The situation and state calculus versus branching. temporal logic? Jaime Ramos, Amlcar Sernadas. Department of Mathematics, IST, Lisbon, Portugal

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1 The situation and state calculus versus branching temporal logic? Jaime Ramos, Amlcar Sernadas Department of Mathematics, IST, Lisbon, Portugal Abstract. The situation calculus (SC) is a formalism for reasoning abo action. Within SC, the notion of state of a given situation is usually characterized by the set of uents that hold in that situation. However, this concept is insucient for system specication. To overcome this limitation, an extension of SC is proposed, the situation and state calculus (SSC), where the concept of state is primitive, just like actions, situations and uents. SSC is then compared with a branching temporal logic (BTL). A representation of BTL in SSC is dened and shown to establish a sound and complete encoding. 1 Introduction The situation calculus (SC) is a specialization of many-sorted rst order logic with equality (MFOL). It was rst proposed in [8] as \a formalism for specifying dynamic systems". Through the years several renements and extensions have been proposed to allow SC to cope with problems like temporal reasoning, concurrency, actions with duration inter alia [6, 4, 13, 10, 11]. The main concepts of SC are situations, actions and uents. A situation represents the \state" of the system that we are specifying. The system changes from one situation to another when an action is performed. The properties of the system are characterized by uents, that may or may not hold at a given situation. Hence, there are three basic sorts in the language: sit for situations, act for actions, and t for uents (there might be other sorts, depending on the system we are specifying). There is a constant symbol of sort sit, S 0, denoting the initial situation. There is a function symbol, do, from actions and situations to situations. There are also two predicate symbols holds and poss. The predicate symbol holds is used to relate uents and situations, i.e., to dene when a uent is true (or false) at a given situation. The predicate symbol poss relates actions and situations, i.e., denes when an action is enabled at a given situation. So, we may write the following sentences: 8x; s poss(drop(x); s)! holds(onfloor(x); do(drop(x); s)) (1) 8x; s poss(drop(x); s) $ holds(onhand(x); s) (2)? We would like to thank our colleagues in the ASPIRE project, and also to Alberto Zanardo, Cristina Sernadas and Javier Pinto their valuable comments and suggestions at some stages of this work. This work was partially supported by the PRAXIS XXI Program and FCT, as well as by PRAXIS XXI Projects 2/2.1/MAT/262/94 SitCalc, PCEX/P/MAT/46/96 ACL plus 2/2.1/TIT/1658/95 LogComp, and ESPRIT IV Working Groups ASPIRE and FIREworks.

2 In the previous sentences, drop is an action symbol (with one parameter), while on- Hand and onfloor are uent symbols (with one parameter). The term do(drop(x),s) (of sort situation) denotes the situation resulting from doing action drop(x) in situation s. The sentence (1) can be understood as \if it is possible to drop an object x then, after dropping x, x will be on the oor". Sentence (2) can be regarded as meaning \an object may only be dropped when it is being held". In [13, 14] the situation calculus is axiomatized in a style similar to the Peano foundational axioms for number theory. The initial situation S 0 plays the role of the number 0. And instead of a single successor function, there is a family of successor functions do(a; ), one for each action term a. There is also a binary relation on situations,, where s s 0 stands for s 0 can be obtained from s by a sequence of possible actions. The axioms, called foundational axioms, are: 8a; s S 0 6= do(a; s) (3) 8a 1 ; a 2 ; s 1 ; s 2 (do(a 1 ; s 1 ) = do(a 2 ; s 2 )! (a 1 = a 2 ^ s 1 = s 2 )) (4) 8s (:s S 0 ) (5) 8a; s 1 ; s 2 ((s 1 do(a; s 2 )) $ (s 1 s 2 ^ poss(a; s 2 ))) (6) ' s S 0! ((8a; s ('! ' s do(a;s) ))! (8s ')) (7) where s s 0 stands for s s 0 _ s = s 0. These axioms establish an order on situations with a tree structure rooted at S 0. For a detailed discussion see [13, 14]. When this axiomatization was rst proposed, the main goal was to nd a monotonic solion to problems like the frame problem, the ramication problem, etc. It is not our intention to study these problems here. Instead, we are going to use this axiomatization with a dierent purpose. It is interesting to observe that we can interpret s s 0 as \the system reached s 0 after s", i.e., we can recognize a temporal component in the situations. This fact was already used in [11] to compare SC with linear temporal logic. Here, we propose a comparison of the SC with a branching temporal logic. However, using this axiomatization bears a price. In section 2, we discuss some problems that arise with this axiomatization of SC. Still in this section, we propose a solion to overcome these problems, the situation and state calculus (SSC). In section 3, SSC is compared with a branching temporal logic (BTL). This comparison is achieved via a map between the underlying logical structures [9] (see the appendix for a denition of logical structure and of map). In order to dene this map, we have to extend SSC with the ability to refer to the lines of the branching structure of situations. We end with a completeness result for this comparison. 2 The situation and state calculus In this section we discuss a problem that arises when we consider SC axiomatized with the foundational axioms. We propose a solion to overcome this problem, by extending SC with a new concept: the state. We call this extension the situation and state calculus (SSC). A preliminary version of SSC was presented in [12]. It is usual to dene the state of a system at a given situation as the set of uents that hold in that given situation. Thus, two situations have the same state when the

