PMON + : A Fluent Logic for Action and. Change. Patrick Doherty. Abstract

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1 PMON + : A Fluent Logic for Action and Change Formal Specication, Version 1.0 Patrick Doherty Abstract This report describes the current state of work with PMON, a logic for reasoning about action and change, and its extensions. PMON has been assessed correct for the K;IA class using Sandewall's Features and Fluents framework which provides tools for assessing the correctness of logics of action and change. A syntactic characterization of PMON has previously been provided in terms of a circumscription axiom which is shown to be reducible to a rst-order formula. This report introduces a number of new extensions which arealso reducible and deal with ramication. The report is intended to provide a formal specication for the PMON family of logics and the surface language L(SD) used to represent action scenario descriptions. It should be considered a working draft. The title of the report has a version number because both the languages and logics used are continually evolving. Since this document isintended as a formal specication which isusedby our group as a reference for research and implementation, it is understandably brief as regards intuitions and applications of the languages and logics dened. We doprovide a set of benchmarks and comments concerning these which can serve as a means of comparing this formalism with others. The set of benchmarks is not complete and is only intended to provide representative examples of the expressivity and use of this particular family of logics. We describe its features and limitations in other publications by our group which can normally be found at Supported in part by the Swedish Research Council for the Engineering Sciences (TFR). IDA Technical Report 1996 LiTH-IDA-R ISSN Department of Computer and Information Science, Linkoping University, S Linkoping, Sweden

2 PMON + : A Fluent Logic for Action and Change Formal Specication, Version 1.0 Patrick Doherty Department of Computer and Information Science Linkoping University S Linkoping, Sweden patdo@ida.liu.se Abstract This report describes the current state of work with PMON, a logic for reasoning about action and change, and its extensions. PMON has been assessed correct for the K;IA class using Sandewall's Features and Fluents framework which provides tools for assessing the correctness of logics of action and change. A syntactic characterization of PMON has previously been provided in terms of a circumscription axiom which is shown to be reducible to a rst-order formula. This report introduces a number of new extensions which arealso reducible and deal with ramication. The report is intended to provide a formal specication for the PMON family of logics and the surface language L(SD) used to represent action scenario descriptions. It should be considered a working draft. The title of the report has a version number because both the languages and logics used are continually evolving. Since this document isintended as a formal specication which isusedby our group as a reference for research and implementation, it is understandably brief as regards intuitions and applications of the languages and logics dened. We doprovide a set of benchmarks and comments concerning these which can serve as a means of comparing this formalism with others. The set of benchmarks is not complete and is only intended to provide representative examples of the expressivity and use of this particular family of logics. We describe its features and limitations in other publications by our group which can normally be found at Supported in part by the Swedish Research Council for Engineering Sciences (TFR). 1

3 Contents 1 Introduction 3 2 The Base Logic FL Syntax The Object Sorts S o The Value Sorts S v The Fluent SortF The Action Sort A The Temporal Sort T Foundational Axioms in FL Axioms for the Value Sorts S v Axioms for the Objects Sorts S o Axioms for the Fluent Sort F Axioms for the Action Sort A Some Useful Notation Reducing Scenario Descriptions to L(FL) 9 A The Surface Language for Scenario Descriptions 9 A.1 Logical Formulas in L(SD) A.1.1 Reducing Logic Formulas in L(SD) tol(fl) A.2 Reassignment Formulas in L(SD) A.2.1 Reducing Reassignment Formulas in L(SD) tol(fl) A.3 Causal Constraint Statements in L(SD) A.3.1 Reducing Causal Constraint Statements in L(SD) tol(fl) A.4 Acausal Constraint Statements in L(SD) A.4.1 Reducing Acausal Constraint Statements in L(SD) tol(fl) A.5 Dynamic Fluent Statements in L(SD) A.5.1 Reducing Dynamic Fluent Statements in L(SD) tol(fl) A.6 Observation Statements in L(SD) A.6.1 Reducing Observation Statements in L(SD) tol(fl) A.7 Action Representation in L(SD) A.8 Action Scenario Descriptions in L(SD) A.8.1 Some Useful Notation

4 B The PMON Family of Logics 29 B.1 Circumscription B.2 The PMON Family of Logics and their Circumscription Policies B.2.1 The Nochange Axiom B.2.2 PMON B.2.3 PMON(RCs): Extending PMON with Causal Constraint Statements B.2.4 PMON + : Extending PMON with Dynamic Fluent Statements C Benchmark Examples 34 C.1 Examples within the K;IA class C.1.1 Yale Shooting Scenario C.1.2 Stanford Murder Mystery Scenario C.1.3 The Red and Yellow Bus Scenario C.1.4 Hiding Turkey Scenario C.1.5 Ferry boat Connection Scenario C.1.6 Furniture Assembly Scenario C.2 Examples outside the K;IA class C.2.1 Jump in a Lake Scenario C.2.2 Extended Electric Circuit Scenario C.2.3 Extended Electric Circuit Example with Device C.2.4 The Delayed Circuit Scenario C.2.5 The Trap Door Scenario C.2.6 Extended Baby Protection Scenario Introduction The following report is intended to provide a formal specication for the PMON family of logics and the surface language L(SD) used to represent action scenario descriptions. The title of the report has a version number because both the languages and logics used are continually evolving. Since this document is intended as a formal specication which is used by our group as a reference for research and implementation, it is understandably brief as regards intuitions and applications of the languages and logics dened. We do provide a set of benchmarks and comments concerning these which can serve as a means of comparing this formalism with others. The set of benchmarks is not complete and is only intended to provide representative examples of the expressivity and use of this particular family of logics. We describe its features and limitations in other publications by our group which can normally be found at The Features and Fluents project began in 1987 and was initially based on a number of technical reports by Sandewall (e.g. [22, 21, 24, 25]) and published articles (e.g. [23, 26,28]) that resulted in a book by Sandewall [27] which covers part of the research done during this period. Two concepts of fundamental importance, the use of ltering and occlusion, were introduced at an early stage in the project [21, 23], and in the current research with PMON are proving to be quite versatile 3

