Towards an Implementation of a Multi
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1. 2000 6 Simons P Niemela I Soininen T Extending and implementing the stable model semantics Artif Intell 138 1 2 2002 181 234 7 Besold T R Schiemann B A Multi Context System Computing Modalities In Proc 23rd Int Workshop on Description Logics DL2010 Waterloo Canada Num ber 573 in CEUR Workshop Proceedings 2010 8 Besold T R Theory and Implementation of Multi Context Systems Contain ing Logical and Sub Symbolic Contexts of Reasoning Master s thesis Depart ment of Mathematics amp Department of Computer Science 8 Artificial Intel ligence FAU Erlangen Nuremberg 2009 The full thesis is available under http www opus ub uni erlangen de opus volltexte 2010 15872. Both are at some points called as external components 4 1 Infrastructure amp Ex Ante Setup As programming language for the first implementation of the MCS framework we used SCALA a general purpose programming language In our MCS framework an MCS is unequivocally defined by an MCS configu ration file In the file all contexts are represented Having read the configuration file and built an internal representation of each context the MCS framework sets up a list of initial knowledge or input bases as well as a list of bridge rule bases each associated with a context Moreover if indicated in the invocation the bridge rule constraint configu ration file is processed This file contains information concerning the grouping of bridge rules into different classes the properties of individual bridge rule classes and the relations between different classes see Section 3 2 For doing so the different bridge rule classes explicitely have to be declared as such in the file afterwards their properties and the relations between the classes may be named T In the meantime a second implementation using CLOJURE has been built Towards an Implementation of a Multi Context System Framework 21 If this functionality shall be used in the bridge rule bases every bridge rule indi vidually has to be labelled with the name of the class it belongs to Unlabelled bridge rules by default will be treated as belonging to class generic After perfo
3. ality of smodels when testing for founded inactivation of a 0 bridge rule The following test will be based on comparing the total number of bridge rules in the MCS with the number of correctly activated inactivated bridge rules If both values are equal the BRM represents an equilibrium of the MCS Given the valid results of the reasoner calls we cycle through the represen tation of the contexts in the BRM If a context s BRM part doesn t contain a 0 we may directly add the number of bridge rules of this context to the number of foundedly activated inactivated bridge rules founded activation has already been assured by the use of the smodels optimization when creating the reasoner input for the context and proceed with the next context s representation Oth erwise we take the context s model which was previously returned as result by the smodels call For every inactivated bridge rule we cycle through its body and add every condition to a list corresponding to the conditions target context e g for 2 not q not q would be added to C s list Having completely traversed the bridge rule s body we add to every context s reasoning result a constraint containing the conjunction of the elements of the context s list E g for the list p not q r s the constraint p not q r s would be added to the model previously returned by the smodels call Having done so for all the contexts of the MCS we per
4. L O H p where is a set of formulae y is a formula and L is a logic t Please note that if r is a bridge rule of the form a then head r a Towards an Implementation of a Multi Context System Framework 15 Equilibria are a fundamental concept of multi context systems as they make the introduction of a notion of semantics for MCS possible and under certain circum stances exhibit properties analogue to those of fixpoints as means of semantics In 3 a restricted class of MCS is introduced for which the authors present a reduction similar to the Gelfond Lifschitz reduction for logic programs For this class of reducible MCS the notion of grounded equilibria is given following ideas and terminology from logic programming Brewka and Eiter finally define a well founded semantics for reducible MCS by constructing an operator involv ing the concepts of reducibility and grounded equilibria and using a fixpoint argument in the style of the Knaster Tarski theorem For the implementation of the proof of concept MCS framework we used a generalized account of the just outlined theory which also allows for the incor poration of sub symbolic contexts of reasoning We will now briefly restate the relevant concepts introduced in 4 where also a more thorough discussion of the now following account may be found which may be understood as direct generalizations of the above stated ideas Definition 6 A reasoner is a 5
