A Foundation for Metareasoning. Part

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1 istituto per la ricerca scientifica e tecnologica Povo (Trento), Italy Tel.: Fax: prdocitc.it url: A Foundation for Metareasoning. Part II: The model theory G. Criscuolo, F. Giunchiglia, L. Serani October 1997 Technical Report # c Istituto Trentino di Cultura, 1997 limited distribution notice This report has been submitted for publication outside of ITC and will probably be copyrighted if accepted for publication. It has been issued as a Technical Report for early dissemination of its contents. In view of thetransfer of copyright to the outside publisher, its distribution outside of ITC prior to publication should be limited to peer communications and specic requests. After outside publication, requests should be lled only by reprints or legally obtained copies of the article.

2 A Foundation for Metareasoning. Part II: The model theory Giovanni Criscuolo (1) Fausto Giunchiglia (2)(3) Luciano Serani (2) (1) ipartimento Scienze Fisiche Mostra d'oltremare pad.19, Napoli, Italy (2) IRST, Istituto Trentino di Cultura Povo, Trento, Italy (3) ISA, University of Trento Trento, Italy Abstract OM pairs are our proposed framework for the formalization of metareasoning. OM pairs allow us to generate deductively the object theory and/or the meta theory. This is done by imposing, via appropriate reection rules, the relation we want to hold between the object theory and the metatheory. In a previous paper we have studied the proof theoretic properties of OM pairs. In this paper we study their model theoretic properties, in particular the relation between the models of the meta theory and the object theory; and use these results to rene the previous analysis. Key words: Foundations of meta reasoning, reection principles, provability and truth, model theory Submitted to The Journal of Logic and Computation 1

3 1 OM pairs A lot of work on metatheoretic reasoning can be found in the literature. As far as we know, all of this work is aimed at dening and studying the properties of specic metatheories. Nobody has ever developed a theory which would allow for a uniform study and comparison of the various metatheories which have been or can be dened. Our overall goal is to make a rst step in this direction. This paper follows the companion paper [2]. [2] provides a general introduction and an explanation of the ideas and motivations underlying our approach. We do not replicate this part in this paper. [2] also introduces, motivates and proposes OM pairs as our proposed framework for the formalization of metareasoning. A basic understanding of OM pairs is necessary for an understanding of the results presented in this paper. Therefore, we provide below the basic denitions, notation, and results. We start from two arbitrary theories O and M, presented as axiomatic formal systems, i.e. O = hlo ; O ; O i and M = hlm ; M ; M i. O is called the object theory, M is called the metatheory; LO (LM) is the language of O (M), O ( M ) is the set of axioms of O (M), and O ( M ) is the deductive machinery (set of inference rules) of O (M). We suppose that O are M are complete for classical propositional logic, and are as dened, e.g., in [5]. `O and `M are the derivability relations dened by O and M, respectively; TH(O)LO and TH(M)LM are the set of theorems provable from O by applying the rules in O, and from M by applying the rules in M, respectively. We write the formulas of the metatheory, also called metaformulas, using the following notation: we write names by surrounding the corresponding element with double quotes (thus, for instance, \A" is the name of the sentential constant A); and have a distinguished unary predicate, \", which, intuitively, represents in the metatheory a corresponding object level property. We make the following simplifying hypotheses: LM contains a single unary metapredicate ; there is a one-toone mapping between names and object level formulas; we only consider the propositional case, we therefore restrict ourselves to the case where LO and LM are propositional and LM is a propositional metalanguage of LO, where a propositional metalanguage is dened as follows: enition 1.1 (Propositional Metalanguage) Given a logical language L, its propositional metalanguage is the propositional language with the set f(\a") : A 2 Lg of atomic ws. We consider a new kind of inter-theory inference rules, called reection rules, whose premises and conclusions belong to dierent languages. We focus on the following reection rules: A (\A") A (\A") Rup :A : (\A") Rupn r :A : (\A") Rupn (\A") A r (\A") A : (\A") :A n r : (\A") :A n Restrictions: A 2 LO, (\A") 2 LM; rules labeled with index r are applicable if and only if the premise does not depend on any assumptions in the same theory. The ules are called reection up rules, the rules, reection down rules. The rules with the r are called restricted, the others unrestricted; the rules with the n are called negative, the others positive. Finally, we dene OM pairs as follows. enition 1.2 (OM pair) An Object-Meta Pair (OM pair) OM is a triple ho; M; (RR)i where O = hlo ; O ; O i, M = hlm ; M ; M i and (RR) is a set of reection rules. Given an object theory O, a metatheory M, and a set of reection rules (RR), we say that OM = ho; M; (RR)i is the OM pair composed of O and M connected by (RR). Notationally, when it 2

4 = J JJ J J^ O R R M Figure 1: Relations among orders. contains more than one bridge rule, we represent (RR) by listing its elements separated by a +. One example of (RR) is Rup + n. An Object-Meta pair OM denes a derivability relation between multiple languages which is a generalization of derivability relation dened in terms of Natural eduction [5]. The key observation is that the inference rules in O ( M ) are restricted to be applicable only to occurrences of formulas which belong to LO (LM). That is, formulas in a deduction are implicitly labeled by the language they belong to, and inference rules are applicable only to formulas with the \correct" label. The context always makes clear the language a formula belongs to. Formally, a deduction of a w A 2 LO [ LM from a set of assumptions LO [ LM is dened in the usual Natural eduction style. A is derivable from a set of assumptions in OM, abbreviated as `OM A, if and only if there is a deduction in OM of A from. A is provable in (a theorem of ) OM, abbreviated as `OM A, if and only if it is derivable from the empty set. TH OM (O)LO is the set of formulas of LO provable in OM. TH OM (M)LM is the set of formulas of LM provable in OM. Trivially TH(O)TH OM (O) and TH(M)TH OM (M). Given an OM pair OM = ho; M; (RR)i we use the term object theory of OM (generated by (RR)) to denote any theory O 0 = hlo ; O 0 ; O i whose set of theorems is TH OM (O) and the term metatheory of OM (generated by (RR)) to denote any theory M 0 = hlm ; M 0; M i whose set of theorems is TH OM (M). Trivially TH(O')= TH OM (O) and TH(M')=TH OM (M). OM pairs can be (partially) ordered according to the provability and derivability relations of the theories they are composed of. enition 1.3 For any two OM pairs OM 1 and OM 2, OM 1 OM 2 (resp. OM 1 O OM 2, OM 1 M OM 2 ) if and only if, `OM1 `OM2 (resp. TH OM1 (O) TH OM2 (O), TH OM1 (M)TH OM2 (M)). Furthermore OM 1 = OM 2 (resp. OM 1 OM 2, OM 1 OM 2 ) if and only if, `OM1 =`OM2 (resp. TH OM1 (O) = TH OM2 (O), TH OM1 (M) = TH OM2 (M)). enition 1.4 For any two combinations of reection rules (RR) 1 and RR2, RR1X(RR) 2 with X 2 f O ; M ; ; ; ; = g if and only if, for any object theory O and any metatheory M, ho; M; (RR) 1 ixho; M; (RR) 2 i. If (RR) 1 = (RR) 2 we say that (RR) 1 and (RR) 2 are equivalent. The three orders dened above give three criteria for comparing sets of reection rules. On the other hand, these orders are not independent of one another. For instance if two reection rules generate the same derivability relation, then they also generate the same object theory and the same metatheory. The relations among orders are summarized in Figure 1. In this gure, an arrow connects X to Y, if and only if (RR) 1 X(RR) 2 implies (RR) 1 Y (RR) 2. Thus, for instance, Figure 1 states that, if (RR) 1 (RR) 2, then (RR) 1 O (RR) 2 and (RR) 1 M (RR) 2. The relations among sets of reection rules are summarized in Figure 2. 3