5 in dealing with many of the problems which arise when reasoning about action and change. For details about the relation between ltering and occlusion and other approaches, see Sandewall [27]. Doherty [4, 3, 5, 6], took an approach somewhat dierent from Sandewall and generated syntactic characterizations of many of the denitions of preferential entailment introduced in Features and Fluents by translating scenario descriptions into a standard sorted rst-order logic with circumscription axioms in an attempt to provide a basis for implementing some of these logics. It turns out that PMON is also well-behaved in this respect any scenario description in PMON is reducible to a rst-order theory. In the current report, we continue the development of PMON and show that it continues to have a number of nice mathematical properties even when extended for ramication [8]. In related work, Karlsson [10, 11, 13, 12] uses PMON as a basis for formal specication of plans. 2 The Base Logic FL There are a number of dierent possibilities for choosing a base logic in which to compile scenario descriptions which are described using the scenario description language L(SD). Sandewall uses a specialized logic called discrete feature logic with a number of interesting non-standard features which can potentially aid in developing ecient implementations for the various action classes. In this section, we introduce an alternative, which we call uent logic and denote FL. FL will be used as a basis for formalizing reasoning about action and change. The language for FL, denoted L(FL), is a many-sorted rst-order language. All scenario descriptions described in L(SD) have a modular translation into L(FL). 2.1 Syntax The language L(FL), is a many-sorted rst-order language with equality. We use the standard connectives : (negation), ^ (conjunction),! (implication), (equivalence), _ (disjunction), and the standard quantiers 8 and 9. In addition, scoping will be indicated by standard use of parentheses or dot notation. We associate a many-sorted signature in L(FL) for each vocabulary and action similarity type in L(SD) as follows. Let = h O V Di be a vocabulary in L(SD), where and In addition, let O = ho 1 ::: O l i V = hv 1 ::: V p i = ff i1 1 : D 1 f i2 2 : D 2 :::fn in : D n g D = hd 1 ::: D n i where D i 2Oor V: = fa i1 1 Ai2 2 :::Aim m g be an action similaritytype for L(SD). The many sorted-signature S =(S ), is a pair where S is a nite set of sort symbols S = ff A Tg[S o [ S v S o = fo 1 ::: O l g and S v = fv 1 ::: V p g and S d = S o [ S v : andisadenumerable set of sort strings describing the type for the function and relation symbols in L(FL) associated with the feature symbols in, the action symbols in, the function and 4

6 relation symbols associated with the temporal sort T, the domain independent relation symbols, and the constant symbols for each of the sorts. contains the following domain dependent strings : For each feature symbol f ik k in with a non-boolean value domain sort dom (f ik k )=D k, L(FL)contains a function symbol f j k of arity j = i k + 1 and contains the string f j k : s 1 ::: s j!f, where s i 2 S o,1 i j ; 1, and s j 2 S d. For each feature symbol f ik k in with a boolean value domain sort dom (f ik k )=D k, L(FL) contains a function symbol f j k of arity j = i k andcontains the string f j k : s 1:::s j!f, where s i 2 S o,1 i j. For each action symbol A ik k in, L(FL)contains a function symbol Aj k of arity j = i k and contains the string A j k : s 1 ::: s j!a, where s i 2 S d,1ij. For each constant c in L(FL)ofsortO i,contains a string c : O i.for each constant c in L(FL)ofsortV i, contains a string c : V i. In addition, contains the following domain independent strings: Holds : TF, Occlude : TF, Observe : TF, Occurs : TTA, <: TT, : TT, +:TT!T, ; : TT!T The Object Sorts S o For each sort O i in S o : We use the letter o i, possibly subscripted for variables of sort O i. For each object name n j in O i,we associate an object name constant n i j of sort O i of the same name in L(FL). We use the letter o i, possibly subscripted for object constants of sort O i. Only constants and variables of sorts O i are object terms. A unique names assumption will be applied to the object name constants, while the object constants will be allowed to vary among object names The Value Sorts S v For each sort V i in S v : We use the letter v i, possibly subscripted for variables of sort V i. For each value name j in V i,we associate an value name constant v i j of sort V i of the same name in L(FL). We use the letter u i, possibly subscripted for value constants of sort V i. 1 For convenience, we may sometimes violate naming conventions for the dierent sorts and use other letters when there is no danger of ambiguity. 5

7 Only constants and variables of sorts V i are value terms. A unique names assumption will be applied to the value name constants, while the value constants will be allowed to vary among value names The Fluent Sort F We use the letters f or z, possibly subscripted for variables of sort F. For each feature symbol fk ik in, we associate a function symbol f k in L(FL) of the same name and having the sort described in. Fluent terms are constructed in the usual manner, but note that embedded terms f(f(;)) are not legal terms in the language. A unique names assumption will be applied to the uents terms in sort F The Action Sort A We use the letter a, possibly subscripted for variables of sort A. For each action symbol A ik k in, we associate a function symbol A k in L(FL) of the same name and having the sort described in. Action terms are constructed in the usual manner, but note that embedded terms A(A(;)) are not legal terms in the language. A unique names assumption will be applied to the actions terms in sort A The Temporal Sort T We use the letter t, possibly subscripted for variables of sort T. We use the letter t, possibly subscripted, and the numerals (0, 1, 2,... ) constants of sort T. for temporal The intended interpretation of T is as a linear discrete time line where T is considered isomorphic to the natural numbers. Although we assume an interpreted theory, using the standard interpretation for the natural numbers, it turns out that when reasoning with K;IA or its current extensions, we use no more than Presburger arithmetic. In fact, we use a sub-theory of Presburger arithmetic, the rst-order theory of point constraints over integers [15]. The relation symbols < and are interpreted as the usual "less than" and "less than or equal to" relations on natural numbers. The function symbols +, and ; are interpreted as the usual "plus" and "minus" functions on natural numbers. In what follows, < 0 < 00, 0 < 00, < 0 00 and 0 00, stand for < 0 ^ 0 < 00, 0 ^ 0 < 00, < 0 ^ 0 00 and 0 ^ 0 00, respectively, where, 0 and 00 are terms of sort T. 3 Foundational Axioms in FL In this section we describe a number of domain independent axioms that are always assumed to be part of any translation of action scenarios into L(FL). In the following, we will use the meta variables 6