5. contexts and an algorithm for finding the equilibria of an MCS is presented In this paper we want to describe how we designed and implemented a proof of concept MCS framework software based on results from 3 and 4 2 Very Crisp Introduction to Multi Context Systems As an introduction to the topic we restate the key definitions given in 3 First the concept of logic is defined in terms of input output conditions 1 As in the following no information containing satisfiability or inference rules within the corresponding logics will be given also the denomination pre logic would be justifiable For the sake of consistency we will keep the original naming calling it a logic 14 T R Besold S Mandl Definition 1 A logic L KB BS ACC_ is composed of the following components 1 KBz is the set of well formed knowledge bases of L We assume each element of KB is a set 2 BSz is the set of possible belief sets 3 ACC KB 2P 2 is a function describing the semantics of the logic by assigning to each element of KBr a set of acceptable sets of beliefs Given several logics bridge rules are used to translate between the logics Definition 2 Let L 1y L be a set of logics An Ly bridge rule over L 1 lt k lt n containing m conditions is of the form s r1 pi rj pj mot r 41 pj41 NOt Tm Pm 1 where j lt m 1 lt rk lt n pk is an element of some beli
6. rules from class or bridge rules from class2 but not bridge rules from both classes at a time are active shall be considered 8 X oneO f ImplicatesExactlyOne Y class oneO f Implicates ExactlyOne class2 expresses that only BRMs in which whenever a bridge rule of class is active ex actly one bridge rule out of class2 is active shall be considered 9 X oneOfImplicatesSome Y Analogously 10 X oneO fImplicatesAll Y Analogously 11 X entireClassImplicatesExactlyOne Y class entireClassImplicatesOne class2 expresses that only BRMs in which whenever the bridge rules of class are all ac tive exactly one bridge rule out of class2 is active shall be considered gt 5 As in general the comparisons between heads and conditions can be performed by magnitudes more time efficient as a reasoner call e g in many logical contexts the comparison may be performed by a simple syntax based matching between heads and conditions the overall computational efficiency of the MCS will be improved In the following using the implemented constraint language where X and Y are variables which can be replaced by concrete classes and numb is a numeric integer variable 20 T R Besold S Mandl 12 X entireClassImplicatesSome Y Analogously 13 X entireClassImplicatesAll Y Analogously The individual bridge rules are explicitly distributed over the different classes by labeling them with the class name e g directly in the brid
7. tuple R Inpp Resp ACC R Condr Updp where 1 Inpp is the set of possible inputs to the reasoner 2 Resp is the set of possible results of the reasoner 3 ACCpr Inpp 28 defines the actual reasoning assigning each input a set of results in a decidable manner 4 Cond is a set of decidable conditions on inputs and results condr Inpp x Resr 0 1 5 Updp is a set of update functions for inputs updr Inprpt Inpp Please note that in contrast to Definition 1 we do not make any assumption about the form of the input to the reasoner The following definitions adapt the basic concepts of multi context reasoning given in Definitions 2 to 5 for the use with reasoners as of Definition 6 Definition 7 Let R Rj Rn be a set of reasoners An Rp bridge rule over R 1 lt k lt n containing m conditions is of the form u ri cr rj c not rj41 Cj 1 NOt Tm Cm 2 where j lt m 1 lt rk lt n and ck is a condition of inputs and results of some Rr and u is an element of the Upd Analogously to Definition 3 we now define a generalized MCS 16 T R Besold S Mandl Definition 8 A generalized multi context system M C1 Cn consists of a collection of contexts Ci Ri inpi bri where R Inp Res ACC Cond Upd is a reasoner inp an input an element of Inp and br is a set of R bridge rules over R Rn as of equation 2 Belief states also have to be adapt
8. 13 Towards an Implementation of a Multi Context System Framework Tarek R Besold and Stefan Mandl University of Erlangen Nuremberg Chair of Computer Science 8 Artificial Intelligence D 91058 Erlangen Germany sntabeso i8 informatik uni erlangen de stefan mand1 informatik uni erlangen de Abstract This system description provides after a limited formal in troduction to the topic an overview of our proof of concept multi context system framework implementation lining out the main functionalities and working principles Moreover two basic techniques for reducing com putational complexity when searching for equilibria of a multi context system are proposed 1 Introduction One of the basic problems in knowledge representation and knowledge engineer ing is the impossibility of writing globally true statements about realistic problem domains Multi context systems MCS are a formalization of simultaneous rea soning in multiple contexts Different contexts are inter linked by bridge rules which allow for a partial mapping between formulas concepts information in different contexts Recently there have been a number of investigations of MCS reasoning for instance see 1 or 2 with 3 and 4 being two of the latest contributions In the former one the authors describe reasoning in multiple con texts that may use different logics locally In the latter one the concept of MCS reasoning is extended to also integrate sub symbolic