5 n r Rup I I n XyXX X I I XXXXXXXXXXXX I Rup Rup n r r ; I I Rup Rup n r 6 Rup n Rup n r r I Rup n Figure 2: Partial ordering of (RR)'s. 4

6 TH(M ) TH(M) TH (M) OM M OM M = TH(O) TH OM (O) TH(O ) O O (RR) M O = Separated object and meta theories Interaction of object and meta theories in an OM pair Characterization of the resulting object and meta theories Figure 3: Meta and object theories deductively generated inside an OM pair. 2 The work plan The idea, already introduced in [2], is to provide a uniform characterization of the relation which exists between the object theory and the metatheory as a function of the chosen set of reection rules (and independently of the specic O, M under consideration). The key observation is that the application of a reection rule may extend the set of theorems of the metatheory, and/or of the object theory. The intuition (described in detail in [2]) is that the extra meta and object theorems are exactly those facts which must hold for a desired property to hold between the object theory and the metatheory. This idea is exemplied in Figure 3, where (RR) is a set of reection rules. (RR) is essentially seen as an operator which maps a pair of theories O and M onto a new pair of theories O 0 = hlo ; O 0 ; O i and M 0 = hlm ; M 0; M i, with TH(O 0 ) = TH OM (O) and TH(M 0 ) = TH OM (M). [2] gives an in depth analysis of the proof theoretic properties of OM pairs. In this paper we provide a model theory of OM pairs and exploit these results to rene the analysis in [2]. The work is organized as follows: 1. In Section 3 we characterize the models of the metatheory. The main idea is to dene them as sets of object level formulas. This allows us to study the relations between the models of the metatheory and the object theory and, therefore, to make precise statements like \the metatheory speaks about the object theory" and \the object theory is the model of the metatheory" which are often made informally in the literature. 2. [2][Section 4] provides a partial ordering characterization of OM pairs based on a comparison of the strength of the derivability relations and the sets of theorems generated. Section 4 renes the analysis in [2] and gives a strict ordering characterization. This work is based on the semantic analysis of Section [2][Section 6] introduces a notion of duality and shows that it preserves derivability. Section 5 gives an analogous result for satisability. 5

7 Following [2], we concentrate on the following fteen combinations of reection rules. 1 ; 2 r Rup 6 + r 7 Rup n r Rup n r n r Rup n n 13 Rup n + 14 Rup + 15 Rup + + Rup n r 3 A model-theoretic characterization We are interested in characterizing the models of the metatheory, and in studying the relation between them and the object theory. M is propositional theory. We therefore take the interpretations of LM to be truth assignments for the atomic ws of LM. Therefore an interpretation m of LM is a function which maps (\A"), for any A 2 LO, to True or False. Notice that the set of interpretations of LM is isomorphic to the powerset of LO. Indeed each interpretation m of LM can be mapped in a subset SmLO and vice versa as follows: A 2 Sm if and only if m((\a")) = True We can therefore consider the set of interpretations of LM to be the powerset of LO. This is the key step which allows us to study the relation between the models of the metatheory and the object theory itself. In the following two subsections, for any set (RR) of reection rules, we characterize: the models of the metatheory generated by (RR) in terms of O. the set of theorems of the object theory generated by (RR) in terms of the models of M. The rst result allows us to make precise in which sense the metatheory speaks about the object theory. The second result allows us to make precise the sense in which the object theory is the model of the metatheory. (Both statements are often made informally in the literature.) In particular, in the cases where we can explicitly dene M 0, O 0 with TH(M 0 ) = TH OM (M) and TH(O 0 ) = TH OM (O), (see [2]) we can make explicit the link between O 0 and the models of M 0. One such (particularly important) case is the metatheory with no axioms. In this case, largely studied in the literature, the metatheory has only a descriptive role and it does not modify the object theory ([2][Section 7.1] shows that O 0 = O for any choice of O and (RR)). This allows us to characterize the models of the metatheory generated by (RR) as a function of O. We use the following notation. Let m be an interpretation of LM. We say that m is a model of, in symbols m j=, to mean that m satises all ws in (that all ws in are true in m). A set of ws L is a classical propositional theory if it contains all the classical tautologies and it is closed under modus ponens. is consistent if and only if 62. is maximal if and only if for any w A 2 L, A 2 or :A 2. We also use the following notation. For any theory T = hl; ; i and any L, T + denotes the extended theory hl; [; i. For any set of ws LO, Rup() denotes the set of ws f(\a") : A 2 g LM. We sometimes use (\") instead of Rup(). Analogously for any set of ws LM, () denotes the set of ws fa : (\A") 2 g LO. 6