3 set of uents that hold is the same for both situations, i.e., samestate(s; s 0 ) $ (8fholds(f; s) $ holds(f; s 0 )) (8) If we adopt this notion of state we face one major problem: the non-determinacy of actions. Consider for instance a stack of natural numbers, with only one uent top of sort nat, one action push with one parameter of sort nat, and one action pop with no parameters. Let s be a (term of sort) situation, and let s 1 = do(push(5); s) and s 2 = do(push(4); s). Clearly, these situations do not have the same state, since the only uent that holds in s 1 is top(5) and in s 2 is top(4). Consider now the situations s 3 = do(push(3); s 1 ) and s 4 = do(push(3); s 2 ). In this case, according to (8), s 3 and s 4 have the same state since top(3) is the only (term of sort) uent that holds in both these situations. Finally, consider the situations s 5 = do(pop; s 3 ) and s 6 = do(pop; s 4 ). If the stack behaves as expected, we would have samestate(s 1 ; s 5 ) and samestate(s 2 ; s 6 ). B this means that we had two situations with the same state (s 3 and s 4 ), performed the same action (pop) on these two situations and got two situations (s 5 and s 6 ) that do not have the same state! Furthermore, we are unable to specify the stack, witho using some auxiliary uents. For instance, we are unable to express that \after a push and a pop we are back at the same state". One attempt would be to write 8n; s do(pop; do(push(n); s)) = s: (9) However, this formula clearly contradicts the foundational axioms. Obviously, we could drop some of the foundational axioms, thus making (9) a legal sentence. B as we said before, we want situations to have a temporal avor. So, the solion is to introduce the concept of state in the language, like situations, actions and uents. 2.1 Syntax We dene the syntax of SSC as a specialization of the syntax of MFOL. We start by dening the set of basic sorts, G bs, composed of the sorts sit for situations, act for actions, t for uents and stt for states. There might be other sorts (e.g. data-types). Denition1. An SSC signature is an MFOL signature = hg; F; P; Xi such that: (i) G bs G; (ii) F is such that F ;sit = fs 0 g; F act sit;sit = fdog; F sit;stt = f[ ]g; F w;s = ; if s is sit or stt, or if w contains an element of G bs ; (iii) P is such that P t stt = fholdsg; P act stt = fpossg; P sit sit = fg; P w = ; if w contains an element of G bs ; (iv) X is a G-indexed family of disjoint sets. Given an SSC signature hg; F; P; Xi, each element a 2 F w;act is said to be an action symbol with parameters sort w, and each element f 2 F w;t is said to a uent symbol with parameters sort w. In comparison with SC, there is a new function symbol [ ]. This function symbol associates a state to each situation. There is also a xed action symbol nil that is present for technical reasons, that will become clear later. Each element of X g is said to be a variable of sort g. We denote variables of sort sit by s; s 0 ; s 1 ; : : :, variables of sort act by a; a 0 ; a 1 ; : : :, variables of sort t by f; f 0 ; f 1 ; : : :, and variables of sort stt by u; u 0 ; u 1 ; : : :. Note that now the predicate symbols holds and poss depend on the states and not on the situations.

