On a generalised modularisation theorem 1 Introduction - CiteSeerX

On a generalised modularisation theorem a

Theodosis Dimitrakosa

and

Tom Maibaumb

Central Laboratory of the Research Councils, Department for Computation and Information, Rutherford Appleton Laboratory, Chilton, Didcot, OX11 0QX, UK. b Department of Computing, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK. [email protected], [email protected]

Abstract

The relation between a metalogical property of entailment (interpolation) and a structural property of categories of theory presentations (stability of faithful morphisms under pushouts) is studied in an abstract \general logic" framework. In particular, a known result for rst order logic (the \modularisation theorem"), stating that the stability of faithful morphisms under pushouts is equivalent to a speci c form of interpolation, is generalised and re-established. Furthermore, the stability under pushouts of the faithfulness of a given theory interpretation is shown to be equivalent to the existence of interpolants on speci c loci of the underlying formalism. The latter neither assumes nor requires that the underlying logic possesses interpolation globally.

1 Introduction In this paper we revise, generalise and re-establish, in a notation independent framework, a strong relation between (a form of) interpolation [5, 20, 29, 1] and the stability of conservative extensions and, more generally, faithful morphisms under pushouts (amalgamation of conservative extensions) [34, 33, 10, 9]. The latter provides the formal basis for some fundamental modularity properties of re nement [21, 22, 35, 32, 34], [33, 10, 9, 8] and databases [24], while some operations of module algebras [2] have been linked directly with a similar form of interpolation ([27] discussing an earlier version of [2]). The paper is structured as follows: In section 2 we review the basic concepts involved and the relevant previous work on this subject. The generalised form of the interpolation property we are interested in, called CRI, is highlighted in section 3, where we also explain why CRI is in general weaker than ordinary interpolation. The strong relation between this form of interpolation and the stability of conservative extensions and faithful morphisms under amalgamation (pushouts), called the \Modularisation property" in [31] and [34], is re-examined in section 4, and re-established in a revised form in the abstract framework of Entailment Systems [25] (called -Institutions earlier in [13]). In addition to its generality, this new proof of the so-called \Modularisation Theorem" [31, 34] exhibits an even deeper relation between the existence of CRI-interpolants and the stability of faithful theory (presentation) morphisms under pushouts. Corollaries 4.3 and 4.4 illustrate that asserting for a given faithful theory (presentation) morphism e that e is stable under pushouts is equivalent to asserting that CRI holds locally on some speci c syntactical loci. This does not assume nor imply that the above mentioned form of interpolation (CRI) is possessed as a global metalogical property. Finally, we compare this result with previous proofs of Modularisation for \concrete" calculi (such as [32] and [34]) and emphasise the essence of our generalisation by means of an example (given in the Appendix) involving a non1

trivial rst order calculus which possesses modularisation, and therefore CRI, while it lacks interpolation in its ordinary form.