8 3.1 The models of the metatheory Let us start by considering the metatheories generated by combinations of reection rules which are characterizable axiomatically, i.e., the metatheories for which TH OM (M) = TH(M + ), with being some schematic metaformula. ([2][Section 5] gives an in depth analysis of the axiomatic characterizability of metatheories.) In this case the (obvious) idea is to look for the subclass of interpretations of LM which satises M and. Theorem 3.1 Let OM be an OM pair composed of O and M connected by the set of reection rules (RR); (i) if (RR) r, then m is a model of TH OM (M), if and only if it is a model of M ; (ii) If (RR), then m is a model of TH OM (M), if and only if it is a model of M, and TH(O) m. Item (i) of Theorem 3.1 states that any model of M is a model of the metatheory of OM and vice versa. m is a model of the metatheory of OM independently of the object theory. Item (ii) states that the models of the metatheory of OM are all the models of M which contains the theorems of the object theory. Proof (Theorem 3.1) These results are a direct consequence of Theorem 3.1 in [2]. This theorem states the (obvious) result that reection down leaves the metatheory unchanged, and we have TH OM (M) = TH(M), while reection up adds to the metatheory all the theorems of the form (\A"), with A 2 TH(O) = TH OM (O). Q.E.. Theorem 3.2 Let OM be an OM pair composed of O and M connected by the set of reection rules (RR). An interpretation m of LM is a model of TH OM (M) if and only if, m j= M and: (i) If (RR) is + then, TH(O)m and m is a classical propositional theory; (ii) If (RR) is ++Rup n r, then TH(O)m and m is a consistent classical propositional theory; (iii) If (RR) is Rup +, then TH(O)m and m is a maximal classical propositional theory; (iv) If (RR) is Rup++Rup n r, then TH(O)m and m is a consistent and maximal classical propositional theory; (v) If (RR) is either Rup n r + or Rup n +, then either O + m is consistent, or m = ;. To understand Theorem 3.2, let us consider the case with M = ;. In this situation the models of TH OM (M) are completely determined by O and (RR). (These observations can be easily generalized. In fact, from Theorem 3.2, M 6= ; only restricts ourselves to the models of M.) Consider item (i). The models of TH OM (M) are the sets of theorems of the extensions of the object theory, i.e., all the theories which can be obtained by adding a set of axioms to O. Therefore if A is a logical consequence of a set of formulas, then (\A") is also a logical consequence of (\"). Consider item (ii). The models of TH OM (M) are the sets of theorems of the consistent extensions of the object theory. Notice that, as there are no consistent extensions of an inconsistent theory, if O is inconsistent then TH OM (M) is inconsistent as well. Consider item (iii). The models of TH OM (M) are the maximal (possibly inconsistent) extensions of the object theory. Consider item (iv). The models of TH OM (M) are the maximal consistent extensions of the object theory. Since the set of maximal consistent sets containing the object theory is isomorphic to the set of models of the object theory, (\A") can be interpreted as \A is true in m". Finally, consider item (v). The models of TH OM (M) are the sets of formulas which can be added to the object theory without making it inconsistent. If, for instance, A is derivable from in O, the object theory estended by and :A is inconsistent. This corresponds to the fact that in any model m of TH OM (M) it is never the case that and :A both belong to m. A further example: if the object theory is inconsistent (i.e. all 7

9 formulas are provable) then it cannot be consistently extended. In this case the only models of the metatheory of OM is the empty set. Proof [Theorem 3.2] This proof is based on the result of Theorem 5.2 and Theorem 5.3 of [2]. In these theorems the metatheory of OM is characterized in terms of the following formulas: (\A B") ((\A") (\B")) : (\A") (\:A") (Comp) : (\") (ntfalse) f(\a1") ^ : : : ^ (\An") : (\A") : `O + A1 ; : : : ; An :Ag (Cons) where A; B are understood as schematic variables (parameters) ranging over LO, and we allow ourselves to substitute uniformly parameters for object level formulas inside quotes. Item (i). In this case TH OM (M) = TH(M + Rup(TH(O)) + (K)), as proved by Theorem 5.2 item (i) in [2]. Then, m j= TH OM (M) if and only if m j= M, m j= Rup(TH(O)) and m j= (K). Suppose that m is a model of TH OM (M), then m j= M and TH(O)m. m is a classical propositional theory as it contains all the propositional tautologies (contained in TH(O)), and it is closed under modus ponens since m j= (K). Vice versa, suppose that m j= M and m is a classical propositional theory which contains TH(O), then m j= M, by hypothesis, m j= Rup(TH(O)) as TH(O)m, and m j= (K) as m is closed under logical consequence. Item (ii). In this case TH OM (M) = TH(M +Rup(TH(O))+(K)+(nTfalse)), as proved by Theorem 5.2 item (ii) in [2]. Then, m j= TH OM (M) if and only if, m j= M, m j= Rup(TH(O)), m j= (K) and m j= (ntfalse). From item (i) if m j= TH OM (M), then m j= m, and m is a classical propositional theory that contains TH(O). Furthermore m j= (ntfalse) implies that 62 m and therefore that m is consistent. Vice versa, if m j= M and m is a consistent classical propositional theory which contains TH(O) then, by item (i), m j= Rup(TH(O)) and m j= (K). The consistency m implies that 62 m and therefore that m j= (ntfalse). Item (iii). In this case TH OM (M) = TH(M +Rup(TH(O))+(K)+(Comp)), as proved by Theorem 5.2 item (iii) in [2]. Then, m j= TH OM (M) if and only if m j= M, m j= Rup(TH(O)), m j= (K) and m j= (Comp). From item (i) of this proof, if m j= TH OM (M), then m j= m, and m is a classical propositional theory that contains TH(O). Furthermore m j= (Comp) implies that, for any object w A, A 2 m or :A 2 m, i.e., m is maximal. Vice versa, if m j= M and m is a maximal classical propositional theory which contains TH(O), then, by item (i) of this proof, m j= Rup(TH(O)) and m j= (K). Furthermore m being maximal implies that A 2 m or :A 2 m and therefore that m j= (Comp). Item (iv). In this case TH OM (M) = TH(M + Rup(TH(O)) + (K) + (Comp) + (ntfalse)), as proved by Theorem 5.2 item (iv) in [2]. Item (iv) is therefore a consequence of item (ii) and (iii) of this theorem. Item (v). In this case TH OM (M) = TH(M + (Cons)), as proved by Theorem 5.3 in [2]. Let us start with the only if direction. Let m be a model of TH OM (M). Then, m is a model of M, and by Theorem 5.3 in [2], for any A1; : : : ; An; A such that A1; : : : ; An `O :A if A1; : : : ; An 2 m, then A 62 m. Suppose that O + m is inconsistent. Then, for any formula A, `O + m :A, This implies that there is a nite number of ws A1; : : : ; An 2 m such that A1; : : : ; An `O :A. This implies that A 62 m. As this is true for any formula A, m must be the empty set. Let us now consider the if direction. In order to prove that m j= TH OM (M) we exploit again Theorem 5.3 in [2] and prove that m j + f(\a1") ^ : : : ^ (\An") : (\A") :`O + A1 ; : : : ; An :Ag. If m = ; then m j= : (\A") for any formula A, and therefore (since m satises m ) m j + f(\a1") ^ : : : ^ (\An") : (\A") :`O + A1 ; : : : ; An :Ag. Suppose instead m 6= ; and m j= M. Then, suppose that m j= (\A1") ^ : : : ^ (\An"). If :A is derivable in O + A1; : : : ; An, then A cannot be in m, otherwise O + m would be inconsistent. This implies m 6j= (\A"), that is m j= : (\A"). Q.E.. Let us now consider the combinations of reection rules which are not characterizable axiomatically. Let us start with + r. We need the following notation. For any set x of interpretations of (K) 8