4 Example 1 Stack. We can dene the following signature for a stack of natural number: = hg; F; P; Xi where nat 2 G, F ;t = ftopg, F ;act = fnil; pop; resetg, F nat;act = fpushg. We dene signature morphisms in the usual way with the proviso that they must preserve the basic sorts. Signatures and signature morphisms constite a category, Sig ssc, which is a subcategory of Sig mfol. Let I : Sig ssc,! Sig mfol be the inclusion functor. The functor Sen ssc : Sig ssc! Set is dened by Sen mfol I. As in rst order logic, we call each element of Sen ssc () a -formula. We denote by T ;g the set of terms of sort g dened over. From now on we assume given an SSC signature. 2.2 Derivation relation In this section we introduce the derivation relation for SSC. We start by presenting the foundational axioms. The formula (6) has to be slightly changed because in SSC the predicate symbol poss depends on states and not on situations. Denition2. The set of foundational axioms for, Ax(F), contains the formulas: 1. 8a; s S 0 6= do(a; s); 2. 8a 1 ; a 2 ; s 1 ; s 2 (do(a 1 ; s 1 ) = do(a 2 ; s 2 )! (a 1 = a 2 ^ s 1 = s 2 )); 3. 8s (:s S 0 ); 4. 8a; s; s 0 (s do(a; s 0 ) $ (poss(a; [s 0 ]) ^ s s 0 )); 5. ' s S 0! ((8a; s ('! ' s do(a;s) ))! (8s ')) for every -formula '. Now we have to introduce axioms for the new objects in the language. We want to ensure that there are no junk states, i.e., a state is always the state of some situation. We also want to guarantee that we have determinacy of actions. Denition3. The set of state axioms for, Ax(S), contains the formulas: 1. 8a; s 1 ; s 2 ([s 1 ] = [s 2 ]! [do(a; s 1 )] = [do(a; s 2 )]); 2. 8u 9s u = [s]. These axioms characterize the states as \equivalence classes" of situations, i.e., a state is the (equivalence) class of all situations that have the same properties. These properties are more than just the set of uents that hold in those situations. Denition4. The set of axioms abo nil for, Ax(N ), contains the formulas: { 8u poss(nil; u) { 8s [do(nil; s)] = [s] Having introduced these axioms, we can dene the derivation relation for SSC. For each signature, the derivation relation `ssc will be dened based on the derivation relation for the same signature in MFOL, `mfol, adding the foundational axioms, the axioms abo nil and the state axioms as axioms of the deductive system. Denition5. Let? be a set of -formulas. We dene the set of derived formulas from? as follows:? `ssc = (? [ Ax(F) [ Ax(S) [ Ax(N ) )`fol : For each, we dene `ssc 2Senssc () Sen ssc () as follows:? `ssc `ssc ' i ' 2?.

5 With this notion of state, we are now in a position to write some formulas abo a stack, like for instance (9). Example 2 Stack. Consider the signature for the stack. We can write the following -formulas: 1. 8s [do(reset; s)] = [S 0 ] 2. 8s; n [do(pop; do(push(n); s))] = [s] 3. 8s; n; m holds(top(n); [do(push(m); s)])! n = m 4. 8u poss(pop; u) $ u 6= [S 0 ] 5. 8u; n; m (holds(top(n); u) ^ holds(top(m); u))! n = m If we look at an equational specication of stack (e.g. [2]), we can observe some similarities, although here we are specifying the behavior of the stack and not the stack as a data type. Within the stack example, consider two situations s 1 and s 2 such that [s 1 ] 6= [s 2 ]. And let s 3 = do(push(3); s 1 ), s 4 = do(push(3); s 2 ), s 5 = do(pop; s 3 ) and s 6 = do(pop; s 4 ), like before. Using axiom (2) of the stack, we derive [s 1 ] = [s 5 ] and [s 2 ] = [s 6 ]. Hence, since [s 1 ] 6= [s 2 ], then [s 5 ] 6= [s 6 ]. Using state axiom (1), we derive [s 3 ] 6= [s 4 ]. However, the only uent that holds in both states [s 3 ] and [s 4 ] is top(3). 2.3 Semantics In this section we introduce the semantics of SSC. In this case however, we cannot dene the SSC interpretation structures as we did for the sentences, i.e., we cannot dene Int ssc : Sig sscop! Cls as Int mfol I op. From these interpretation structures, for each signature, we choose the ones that satisfy all the foundational axioms for, Ax(F), all the state axioms for, Ax(S), and all the axioms abo nil, Ax(N ). Denition6. We dene Int ssc () as the class of all interpretation structures I 2 Int mfol I op () such that: { I satises all the foundational axioms for, Ax(F) ; { I satises all the state axioms for, Ax(S) ; { I satises all the axioms abo nil for, Ax(N ). For each signature morphism :! 0, Int ssc () is the restriction of Int mfol () to Int ssc ( 0 ). Given an SSC interpretation structure for, I = hd; F ; P i, we denote by o F the interpretation of the operation symbol o in I, and by p P the interpretation of the predicate symbol p in I (omitting any explicit reference to the arity of the symbols). It is easy to check that Int ssc is still a functor. In what follows, we are going to need the notion of standard interpretation structure. This concept is closely related with the notion of standard interpretation structure in number theory. Denition 7. An SSC interpretation structure I = hd; interpretation structure when: F ; P i is called a standard