2 Preliminaries

Ordinary interpolation for rst order logic states that if ! ' is a ( rst order) theorem, then there is an \interpolant" sentence # such that (1) both ! # and # ! ' are rst order theorems, and (2) every extralogical symbol that appears in the interpolant # appears in both the assumption and the conclusion '. One can abstract away a little from the traditional formulation of ordinary interpolation and rewrite the ordinary interpolation property in the following (more general) form: for all sentences and ' such that entails ', in symbols f g`', there is an interpolant sentence # such that (1) f g`# and f#g`', and (2) every extralogical symbol that appears in # appears in both and '. Note that interpolation neither assumes nor implies that all derivations of ' from are through the interpolant sentence # built entirely from symbols that are common in the assumption and the conclusion '. Instead, interpolation states that, if ' is derivable from (in symbols f g`'), then there is at least one derivation of ' from through an interpolant #. Also note that,in general, the choice of the interpolant sentence # depends on both the assumption and the conclusion '. A theory (presentation) morphism i:A!B from a source theory (presentation) A to a target theory (presentation) B is called faithful i it is theorem conserving, ie., for every sentence a in the language of A, the sentence a is a theorem of A if and only if the translation ai of a via i is a theorem of B. The morphism i is called a conservative extension i i is faithful and the language of B expands the language of A. The Modularisation property [22, 31, 34], originally formulated for logical speci cations ( rst order theory presentations), states that conservative extensions are stable under amalgamation. The rst attempt to analyse this property, based on an observation of M. Sadler [22], indicated a strong interconnection between the Modularisation property and interpolation: The requirement that any diagram he; ii with e conservative can be completed to a rectangle he; i ; i; e i so that e is conservative, was equivalent { in rst order logic { to ordinary interpolation. Some other logics with (a rst order grammar and) poor interpolation properties failed to meet this requirement and this accounted for the negative results in [11]. In particular the Modularisation property fails for the combination of equational logic with initial algebraic semantics on which [11] had focused. The initial analysis indicated that, within rst order logic, Modularisation was equivalent to the Craig Interpolation Lemma. Detailed proofs for the classical rst order case were given later in [32] and [34]. As we explain in this paper, the actual reason that makes ordinary interpolation to be equivalent to modularisation in classical rst order logic is that ordinary interpolation and the instance of the general interpolation property CRI (de ned in the next section) con ate in this logic. As we elaborate in the sequel, this is mainly because the classical rst order calculus is compact, there is a deduction theorem for this calculus, this calculus admits internalisation of equality/equivalence, and also because rst order language expansions can be amalgamated and their amalgamation produces another language expansion. In fact, in the Appendix we provide an interesting rst order logic, namely the predicate logic over rst order arithmetic, which possesses modularisation, but apparently lacks ordinary interpolation. Meanwhile, in the next two sections, we provide a generalised version of interpolation which we prove to be equivalent to a general form of modularisation for virtually every monotonic, transitive and re exive formal calculus. Concrete instances of this generalised interpolation may not con ate with ordinary interpolation unless other meta-logical properties, which are not directly related to interpolation, also hold. 0

0

0

2

3 A generalised pushout version of interpolation In order to state the critical generalisation of interpolation in an abstract (\notation and semantics independent") framework, we have used the concept of an Entailment System [25], (called a -Institution in [13] and [15]).

De nition 3.1

The concept of an Entailment System An Entailment System E (-Institution) is a triple hSign; gram; È i where Sign is a category of signatures, gram:Sign!Set is the grammar functor (that assigns to each signature , the set of well formed formulae on ) and È is a Sign-indexed family of binary relations È 2gram( ) gram( ) such that, for every in Sign, È is re exive: f'gÈ '; monotonic: if ? and ?È ' then È '; transitive: if È ' and ?È , for all 2 , then ?È '; stable under translation: if ?È1 ' and i:1 !2 is a morphism in Sign, then gram(i)(?)È2 gram(i)('). In addition, E is called compact i for every signature and every A gram( ), ' 2 gram( ) such that AÈ ' there is a nite ? A such that ?È '. Finally, the category of theory presentations over E is denoted by Pres[E ]. The following generalised pushout version of Craig-Robinson Interpolation1, labelled by CRI, provides, as we prove in section 4, a necessary and sucient condition for Modularisation over an arbitrary Entailment System.

Note: In the following statements, and for each translation i:1 !2 and 2 gram(1 ), the writing of i 2 gram(2 ) denotes the image of the formula via the translation induced by i. Analogously, if gram (1 ) then i = f i : 2 g. Finally, E E ?` for ?; gram( ) means ?` for each 2 . CRI: a generalised pushout version of Craig-Robinson interpolation De nition 3.2 An Entailment System E possesses CRI i for every pushout diagram D in Sign, depicted below, and every A gram(A ), B gram(B ), ' 2 gram(B ), such that Ai [Be ÈC 'e , there is a set I A;B;'; gram(R ) of interpolants such that 0

0

0

(

D)

1. AÈA Ie (A;B;'; ), D

2. Ii (A;B;'; )[BÈB '. D

A

i

e

D

R

0

i

C e

0

B

The writing of CRI[A; B; D] denotes the specialisation of the above stated CRI-condition to a given diagram D presenting the pushout completion of a span A e ? R ? i!B , and for some given A gram(A ) and B gram(B ). An analogous generalisation of Craig Interpolation to an arbitrary Entailment System E

results in the following pushout version of interpolation:

De nition 3.3 CI: a generalised pushout-version of Craig-Interpolation E possesses CI when the above de ned CRI property (De nition 3.2) holds (at least) for B = ?.