8 d 1 ::: possibly superscripted, to denote object or value variables of sorts ins o,ors v. d 1 ::: possibly superscripted, to denote object or value constants, or name constants of sorts S o or S v. 3.1 Axioms for the Value Sorts S v For each uent fk ik with value domain D k 2 S d, the following unique values axioms are assumed which state that at a specic time-point t, auent fk ik can only be assigned one value from its value domain for a particular set of object arguments o 1 ::: o ik ; 1: 8o t d k 1 d k 2:(d k 1 6= d k 2! (1) :(Holds(t f ik k (o 1 ::: o ik;1 d k ik 1 )) ^ Holds(t fk (o 1 ::: o ik;1 d k 2 )))) where o = o 1 ::: o ik;1.we will denote the set of unique values axioms by ; uva. For each uent f ik k with value domain D k 2 S d, the following value existence axioms are assumed which state that at a specic time-point t, a uent fk ik must be assigned at least one value from its value domain for a particular set of object arguments o 1 ::: o ik : 8o8t9d k 1:Holds(t f ik k (o 1 ::: o ik;1 d k 1)) (2) where o = o 1 ::: o ik;1.we will denote the set of value existence axioms by ; vea. For theories with nite value domains, the following unique name axioms are assumed for each of the value sorts V j 2 S v : ^ v j i 6= vj k : (3) 1i<kjVjj Note that this axiom applies to value name constants and not value constants, which may be bound to the interpretation of any value name constant. We will denote the set of unique names axioms for value domains by ; unav. For theories with nite value domains, the following domain closure axioms are assumed for each of the value sorts V j 2 S v : jvjj _ 8v j : v j = v j i : (4) i=1 Similar axioms for value domains D i of type S o will be dened in Section 3.2 We will denote the set of domain closure axioms for value domains by ; dcav. 3.2 Axioms for the Objects Sorts S o For theories with nite object domains, the following unique name axioms are assumed for each of the object sorts O j 2 S o : ^ n j i 6= nj k : (5) 1i<kjOjj We will denote the set of unique names axioms for object domains by ; unao. 7

9 For theories with nite object domains, the following domain closure axioms are assumed for each of the object sorts O j 2 S o : 8o j : jojj _ i=1 o j = n j i : (6) We will denote the set of domain closure axioms for object domains by ; dcao. 3.3 Axioms for the Fluent Sort F For theories with nite uent domains, the following unique names axioms are assumed: ^ 8o d ij d ik :f ij j (o 1 ::: o ij;1 d ij ) 6= f ik k (o0 1 ::: o0 i k;1 d 0 i k ) (7) 1j<kn where d ij d 0 i k are either object or value variables. i k;1 ^ 8o d ik d 0 i k :(fk ik (o 1 ::: o ik;1 d ik )=fk ik (o0 1 ::: o 0 i k;1 d 0 i k ))! ( o j = o 0 j) ^ d ik = d 0 i k : (8) We will assume that similar axioms exist for boolean value domains and will denote the set of unique names axioms for the uent domainby; unaf. 3.4 Axioms for the Action Sort A For theories with nite action domains, the following unique names axioms are assumed: ^ 8d:A ij j (d 1 ::: d ij ) 6= A ik k (d0 1 ::: d 0 i k ): (9) 1j<km 8d:(A i k^ ik k (d 1 ::: d ik )=A ik k (d0 1 ::: d 0 i k ))! ( d j = d 0 j) (10) where d = d 1 ::: d ik d 0 1 ::: d 0 i k are either object or value variables. We will denote the set of unique names axioms for the action domain by ; unaa. 3.5 Some Useful Notation The following notation will be useful when describing circumscription policies in a later section. ; UNA =; unav [ ; unao [ ; unaf [ ; unaa : (11) ; DCA =; dcav [ ; dcao : (12) The following collection of Foundational Axioms will be used in the circumscription policies. ; FA =; UNA [ ; DCA [ ; uva [ ; vea : (13) 8

10 4 Reducing Scenario Descriptions to L(FL) Given a scenario description, consisting of statements in the surface language L(SD), these statements can be translated into formulas in the language L(FL) via a two-step process. In the rst step: Action schemas in are instantiated with action occurrence statements, resulting in what are called schedule statements. The resulting schedule statements replace the action schemas. Observation statements are translated into xed observation statements. The result is an expanded (action) scenario description 0, consisting of schedule, xed observation, causal constraint, acausal constraint and dynamic uent statements. Each of these are statements in L(SD). In the second step, translation operators mapping statements from L(SD) into L(FL) are used to translate statements in 0 into well-formed formulas in L(FL). The details are provided in the appendices. A The Surface Language for Scenario Descriptions In this appendix, we will dene each of the dierent types of statements in L(SD) together with the translation operators which map statements from L(SD) into L(FL). In addition, we provide a number of Lemmas and Theorems related to the reduction of circumscriptive theories to the rst-order case. In Section 2, we dened the relationship between symbols in L(SD) and those in L(FL). In the following, we will additionally assume that there is a straightforward and unambiguous mapping from all logical connectives, quantiers, delimiters, and variable symbols used in L(SD) to their correlates in L(FL). The translation operators Tran EOE, Tran EV E, and Tran ETE dene part of this mapping, but we do not always explicitly apply these operators when the application is clear from the context. This will avoid a certain amount of notational overhead. The legal syntax for representing scenario descriptions is dened in terms of a surface language L(SD) consisting of: action occurrence statements, action law schemas, observation statements and xed observation statements, schedule statements, causal constraint statements, acausal constraint statements, and dynamic uent statements. All statements in L(SD), with the exception of occurrence statements, are constructed using logical and reassignment formulas in L(SD). Both logical and reassignment formulas, together with occurrence statements, can be translated to formulas in the base logic L(FL). Denition A.1 (Object Domain for L(SD)) An object domain O = ho 1 ::: O n i is a nite tuple of object sorts. Each object sort is dened as a non-empty nite set of object names. Denition A.2(Value Domain for L(SD)) A value domain D = hd 1 ::: D n i isanitetupleofvalue or object sorts from O V. Each object or value sort is dened as a non-empty nite set of object or value names. 9