6 { D sit is inductively dened as follows: S 0 2 D sit do(a; s) 2 D sit, provided that s 2 D sit and a 2 D act ; { S F 0 = S 0 { do F = as:do(a; s) { P is the transitive closure of f(s; do F (a; s)) : s 2 D sit ; a 2 D act ; poss P (a; [s] F )g Any interpretation structure isomorphic to a standard interpretation structure will also be called standard. 2.4 Satisfaction relation Like for the derivation relation, for each SSC signature, we dene the satisfaction relation for SSC based on the satisfaction relation for MFOL. Denition8. We dene the satisfaction relation j= ssc Intssc () Sen ssc () as I j= ssc ' i I j=mfol ' 2.5 Logical structure Based on the previous denitions we can p together the logical structure for SSC. Proposition9. The tuple L ssc = hsig ssc ; Sen ssc ; Int ssc ; j= ssc ; `ssc i is a logical structure. This logical structure is obviously sound, i.e., if? `ssc ' then? j=ssc ', and is also complete, i.e., if? j= ssc ' then? `ssc '. The proof of this results as well as the fact that L ssc is indeed a logical structure is straightforward. 3 SSC versus BTL In this section we compare the situation and state calculus with a branching temporal logic (BTL). This comparison will be established via a map between the underlying logical structures [9]. In order to dene this map we have to establish a relation between the two logical structures at the syntactical level (signatures and formulas), and at the semantic level (interpretation structures). This is closely related to van Benthem's Correspondence Theory, which studies connections between modal and classical logics [18]. We start by presenting the logical structure for the intended branching temporal logic. Then we extended SSC with the ability to refer to the lines of the tree of situations as objects of the language. And nally we dene a map between the two logical structures and study some of its properties.

7 3.1 Branching Temporal Logic In this section we present a logical structure for BTL. All the denitions and results are taken from [17]. A BTL signature is just a set of propositional symbols. Hence, Sig btl is Set. For each BTL signature the set of -formulas, L, is inductively dened as follows: { p 2 L provided that p 2 ; { (:') 2 L provided that ' 2 L ; { ('! ) 2 L provided that '; 2 L ; { (' U ) 2 L provided that '; 2 L ; { (A') 2 L provided that ' 2 L ; Then, the functor Sen btl can be dened in the usual way, i.e., for each signature, Sen btl () = L, and for each signature morphism :! 0, Sen btl () is dened by Sen btl ()(p) = (p) for all p 2, and is structural for all the other formulas. We assume as given the usual abbreviations: the logic connectives _, ^, $, the temporal operators X (next), F (sometime in the fure), G (always in the fure), and the path operator E (exists a path). 2 An interpretation structure for a given a signature is a tuple M = hht; P i; V i where: { T = hw; Ri is a total transition system, i.e., for every w 2 W there is w 0 2 W such that (w; w 0 ) 2 R; { P is a non-empty set of innite paths (also called runs) over T such that every w 2 W occurs in some path 2 P (where a path is a map : IN o! W such that ((n); (n + 1)) 2 R); { V :! 2 W. We denote by (n) the n-th position of the path. The functor Int btl : Sig btlop! Cls is such that, for each signature, Int btl () is the class of all interpretation structures for, and for each signature morphism :! 0, Int btl () : Int btl ( 0 )! Int btl () is dened by Int btl ()(hht 0 ; P 0 i; V 0 i) = hht 0 ; P 0 i; V i where V (p) = V 0 ((p)) for every p 2. The satisfaction of a formula by a given interpretation structure M = hht; P i; V i at index n 2 IN o of path 2 P is inductively dened as follows: { M; ; n j= btl p i (n) 2 V (p) { M; ; n j= btl (:') i M; ; n 6j=btl ' { M; ; n j= btl ('! ) i M; ; n 6j=btl ' or M; ; n j=btl { M; ; n j= btl (' U ) i there is m > n such that M; ; m j=btl and for all k such that n < k < m, M; ; k j= btl ' { M; ; n j= btl (A') i for all 0 2 P such that (n) = 0 (n), M; 0 ; n j= btl ' A formula ' is M-true, M j= btl ', if for all and i we have M; ; i j=btl '. An axiomatization for this logic is proposed in [17]. We are not going to detail it here, b rather use the fact that such an axiomatization exists and denote by 2 The operators A and E are sometimes represented by 8 and 9. However, we adopted A and E to avoid confusion with the rst order quantiers.