9. anded over to the findEquilibria subroutine findEquilibria returns pairs of equilibria and corresponding BRMs The output may be written out directly or postprocessing steps may be performed 4 2 The Equilibria Mechanism findEquilibria Given the MCS s knowledge bases the bridge rules a list of BRMs and the contexts findEquilibria lists the equilibria amongst the BRMs and the cor responding belief states of the respective contexts Given a BRM the heads and bodies of the corresponding bridge rules the contexts and the knowledge bases findEquilibria performs all the updates which are indicated by the heads of bridge rules set to 1 in the given BRM This is done by a subroutine called createReasonerInput Then it invokes the corresponding reasoners indicated by the information stored in the con texts thereby performing a global reasoner call over all the possibly updated knowledge bases If the reasoner calls yield consistent results for all knowledge bases for every inactivated bridge rule findEquilibria has to test whether the bridge rule is correctly set to 0 in the BRM To do so it has to find in every inactivated bridge rule s body at least one condition that is not fulfilled The same has to be done for all the bridge rules set to 1 All the conditions in every bridge rule body have to be fulfilled If this tests may successfully be completed the BRM represents an equilibrium for the g
10. bridge rules br we can initially compute for every br br the corresponding set of i conflicting bridge rules in br As every active bridge rule br br may directly inactivate all i conflicting bridge rules we may further reduce the number of BRMs tested on being an equilibrium by setting all 7 conflicting bridge rules to 0 when setting br to 1 The additional effort spent on initially computing the conflicting bridge rules within the set br in most cases pays off as in total aaa 2r C n the total number of bridge rules of the MCS C the total number of 7 conflicting bridge rules for all br br BRMs are pruned out Hence at least DREN 27 0 m N N the number of contexts of the MCS reasoner calls less have to be performed An upper boundary for the extra costs of computing the conflicting bridge rules within br is given by n l n the total number of bridge rules of the MCS l the Towards an Implementation of a Multi Context System Framework 19 maximum of the numbers of conditions in the body of a bridge rule from br comparisons between bridge rule heads and bridge rule conditions from bridge rules within the set br 3 2 Constraints on Bridge Rules Moreover the wish to reduce some complexity by making knowledge of the combinatorical structure of bridge rules that is already implicitly contained in the MCS explicit may arise In order to do so constraints on bridge rules or groups of bridge rules may be used re
11. ed Definition 9 Let M C1 Cn be a generalized MCS A generalized belief state is a sequence S S1 Sn such that each S is of the form inpi resi with inp Inp and res Resi We say a bridge rule r of form 2 is applicable in a generalized belief state S S1 S iff for 1 lt i lt j cilinpi resi 1 in S and for j 1 lt k lt m cklinppg resk 0 in Sk Now we prepare for the concept of equilibrium in the generalized setting Definition 10 The set of context local update functions with respect to a corresponding element S of a belief state S is given by US MCS S head r r br applicable in S where br denotes the set of bridge rules of S s corresponding context Ci In general a set of update functions may yield different results when the func tions are applied multiple times or in different orders We do not allow such sets of update functions Definition 11 An applicable set of update functions US MCS S is station ary for an input inp iff the following two conditions hold Vu US MCS S u inp u inp for m gt 1 idempotency Vu u US MCS S u u inp u u inp commutativity Definition 12 The update of a belief state element S of a belief state S with respect to a set of update functions US with k elements is given by u ua Ug inp ee if US is stationary for inp and undefined otherwise Please note that s
12. ef set of Ly and s is a syntactically valid element of a knowledge base from KB and for each kb KB kbU s KB A configuration of logics and bridge rules comprises a multi context system Definition 3 A multi context system M C1 Cn consists of a collection of contexts C Li kbi bri where L KB BS ACC is a logic kb a knowledge base an element of KB and br is a set of Li bridge rules over Ree oa Ln A belief state is the combination of the belief sets of all contexts of the MCS Definition 4 Let M C1 Cn be a MCS A belief state is a sequence S S1 Sn such that each S is an element of BS Now we can clarify the applicability of a bridge rule in the context of a belief state We say a bridge rule r of form 1 is applicable in a belief state S S1 S iff for 1 lt i lt j pi E TR S and for j 1 lt k lt m pk TiS ye A belief state is an equilibrium if the consequences of all applicable bridge rules are given or derivable hence each context has an acceptable belief set given the belief sets of the other contexts Definition 5 A belief state S S1 Sn of M is an equilibrium iff for 1 lt i lt n the following condition holds Si E ACC kb U head r r br applicable in S 4 In contrast to 3 where a similar constraint concerning the nature of s is imposed only implicitly 3 Th indicates the deductive closure i e Th yp