10 LM, T S x = fa 2 LO : for all m 2 x; A 2 mg x = fa 2 LO : there exists an m 2 x A 2 mg Theorem 3.3 Let OM be an OM pair composed of O and M connected by + r. The set of models of TH OM (M) is the largest subset of interpretations of LM that satises equation (1) in x: x = fm j= M : TH(O + T x) mg (1) Theorem 3.3 states that m is a model of TH OM (M) i it satises M and contains the set of formulas which are logical consequence of the intersection of all models of TH OM (M). Notice that this theorem does not provide an explicit denition of the set of models of TH OM (M). 1 Indeed, to prove that a certain interpretation m is a model of TH OM (M) by exploiting Theorem 3.3, one needs, rst to prove that a certain set M of interpretations is contained in the set of models of TH OM (M) (by showing that M is a solution of equation (1)), and then to prove that m 2 M. Let us see some examples: Example 3.1 If M is the empty set, then a set M which satises equation (1), is the set of mlo which contains TH(O). Indeed any m 2 M satises an empty set of axioms M. Furthermore we have T M = TH(O) ( r has no eect) and therefore TH(O + M) = TH(O). If M is the axiom (\p"), then the set M that satises equation (1) is the set of mlo which contains TH(O + p). If M is the axiom (\p") _ (\q"), then a set M that satises equation (1) is the set of mlo such that TH(O) [ pm or TH(O) [ qm. Notice that the models of TH OM (M) might not be closed under logical consequence. Notice that M is dierent from the set of subsets of LO which contains TH(O + p) or TH(O + p). This set is indeed the set of models of the metatheory generated by +. If M is the axiom : (\p"), then a set of interpretations M which satisfy equation (1) is the set of subsets of LO which contain TH(O) and don't contain p. Proof (Theorem 3.3) The proof is in two steps. In the rst step we prove that the set M of models of the metatheory generated by + r satises equation (1). In the second step we prove that any other solution of equation (1) is contained in M. First step. From Theorem 5.4 in [2] we have that TH OM (O) and TH OM (M) are the smallest sets of ws which satisfy the following equations: We have therefore that xo = TH(O + (xm)) (2) xm = TH(M + Rup(xO)) (3) TH OM (M) = TH(M + (TH(O + r (TH OM (M))))): Let M be the set of models of TH OM (M). m 2 M if and only if it satises M and satises all the formulas in Rup(TH(O+(TH OM (M)))). We have therefore that TH(O+(TH OM (M)))m, and that M = fm j= M : TH(O + (TH OM (M)))mg (\A") belongs to TH OM (M) if and only if for all m 2 M, A 2 m. Hence, (TH OM (M)) = T M. We have therefore: M = fm j= M : TH(O + \ M)mg: 1 This is the model theoretic counterpart of the xpoint characterization of the metatheory generated by + r. 9

11 Second Step. Let us prove that the M is the largest set of interpretations of LM which satisfy (1), namely, that if N is a solution of equation (1), then NM. To do this let us consider the following sequence of theories M0; M1; : : : (originally dened in the proof of Theorem 5.4 in [2]): M0 (4) O0 (5) Mi+1 i + (TH(Oi)) (6) Oi+1 i + r (TH(Mi)) (7) For any i 0, let M i be the set of models of Mi. We prove by induction on i that for all i 0, NM i. (Base Case) Nfm : m j= M g 0. (Step Case) Since N satises (1), N = fm j= M : TH(O + \ N )mg The induction hypothesis NM i implies that T M i T N, and therefore that Nfm j= M : TH(O + \ M i )mg Since M i is the set of models of Mi, we have that T M i = (TH(Mi)). If we substitute T M i with (TH(Mi)) in the previous equation we obtain: Nfm j= M : TH(O + (TH(Mi)))mg Notice that any interpretation m which satises M and satises (\O+(TH(Mi))") is a model of Mi+1 and therefore we have that NM i+1 Thus we have shown that for any i 0, NM i. This implies that: N \ i0m i Since TH OM T S (M) = i0 M i, the set of models of M is the intersection of the models of each Mi. i.e. M = i0 M i. Thus we conclude that NM. Q.E.. The set of models of the metatheories generated by Rup n r + r and + + r n can be characterized similarly to + r. For a set of w, let : be dened as : = f:a : A 2 g. Theorem 3.4 Let OM be an OM pair composed of O, M connected by Rup n r + r. The set of models of TH OM (M) is the largest subset of interpretations of LM which satises equation (8) in x: x = fm j= M : TH(O + T x) \ :m 6= ;g (8) Theorem 3.5 Let OM be an OM pair composed of O, M connected by + + r n. The set of models of TH OM (M) is the largest subset of interpretations of LM which satisfy equation (9) in x: x = m j= M : m is the set of theorems of an extension of O If A 62 S x, then :A 2 m The proofs of these two theorems are analogous to that of Theorem 3.3. n r implements a weak form of negation as failure. The models of the metatheory generated by + + n r indeed are the extensions of O, which satisfy M and contain :A whenever A is not provable in any extension of O. (9) 10