1 Craig-Robinson interpolation originates in a combination of Craig's Interpolation Lemma with Robinson's Joint Consistency Theorem that is stated in [29] for rst order logic. Both Craig's Interpolation Lemma [5] and Robinson's joint Consistency Theorem [29, 12, 26] have also been stated originally for rst order predicate calculus with equality. See also Remark 3.2[2].

3

Note the use of the pushout construction to capture the idea that the interpolants are in a \shared" language, which is required for stating the property (as in [14] and [30]). The above de nition of CRI is very similar to the one that Fiadeiro and Maibaum use in [14]. Tarlecki proposes in [30] a version, weaker in the general case, where the interpolation condition is required only for B = ? as in CI. It is often the case that the instances of dierent, non equivalent in general, metalogical properties con ate on a structurally rich calculus such as the classical rst order calculus with equality. In our case, the instances of CI and CRI are equivalent to ordinary interpolation in rst order logic with equality. This is basically for the following reasons: (a) all pushout squares in a category of rst order signatures (alphabets) are also pullback squares; (b) equality (of language translations) can be internalised. (Hence, some structural properties that are implicitly introduced via a translation gram(i):gram(S )!gram(T ) can be axiomatised by a set of de nitions in the ( rst order) language of a suciently large common extension of S and T . See [33] for examples of such an internalisation.); (c) the entailment of (many sorted) classical rst order logic is compact; (d) the entailment of (many sorted) classical rst order logic enjoys a deduction detachment property: for every set A of rst order sentences and all rst order sentences ; in the language of , A[fg` ' i A` ! '.

CRI, compactness and deduction detachment Proposition 3.1 If E is compact then, for each ', the set of interpolants is nite. If, in addition, E admits deduction detachment, ie. there is a deduction theorem for E , then the set of interpolants I A;B;'; is independent of B. Proof: By the de nition of a compact Entailment System E , for each A, B and ', one can nd a nite subset Io A;B;'; of the interpolants I A;B;'; , and a nite subsect Bo A;B;'; of B such that Iio A;B;'; [Bo A;B;'; ÈB '. Furthermore, if E admits deduction detachment, one can internalise the nite set of assertions Bo A;B;'; , togetherE with the conclusion ', into a sentence B;' 2 gram(B) such that Iio A;B;'; `B B;' . (

D)

(

(

D)

D)

(

(

(

D)

(

D)

(

D)

D)

(

D)

(

)

)

In general, for compact Entailment Systems that admit deduction detachment and internalisation of equality/equivalence, such as (both classical and intuitionistic) propositional and rst order logics, CRI and CI con ate.

CRI and CI under compactness and deduction detachment Corollary 3.1 CRI and CI con ate for compact Entailment Systems that admit deduction detachment. Proof: Immediate by the above de nitions of CRI and CI and the Proposition 3.1.

3.1 Calculi that possess CRI

Among the concrete calculi that appear in the computing literature, those that possess CRI include: the Minimal Calculus [3], Heyting's Intuitionistic propositional calculus [16], the Classical Propositional Calculus [5, 12, 1, 20], the In nitary Propositional Calculus [29, 12, 18], the Unary First Order calculus (and its many-sorted variants), the modal K, K4 and T [17], S4, S4:1 [17, 3], S4 fp $ p : p 2 gramS4 ( )g [3], Lob's calculus(the modal GL) [28, 3], Grzegorczyk's calculus (the modal S4Grz) [3], propositional Polymodal calculus (the Hennessy-Milner logic) [36, 6], the -calculus [6], the Intuitionistic First Order calculus with equality [16, 23], the Classical First Order calculus with equality [5, 12, 1, 20], the In nitary First Order calculus L!1 ! with equality [4, 1, 18], the Intuitionistic Second Order Calculus with (predicate) variables of all arities [6], the Classical Weak Second Order Calculus with (predicate) variables of all arities [6], the Classical Standard Second Order Calculus with (predicate) variables of all arities [6]. 4

The calculi that do not possess CRI include: the (conditional) equational logic (and its many-sorted variants) [27, 7], Horn-clause logic, classical monadic second order logic and intuitionistic monadic second order logic [6], amongst others. In fact, based on [3], one should expect a continuum of (proper) extensions of the modal S4, all of which but for a nite number do not possess CRI.