13. erformed Given an MCS with m contexts and n BRMs In the worst case at least m n reasoner calls have to be performed Therefore in order to identify all equilibria of an MCS in the worst case at least m 2 reasoner calls k the number of bridge rules in the MCS are needed In real world applications when all equilibria of an MCS ought to be found the dominating element in this estimation is the number of bridge rules k For the naive generate and test approach in the most general case the number of m 2 reasoner calls is a hard lower boundary All BRMs have to be tested and in every test for every context a reasoner call has to be performed 8 We call an equilibrium self sustaining iff the application of all bridge rules may only be based on the application of other bridge rules and thus be independent of the concrete knowledge bases of the MCS Always assuming that per context per BRM one reasoner call is in fact needed and is sufficient Here sufficiency may be assured for the different types of contexts of finite generalized MCS as admissible for the algorithm presented in 4 24 T R Besold S Mandl For the findSstEquilibria mechanism in the most general case m 2 1 is a hard lower boundary for the number of reasoner calls m the number of contexts of the MCS k the number of bridge rules In the average case the number of reasoner calls will be higher For every BRM which in fact represents a s st equil
14. form another global reasoner call invoking smodels for every single context handing over the modified reasoning results of the earlier calls If there is at least one context with modified reasoning result as input for which the second reasoner call returns a valid model i e the con junction constraint is not applied we may augment the number of foundedly activated inactivated bridge rules by one and proceed with the next bridge rule If the models of all contexts with modified input are invalid the 0 bridge rule has been inactivated unfoundedly and the BRM does not represent an equilibrium Unfortunately we have to perform this process for every single bridge rule and cannot include all constraints from bridge rules into one global reasoner Towards an Implementation of a Multi Context System Framework 23 call as bridge rules which overlap in their conditions e g one containing 2 not s the other containing 2 s may otherwise cause wrong results 4 3 Identifying Self sustaining Equilibria For MCS consisting only of logical contexts using smodels as a reasoner we added findSstEquilibria used to list the self sustaining s st equilibria The findSstEquilibria function is given a BRM the knowledge bases the bridge rules and the contexts The createReasonerInput subroutine is called with empty knowledge bases again performing the smodels optimization already mentioned and the res
15. ge rule base by this establishing some kind of instanceOf relation between the individual bridge rules and the classes they belong to the class being the concept the bridge rule the instance Every unlabeled bridge rule is implicitly added to a generic class of bridge rules classgeneric which is the superclass of all other classes On ClaS8 generic NO Constraints in form of properties as listed above may be imposed 4 A Proof of concept MCS Framework Implementation The proof of concept implementation is based on the principles of the aforemen tioned algorithm for finding the equilibria of an MCS presented in 4 After some steps of reading the MCS specification getting input and creating the lo cal representations of the knowledge bases and the sets of bridge rules of the MCS we generate all possible BRMs Then for each BRM we perform the up dates it indicates and check whether the result is an equilibrium or not When doing so we also perform a test if the conditions of all applied bridge rules are really fulfilled and the bridge rules have been applied justifiedly According to the result of this tests we add the BRM to the list of equilibria BRMs or we directly proceed with the next bridge rule model Inter alia for the implementation of the constraint mechanism for bridge rules and bridge rule classes we make use of the lparse smodels combination see 5 and 6 as implementation of the stable model semantics for logic programmes