12 3.2 The object theory in terms of the models of M Our goal here is to understand the meaning of in terms of properties of the object theory, i.e., which is the property of the object theory represented by. This allows us to make precise the sense in which the object theory can be considered a model of the metatheory. We start from the weakest combinations of reection rules and then move to those of increasing strength, following the graph reported in Figure 2. We have the following results. Theorem 3.6 Let OM be an OM pair composed of O and M connected by the set of reection rules (RR). Let M be the set of models of TH OM (M). Then: (i) If (RR) r, then TH OM (O) = TH(O + T M) (ii) If (RR), then TH OM (O) = TH(O). The underlying intuitions are the same as those described above for Theorem 3.1. Theorem 3.7 Let OM be an OM pair composed of O and M connected by a set of reection rules (RR). Let M be the set of models of TH OM (M). Then: (i) If r O (RR), then T M TH OM (O); (ii) If M (RR), then TH OM (O) T M. (iii) if r O (RR) and M (RR), then TH OM (O) = T M. Proof (Theorem 3.7) Item (i) is a straightforward consequence of item (i) of Theorem 3.6. Item (ii) is provable T as follows. Suppose that A 2 TH OM (O), then, by, (\A") 2 TH OM (M) and therefore A 2 M. Item (iii) is a consequence of item (i) and (ii). Q.E.. Notice that if the object theory is generated by a set of reection rules weaker than or r (in the sense of Theorem 3.7), then the set of object theorems is not completely determined by the set of models (theorems) of the metatheory. This means that the object-metarelation generated, (that is, the set fhth OM (O); TH OM (M)i : OM = ho; M; (RR)i for any O and Mg) is not one-to-one, but it associates more than one object theory to the same metatheory. Vice versa, from Theorem 3.7 we can conclude that the object theory (more precisely, its set of theorems) generated by a set of reection rules which are at least as strong as and r, is completely determined by the set of models (theorems) of the metatheory. This means that the object-metarelation generated is one-to-one, i.e., to each metatheory there corresponds only one object theory and vice versa. Finally notice that in general, according to Theorem 3.7, an interpretation m of LM with m = TH OM (O) is not necessarily a model of TH OM (M). Consider the following example. Example 3.2 Let O and M be such that M = f(\p")_(\q")g and O = ;. Let OM be obtained by making O and M interact by +. Any model of the metatheory of OM contains either p or q. In particular m1 = TH(O + p) and m2 = TH(O + q) are both models of the metatheory of OM. On the other hand the intersection of these models (which is TH OM (O)) does not contain p, nor q. Therefore TH OM (O) is not a model of TH OM (M). Theorem 3.7 provides the characterization of the object theory for all the metatheories which are in relation O and M with and r respectively. As shown in Figure 2, this covers all the combinations of reection rules, with the exception of Rup n r + and Rup n +. We have the following result. Theorem 3.8 Let OM be an OM pair composed of O and M connected by the set of reection rules (RR). If M is the set of models of TH OM (M), then: (i) If (RR) is Rup n r +, then TH OM (O) = TH(O + T M); 11

13 T (ii) If (RR) is Rup n +, then TH OM (O) = m2m TH(O + m). Proof (Theorem 3.8) Item (i). From Theorem 5.3 in [2] we have TH OM (O) = TH(O + (TH OM (M))) (10) Item (i) is a consequence of equation (10) and the fact that (TH OM (M)) = T M. Concerning item (ii), from Theorem 5.3 in [2] we have TH OM (O) = TH(O + fa1 _ : : : _ An : (\A1") _ : : : _ (\An") 2 TH OM (M)) (11) T Let us prove TH OM (O) m2m TH(O + m). For each m 2 M, if (\A 1") _ : : : _ (\An") 2 TH OM (M) then m j= (\A1")_: : :_(\An"), which implies that there is an 1 i n T such that Ai 2 m. Therefore for each m 2 M, TH OM (O)TH(O + m); and therefore TH OM (O) m2m TH(O + m). T T Vice versa, let us prove m2m TH(O + m)th OM(O). Let A 2 m2m TH(O + m), and for each m 2 M let m be a proof of A in O + m. Let M 0 be the following theory: M 0 = TH OM (M) [ :(^ (\m ")) : m 2 M, and m m is the nite (possibly empty) set of formulas occurring as axioms in m where ^ (\ m ") is the conjunction of the formulas in (\ m "). M 0 is unsatisable. Indeed any model m of M 0 is a model of TH OM (M) and therefore m m. This implies that m j= (^(\ m ")) and therefore that m 6j 0. If M 0 is unsatisable, then it is nitely unsatisable, i.e., there is a nite sequence of sets m1 ; : : : ; mn such that TH OM (M) + f:(^ (\ m1 ")); : : : ; :(^ (\ mn "))g is unsatisable. This implies that (^ (\ m1 ")) _ : : : _ (^ (\ mn ")) 2 TH OM (M) and therefore that for each 1; : : : ; n, with i 2 mi, (\1") _ : : : _ (\n") 2 TH OM (M) We have therefore that 1 _ : : : _ n 2 TH OM (O) A possible proof of A in TH OM (O) is therefore the following: V. i2 i (1 _ : : : _ n) (^ 1 ) _ : : : _ (^ n ) A [^ 1 ] 1 A : : : [^ n ] n A _E We can therefore conclude that A 2 TH OM (O). Q.E.. Notice that, while Rup n r + and Rup n + generate the same metatheory, this is not the case for the object theory. Consider the following example. Example 3.3 Let OM r and OM be two OM pairs composed of the empty object theory O and the metatheory M with axiom (\p") _ (\q"), connected by the reection rules Rup n r + and Rup n + respectively. Since they have the same set of metatheorems, OM r and OM have the same set of models M dened as follows: M = fm : m is consistent and p 2 m or q 2 mg (12) 12

