalue If all uentvalues are needed in exactly the situations in which they are given or arise then the inferential Frame Problem causes no computational burden at all The more uents have to b e carried unchanged through manyintermediate sit uations or event o ccurrences however the more valuable can a solution to the inferential Frame Problem b e The Fluent Calculus which ro ots in the logic programming formalism of addresses the inferential Frame Problem by sp ecifying the eect of actions in terms of how an action mo dies a state The application of a single state up date axiom always suces to derivetheentire change caused by the action in question Central to the axiomatizationtechnique of the Fluent Calculus is a function State swhich relates a situation s to the state of the world in that situation In turn these world states are collections of uents which are reied to this end ie treated as terms Fluents that are known to hold in a state are joined together using the binary function symbol This function is assumed to b e b oth asso ciativeand commutative It is illustratively written in inx notation As an example supp ose that ab out the initial state in some Blo cks World scenario it is known that blo ck A is on some blo ck x whichinturnstands on the table and that nothing is on top of blo ck A or blo ck B In the Fluent Calculus this incomplete knowledge can b e axiomatized as follows x z State S On A x On x Table z y z z On y A z z On y B z Put in words of state State S it is known that for some x both On A x and On x Table are true and p ossibly some other facts z hold to owith the restriction that z do es not include a uentoftheform On y A nor On y B of whichwe know they are false State up date axioms sp ecify how the states at two consecutive situations are related to each other The universal form of these axioms is s State Do A s State s where s states conditions on s or rather on the corresp onding state under which denes how the successor state State Do A s is obtained by mo difying the current state S tates For example let the eect of an action denoted by Move u v w bethatthe k w blo ck u is moved away from the top of blo ck v onto the top of blo c A suitable state up date axiom is Poss Move u v w s State Do Move u v w s On u v State s On u w That is if Move u v w is p ossible and p erformed in s then the new state plus On u v equals the old state plus On u w In other words the initial situation A A B C B C goal situation A A B C B C Figure A simple planning problem with incomplete information only negative eect of this action is On u v and the only p ositive eect is On u w The preconditions of our action Move u v w are that the blo cktobe relo cated u is currently on v and that b oth u and w are clear ie not obstructed byany other blo ck Formally Poss Move u v w s Holds On u v s x Holds On x us Holds On x w s where Holds f s is a macro which abbreviates z State sf z Recall from ab ove the partial initial sp ecication given byformula and supp ose blo ck A shall b e moved away from its current lo cation onto blo ck B Then the term State S in the instance fuA v x w B sS g of state up date axiom can b e replaced by a term which equals State S according to So doing yields after evaluating Poss Move A x B S x z State Do Move A x B S On A x On A x On x Table z On A B This formula can b e simplied to A B x z State Do Move A x B S On x Table z On In this way one obtains from an incomplete initial sp ecication a still partial description of the successor state which in particular includes the unaected uent On x Table This prop ertythus survived the computation of the eect of the action and so needs not b e carried over by separate application of an axiom Providing a solution to the inferential Frame Problem the merits of state up date axioms b ecome apparent as so on as larger reasoning problems are considered Take for example the planning problem sketched in Fig ure Of the starting situation it is known that eachblock A is on top i of the corresp onding blo ck B and that all blo cks A and C are clear i i i The goal is to reshue the conguration so that each blo ck A is on the i corresp onding C i We rst enco de this planning problem by means of the Situation Calculus formalism as describ ed by Let On u v s denote that blo ck u is on v in situation s then the partial knowledge of the initial situation can b e formalized as On A B S On A B S x On x A S On x A S On x C S On x C S The goal is to reach a situation S which satises On A C S On A C S the only Assuming that On is the only relevantuentand Move u v w relevant action a suitable eect sp ecication is given by the successor state axiom Poss a s On u w Do a s v a Move u v w On u w s v a Move u w v along with the precondition axiom Poss Move u v w s On u v s x On x u s On x w s Now a straightforward solution to the planning problem is to movein succession the blo cks A A away from their initial lo cation onto blo cks C C that is S Do Move A B C Do Move A B C S In order to formally verify this action sequence a solution let UNA be a suitable collection of axioms expressing uniqueness of names Then f gUNA j A pro of of this theorem requires at least instances of the successor state axiom As many as of these instances are used to conclude that some uentisnot changed bysome action The corresp onding Fluent Calculus formalization of the planning prob lem consists of the initial sp ecication z State S On A B On A B z x z z On x A z z On x A z z On x C z z On x C z and the goal sp ecication z State S On A C On A C z As ab ove let S b e the situation which corresp onds to the plan of moving in succession the blo cks A from B onto C Let b e the foundational i i i axioms of the Fluent Calculus see b elow then a pro of for the theorem f gj requires just instances of the state up date axiom one for each p erformed action The computational value of the Fluent Calculus is crucially dep endent on an ecient treatment of equality While the simple addition of equality axioms may constitute a considerable handicap for theorem proving a va riet y of ecient constraint solving

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