16. ibrium m reasoner calls have to be performed 5 Conclusion When applying the MCS framework to different testing examples it has shown to be applicable and appropriate see 7 and 8 for examples For the future more sophisticated techniques for reducing computational complexity will be needed Nevertheless the approach to implementing an MCS has proven to be effective and an important step towards the implementation of an MCS framework has been made References 1 Roelofsen F Serafini L Minimal and Absent Information in Contexts In Inter national Joint Conference on Artificial Intelligence Volume 19 Lawrence Erlbaum Associates LTD 2005 558 2 Brewka G Roelofsen F Serafini L Contextual Default Reasoning Proc IJCAI 07 Hyderabad India 2007 3 Brewka G Eiter T Equilibria in Heterogeneous Nonmonotonic Multi Context Systems In Proceedings of the National Conference on Artificial Intelligence Vol ume 22 Menlo Park CA Cambridge MA London AAAI Press MIT Press 1999 2007 385 390 4 Besold T R Mandl S Integrating Logical and Sub Symbolic Contexts of Rea soning In Filipa J Fred A Sharp B eds Proceedings of ICAART 2010 Second International Conference on Agents and Artifcial Intelligence INSTICC Press 2010 For a full version of the paper see also http www8 informatik uni erlangen de inf8 Publications bridging mcs original pdf 5 Syrjnen T Lparse 1 0 User s Manual
17. ing compu tational complexity we added to the MCS algorithm from 4 18 T R Besold S Mandl Input MCS C1 Cn a finite multi context system Output answer answer lt br Us lt icn bri ums UPDATEMONOTONICBRIDGERULES MCS br cand brm 2 ums C brm for brm cand do S inpi resi inpn resn for b brm do S APPLY head b S for 1 lt i lt n do S i nP S i ACC NP Sfi for 1 lt i1 lt RES S1 1 lt in lt RES Sn do S INP S1 RES S1 i1 NP S1 RES S1 lin if EQUILIBRIUM S then answer answerU S Algorithm 1 Algorithm for computing all equilibria of an MCS 3 1 The Notion of Conflicting Bridge Rules The first refinement comprises a method for pruning out some bridge rule models BRMs already before they are falsely considered as possible equilibria We exploit the within every context locally valid demand for consistency in order to reduce the equilibria search space via conflicting bridge rules Definition 17 Two bridge rules br br br where br is a set of bridge rules of an MCS are called i conflicting if applying br would directly inhibit the ap plication of br due to a condition cj body br which contains where appli cable chead br or not head br or any other statement contradicting head br This allows for an extension of the algorithm For a given set of
18. iven MCS and the BRM as well as the the belief state are linked and jointly added to the list of equilibria Then findEquilibria proceeds with the next BRM if there is any 22 T R Besold S Mandl As long as we are dealing with logical contexts for which we may use the lparse smodels answer set solver as a reasoner we may make the test for founded activation of the 1 bridge rules superfluous by modifying the createReasoner Input mechanism For every knowledge base element a demanded in a condi tion in the body of an activated bridge rule we integrate a constraint not a into the answer set solver input which in its result inhibits the generation of a model in that a is not present and for every knowledge base element b listed in an inhibitive condition we add b preventing the occurrence of models in which b is present Thus we assure that if a model is returned for every l bridge rule the conjunction of the conditions in its body holds For the inactivated bridge rules this technique may not be applied For a bridge rule set to 0 it is sufficient that only one of its conditions fails Thus doing as we did before with the 1 bridge rules would be by far too restrictive But nevertheless as long as we are dealing with contexts allowing for smod els as a reasoner and yielding unique reasoning results i e only one model is returned when smodels is called as a reasoner we may exploit some function
19. ng ACC inp yielding a set of results res for each i being of finite cardinality as MCS was said to be finite Thus testing whether inp res is an equilibrium for all 7 we obtain the set of equilibria for the given bridge rule model Iterating the procedure over the finite set of all bridge rule models and joining the resulting sets of equilibria finally yields the set of all equilibria Definition 16 Given an MCS with a global set of bridge rules br bri A set of bridge rules br C br is called update monotonic iff for all belief states S the following condition holds S update MCS S gt VC MCS S C VC MCS S where VC MCS S J cond R cond inp resi 1 and update MCS S is the global update over all S S As bridge rules in the update monotonic subset of bridge rules of the MCS are guaranteed to remain active after any update the update monotonic bridge rules that are initially active in the MCS when searching for equilibria have to be active in any equilibrium Hence when iterating over all bridge rule models only those bridge rule models that comply with the initially active update monotonic bridge rules have to be considered Proposition 2 For finite MCS the above sketched algorithm for a pseudo code representation see Algorithm 1 is both complete and correct 3 Refinements of the Basic Algorithm In the following we present two refinements mainly used for reduc