14 (RR) Models of TH OM(M) Thm. TH OM(O) Thm. 1: ; Any set of LO-ws 3:1 TH(O) 3:6 2: r Any set of LO-ws 3:1 TH(O + T M) 3:6 3: TH(O)m 3:1 TH(O) 3:6 4: Any set of LO-ws 3:1 TH(O + T M) 3:6 5: Rup TH(O)m 3:1 TH(O) 3:6 6: + r Fixpoint equation (1) 3:3 T M 3:7 7: Rup n r + m is consistent with TH(O) 3:2 TH(O + T M) 3:8 8: + Any extension of O 3:2 T M 3:7 9: + + Rup n r Any consistent extension of O 3:2 T M 3:7 10: + + n r xpoint equation (9) 3:5 T M 3:7 11: + + Rup n Any consistent extension of O 3:2 T M 3:7 12: + + n Any maximal extension of O 3:2 T M 3:7 13: Rup n + m is consistent with TH(O) 3:2 T m2m TH(O + m) 3:8 14: Rup + Any maximal extension of O 3:2 T M 3:7 15: Rup + + Rup n r Any maximal consistent extension of O 3:2 T M 3:7 Table 1: Metamodels and object theorems generated by a set of reection rules The set of object theorems of OM r diers from that of OM. Indeed, according to Theorem 3.8, we have the following. The set of object level theorems of OM r is equal to the theorems provable from O extended with the intersection of all the elements of M, which is the empty set. Therefore TH OMr (O) = TH(O) is the set of the propositional tautologies. On the other hand the set of object level theorems of OM is equal to the intersection of the theorems provable from O extended with m for some m 2 M. This means, for instance, that, since either p or q belongs to any m, p _ q is provable O + m for all m. This implies that p _ q 2 TH OM (O). Table 1 summarizes the results proved in this section. 4 A strict ordering characterization In Figure 2 (Section 1) reection rules have been partially ordered; however we still do not have explicit results stating that the object and meta relations generated by two sets of reection rules are dierent. In this section we show that the partial orders in Figure 2 are indeed strict orders. enition 4.1 For any two OM pairs OM 1 and OM 2, OM 1 OM2 (resp. OM 1 <O OM2, OM 1 <M OM2) if and only if OM 1 OM 2 and not OM 1 = OM 2 (resp, OM 1 O OM 2 and not OM 1 OM 2, OM 1 M OM 2 and not OM 1 OM 2 ). enition 4.2 For any two combinations of reection rules (RR) 1 and (RR) 2, (RR) 1 (RR) 2 (resp, (RR) 1 <O (RR) 2, (RR) 1 <M (RR) 2 ) if and only if (RR) 1 (RR) 2 and not (RR) 1 = (RR) 2, (resp, (RR) 1 O (RR) 2 and not (RR) 1 (RR) 2, (RR) 1 M (RR) 2 and not (RR) 1 (RR) 2 ). The graph of the partial orders given in Figure 1 can be extended obtaining the graph depicted in Figure 4. The correctness of this latter graph is straightforward. Theorem 4.1 (Correctness of the graph in Figure 5) If in the graph of Figure 5 there is an arc labeled with X from (RR) 1 to (RR) 2, then (RR) 1 X(RR) 2. To prove the previous theorem we need the following lemma: 13

15 = J JJ J^ <O <M P PP Pq R R ) O M Figure 4: Relations among orders. Lemma 4.1 (Eective Assumptions) Let OM be an OM pair with the set (RR) of reection rules. Let O ; AO and M ; AM be a set of ws of LO and LM respectively. Then (i) If the reection up rules in (RR) are restricted, then O ; M `OM AM if and only if M `OM AM; (ii) If the reection down rules in (RR) are restricted, then O ; M `OM AO if and only if O `OM AO. Proof (Lemma 4.1) The proof is a generalization of the proof of the analogous lemma in [3]. The idea is the following. In item (i), the only way to \switch" from object to meta in order to infer a meta formula from object assumptions, is by applying restricted reection rules. However these rules are applicable only if all the assumptions in the object theory are discharged. This implies that any meta formula cannot depend from assumptions in the object theory. Which is equivalent to item (i). A similar argument can be used to prove item (ii). Q.E.. Proof (Theorem 4.1) The proof is based and further renes the graph in Figure 2. Indeed the graph in Figure 5 is obtained from the graph in Figure 2 by replacing all the occurrences of with a combination of <O, <M, and. Since <O (resp. <M and ) can be dened as the conjunction of and 6 (resp. the conjunction of and 6 and the conjunction of and 6= ), the additional information encoded in the graph in Figure 5, with respect to the graph in Figure 2, consists of statements asserting that two combinations of reection rules dier either in the object and/or metatheory they generate, or in their derivability relation. Following enition 1.4 in Section 1, we prove (RR) 1 6 (RR) 2 (resp. (RR) 1 6 (RR) 2 ), by showing that there are two OM pairs OM 1 and OM 2 composed of the same object and metatheories and connected by (RR) 1 and (RR) 2 respectively, which generate a dierent object (resp. meta) theory. In practice, we look for an object (resp. meta) w A not provable in OM 1 and provable in OM 2. (RR) 1 6= (RR) 2 is automatically proved as a consequence of (RR) 1 6 (RR) 2 or (RR) 1 6 (RR) 2. We prove 6= without proving (RR) 1 6 (RR) 2 or (RR) 1 6 (RR) 2 by providing two OM pairs OM 1 and OM 2 and a set of object and meta ws ; A such that A is not derivable from in OM 1, and A is derivable from in OM 2. We proceed by considering the arcs of the graph in Figure 5 from the bottom up. We say that a (meta or object) theory is an empty (meta or object) theory meaning that it has no axioms. ; <M. (We prove ; 6.) Consider the OM pairs OM 1 and OM 2 composed of the empty object theory and the empty metatheory connected by no reection rules and by respectively. No w of the form (\A") is a theorem of the metatheory of OM 1, while, for instance, (\:") is a meta theorem of OM 2. ; <O r. (We prove ; 6 r.) Consider the OM pairs OM 1 and OM 2 composed of the empty object theory and the metatheory with the only axiom (\") connected by no reection rules and by r respectively. Since the object theory of OM 1 is equal to O, is not provable, Instead is provable (by r ) in the object theory of OM 2. 14

16 n r Rup I I n < XyXX X I I < XXXXXXXXXXXX I Rup Rup n r r ; I I < Rup Rup n r 6 Rup n Rup n r r I Rup n Figure 5: Strict ordering of (RR)'s. 15