11 Denition A.3 (Similarity Type for L(SD)) A similarity type for L(SD) is a mapping ff i1 1 : D 1 f i2 2 : D 2 :::fn in : D ng where i k 0foreach k and each fk ik is a feature symbol with i k arguments. An object domain sort O j from the object domain O is associated with each of the i k arguments of f ik k. Each D k is a non-empty nite set of feature values for f k of value domain sorts O i or V i. Dom (f k ) will be used to denote the value domain sort for the feature f k. Dom (f j k ) will be used to denote the object domain sort O i in O associated with the jth argument of the feature f k. A vocabulary = h O V Di is a tuple consisting of a similarity type, an object domain, value sorts, and a value domain. The value domain for, D = hd 1 ::: D n i, is the nite tuple of value sorts in O or V associated with the feature symbols in. Remark A.1 The surface language will be set up so that the denitions can be extended asneeded when modeling additional concepts associated with action and change. For example, currently, both the object and value domains are dened as being nite. At a later date, this restriction may be relaxed. A.1 Logical Formulas in L(SD) Let = h O V Di be a vocabulary. Denition A.4 (Elementary Object Expression (EOE)) An elementary object expression is any one of the following: 1. An object constant symbol. 2. An object name symbol. 3. An object variable symbol. Denition A.5 (Elementary Value Expression (EVE)) An elementary value expression is any one of the following: 1. A value constant symbol. 2. A value name symbol. 3. A value variable symbol. Denition A.6 (Elementary Feature Expression (EFE)) An elementary feature expression is any one of the following: 1. A feature symbol f 0 k from the vocabulary. 2. f ik k (! 1 :::! ik ) where f ik k is from the vocabulary and! 1 :::! ik are elementary object expressions. Denition A.7 (Elementary Fluent Formula (EFlF)) An elementary uent formula has the form f ^=X, where f is an elementary feature expression and XDom (f). Denition A.8 (Fluent Formula (FlF)) The set of uent formulas FlF is dened as follows : 1. If is an elementary uent formula then is in FlF. 10

12 2. If o is an object variable and is in FlF, then 8o: and 9o: are in FlF. 3. If and are in FlF then : is in FlF and any boolean combination of and are in FlF. 4. Nothing else is in FlF. Denition A.9 (Elementary Time-point Expression (ETE)) An elementary time-point expression is any one of the following: 1. A temporal constant in L(FL). 2. A temporal variable in L(FL). 3. Any temporal term constructed from temporal functions in L(FL). Remark A.2 Note that as regards ETE's, it is generally assumed that any term in L(FL) can be used in L(SD). We set up the correspondence in this manner because it should be relatively straightforward to change temporal structures when using Fluent Logic. Denition A.10 (Elementary Fixed Fluent Formula (EFFF)) Let denote an elementary time-point expression, and denote an elementary uent formula. An elementary xed uent formula has the following form: 1. []. Denition A.11 (Elementary Fixed Formula (EFF)) Let and 0 denote elementary time-point expressions,! and! 0 denote elementary object expressions. An elementary xed formula is any one of the following: 1. An elementary xed uent formula. 2. = 0. 3.! =! , where is any temporal relation dened in L(FL). Denition A.12 (Fixed Fluent Formula (FFF)) Let denote an elementary time-point expression, and denote a uent formula. A xed uent formula has the following form: 1. []. Denition A.13 (Logic Formula (LF)) The set of logic formulas LF is dened as follows : 1. If is an elementary xed formula then is in LF. 2. If t and o are temporal and object variables, respectively, and is in LF, then 8t: 9t: 8o: 9o: are in LF. 3. If and areinlfthen: is in LF and any boolean combination of and is in LF. 4. Nothing else is in LF. Denition A.14 (Restricted Logic Formula (RLF)) The set of restricted logic formulas RLF is dened as follows : 1. If is a xed uent formula then is in RLF. 2. If t and o are temporal and object variables, respectively, and is in RLF, then 8t: 9t: 8o: 9o: are in RLF. 3. If and are in RLF then : is in RLF and any boolean combination of and is in RLF. 11

13 4. Nothing else is in RLF. Denition A.15 (Time-Point Formula (TPF)) A time-point formula is a logic formula where the elementary xed formulas from which it is constructed only contain types 2 and 4 in Denition A.11. Denition A.16 (Object Formula (OF)) An object formula is a logic formula where the elementary xed formulas from which it is constructed only contain type 3 in Denition A.11. Any xed uent formula can be represented as a composition of elementary xed uent formulas (EFFF) using the logical connectives and quantication over objects in the usual fashion. Denition A.17 (Fixed Fluent Formula Abbreviations) Let be an ETE, o an object variable and, 0,uentformulas. The following reduction rules can be used to translate any FFF into a composition of elementary xed uent formulas: 2 []:( ^ 0 ) def =[]: _: 0. []:( _ 0 ) def =[]: ^: 0. [] ^ 0 def =[] ^ [] 0. [] _ 0 def =[] _ [] 0. []! 0 def =[]: _ 0. [] 0 def =[] [] 0. []:: def =[]. []: def = :[]. []8o: def = 8o:([]) []9o: def = 9o:([]) Denition A.18(Interval Fixed Fluent Formula (IFFF)) Let, 0 denote elementary time-point expressions, and denote a uent formula. An interval xed uent formula has any of the following forms: 1. [ 0 ], ( 0 ], [ 0 ), ( 0 ). 2. (;1 0 ], (;1 0 ), [ 1), ( 1), (;1 1). Denition A.19 (Abbreviations for IFFF's) Let, 0 be ETEs, and be a uent formula. The following reduction rules can be used to translate any IFFF into a logic formula [ 0 ] def = 8t: t 0! [t]. [ 0 ) def = 8t: t< 0! [t]. ( 0 ] def = 8t: < t 0! [t]. ( 0 ) def = 8t: < t < 0! [t]. 2 Note that the standard application of these rules is assumed, where all negations are rst driven inward before applying the double negation elimination and then any single negation left can be driven to the left outside the scope of the [t] notation. 12