3.2 Remarks

1. An instance of CRI on compact calculi with a rst order syntax may appear as the following variant of interpolation, called \splitting interpolation" in [27]: For a ( rst order) formula ' and sets of ( rst order) formulae A; B, if A[B`', then there are ( rst order) formulae #1; : : :; #n in the common language of A[B and ' such that: (a) A`#i , 1 i n, and (b) B [f#1; : : :; #ng`'. 2. A property similar to CRI rst appeared explicitly in the literature in [20] where it was established for the intuitionistic propositional calculus. (See also [1].) \Maehara Interpolation" { a term now commonly used in sentential logic{ and CRI con ate for compact calculi on propositional and predicate languages. In the computer science literature, the terms \Splitting", \Strong" and \Pushout" Interpolation have been used to designate properties equivalent to the corresponding instances of CRI for dierent calculi on propositional and predicate languages. (See [27, 30, 14, 7] and [15], amongst others.) Finally, the term \Craig-Robinson interpolation" originally appeared in Shoen eld's book [29] and recently reappeared in [33] and [9]. 3. Note that if, for some appropriate grammar, one can state CI using unions and intersections of signatures (as for example is done in [7]), then this would be in a sense the most basic generalisation of the original formulation of the property that is stated in Craig's Interpolation Lemma [5] for classical predicate logic.

4

CRI

and Modularisation

The Modularisation property is captured in the Entailment Systems framework by the following property MP which asserts the stability of the class of faithful morphisms (and their subclass of conservative extensions) under pushouts:

De nition 4.1

MP: an abstract formulation of modularisation An Entailment SystemE possesses MP i the pushout eBAi +Be :hB ; Bi!hC; Ai [Be i of any faithful morphism (resp. conservative extension) eRA :hR ; Ri!hA; Ai along any morphism iRB :hR ; Ri!hB; Bi is also faithful (resp. conservative).

Sign-diagram

A

i

e

D

R

0

0

0

i

0

0

Pres-diagram hA ; Ai

C e

B

eRA 0

hR ; Ri 5

iAC 0

Dpres iRB

hC ; Ai [Be i 0

0

eBC 0

hB ; Bi

0

Writing MP[A; B; R; D] denotes the specialisation of the above stated MP-condition to a given diagram Dpres presenting the pushout completion of a span hA ; Ai eRA ?hR ; Ri? iRB !hB ; Bi in Pres, and having D as the underlying square of Sign. A proof that CRI is necessary and sucient for MP on any arbitrary Entailment System

E (Corollary 4.1, inspired from [14]) follows from our main contribution, Proposition 4.1, together with two novel, critically stronger results (Corollaries 4.3 and 4.4). The rst of these shows the above equivalence to also hold locally in the locus of all theory presentations and theory presentation morphisms on an arbitrary but xed Sign-diagram D, while the second provides a necessary and sucient condition under which all pushouts of a given faithful theory (presentation) morphism are faithful.

Interpolation and Modularisation Proposition 4.1 CRI[A; B; D] is equivalent to MP[A; B; R; D] for every diagram D depicting the pushout completion of the span A e ? R ? i!B , all sets of sentences A gram(A ), B gram(B) and R gram(R ). Proof: ) To prove that CRI[A; B; D] implies MP[A; B; R; D] is the same as proving that CRI[A; B; D] together with the assumption that eRA :hR ; Ri!hA ; Ai is faithful implies that eBC :hB ; Bi!hC ; Ai [Be i is faithful. For the latter, it suces to show that for every ' in gram(B ), if Ai [Be ÈC 'e then BÈB '. By CRI[A; B; D], for every ' in gram(B ), if Ai [Be ÈC 'e , there is a set I A;B;'; of interpolants such that (i) AÈA Ie A;B;'; , and (ii) Ii A;B;'; [BÈB '. By the faithfulness of e, the above statement (i) implies RÈR I A;B;'; , and, hence, BÈB Ii A;B;'; , since hB; Bi interprets hR ; Ri via i. The latter, together with statement (ii) above and the transitivity of È imply that BÈB '. ( To show the inverse implication, it suces to invent an interpolant presentation on R 0