17 Rup. (We prove 6= Rup.) Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O and the empty metatheory M, connected by and Rup respectively. The metatheory of OM 1 is equal to M and therefore (\") is not provable in it. Furthermore, by Lemma 4.1 (\") is not derivable in the metatheory from the assumption of the object theory. In OM 2 instead, (\") is derivable (by Rup) from the assumption in the object theory. r. Analogous to the previous case. <M + r. (We prove 6 + r.) Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O and the metatheory M with the axiom (\") connected by and + r respectively. By Theorem 3.1, the interpretation m = TH(O)[fg is a model of the metatheory of OM 1. As O is consistent, there is an object w A distinct from such that m 6j= (\A") and therefore (\A") is not a theorem of the metatheory of OM 1. In OM 2, instead, each formula (\A") is provable as follows: (\") r A (\A") r <O + r. (We prove r 6 + r.) Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O and the metatheory M with the axiom : (\:") connected by r and + r respectively. From theorem 5.1 in [2] we have that the set of object theorems of OM 1 is TH(O + (TH(M))). Since M does not prove any theorem of the form (\A"), the set of object theorems of OM 1 is TH(O). In particular is not provable in O and therefore it is not provable in OM 1. In OM 2 instead is a theorem of the object theory. Indeed: : I (\:") : (\:") E (\") r Rup Rup +. (We prove Rup 6= Rup +.) Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O and the empty metatheory M, connected by Rup and Rup + respectively. From Theorem 5.1 in [2] we have that the set of theorems of the object theory of OM 1 is TH(O) and therefore is not provable in the object theory of OM 1. Therefore, by Lemma 4.1, in OM 1 is not derivable in the object theory from (\"). In OM 2, instead, can be derived in the object theory from (\") in the metatheory (via a single application of ). + r <M +. (We prove + r 6 +.) Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O and the empty metatheory M, connected by + r and + respectively. By Theorem 3.3 the set of models of the metatheory of OM 1 is fm : TH(O)mg. Consider the model of the metatheory of OM 1 m = TH(O) [ fg. Clearly m 6j= (\") (\p"), and therefore (\") (\p") is not a theorem of OM 1. However, (\") (\p") is a theorem of OM 2. + r <O +. (We prove + r 6 +.) Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O and the metatheory M with axiom 16

18 (\p") _ (\q"), connected by + r and +, respectively. By Theorem 3.3 the set of models of the metatheory of OM 1 is: fm : TH(O)m and p 2 m or q 2 mg (13) By Theorem 3.7 the set of theorems of the object theory of OM 1 is the intersection of m for each m in (13), which is TH(O). The formula p _ q is not provable in TH(O) as O, being empty, proves only classical tautologies. This implies that p _ q is not a theorem of the object theory of OM 1. In OM 2, instead, p _ q is provable as follows: (\p") _ (\q") (\p") p p I q (\p _ q") (\p _ q") p _ q (\q") q p I q (\p _ q") _E <M Rup n r +. (We prove 6 Rup n r + ). Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O and the empty metatheory M. From Theorem 5.1 in [2] the set of theorems of the metatheory of OM 1 is TH(M) (i.e. the set of tautologies of LM), and by Theorem 5.3 in [2] the set of metatheorems of OM 2 is TH(M + (Cons)) which contains ws which are not tautologies of LM (e.g. (\p") ^ (\q") : (\:p _ :q")). <O Rup n r +. (We prove 6 Rup n r + ). Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O and the metatheory M with the axiom (\")_(\p"). By Theorem 3.1 the set of models of the metatheory of OM 1 is the set of models of M, i.e. the set of models m such that 2 m or p 2 m. By Theorem 3.7 the set of object theorems of OM 1 is the set of theorems of O extended with the intersection of the models of M. Since this intersection is empty, the object theory of OM 1 is equal to O and, therefore, it does not prove p. In the object theory of OM 2, instead, p is provable as follows: (\") _ (\p") : I (\") : (\") Rupn r E (\p") (\p") p (\p") + <M + +r n. (We prove r n ). Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O and the metatheory M, with the axiom : (\p"). Since p is not provable form the empty object theory, the interpretation m = TH(O) is a model of the metatheory (according to theorem 3.2). Since also :p is not provable in the empty object theory (i.e. :p 62 TH(O)), m 6j= (\:p") and therefore (\:p") is not provable in the metatheory of OM 1. In OM 2 instead (\:p") is provable. A proof of (\:p") is the following: : (\p") :p n r (\:p") + <O + + n r. (We prove n r ). Consider the two OM pairs of the previous case. The object theory of OM 1 does not prove :p, Indeed by Theorem 3.1 the object theory of OM 1 is the intersection of the models of the metatheory _E 17

19 of OM 1. The model m considered in the previous case is such that :p 62 m. Therefore :p is not provable in the object theory of OM 1. In OM 2 instead :p is provable (by applying n r to : (\p")). + <M + + Rup n r. (We prove that Rup n r ). Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O and the empty metatheory M. The metatheory of OM 1 does not prove :(\"). Indeed the model m which interprets in the set of ws LO, is a model of TH OM1 (M). The metatheory of OM 2 instead proves : (\"). A proof is the following. : I : (\") Rupn r (14) Rup n r + <O Rup n +. See Example n r + + n. (We prove + + n r n ). Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O and the empty metatheory M, connected by the reection rules + + n r and + + n respectively. By Theorem 3.5 it can be proved that the set of models of the metatheory of OM 1 is equal to the set of interpretations which are the sets of theorems of an extension of O. The interpretation m = TH(O) is, therefore, a model of the metatheory of OM 1. Since neither p nor :p is provable in O, then neither m j= (\p") nor m j= (\:p") and therefore m 6j= (\p")_(\:p"). This implies that (\p")_(\:p") is not valid, and therefore not provable, in the metatheory of OM 1. In the metatheory of OM 2, instead, (\p")_(\:p") is provable as follows: (\p") _ : (\p") (\p") (\p") _ (\:p") _I (\p") _ (\:p") : (\p") n :p (\:p") (\p") _ (\:p") _I _E + + n r + + n. (We prove + + n r n ). Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O, and the metatheory M with the axiom (\p")_(\:p") (\q"), connected by the reection rules + + n r and + + n respectively. By Theorem 3.5 it can be proved that the set of models of the metatheory of OM 1 is equal to the set of interpretations m which is the set of theorems of an extension of O, and q 2 m if p 2 m or :p 2 m. Since neither p nor :p is provable in O, then the interpretation m = TH(O) is a model of the metatheory of OM 1. Theorem 3.7 implies that the set of theorems of the object theory of OM 1 is equal to the intersection of each model of the metatheory. Since q does not belong to a model m, then p is not provable in the object theory of OM 1. In OM 2, instead, q is provable as (\p") _ (\:p") is a theorem of the metatheory of OM 2 (see the previous example) and q can be derived by E and. + + Rup n r + + Rup n. (We prove + + Rup n r 6= + + Rup n ). Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O, and the empty metatheory M, connected by the reection rules + + Rup n r and + + Rup n respectively. By Theorem 3.2 the metatheory of OM 1 has at least a model, (e.g. m = TH(O)), and therefore it is consistent. By Lemma 4.1, the non derivability of in the metatheory implies the non derivability of in the metatheory starting from in 18