8 ? `btl ' the fact that the formula ' can be derived from? using the proposed consequence relation. So, we can dene a logical structure for BTL: L btl = hsig btl ; Sen btl ; Int btl ; j= btl ; `btl i This logical structure is sound and (weakly) complete [17]. In what follows we are going to need some notation and technical results. Let be a path over a transition system and n 2 IN o. We denote by n the suf- x of starting at n, i.e., satisfying the condition n (k) = (n + k), for every k 2 IN o. Let M = hhw; R; P i; V i be a BTL interpretation structure (for a given signature ), and consider 2 P and n 2 IN o. Dene the set P ;n = f n : 2 P and (n) = (n)g, and using this set, dene the BTL interpretation structure M ;n = hhw ;n ; R ;n ; P ;n i; V ;n i such that: { W ;n = fw 2 W : w occurs in some 2 P ;n g { R ;n = R \ (W ;n W ;n ) { V ;n (p) = V (p) \ W ;n. Then, M; ; n j= btl ' i M ;n ; n ; 0 j= btl Generation Theorem [18]. '. This result is a particular case of the 3.2 Branching situation and state calculus Like BTL, SSC has an underlying branching structure. The main idea when comparing these two logics is to use the temporal component of situations as time instants, and the lines over the tree of situations as the paths of BTL. However, to do this we need to be able to refer to lines as objects of the language. So, we need to extend SSC with the ability to refer to lines. We call this extension, branching situation and state calculus (BSSC). The rst step is to p the lines in the language. So we introduce a new sort, lin, for lines. Furthermore, we need to be able to relate lines with situations, i.e., we need to be able to express which situations occur in a line. For this, we introduce a new predicate symbol in, with two arguments, a situation and a line. Denition10. A BSSC signature is an SSC signature = hg; F; P; Xi such that (i) lin 2 G; (ii) P sit lin = fing; (iii) F w;s = ;, P w = ; if w contains lin or s = lin. A BSSC signature morphism is an SSC signature morphism such that G (lin) = lin. BSSC signatures and morphisms constite a category Sig bssc. Again, there is an inclusion functor from Sig bssc into Sig ssc, J : Sig bssc,! Sig ssc, and so the functor Sen bssc : Sig bssc! Set is dened by Sen ssc J. From now on we assume that denotes a BSSC signature. Having dened the language, we need to characterize lines in terms of situations. A similar idea was proposed in [11] to compare SC with linear temporal logic, where a predicate, actual, is dened to select a single line from the tree of situations. Denition11. The set of line axioms for, Ax(L), contains the formulas: 1. 8l in(s 0 ; l);

9 2. 8l; a; s in(do(a; s); l)! (in(s; l) ^ poss(a; [s])); 3. 8l; a 1 ; a 2 ; s (in(do(a 1 ; s); l) ^ in(do(a 2 ; s); l))! a 1 = a 2 ; 4. 8l; s in(s; l)! (9a in(do(a; s); l)); 5. 8s S 0 s! (9l in(s; l)) The rst axiom expresses the fact that every line passes through the initial situation. The second axiom expresses the fact that there are no gaps in a line. The third and fourth axioms impose that at each point of the line there is exactly one successor. The last axiom expresses that there are enough lines, i.e., there is at least one line passing through each (reachable) situation. These axioms are true in every interpretation structure whose domain for sort lin (D lin ) is a set of paths (over the tree of situations) fullling suitable closure conditions; in the literature on branching temporal logic these sets of paths are called bundles [1]. In [19, 20] it is shown that there is no rst order formula that is true in (exactly) all the interpretation structures in which D lin is the set of all paths. We dene the derivation relation for BSSC, by adding the line axioms to the derivation relation of SSC. Denition12. Let? be a set of -formulas. We dene the set of derived formulas from? as follows:? `bssc = (? [ Ax(L) )`ssc : We dene `bssc 2Senbssc () Sen bssc () as follows:? `bssc `bssc ' i ' 2? Denition13. We dene Int bssc () as the class of all SSC interpretation structures I 2 Int ssc J that satisfy all the line axioms for, Ax(L). For each signature morphism :! 0, Int bssc () is the restriction of Int ssc () to Int bssc ( 0 ). The (family of) satisfaction relation(s), j= bssc, is dened as for SSC. Proposition14. The tuple L bssc = hsig bssc ; Sen bssc ; Int bssc ; `bssc ; j= bssc i is a logical structure. In what follows we consider only standard interpretation structures, i.e., from now on, Int bssc () denotes the class of all standard interpretation structures. These are dened as for SSC. 3.3 A map between BTL and BSSC We are now going to dene a map between L btl and L bssc. The rst step is to translate signatures. For this, we dene a functor between Sig btl and Sig bssc. We translate each propositional symbol to a uent (with no parameters). Denition15. We dene the functor : Sig btl! Sig bssc as follows: { for each 2 jsig btl j, () = hg; F; P; Xi is such that F bssc ;t = ; { for each :! 0 in Sig btl, () is such that ()(p) = (p), for every p 2.