14 (;1 0 ] def = 8t: t 0! [t]. (;1 0 ) def = 8t: t < 0! [t]. [ 1) def = 8t: t! [t]. ( 1) def = 8t: < t! [t]. (;1 1) def = 8t: [t]. A.1.1 Reducing Logic Formulas in L(SD) to L(FL) In this section, we will provide an algorithm which takes as input a formula in L(SD) and returns a w in L(FL). We will assume that the relation between vocabularies described in Section 2 holds. Denition A.20(Tran EOE ) Let! be an elementary object expression (EOE). 1. If! is an object constant symbol or object name symbol of sort O i then Tran EOE (!) =c i, where c i is a constant symbol in L(FL) of sort O i, which is the correlate to O i and assumed to exist in L(FL)'s similarity type. 2. If! is a variable symbol of sort O i then Tran EOE (!) =o i, where o i is a variable symbol in L(FL)ofsortO i. Denition A.21(Tran EV E ) Let! be an elementary value expression (EVE). 1. If! isavalue constant symbol or value name symbolofsortv i then Tran EOE (!) =c i, where c i is a constant symbol in L(FL) of sort V i,which is the correlate to V i and assumed to exist in L(FL)'s signature. 2. If! is a variable symbol of sort V i then Tran EV E (!) =v i, where v i is a variable symbol in L(FL)ofsortV i. Denition A.22(Tran E(OV )E ) Let! be an EOE or EVE. We will often use the translation operator Tran E(OV )E which is dened as TranEOE (!) if! is of sort O Tran E(OV )E (!) = Tran EV E (!) if! is of sort V Denition A.23(Tran ETE ) It is assumed that there is a straightforward mapping from temporal constants, variables, and expressions used in L(SD) into L(FL). Denition A.24(Tran EFFF ) Let be an elementary xed uent formula []f k (! 1 :::! ik )^=X, where X = f 1 ::: n g, Dom (f k ) is non-boolean, and f k has arity i k and sort Dom (f 1 k ) ::: Dom (f ik k )! Dom (f k ): Tran EFFF () = n_ Holds( f k ( 1 ::: ik d k j )) where = Tran ETE (), each d k j = Tran E(OV )E( j ) is a name constant ofsortd k associated with each name in Dom (f k ), each l = Tran EOE (! l ) is either a constant orvariable of the appropriate 13

15 sorts O l associated with sorts Dom (f l k ), and f k is the function symbol in L(FL)ofarity i k +1 associated with f k in L(SD). If Dom (f k ) is Boolean and X = ft Fg then Tran EFFF () = If Dom (f k ) is Boolean and X = ft g then If Dom (f k ) is Boolean and X = ff g then Holds( f k ( 1 ::: ik )) _:Holds( f k ( 1 ::: ik )): Tran EFFF () =Holds( f k ( 1 ::: ik )): Tran EFFF () =:Holds( f k ( 1 ::: ik )): Lemma A.1 Let be an arbitrary logic formula in the surface language for scenario descriptions. There is an algorithm that transforms intoawinl(fl). Proof Since a logic formula is composed of elementary xed formulas (EFF) using logical connectives and quantiers in the usual fashion, and we assume the vocabulary of L(FL) contains the necessary sorts, and logical and non-logical symbols, it suces to show that elementary xed formulas can be translated. Let be a logical formula in L(SD) and be an arbitrary elementary xed formula in.according to Denition A.11, there are four cases: 1. If [] is an elementary xed uent formula then replace in with Tran EFFF (). 2. If = 0 then replace in with Tran ETE () =Tran ETE ( 0 ). 3. If! =! 0 then replace in with Tran EOE (!) =Tran EOE (! 0 ). 4. If 0 then replace in with Tran ETE () Tran ETE ( 0 ). Under the assumption that there is a straightforward mapping of the logical connectives, quantiers, and delimiters, then after applying all possible substitutions, we are left with a w in L(FL). Remark A.3 We will assume that Tran LF ( ) denotes the corresponding w in L(FL). In addition, if Tran LF ( ), where might include sub-formulas where one of the dened transformation operators Tran is applied, then it is assumed that the outer Tran LF simply leaves the result of Tran unchanged. A.2 Reassignment Formulas in L(SD) Reassignment formulas play a very important role in PMON and represent the assignment of a new value to a uent. The reassignment operator is denoted by \:=". Denition A.25 (Elementary Reassignment Expression (ERE)) An elementary reassignment expression has the form f i := fg where f i is an EFE and v 2 Dom (f i ). The following abbreviations will be used. The rst two apply to any value domain for f i while the last two apply when Dom (f i ) is Boolean: 14

16 f i := def = f i := fg. :f i := def = f i := Dom (f i ) nfg. :f i := ft g def = f i := ff g. :f i := ff g def = f i := ft g. Denition A.26 (Reassignment Expression (RE)) A reassignment expression is an expression of the form f i := X, where f i is an EFE and X is a non-empty subset of Dom (f i ). A reassignment expression is an abbreviation for a disjunction of ERE's. Let be the RE f i := f 1 ::: n g, then n_ def = f i := j : The following abbreviation will be used. :f i := X def = f i := Dom (f i ) nx. Remark A.4 For the boolean value domain B, there are three cases: 1. f i := ft g. 2. f i := ff g. 3. f i := ft Fg def = f i := ft g_f i := ff g. Denition A.27 (Elementary Reassignment Formula (ERF)) An elementary reassignment formula is a formula of type [ 0 ]f i := X, where both 0 are elementary time-point expressions and f i := X is a RE. The following abbreviation will be used: n_ n_ [ 0 ]f i := f 1 ::: n g =[ 0 def ] f i := j = [ 0 ]f i := j : When X is a singleton set, [ 0 ]f i := X will be referred to as a simple ERF. It is easily observed that ERF's are simply abbreviations for a disjunction of simple ERF's. Denition A.28 (Restricted Reassignment Formula (RRF)) A restricted reassignment formula is a conjunction of ERF's n^ 1 [ 0 ]f j := X j A ^ ( t o) together with a logic formula (t o) representing duration constraints for the interval [ 0 ], where [ 0 ] is the same for each conjunct and no two EFE's,f j and f j 0, are the same. In addition, t and o, denoting the free temporal and object variables in, are restricted to be only those variables mentioned freely in any ofthef j,, or 0, and constraints in may only be placed on the elementary feature expressions f j between +1and 0, inclusively. The following abbreviation will be used: n^ [ 0 def ] f j := X j = n^ [ 0 ]f j := X j 15