0

0

0

0

(

0

0

D)

0

(

(

D)

D)

(

(

0

D)

D)

or to establish the existence of such a presentation. We claim that the axiomatisation R of hR ; Ri is one such, given that (a) the morphism eRA:hR ; Ri!hA ; Ai is faithful, and (b) the faithfulness of eRA implies the faithfulness of its pushout eBC along i. Indeed, AÈA Re and, since the pushout e of e along i is faithful, then Ri [BÈB ', for any ' in gram(B ) such that Ai [Be ÈC 'e . This is because Ai [Be ÈC 'e , implies BÈB ' by faithfulness for e and then Ai [BÈB ' by monotonicity. Furthermore, if E is compact, then, for every A, B and ' Eas above, there is a nite subset Io (A;B;'; ) of I(A;B;'; ), such that Iio (A;B;'; )[B`B '. 0

0

0

0

0

0

0

D

0

D

0

0

D

If E possesses CRI (respectively, possesses MP), then, the above Proposition 4.1 implies that E possesses MP (respectively, possesses CRI). Hence, we obtain a generalisation of a \Modularisation Theorem" for faithful morphisms on an arbitrary Entailment System.

A generalised Modularisation Theorem Corollary 4.1 E possesses CRI i faithful theory (presentation) morphisms are stable under pushouts in Pres.

Moreover, if subsignature inclusions are stable under pushouts then the specialisation to conservative extensions is immediate: 6

Corollary 4.2 CRI guarantees the stability of conservative extensions under pushouts over Entailment Systems where subsignature inclusions are stable under pushouts.

The following two corollaries of Proposition 4.1 are easily derived by generalising the statement to all theory presentations over a given pushout diagram D of Sign. That is, by shifting the universal [meta]quanti cation on sets of sentences (theory presentations) inside the equivalence.

CRI and Modularisation over a locus Corollary 4.3 Let D be an arbitrary but xed pushout diagram of Sign as is depicted in 4.1. Then CRI holds locally in the (grammatical) locus of all sentences/theory presentations on D, i all

faithful morphisms on e are stable under the pushout along a morphism on i.

Corollary 4.4

CRI and stability of property conservation A faithful theory (presentation) morphism eRA :hR ; Ri!hA; Ai on e:R !A, is stable under pushouts in Pres i for every signature morphism i:R !B over R in Sign, such that the pushout of e along i exists, and every B gram(B ), CRI holds locally for A, B and D where D is the Sign-diagram presenting the pushout of e along i.

4.1 Remarks

1. Note that CI is in general weaker than CRI and, hence, in itself CI is necessary but not sucient for modularisation. Also remember that its instances over concrete calculi may not identify with Craig-interpolation in its ordinary form. In general on should expect a concrete instance of CI to be weaker than Craig-interpolation in its ordinary form. In the particular case of rst order logic, though, the concrete instances of CI and CRI with each other and with ordinary interpolation. 2. CI generalises and essentially weakens ordinary Craig-interpolation by (i) abstracting the implication to a class of derivations (an entailment), (ii) generalising the assumption of an implication to the set of assertions in an entailment, (iii) generalises the interpolant sentence to a set of interpolants on the shared language (an interpolant presentation), (iv) generalises the (extra-logical) langauge sharing essentially to a pushout in an appropriate category of languages. CRI, on the other hand, strengthens CI by (i) splitting the set A of primary and a set B of secondary assertions in dierent and possibly overlapping languages and (ii) requiring that an interpolant presentation exists not only for every set A of primary but also for every set B of secondary assertions. MP generalises Modularisation up to faithful morphisms and draws a line between the \logical" and the \linguistic" assertions underlying modularisation. 3. Finally, note that the above Corollaries 4.3 and 4.4 neither assume nor imply that the underlying Entailment System possesses CRI (and therefore MP) globally.