10 The next step is to translate formulas. For this we have to dene a natural transformation from Sen btl to Sen bssc. B rst we need to dene an auxiliary function, for each signature. The motivation for this function, is that it translates each BTL formula to a BSSC formula at a given line and a given situation, following the semantics of BTL formulas. Denition16. The function : Sen btl ()T ();lin T ();sit! Sen bssc (()) is inductively dened by: { (p; l; s) = holds(p; [s]) { (:'; l; s) = : ('; l; s) { ('! ; l; s) = ('; l; s)! ( ; l; s) { ('U ; l; s) = 9s 0 s s 0 ^in(s 0 ; l)^ ( ; l; s 0 )^8s 00 s s 00 s 0! ('; l; s 00 ) { ((A'); l; s) = 8l 0 in(s; l 0 )! ('; l 0 ; s). Having dened this function, we can now dene the natural transformation. Again, following the semantics of BTL, a formula is satised by an interpretation structure i it is satised at all points of all lines. Proposition17. We dene the natural transformation : Sen btl! Sen bssc as follows: (') = 8l; s (in(s; l)! ('; l; s)) Proof. To prove that is indeed a natural transformation from Sen btl to Sen bssc it is sucient to show that, for every signature morphism :! 0, -formula ', l 2 T ;lin, and s 2 T ;sit the following condition holds: (Sen bssc )()( ('; l; s)) = 0(Sen btl ()('); l; s): This can be easily proven by induction on the structure of '. To prove that we are dening a map we need to show that preserves theorems. Lemma 18. Let f'g [? Sen btl (). Then? `btl ' implies (? ) `bssc ('). Proof. It is enough to show that the translation of each of the BTL axioms is a BSSC theorem and that the rules are preserved by. This is straightforward. Finally, we dene the natural transformation, for translating BSSC interpretation structures into BTL interpretation structures. For each signature, assigns to each BSSC interpretation structure a BTL interpretation structure over the corresponding signature. We dene the transition system taking as states the set of reachable situations and as transition relation the immediate transition relation induced by the do over these situations. The set of paths is dened in the usual way, choosing only the ones that correspond to lines. Proposition19. We dene the natural transformation : Int bssc op! Int btl as follows: for each I 2 Int bssc (()), (I) = hhw; R; P i; V i where { W = fs 2 D sit : S F 0 P sg { R = f(s; s 0 ) : s 0 = do F (a; s); for some a 2 D act such that poss P (a; [s] F )g

11 { P = : IN o! W (0) = SF 0 ; ((i); (i + 1)) 2 R; and there is d l 2 D lin such that in P ((i); d l ); for every i 2 IN o { V (p) = fs 2 W : holds P (p; [s] F )g Proof. We have to show that for each and I, (I) is in fact a BTL interpretation structure for. B this an immediate consequence of I being a BSSC interpretation structure which ensures that the transition system is total (due to the axioms abo nil) and P is in fact a set of paths over T satisfying the condition on paths (due to the line axioms). We also have to show that is a natural transformation from Int bssc op to Int btl. It is enough to show that, for each morphism :! 0 and BSSC interpretation structure, I 0, (for ( 0 )) the following condition holds: (Int bssc op ()(I 0 )) = Int btl ()( 0(I 0 )): Having established this result, we now prove the second condition in the denition of map between logical structures. B before we have to prove some technical results. Let us start by introducing some notation. Given a BSSC interpretation structure I (and with (I) = hht; P i; V i), we dene the relation : = P D lin by : = d l i in P ((n); d l ), for every n 2 IN o. Lemma 20. Let I be a BSSC interpretation structure and d s 2 D sit. Then, there are n 2 IN o and d a1 ; : : : ; d an 2 D act such that d s = do F (d a1 : : : d an ; S F 0 ). Proof. A simple induction on the structure of d s, taking into account that I is a standard interpretation structure, and satises the line axiom (4). Lemma 21. Let I be a BSSC interpretation structure (with (I) = hht; P i; V i) and d l 2 D lin. Then, there is 2 P such that : = d l. Proof. Take to be the following path: { (0) = S F 0 { (n + 1) = do F (d a ; (n)), for every n 2 IN o, where d a 2 D act results from I satisfying line axiom (4) (take the assignment v such that v(s) = (n) and v(l) = d l ). Then, prove by induction on the length of that : = d l. Lemma 22. Let ' be a -formula, I a BSSC interpretation structure for (), 2 P, n 2 IN o, and v an assignment such that : = v(l) and v(s) = (n). Then, the following condition holds: (I); ; n j= btl ' i I; v j=bssc () ('; l; s) Proof. We prove this by induction on the structure of '. We just sketch the proof for the base. Let p 2. Then (I); ; n j= btl p i (n) 2 V (p) i holdsp (p; [(n)] F ) i I; v j= bssc () holds(p; [s]) i I; v j=bssc () (p; l; s). The cases of the logical connectives are an immediate consequence of the induction hypothesis. The proof for the temporal operator and for the path operator are a consequence of the induction hypothesis and lemmas 20 and 21, respectively.