17 Proposition A.1 Let be an arbitrary restricted reassignment formula in the surface language for scenario descriptions. There is an eective algorithm that transforms into a formula in conjunctive normal form, where each conjunct is a disjunction of simple elementary reassignment formulas. Proof Straightforward. Follows from the denitions. Denition A.29 (Reassignment Formula (RF)) A reassignment formula is a disjunction of restricted reassignment formulas (RRF's) k_ ni [ 0 ]f ij := X ij A ^ i (o t) A i=1 where the interval [ 0 ] is the same for each RRF. The following abbreviation will be used: k_ ^ [ 0 ni f ij := X ij A ^ i (o t) A def = i= k_ ni [ 0 ]f ij := X ij A ^ i (o t) A Remark A.5 An additional restriction will be placed on reassignment formulas for the class of action theories we deal with. If the elementary feature expression f mj is mentioned inthemth disjunct of a reassignment formula, then it must be mentioned inallthekdisjuncts. If this is not the case when using a reassignment formula in an action law specication then it is assumed that during the translation process the elementary reassignment formula [ 0 ]f mj := X,where X = D mj (the whole value domain for f mj ), is added toeach disjunct where f mj is not mentioned. Note that this does not inuence the semantics of the reassignment formula other than to ensure that the same feature expressions are occluded ineach disjunct of the reassignment expression. In other words, legally the occlusion specication can not be any other way. Alternatively, when translating a reassignment formula into a w in L(FL), one need onlymodify the occlusion assertions so that any feature expression mentioned inareassignment formula is occluded. One simply denes a set of inuenced features as the union of the sets of feature expressions mentioned ineach disjunct. When doing the translation, it is assumed thateach feature expression in the set is occluded. Denition A.30 (Conditional Reassignment Formula (CRF)) A conditional reassignment formula is a formula of the following type 8t8o:([](o)! [ 0 ](o)), where [](o) is a xed uent formula and [ 0 ](o) is a reassignment formula. The variables t o of sorts O i and T, respectively, are free in and. The variables t are restricted to those that are free in the elementary time-point expressions and 0. A restricted conditional reassignment formula is a CRF where [ 0 ] is a restricted reassignment formula. i=1 A.2.1 Reducing Reassignment Formulas in L(SD) to L(FL) 16

18 Denition A.31(Tran SERF, Tran ERF ) Let be a simple elementary reassignment formula [ 0 ]f k (! 1 :::! ik ):=X, where X = f j g, Dom (f k ) is non-boolean, and f k has arity i k and sort Dom (f 1 k ) ::: Dom (f ik k )! Dom (f k ): Tran SERF () =Holds( 0 f k ( 1 ::: n d k j ))^ 8t8d k (<t 0! Occlude(t f k ( 1 ::: n d k ))) where = Tran ETE () 0 = Tran ETE ( 0 ), d k j = Tran E(OV )E ( j )isavalue or object name constant of sort D k associated with each object of sort Dom (f k )=D k, d k is a variable of sort D k, each m = Tran EOE (! m ) is either a constantorvariable of the appropriate sorts O l associated with sorts Dom (f l k ), and f k is the function symbol of arity i k +1inL(FL) associated with f k in L(SD). If Dom (f k ) is Boolean and X = ft g then If Dom (f k ) is Boolean and X = ff g then Tran SERF () =Holds( 0 f k ( 1 ::: n ))^ 8t( <t 0! Occlude(t f k ( 1 ::: n ))) Tran SERF () =:Holds( 0 f k ( 1 ::: n ))^ 8t( <t 0! Occlude(t f k ( 1 ::: n ))) Let be an elementary reassignment formula [ 0 ]f k (! 1 :::! ik ):=X,asabove, but where X = f 1 ::: n g. Tran ERF () = _ j2x Holds( 0 f( 1 ::: n d k j ))^ 8t8d k (<t 0! Occlude(t f( 1 ::: n d k ))) Let be an elementary reassignment formula [ 0 ]:f k (! 1 :::! ik ):=X,asabove, but where X = f 1 ::: n g. Tran ERF () = ^ j2x :Holds( 0 f( 1 ::: n d k j ))^ 8t8d k (<t 0! Occlude(t f( 1 ::: n d k ))) Remark A.6 Note that both [ 0 ]f k (! 1 :::! ik ):=X and [ 0 ]:f k (! 1 :::! ik ):=X are de- ned as abbreviations for disjunctions of simple elementary reassignment formulas. We dene specic translations for the unabbreviated forms because they are equivalent to the translations for the abbreviated forms when the unique values and existence of values axioms are taken into account, and these translations will also be useful when we choose to use innite value domains. Proposition A.2 (Tran CRF ) Let = 8t8o:([](o)! [ 0 ](o)) be a conditional reassignment formula in L(SD). There is a transformation operator Tran CRF that transforms into a w = Tran CRF ( ) inl(fl). Proof Since [](o) is a logical formula, Lemma (A.1) shows that [](o) can be eectively transformed into a w in L(FL). Since [ 0 ](o) is a reassignment formula, Lemma (A.1) together with Denition A.31 shows that [ 0 ](o) can be eectively transformed into a w 0 in L(FL). Let = 8t8o:(! 0 ). isawinl(fl). 17

19 Proposition A.3 Vk Let= i=1 Tran ERF( i ) ^ Tran LF ( (o t)) be a formula in L(FL)whichistheresultof translating a restricted reassignment formula consisting of a conjunction V k i=1 i of elementary reassignment formulas together with a duration constraint (o t) inl(sd) into L(FL). can be transformed into the logically equivalent form, ; ^8t8z8 d:(t z d o t)! Occlude(t z): Proof Any Tran ERF ( i ) has the following form, ( _ Holds( 0 f i ( 1 ::: in d ij )))^ j2x i 8t8d i (<t 0! Occlude(t f( 1 ::: n d i ))) which can be transformed into the logically equivalent form, ( _ Holds( 0 f i ( 1 ::: n d ij )))^ j2x i 8t8z8d i (( <t 0 ^ z = f i ( 1 ::: n d i ))! Occlude(t z)): It is easily observed that V k i=1 Tran ERF( i ) has the following form 3, k^ _ ( Holds( 0 f i ( 1 ::: n d ij )))^ Let i=1 j2x i 8t8z( V k i=1 8d i)(( <t 0 ^ ( W k i=1 z = f i( 1 ::: n d i )))! Occlude(t z)): ;= k^ _ i=1 j2x i Holds( 0 f i ( 1 ::: n d ij ))) A ^ TranLF ( (o t)) where (o t) is the duration constraint for the restricted reassignment formula being transformed, and k_ (t z d o t) =((<t 0 ^ ( z = f i ( 1 ::: n d i ))): Remark A.7 Note that a restricted reassignment formula is used in the specication of an action law where the postconditions to an action in a particular context can be a disjunction of restricted reassignment formulas. The proposition above provides a very nice characterization of one alternative in a nondeterministic action. ; species the eects of the alternative, where we observe that even in a specic alternative there maybe new value ambiguity. (t z d o t) speci- es the sucient conditions for locally releasing the global inertia constraint built into the logic's minimization policy. Proposition A.4 Let = W m Vkl l=1 i=1 Tran ERF( i ) ^ Tran LF ( l (o t)) be a formula in L(FL) satisfying the restriction stated in Remark A.5, which is the result of translating a disjunction of restricted reassignment formulas (i.e. reassignment formula) in L(SD) into L(FL). can be transformed into the logically equivalent form, m_ ( ; l ) ^8t8z8d:(t z d o t)! Occlude(t z): l=1 V 3 Note that in the following, we will sometimes V abuse the notation and common meaning of the operator by k applying it to quantiers. For example, ( i=1 8d i) denotes 8d 1 ::: 8dk or equivalently 8d 1 ::: dk. 18 i=1 1