5 Conclusion The Modularisation property [22, 31, 34], originally formulated for logical speci cations ( rst order theory presentations), essentially states that property conservation is stable under the amalgamation of theory interpretations. The rst attempt to analyse this property, based on an observation of M. Sadler [22], indicated a strong interconnection between the Modularisation property and interpolation: The stability of conservative extensions under amalgamation was equivalent { in rst order logic { to ordinary interpolation. The important role of interpolation for various properties related to the modularity of a speci cation formalism was also observed by a group working with Bergstra [2], who attributed 7

the lack of certain modularity properties in their formalisms to the absence of the interpolation property. Further work on this, revealed the important observation, put forward by Rodenburg and van Glabbeek in [27], that in other logics, such as (conditional) equational logic, the various formulations of Craig interpolation are not equivalent and only some of them are relevant to Modularisation. As we elaborated in this paper, the CRI (De nition 3.2) is the critical form of interpolation which is relevant to Modularisation. Of even greater import than the above, it was noted that the original Modularisation property was actually asserting the stability of faithful morphisms, and in particular conservative extensions, under pushouts in the category of rst order theories. The observation having been made, there was an obvious route to generalisation [14, 15] to ?Institutions and Entailment Systems which lead to Corollary 4.1. The proof of Proposition 4.1 indicated that the relation between CRI and the stability of property conservation under pushouts could also be studied as a phenomenon over appropriate loci of an Entailment System. This new observation lead to Corollaries 4.3 and 4.4. As we illustrated in this paper, these results appear to have the status of general laws of speci cation theory. Corollary 4.4 in particular links a suitable localisation of CRI with a necessary and sucient condition for all pushouts of a given faithful theory (presentation) morphism to be faithful morphisms. Hence, an appropriate specialisation of Corollary 4.4 to the Entailment System of (conditional) equational logic with algebraic initial semantics may indicate a connection between localisations of CRI and persistence. Recall that Proposition 4.1 and both Corollaries 4.3 and 4.4 do not assume nor imply that the underlying Entailment System E possesses CRI as a global metalogical property.

Appendix: predicate extensions of rst order arithmetic The calculus of conditional equational logic with initial algebra semantics is a well-known example [11] of a concrete Entailment System which lacks CRI and therefore lacks Modularisation [27] (although it possesses CI as is illustrated in [27]). In order to emphasise the strength of the Proposition 4.1 and Corollary 4.1 further, we indicate another interesting calculus, namely the calculus of predicate extensions of rst order arithmetic, which possesses CRI, and therefore MP, but lacks ordinary interpolation. The example is based on a note given to the authors by Professor Grigori Mints in the summer of 1998. (Hints that ordinary interpolation does not hold for the predicate extensions of rst order arithmetic may also be found in [19].) Let E = hSign[E ]; gram[E ]; È i be the Entailment System presenting the calculus of the predicate extensions of the rst order theory of arithmetic. Sign[E ] is the category of those rst order signatures that expand the signature FOA of rst order arithmetic with new predicate symbols,E gram[E ] is the restriction of the grammar functor of rst order logic to Sign[E ] and ` is the Sign[E ]-indexed family of entailment relations that is produced by extending the rst order entailment `FOL by hiding the axioms of rst order arithmetic (including the rst order induction schema) inside the entailment relation. The standard interpolation theorem for rst order logic does not apply for E because there can be sentences and ' such that ! ' is established by induction, in which case the induction axiom that is used may depend on the union of the extra-logical (extension) predicate symbols that appear in and '. Applying the standard deductive proof of the ordinary interpolation theorem using a cut free Gentzen-style proof system fails, on : : : ! (n) : : : Indeed, when an interpolant #n for the other hand, at the !-rule: ! 8x (x) each premise ! (n), where n 2 N, has been determined, one has to take an in nite 8