20. rming these steps the BRMs which afterwards will be tested for representing an equilibrium are generated This may be done by creat ing a bitstring representing a BRM for each possible combination of activa tion inactivation of bridge rules Thereby in total 2 n the total number of bridge rules in the MCS bitstrings are generated Due to performance considerations we implemented another mechanism ex ploiting whenever indicated and possible constraints on bridge rules and conflicting bridge rules In this scenario we are using the lparse smodels an swer set solver to generate the BRM bitstrings This offers the possibility to directly include the information given in the constraints on the bridge rules into the model generation smodels is used for generating a complete list of mod els of bridge rule combinations complying with the constraints stated via the constraints on bridge rules formalism which then are transformed into BRM bitstrings By this no a priori contradicting bridge rule combinations are con sidered possible BRMs At least for logical contexts we may also use the notion of conflicting bridge rules further sparsening the field of BRMs The bridge rule head bases and the bridge rule body bases are scanned for conflicting structures adding information of any conflict found to the input for the answer set solver To start the equilibria mechanism the MCS s knowledge bases the bridge rule bases and the BRMs are h
21. sulting in a formalism to impose constraints on the BRMs to be considered The constraint formalism shall offer possibilities to group bridge rules into classes and impose cardinality restrictions on these classes up to uniqueness of a rule as a representant of a certain rule class It may be used to model inter connectedness and hierarchy between the classes of bridge rules by establishing a concept of subclasses and superclasses as well as introducing complementary classes Moreover features have been added to provide some kind of rudimentary inference mechanism amongst bridge rules bridge rule classes 1 X maxCard numb class maxCard a with 0 lt a lt max where maz is the total number of bridge rules in the model expresses that only BRMs in which at most a bridge rules from the class class are active shall be considered 2 X minCard numb Analogously 3 X unique class unique expresses that only BRMs in which exactly one bridge rule from the class class is active shall be considered 4 X bundled class bundled expresses that only BRMs in which all bridge rules from the class class are active or BRMs in which all bridge rules from the class class are inactivated shall be considered 5 X superClass Y class1 superClass class2 expresses that class has class2 as a superclass i e class C classe X subClass Y Analogously 7 X complementary Y class complementary class2 expresses that only BRMs in which bridge
22. tationarity is only required for the set of update functions that is actually applied to belief state elements at a time We now can give the definition of the generalized concept of equilibrium Definition 13 A generalized belief state S inpi res1 npn resn of M is an equilibrium iff for 1 lt i lt n the following condition holds update inp US inp and res E ACC inp where update inp US denotes the update of S with respect to US which in turn has to be stationary for the corresponding inp Towards an Implementation of a Multi Context System Framework 17 The remainder of this section describes a procedure to compute all equilibria of a finite MCS based on complete enumeration which served as basis for our proof of concept implementation Definition 14 An MCS M C1 Cn is said to be finite iff fori lt i lt n following condition holds ACC inp lt co and br lt co For the implementation we consider finite MCS only Definition 15 Let Br be a set of n bridge rules of an MCS A bridge rule model is an assignment Br gt 0 1 that represents for each bridge rule in Br whether it is active or not Proposition 1 For each equilibrium there is exactly one bridge rule model For a given bridge rule model and an MCS we first apply all the bridge rules activated in the bridge rule model yielding inp inp Then we compute the set of results for each context i given inp by applyi
23. ult is handed over to the contexts reasoners If a reasoner returns an invalid e g inconsistent reasoning result the BRM does not represent an s st equilibrium of the given MCS In order to eliminate all equilibria other than the s st ones additional tests are performed on the reasoning results For every bridge rule set to 1 in the BRM a test is performed whether one of its conditions contains a not statement If this test yields a positive result the found equilibrium is not an s st equilibrium as its status as equilibrium is based on the absence of the knowledge base element contained in the not condition Then a test for unfoundedly inactivated 0 bridge rules in the BRM has to be performed analogously as in findEquilibria Finally we add the content of the knowledge bases initially handed over when invoking findSstEquilibria to the belief state obtained as equilibrium and perform another reasoner call for every context assuring the consistency of the expanded belief state If all reasoner calls yield a valid result the equi librium found is a s st equilibrium of the MCS The BRM and the expanded belief state are linked and jointly added to the list of s st equilibria Then findS stEquilibria proceeds with the next BRM if there is any 4 4 Comments Concerning Algorithmic Complexity As a measure for the complexity of the findEquilibria routine we may use the number of reasoner calls p
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