20 Proof By Proposition A.3, any two restricted reassignment formulas in have the following form: ; i ^8t8z8 d i : i (t z d i o i t)! Occlude(t z): ; j ^8t8z8 d j : j (t z d j o j t)! Occlude(t z): Based on Remark A.5, di = dj and i (t z di o i t) = j (t z dj o j t). It is easily observed that the reassignment formula has the following form, which is equivalent to m_ (; l ^8t8z8d:(t z d o t)! Occlude(t z)) l=1 m_ ( ; l ) ^8t8z8d:(t z d o t)! Occlude(t z): l=1 Remark A.8 Note that a reassignment formula is used in the specication of an action law where the postcondition to an action in a particular context is a reassignment formula. The proposition above provides a very nice characterization of the postcondition of a nondeterministic action. Each of the ; i species the eects of an alternative, while (t z d o t) species the sucient conditions for locally releasing the global inertia constraint built into the logic's minimization policy. In this case, (t z d o t) is the same for each alternative in an action with precondition. Proposition A.5 Let be a formula in L(FL) which is the result of transforming a conditional reassignment formula in L(SD) satisfying the restriction stated in Remark A.5 using the appropriate transformation operators. can be transformed into the logically equivalent form, ; ^8t8z8t8o:( 1 ^ (8 d:(t z d t o)))! Occlude(t z)): Proof A conditional reassignment formula has the following form, 8t8o:([](o)! [ 0 ](o)): After translating this into a formula in L(FL) and applying the transformation in Proposition A.4, it has the following form,! m_ 8t8o: 1! ( ; l ) ^8t8z8d:(t z d o t)! Occlude(t z) : l=1 where 1 = Tran LF ([](o)), and we assume that all variables, logical connectives, and delimiters have been translated appropriately. It is easily observed that this formula is equivalent to the following conjunction, 8t8o:[ 1! m_ ; l ]^ l=1 8t8z8t8o:[( 1 ^ (8 d:(t z d o t)))! Occlude(t z)]: Let m_ ;=8t8o:[ 1! ; l ]: l=1 19

21 Remark A.9 Proposition A.5 provides a nice characterization of one context in a context dependent action. 1 is the precondition, which if satised has the eect m_ ; l l=1 where W m l=1 ; l describes the eects of the action after it is executed, including the duration constraints. If m>1 then the action is nondeterministic in context 1 and has alternative eects. Lemma A.2 Let = V k i=1 Tran CR( i )beaformula in L(FL) satisfying the restriction stated in Remark A.5, which is the result of translating a conjunction V k i=1 i of conditional reassignment formulas in L(SD) into L(FL). can be transformed into the logically equivalent form, k^ k^ k_ ( ; j ) ^8t8z8t( 8o j 8d j ):[ ( j ^ j (t z d j t o j ))]! Occlude(t z)): Proof By Proposition A.5, any translation of a conditional reassignment formula can be transformed into the following form, ; j ^8t8z8t8o j :( j ^ (8 d j :(t z d j t o j )))! Occlude(t z)): A conjunction of such formulas has the form, k^ (; j ^8t8z8t8o j :( j ^ (8d j : j (t z d j t o j )))! Occlude(t z))): It is easily observed that this is equivalent to k^ k^ k_ ( ; j ) ^8t8z8t( 8o j 8d j ):[ ( j ^ j (t z d j t o j ))]! Occlude(t z)): Remark A.10 The result of instantiating an action law schema is a conjunction of conditional reassignment formulas, Proposition A.4 provides a nice modular characterization of an action law which separates the specication of the eects of the action for each context j, k^ ; j from the specication j (t z d j t o j ), of the sucient conditions for relaxing the inertia policy for the specied set of uents inuenced by the action in each context j, 8t8z8t( k^ k_ 8o j 8d j ):[ ( j ^ j (t z d j t o j ))]! Occlude(t z)): Any expanded action scenario description contains a nite set of instantiations of action law schemas. The set of instantiations is denoted by ; SCD. The next Lemma will show that a conjunction of schedule statements can be put into a form which will be useful when reducing the circumscription of ; SCD to the rst-order case. 20

22 Lemma A.3 Let = V m V kl l=1 i=1 Tran CR( i )beaformula in L(FL) satisfying the restriction stated in Remark A.5, which is the result of translating a conjunction V m V kl l=1 i=1 i of conjunctions of conditional reassignment formulas in L(SD) into L(FL). can be transformed into the logically equivalent form, ; ^8t8z:(! Occlude(t z)): Proof By Lemma A.2, a conjunction of conditional reassignment formulas (i.e. schedule statement) has the following form, k^ k^ k_ ( ; j ) ^8t8z8t( 8o j 8d j ):[ ( j ^ j (t z d j t o j ))]! Occlude(t z)): It is easily observed that a conjunction of such formulas has the following form, Let and m^ ( ^k l l=1 ; jl )^ (8t8z( V m l=1 8 t l ( V k l 8o j l 8 d jl )):[ W m l=1! Occlude(t z)): m^ ;=( ^k l l=1 W kl ( j l ^ jl (t z d jl t l o jl ))] ; jl ) m^ ^k l m_ =( 8t l ( 8o jl 8d _ k l jl )):[ ( jl ^ jl (t z d jl t l o jl ))]: l=1 l=1 Theorem A.1 Let = V m V kl l=1 i=1 Tran CR( i )beaformula in L(FL) satisfying the restriction stated in Remark A.5, which is the result of translating a conjunction V m V kl l=1 i=1 i of conjunctions of conditional reassignment formulas in L(SD) into L(FL). The circumscription of with Occlude minimized and all other predicates xed Circ SO ( Occlude), is equivalent to the following rst-order formula where and ; ^8t8z:( Occlude(t z)) m^ ;=( ^k l l=1 ; jl ) m^ ^k l m_ =( 8t l ( 8o jl 8d _ k l jl )):[ ( jl ^ jl (t z d jl t l o jl ))]: l=1 l=1 Proof By Lemma A.3, can be transformed into the form ; ^8t8z:(! Occlude(t z)). Theorem B.1 by Lifschitz allows us to factor out ;when circumscribing with Occlude minimized and all other predicates xed, since ;contains no occurrences of Occlude. Theorem B.2 by Lifschitz shows us that Circ SO (f8t z:(! Occlude(t z))g Occlude) ; ^8t z:( Occlude(t z)): The proof of Lemma A.3 can be used to show that;and have the values stated in the theorem. 21