conjunction ^n#n as the overall interpolant and this is not allowed by the syntax. Whereas one can consider the possibly in nite set f#n : n 2 Ng as the interpolant presentation for CRI. In fact, the assumption that ordinary interpolation holds in the predicate extensions of rst order arithmetic may easily lead to a contradiction: Let T(P) be a formula in the language of FOA +fP g that expands the language of rst order arithmetic with a new predicate symbol P and assume that T(P) states that P satis es the standard inductive truth de nition T(P) 8x(SentenceCode(x) ! [x; P(x)]^[x; P(x)]^[x; P(x)] for the prenex sentences of arithmetic, where [x; P(x)] QuantifierFree(x) ! (P(x) $ To (x)), and [x; P(x)] 9z 9(x = GodelNumber(9z)) ! (P(x) $ 9nP(subst(; z; n))), and [x; P(x)] 9z(x = GodelNumber(8z)) ! (P(x) $ 8nP(subst(; z; n))), and To is a standard truth de nition for the quanti er-free sentences. One can prove (by induction on the construction of the formula with code x) that T(P) uniquely de nes P, and so, for any other extension predicate symbol Q, T(P)^T(Q) ! (P(x) $ Q(x)). Hence, T(P)^P(x) ! (T(Q) ! Q(x)). If we assume that ordinary interpolation holds for the predicate extensions of rst order arithmetic, then there should be an interpolant sentence #(x) (in the shared language of FOA ) such that (T(P)^P(x) ! #(x))^(#(x) ! (T(Q) ! Q(x))) is a theorem of the theory of rst order arithmetic with two additional predicate symbols. Since T(P) uniquely de nes P, one can substitute P for Q in the above and obtain T(P) ! 8x(P(x) $ #(x)) which can then be interpreted into the standard model of arithmetic and, by substituting for P the truth predicate for arithmetic, yield that T(#) is true in the standard model of arithmetic. But this contradicts Tarski's unde nability theorem because # is in the language of rst order arithmetic and, by Tarski's unde nability theorem, there is no arithmetical formula # such that T(#) is true in the standard model of rst order arithmetic.

References

[1] G. Takeuti. Proof Theory. North-Holland Publishing Co., Amsterdam, 1976. [2] J.A. Bergstra, J. Heering, and P. Klint. Module algebra. ACM, 37(2):335{372, 1990. [3] Alexander Chagrov and Michael Zacharyasev. Modal Logic. Number 35 in Oxford Logic Guides. Clarendon Press, Oxford, Oxford University Press, Great Clarondon Street, Oxford OX2 6DP, 1997. [4] C.C. Chang and H.J. Keisler. Model Theory, volume 73 of Studies in Logic (and the foundations of Mathematics). North-Holland publishing company, 1973. [5] W. Craig. Three uses of the Herbrand-Getzen theorem in relating model theory and proof theory. Journal of Symbolic Logic XXII, pages 269{285, 1957. [6] Giovanna D'Agostino and Marco Hollenberg. Uniform interpolation, automata and the modal calculus, May 1996. Available at http://www.phil.ruu.nl/home/marco/. [7] Razvan Diaconnescu, Joseph Goguen, and Petros Stefaneas. Logical support for modularisation. In Gerard Huet and Gordon Plotkin, editors, Logical Environments, pages 83{130. Cambridge University Press, 1993. [8] T. Dimitrakos. Parameterising (algebraic) speci cations on diagrams. In Automated Software Engineering{ASE'98, 13th IEEE International Conference, pages 221{224, 1998. An extended revised version available at http:www-tfm.doc.ic.ac.uk/td/publications. [9] T. Dimitrakosand T.S.E. Maibaum. Notes on re nement,interpolationand uniformity. In Automated Software Engineering{ASE'97, 12th IEEE International Conference, 1997. [10] T. Dimitrakos and T.S.E. Maibaum. On the role of interpolation in stepwise re nement. In Proceedings of the First Panhellenic Symposium on Logic, July 1997. [11] H.-D. Ehrich. On the theory of speci cation, implementation and parametrization of abstract data types. J.ACM, 29(1), 1982. [12] H. B. Enderton. A Mathematical Introduction to Logic. Academic Press, 1972.