12 Having all these results, we can now prove the second condition in the denition of map between logical structures. Lemma 23. Let ' 2 Sen btl () and I 2 Int bssc (()). Then, (I) j= btl ' i I j= bssc () ('). Proof. It follows from the previous lemmas. Proposition24. The tuple h; ; i is a map between L btl and L bssc. Proof. Lemmas 17 and 19 prove that and are natural transformations. Lemmas 18 and 23 prove that these natural transformations satisfy the two conditions in the denition of map. With this result we can represent the considered BTL in BSSC. So, we can adopt BSSC as a semantics for BTL, knowing that the consequence relation is sound w.r.t. this semantics, i.e., all the temporal theorems can be proved in BSSC. Then, the natural question of completeness arises: is BTL complete w.r.t. to BSSC semantics? The answer to this question is given below. B, rst we need some auxiliary results. Lemma 25. Let ' be a -formula. Then, j= bssc () (') implies j= btl '. Proof. Assume that 6j= btl '. Then, there is a BTL interpretation structure for, M = hhw; R; P i; V i, such that M 6j= btl ', i.e., there are 2 P and n 2 IN o such that M; ; n 6j= btl '. Consider M ;n as dened above. Let us dene the BSSC interpretation structure I M = hd; F ; P i satisfying the following conditions: { D act = R ;n { D sit is inductively dened as follows: (n) 2 D sit w w 0 2 D sit provided that w 2 D sit and (w 00 ; w 0 ) 2 R ;n for some w 00 { D lin = P ;n { D stt = W ;n { S F 0 = (n) { do F (w 0 ; w 00 ; w) = w w 00 { [w 0 : : : w k ] F = w k { poss P ((w; w 0 ); w 00 ) i w = w 00 { holds P (p; w) i w 2 V (p) { s P s 0 i there are w 1 ; : : : ; w k such that s 0 = s w 1 : : : w k, for some k 2 IN; { in P (w; ) i w is a is a prex of In I M, situations are sequences of states, states are just the states (of M ;n ), actions are the pairs in R ;n, and lines are the paths. The initial situation is (n) because all the paths in P ;n start at this state. We need to generate D sit in order to have a standard interpretation structure. It is intuitive to conclude that the state of a situation is the last state of the sequence, and that an action is only enabled in a state if the rst component of the action is exactly that state. Having dened this interpretation structure, we have to prove that this is in fact a BSSC (standard) interpretation structure. It satises all the foundational axioms, all the state axioms and all the line axioms, due to the construction and the properties of BTL interpretation structures.

13 Lemma 26. Let 2 P ;n, k 2 IN o and let v be an assignment such that v(l) = and v(s) = h(0) : : : (k)i. Then, the following condition holds: M ;n ; ; k j= btl ' i IM ; v j= bssc () ('; l; s): Proof. The proof is similar to the proof of lemma 22. Using the fact that M ;n ; n ; 0 6j= btl ', we conclude that IM ; v 6j= bssc () ('; l; s), for an assignment v such that v(s) = (n) and v(l) = n. Hence, I M 6j= bssc () ('). Which proves that 6j= bssc () ('). Using this result, we can now answer the question of whether BTL is complete w.r.t. to BSSC semantics. This is equivalent to showing that the map is (weakly) conservative. Proposition27. The map h; ; i is weakly conservative, i.e., `bssc () (') implies `btl '. Proof. Assume that 6`btl '. Since BTL is (weakly) complete, we have 6j=btl '. Using lemma 25 we have 6j= bssc () ('). And nally, using the soundness of BSSC, we may conclude that 6`bssc () ('). With this result, we may conclude that BTL is (weakly) complete w.r.t. BSSC semantics. The proof of this result depends on the existence of a complete axiomatization for BTL. So, this technique may be used for other temporal logics for which there are complete axiomatizations. 4 Concluding remarks We started from the situation calculus with an axiomatization proposed by Reiter. The main reason for adopting this axiomatization is that it enriches situations with a temporal component. B, with this axiomatization we loose the ability to go back to a previous situation, i.e., we loose the notion of situation as state that was present in the earlier versions of SC. Some ahors try to solve this problem by dening a state as a set of uents that hold in a given situation. However, we showed that this notion is not enough to specify some systems. In order to solve this problem, we proposed a new extension of SC, the situation and state calculus (SSC). We also compared SSC with a branching temporal logic (BTL). We encoded BTL in an extended version of SSC (BSSC), and showed that this coding is sound and complete, i.e., all the theorems of BTL can be translated into BSSC theorems. Furthermore, we proved that we gain no extra theorems. We have similar results for linear temporal logic, that extend [11] (we detail the translation of interpretation structures and prove similar completeness result). The temporal logic considered here is not full CTL, b a weaker version (where the interpretation structures are transition systems with a distinguished set of paths). However, it is important to stress o that we considered BTL instead of CTL not because of limitations in BSSC b because the conservativeness result