20 the object theory. This derivation is instead possible in OM 2 : :: I : : (\:") Rupn (\:") E Rup n + <M + + Rup n. (We prove Rup n Rup n ). Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O, and the empty metatheory M, connected by the reection rules Rup n + and + + Rup n respectively. According to Theorem 3.2 the interpretation m = ;, is a model of the metatheory of OM 1. This model does not satisfy (\:"). This implies that (\:") is not a theorem of the metatheory of OM 1. In OM 2 instead, (\:") is provable (by ). Rup n + <O + + Rup n. (We prove Rup n Rup n ). Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O, and the metatheory M with axiom (\:") (\"), connected by the reection rules Rup n + and + + Rup n respectively. By Theorem 3.8 we have that the set of theorems of the object theory of OM 1 is equal to the intersection of the sets TH(O + m) for each model m of M. According to Theorem 3.2 the intrepretation m = ;, is a model of the metatheory of OM 1. This implies that the set of theorems of the object theory of OM 1 is TH(O). Note that O is empty and therefore consistent. The object theory of OM 2, instead, is inconsistent, as is provable in the object theory of OM. + + n Rup +. (We prove + + n 6= Rup + ). The reasoning is similar to that done for the case Rup. + + Rup n <M Rup + + Rup n r. (We prove that + + Rupn 6 Rup + + Rup n r ). Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O, and the empty metatheory M, connected by the reection rules + + Rup n and Rup + +Rup n r respectively. According to Theorem 3.2 the interpretation m = TH(O) is a model of the metatheory of OM 1. Note that, for any proposition p, neither p nor :p is provable in O. From this we can infer that m 6j= (\p") and m 6j= (\:p"), i.e. m 6j= (\p") _ (\:p"). This implies that (\p") _ (\:p") is not provable in the metatheory of OM 1. The same formula, however, is provable in the metatheory of OM 2 (see proof above). Rup + <M Rup + + Rup n r. (We prove that Rup + 6 Rup + + Rup n r ). Consider two OM pairs OM 1 and OM 2 composed of the empty object theory O, and the empty metatheory M, connected by the reection rules Rup + and Rup + + Rup n r, respectively. According to Theorem 3.2 the interpretation mm = LO is a model of the metatheory of OM 1. Since this model does not satisfy :(\"), :(\") is not provable in the metatheory of OM 1. As showed before : (\") is a Theorem of the metatheory of OM Note that r and r <O + r, implies that <O + r. From the fact that + r <O +, we have that Rup <O +. Since (RR) 1 (RR) 2 and (RR) 1 <O (RR) 2 implies (RR) 1 (RR) 2, we conclude that Rup +. Rup n r + <O + + Rup n r. Suppose, by contradiction, that Rup n r Rup n r. From the fact that + + Rupn r + + Rup n, we have that Rup n r Rup n. However, from the fact that Rup n r + Rup n +, and from the fact that Rup n + <O + + Rup n, we have that Rup n r + <O ++Rup n and, therefore, that Rup n r Rup n. This contradicts what we have proved before. We conclude therefore that Rup n r Rup n r. 19

21 Rup n r + <M + + Rup n r. We reason by contradiction as before. Suppose that Rup n r Rup n r. Then from the fact that + + Rupn r ++Rup n, we have that Rup n r + ++Rup n. On the other hand from the fact that Rup n r + Rup n + and that Rup n + <M + + Rup n, we have that Rup n r + <M + + Rup n and therefore that Rup n r Rup n. This contradicts what we have proved before. We conclude therefore that Rup n r Rup n r. Q.E.. Further relations existing between sets of reection rules can be inferred from the structure of the graph in Figure 5. Indeed we can exploit the following properties. Proposition 4.1 Let X and Y be two relational symbols describing a relation among two sets of reection rules. Let x stand for, M, or O. Let (RR) 1, (RR) 2 and (RR) 3 be sets of reection rules. Then the following properties hold: Implication If (RR) 1 X(RR) 2, and Y is reachable from X in graph of gure 4, then (RR) 1 Y (RR) 2. Transitivity-1 If (RR) 1 x (RR) 2 and (RR) 2 x (RR) 3, then (RR) 1 x (RR) 3. Transitivity-2 If (RR) 1 <x (RR) 2 and (RR) 2 x (RR) 3, then (RR) 1 <x (RR) 3. Identity (RR) = x (RR). Substitutivity If (RR) 1 = x (RR) 2 and (RR) 2 X(RR) 3, then (RR) 1 X(RR) 3. These properties follow immediately from the denitions of = x ; x and <x, with x 2 fo; M; g. Table 2 links the holding of = ; ; ; ; <O; <M of any two sets of reection rules. A relational symbol X occurs in position (RR) 1 (RR) 2 if and only if (RR) 1 X(RR) 2. The results in Table 2 can be derived from the facts explicitly stated in the graph in Figure 5 plus the properties Implication, transitivity-1, transitivity-2, Identity and Substitutivity. Theorem 4.2 (Completeness of the graph in Figure 5) The graph in Figure 5 is complete. That is, for each two sets of reection rules (RR) 1 and (RR) 2 and for each relational symbol X 2 f= ; ; ; ; <O; <Mg, (RR) 1 X(RR) 2 i (RR) 1 X(RR) 2 is derivable from the facts explicitly stated in the graph in Figure 5, and from Implication, transitivity-1, transitivity-2, Identity and Substitutivity. The proof of Theorem 4.2 can be done by showing that, if <x [= x ] does not occur in position (RR) 1 (RR) 2 of Table 2, then there are two OM pairs OM 1 and OM 2, such that OM 1 6<x OM 2 [OM 1 6= x OM 2 ]. The proof, not reported because routine and tedious. exploits the same techniques exploited in the proof of Theorem 4.1. Let us discuss, top to bottom, left to right, some of the results in Table 2. The results in row 1 ((RR) 1 = ;) are obvious. It is sucient to notice that a single reection up rule does not modify the object theory. Trivially, the dual result holds in the case of reection down only. The results in row 2 and row 3 have similar explanations. The relations generated by a reection up rule cannot be compared with those generated by a reection down rule. A restricted rule generates the same meta and object theories as its corresponding unrestricted rule. It only allows more derivations, all those which (start in one theory and nish in another and) contain at least an application of a reection rule in presence of open assumptions. The eects of dropping the restriction appear when we have at least one reection up rule and one reection down rule. In row 4 the only not trivial result is in column 6. As from row 3 column 4, and r generate the same meta an object theories. Adding to r increases the theorems of the metatheory, this proves <M + r. To prove that <O + r it is sucient to consider the case 20