9

[13] J.L. Fiadeiro and A.Sernadas. Structuring theories on consequence. In D. Sannella and A. Tarlecki, editors, Recent Trends in Data Type Speci cation, pages 44{72, 1988. [14] J.L. Fiadeiro and T.S.E. Maibaum. Stepwise program development in -institutions. Technical report, Imperial College, 1990. [15] J.L. Fiadeiro and T.S.E. Maibaum. Generalising interpretations between Theories in the Context of (-)institutions. In S. Gay G. Burn and M. Ryan, editors, Theory and Formal Methods, pages 126{147. Springer-Verlag, 1993. [16] D.M Gabbay. Sematnicproof of the craig interpolationtheorem for intuitionisticlogic and extensions. In R.O. Gandy and C.M.Y. Yates, editors, Logic Colloquium'69, pages 391{401,496{503,Amsterdam, 1971. North-Holland. [17] D.M Gabbay. Craig's interpolation theorem for modal logics. In W. Hodges, editor, Porceedings of Logic Conference, London, 1970, volume 255 of Lecture Notes in Mathematics, pages 11{127, Berlin, 1972. Springer-Verlag. [18] Jean-Yves Girard. Proof Theory and Logical Complexity, volume 1. Bibliopolis, 1987. [19] G. Kreisel. Survey of proof theory. JSL, 1968. [20] S. Maehara. Craig's interpolation theorem. Sugaku, pages 235{237, 1961. In Japanese. [21] T.S.E. Maibaum. The role of abstraction in program development. Information processing '86, 1986. [22] T.S.E. Maibaum and M.R. Sadler. Axiomatising speci cation theory. In H.J. Kreowski, editor, Recent Trends in Data Type Speci cation, pages 171{177. Springer-Verlag, Berlin, 1985. [23] Michael Makkai. On Gabbay's proof of the Craig interpolation theorem for intuitionistic predicate logic. Notre Dame Journal of Formal Logic, 36(3):364{381, Summer 1995. [24] M. Marx. Interpolation, modularization and knowledge representation. In C. Bioch and Y.H. Tan, editors, Proceedings of Dutch Conference on AI (NAIC'95), June 1995. [25] Jose Meseguer. General logics. In H.D. Ebbinghaus, editor, Logic Colloquium'87, pages 275{329, 1989. [26] A. Robinson. Complete Theories. North-Holland, Amsterdam, 2 edition, 1977. [27] P.H. Rodenburg and R.J. van Galbbeek. An Interpolation Theorem in Equational Logic. Technical Report CS-R8838, Centre for Mathematics and Computer Science, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands, 1988. [28] V. Yu. Shavrukov. Subalgebras of diagonalizable algebras of theories containing arithmetic. Dissertationes Mathematicae, Polska Akademia Nauk, Mathematical Institute, Warszawa, 1993. [29] J.R. Shoen eld. Mathematical Logic. Addison-Wesley, Publishing Company, 1967. [30] A. Tarlecki. Bits and pieces of the theory of institutions. In Workshop on Category Theory and Computer Science, LNCS 240. Springer-Verlag, 1986. [31] Wladyslaw M. Turski and Thomas S. E. Maibaum. The Speci cation of Computer Programs. Addison-Wesley, 1987. [32] P.A.S. Veloso. A new, simpler proof of the modularisation theorem for logical speci cations. Bulletin of the IGPL, 1(1):3{12, July 1993. [33] P.A.S. Veloso. From extensions to interpretations: Pushout consistency, modularity and interpolation. Technical report, Departamento de inforamtica Ponti cia Universidade Catolica do Rio de Janeiro, 1996. (To appear in Information Processing Letters 1997). [34] P.A.S. Veloso and T.S.E. Maibaum. On the modularisation theorem for logical speci cations. Information Processing Letters 53, pages 287{293, 1995. [35] P.A.S. Veloso, T.S.E. Maibaum, and M.R. Sadler. Program development as theory manipulations. In 3rd International Workshop on Software Speci cation and Design, 1985. [36] A. Visser. Bisimulations, model descriptions and propositional quanti ers. Technical report, Department of Philosophy, Utrecht University, 1996.

10