14 we wanted required a complete axiomatization, which is still an open problem for full CTL. If such an axiomatization is found, and if it is possible to dene the map between the underlying logical structures, then this map will be conservative. The next step is to use these codications of temporal logics into SSC and import reasoning techniques and tools that already exist for temporal logic like for instance to prove safety properties, liveness properties, etc [7]. Furthermore, we also want to extend SSC to an n-agent SSC, where it possible to specify a community of interacting agents, in the line of what is done in [15, 16] for temporal logic. References 1. J. P. Burgess. Logic and time. Journal of Symbolic Logic, 44:566{582, H. Ehrig and B. Mahr. Fundamentals of Algebraic Specication 1: Initial Semantics, volume 6 of EATCS Monographs on Theoretical Comper Science. Springer-Verlag, New York, N.Y., J. Fiadeiro and A. Sernadas. Structuring theories on consequence. In D. Sannella and A. Tarlecki, editors, Proceedings of the 5th Workshop on Recent Trends in Data Type Specication, volume 332 of LNCS, pages 44{72, Berlin, Springer. 4. M. Gelfond, V. Lifschitz, and A. Rabinov. What are the limitations of the situation calculus? In Robert Boyer, editor, Aomated Reasoning: Essays in Honor of Woody Bledsoe, pages 167{179. Kluwer Academic, Dordrecht, J. Goguen and R. M. Burstall. Introducing institions. In E. Clarke and D. Kozen, editors, Logics of Programs: Workshop, Carnegie Mellon University, June 1983, volume 164 of LNCS, pages 221{256. Springer-Verlag, New York, N.Y., V. Lifschitz and A. Rabinov. Miracles in formal theories of action. Articial Intelligence, 38:225{237, Z. Manna and A. Pnueli. Completing the temporal picture. Theoretical Comper Science, 93:97{130, J. McCarthy and P. Hayes. Some philosophical problems from the standpoint of arti- cial intelligence. In B. Meltzer and D. Michie, editors, Machine Intelligence 4, pages 463{502. Edinburgh University Press, Scotland, J. Meseguer. General logics. In H.-D. Ebbinghaus et al, editor, Proc. Logic Colloquium'87. North-Holland, R. Miller and M. Shanahan. Narratives in the situation calculus. Journal of Logic and Compation { Special Issue on Actions and Processes, 4(5):513{530, J. Pinto and R. Reiter. Reasoning abo time in the situation calculus. Annals of Mathematics and Articial Intelligence: Papers in Honour of Jack Minker, 14(2-4):251{ 268, J. Ramos. The situation and state calculus. In A. Drewery, G.-J. Kruij, and R. Zuber, editors, Proceedings of the Second ESSLLI Student Session, R. Reiter. The frame problem in the situation calculus: a simple solion (sometimes) and a completeness result for goal regression. In V. Lifschitz, editor, Articial Intelligence and Mathematical Theory of Compation: Papers in Honor of John McCarthy, pages 359{380. Academic Press, San Diego, CA, R. Reiter. Proving properties of states in the situation calculus. Articial Intelligence, 64:337{351, A. Sernadas, C. Sernadas, and J. Costa. Object specication logic. Journal of Logic and Compation, 1:7{25, A. Sernadas, C. Sernadas, and J. Ramos. A temporal logic approach to object certication. Data and Knowledge Engineering, 19:267{294, 1996.

15 17. C. Stirling. Modal and temporal logics. In S. Abramsky, D. Gabbay, and T. Maibaum, editors, Handbook of Logic in Comper Science. Volume 2. Background: Compational Structures, pages 477{563. Oxford University Press, J. van Benthem. Correspondence theory. In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, Volume II: Extensions of Classical Logic, volume 165 of Synthese Library, chapter II.4, pages 167{247. D. Reidel Publ. Co., Dordrecht, A. Zanardo. Branching-time logic with quantication over branches: The point of view of modal logic. Journal of Symbolic Logic, 61(1):1{39, A. Zanardo, B. Barcellan, and M. Reynolds. Non-denability of the class of complete bundled trees To appear. A Logical structures The concept of logical structure is proposed in [9] (where it is called logic). Other related concepts are institions [5] and -institions [3]. Denition28. A logical structure is a tuple hsig; Sen; Int; j=; `i such that: { Sig is a category; { Sen : Sig! Set is a functor; { Int : Sig op! Cls is a functor; { j= is a jsigj-indexed family of relations fj= Int() Sen()g 2jSigj ; { ` is a jsigj-indexed family of relations f` 2 Sen() Sen()g 2jSigj ; satisfying the following conditions: { for each 2 jsigj: f'g ` ', for ' 2 Sen();? ` ', provided that? 0 ` ' and? 0?, for ' 2 Sen(),?;? 0 Sen();? ` ', provided that? ` for each 2? 0 and? 0 ` ', for ' 2 Sen(),?;? 0 Sen(); { for each morphism :! 0 in Sig: if? ` ' then Sen()(? ) 0 ` Sen()('), for ' 2 Sen() and? Sen(); I 0 j= 0 Sen()(') i Int()(I 0 ) j= ', for ' 2 Sen() and I 0 2 Int( 0 ). A logical structure is sound if? ` ' implies? j= '. A logical structure is complete if? j= ' implies? ` '. Denition29. Given two logical structures L and L 0, a map between L and L 0 is a triple h; ; i such that: { : Sig! Sig 0 is a functor; { : Sen! Sen 0 is a natural transformation; { : Int 0 op! Int is a natural transformation; satisfying the following conditions: { if? ` ' then (? ) `0() ('); { (I 0 ) j= ' i I 0 j= 0 () ('). A map between logic structures is said to be conservative i (? ) `0() (') implies? ` ': This article was processed using the LaT E X macro package with LLNCS style