23 A.3 Causal Constraint Statements in L(SD) Denition A.32 A causal constraint statement in L(SD) has the following form: 8t 1 8o:[Q o0 :(o 0 o)! ([t 1 ]Q o1 :(o 1 o) []Q o2 :(o 2 o))] where o 0 o 1 o 2 o are disjoint tuples of variables of sorts O, andq o1 Q o1 Q o2 are nite sequences of quantiers binding the variables o 0 o 1 o 2, respectively, and are quantier free uent formulas (FlF's), and is a quantier free logic formula where any elementary time-point expression 0 in is equal to a function g 0 of t 1, where g 0(t 1 ) t 1. In addition the temporal expression is a function g of t 1 where g (t 1 ) t 1. We will use the label cc to label causal constraint statements in action scenarios. Remark A.11 Observe that it is very dicult to provide a strict denition of a causal constraint statement. This denition allows for certain anomalies which certain temporal structures would rule as illegal. For example, in, one has to be careful that the interpretation of any temporal expression is greater than 0 if a linear discrete time structure with begin point is being used. In addition, it may be necessary to add additional constraints when dening the translation Tran CC (see below) which translates causal constraint statements to ws in L(FL). Currently, there are also a number of restrictions placed on the relation between the dierent time-points mentioned in a causal constraint statement which are most probably over restrictive. For example, it is sometimes useful to mention time-points greater than t 1 in the context. The restrictions placed onthecontext Q o0 :(o 0 o), can be relaxed without diculty so that a context is simply a logic formula in L(SD). A.3.1 Reducing Causal Constraint Statements in L(SD) to L(FL) Denition A.33(Tran EFE ) Let = f k (! 1 :::! ik )^=X be an elementary uent formula, where X = f 1 ::: n g, Dom (f k )is non-boolean, and f k has arity i k and sort Dom (fk 1):::Dom (f ik k )! Dom (f k ). Tran EFE () is dened as follows: Tran EFE () =ffk( 0 1 ::: ik d k )g where f 0 k ( 1 ::: ik d k ) is a parameterized term in L(FL), and d k is a variable of sort D k of L(FL), associated with the sort Dom (f k ). If Dom (f k ) is boolean, then Tran EFE () =ff 0 k( 1 ::: ik )g: If is a logic formula, we dene Tran EFE () asfollows: Tran EFE () = [ Tran EFE (f) f2 where is the set of elementary uent formula in. Denition A.34(Tran X ) Let =[]Q o2 :(o 2 o) be a logical formula (LF), where (o 2 o) isaquantier free uent formula. ^ Tran X () =Q X : Occlude( f i ( 1 ::: in d i )) where, f i(1 ::: in d i )2Tran EFE () 22

24 1. = Tran ETE () is the translation of the ETE into L(FL). 2. Q X is a sequence of universally quantied variables constructed from Q o2 by replacing each existential quantier in Q o2 with a universal quantier, renaming all variables o 2 and d, where d are the free variables of sorts D i in Tran EFE (), with new and unique variables not yet used in the formula of which is a part, and prexing 8 d to Q o2. Denition A.35(Tran CC ) Let = 8t 1 8o:[Q o0 :(o 0 o)! ([t 1 ]Q o1 :(o 1 o) []Q o2 :(o 2 o))]. Tran CC () = Tran LF (8t 1 8o:[Q o0 :(o 0 o)! ([t 1 ]Q o1 :(o 1 o)! []Q o2 :(o 2 o))]) ^ Tran LF (8t 1 8o:[(0 <t 1 ^ Q o0 :(o 0 o))! [([t 1 ; 1]:Q o1 :(o 1 o) ^ [t 1 ]Q o1 :(o 1 o))! Tran X ([]Q o2 :(o 2 o))]]) Remark A.12 Note the insertion of 0 <t 1 in the context of the Occlude formula. This is inserted to avoid anomalies at the initial point in the time-line. Another means of avoiding this problem would be to axiomatize change from t 1 to t 1 +1instead of t 1 ; 1 to t 1, but this would necessitate change in the temporal terms for the domain constraint. Semantically, there is no distinction between these two approaches. In the report, we will sometimes assume 0 <t 1 without actually stating it explicitly in the axioms. Lemma A.4 Let = 8t 1 8o:[(Q o0 :(o 0 o))! ([t 1 ]Q o1 :(o 1 o) []Q o2 :(o 2 o))], be a causal constraint statement inl(sd). There is an eective algorithm that transforms into a w in L(FL). Proof Since Tran X returns a formula in L(FL), it is easily observable that Tran CC ( ) is a boolean combination of logic formulas, possibly quantied, together with the reduced argument totran X. Application of Tran LF in Lemma A.1 completes the reduction. The following Lemmas and Propositions will prove to be useful when circumscribing theories containing causal constraints. They provide a basis for rst-order reductions of circumscriptive theories. Lemma A.5 Let ;= V k i=1 Tran CC( i )beaformula in L(FL)which is the result of translating a conjunction V k i=1 i of causal constraint statements in L(SD) into L(FL). The predicate Occlude only occurs positivelyin;. Proof Each Tran CC ( i ) has the form ; 1 ^ (8t 1 8o:(o)! Q X : ; 1 contains no occurrences of Occlude and ^ (8t 1 8o:(o)! Q X : can be rewritten equivalently as 8t 1 8oQ X :(o)! ^ f i(1 ::: in d i )2Tran EFE () f i(1 ::: in d i )2Tran EFE () ^ f i(1 ::: in d i )2Tran EFE () 23 Occlude( f i ( 1 ::: in d i )): Occlude( f i ( 1 ::: in d i )) Occlude( f i ( 1 ::: in d i )):