This article was downloaded by: [Hacettepe University]On: 19 December 2014, At: 19:10Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

Journal of Applied Non-Classical LogicsPublication details, including instructionsfor authors and subscription information:http://www.tandfonline.com/loi/tncl20

A multimodal logicfor reasoning aboutcomplementarityIvo Düntsch a & Beata Konikowska ba School of Information and SoftwareEngineering University of Ulster atJordanstown , Newtownabbey , BT , 37OQB , N. Ireland E-mail:b Institute of Computer Science PolishAcademy of Sciences , Ordona 21, 01-237 ,Warsaw E-mail:Published online: 30 May 2012.

To cite this article: Ivo Düntsch & Beata Konikowska (2000) A multimodallogic for reasoning about complementarity, Journal of Applied Non-ClassicalLogics, 10:3-4, 273-301, DOI: 10.1080/11663081.2000.10511000

To link to this article: http://dx.doi.org/10.1080/11663081.2000.10511000

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy ofall the information (the “Content”) contained in the publicationson our platform. However, Taylor & Francis, our agents, and ourlicensors make no representations or warranties whatsoever as to theaccuracy, completeness, or suitability for any purpose of the Content.Any opinions and views expressed in this publication are the opinionsand views of the authors, and are not the views of or endorsed byTaylor & Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources of

information. Taylor and Francis shall not be liable for any losses,actions, claims, proceedings, demands, costs, expenses, damages,and other liabilities whatsoever or howsoever caused arising directlyor indirectly in connection with, in relation to or arising out of the useof the Content.

This article may be used for research, teaching, and private studypurposes. Any substantial or systematic reproduction, redistribution,reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of accessand use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multimodallogic for reasoning about complementarity

Ivo Dtintsch* School of Information and Software Engineering University of Ulster at Jordanstown Newtownabbey, BT 37 OQB, N.Ireland I.Duentsch@ulst.ac.uk

Beata Konikowska* Institute of Computer Science Polish Academy of Sciences Ordona 21, 01-237 Warsaw beatak@ipipan.waw.pl

ABSTRACT Two objects o1 , o2 of an information system are said to be complementary with respectto attribute a if a( o1) = -a( 02 ), where a( o) is the set of values of attribute a assigned too. They are said to be complementary with respect to a set of attributes A if they are com­plementary with respect to each attribute a E A. A multi-modallogicallanguage for reason­ing about complementarity relations is presented, with modalities [A] and (A) parameterised by subsets of a given set ATTR of attributes. Two complete deduction system for the language are presented: a Rasiowa-Sikorski style system using signed formulae, and an equivalent sequent calculus system in Gentzen style complete for theories. KEYWORDS: Information relation, Complementarity, Rasiowa-Sikorski System

1 Introduction

In the world around us, entities are perceived by the presence or absence of features, and the

OBJECT >--+ ATTRIBUTE

relationship serves in many cases as a basis for knowledge representation. More for­mally, an information system [Paw73, Lip76, OP87] is a tuple

I= (OBJ, ATTR, {Va}aEATTR),

such that

• OBJ is a nonempty set of objects,

• ATTR is a nonempty set of attributes or features, such that

- Each a E ATTR is a mapping from OBJ to 2v", where Va is a nonempty set of possible values of attribute a.

*The authors gratefully acknowledge partial support by the KEN/British Council Grant No WAR/992/174

Journal of Applied Non-Classical Logics. Volume 10- No. 3-4/2000, pages 273 to 301

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

274 Journal of Applied Non-Classical Logics. Volume 10- No. 3-4/2000

If both OBJ and ATTR are finite, we can, in principle, depict the system as a data table, similar to a table of a relational database.

With each set A of attributes one can associate various relations on the object set, the most widely used being the relationship ind(A) of indiscernability: here, two ob­jects are related if they cannot be distinguished with the knowledge given by the at­tributes in A. In other words, for all x, y E 0 BJ,

(1) (x, y) E ind(A) if and only if a(x) = a(y) for all a EA.

Indeed, there are many types of such relations which can be sensibly derived from an information system, and the reader is invited to consult [Orl95] for more details, and [DGOOO] for a general form of relational representation.

While relations of similarity have been frequently studied and are well understood, the situation with regard to relations which describe the differences of objects is much less clear. One dissimilarity relation which has been studied in some detail is that of (strong) complementarity, where, for a nonempty set of attributes A,

(x,y) E comp(A) ifandonlyifa(x) = Va \ a(y) for all a EA.

[D098] show that camp( A) is

C 1. irrefiexive,

C2. symmetric, and

C3. 3-transitive, i.e. for all w, x, y, z E OBJ,

(w, x), (x, y), (y, z) E comp(A) implies (w, z) E comp(A).

Conversely, they show that for each relation R on OBJ with these properties there are an information system I and a set of attributes A such that R = camp( A).

These relations are not independent, since

(2) comp(u A;)= n comp(A;) iEI iEI

for any family {A; : i E I} of nonempty subsets of ATTR, so, for example,

(3) comp(A U B)= comp(A) n camp( B),

for all nonempty A, B ~ ATTR.

Condition (2) says that two objects are complementary with respect to the attributes in U;EI A; iff they are complementary with respect to all attributes in each A; fori E /. In other words, the larger the nonempty set of attributes, the smaller the corresponding complementarity relation.

It was not without reason that above we limited our considerations to nonempty sets of attributes only. Indeed, it is not obvious how complementarity corresponding to an empty set of attributes is to be understood. One approach is based on remarking that

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multi modal logic 275

in case of an empty set of attributes every object is assigned an empty set of attribute values, and the overall &et of attribute values is also empty, so 0 \ 0 = 0 implies that we could formally assume comp(0) = OBJ x OBJ. However, this is counterintuitive, since the whole idea behind the complementarity relation is that the attribute values for the considered objects are totally different- and in case of an empty universe of values this cannot happen for any two attributes. Hence from the intuitive viewpoint is natural to assume an additional condition

(4) comp(0) = 0.

- which is what we shall do here. The relations thus obtained are parameterised by the sets of attributes, and one

speaks of the family (OBJ, comp(A)A~ATTR) as a family of complementarity relations parameterised by ATTR.

In this paper we will develop two complete formal systems of modal logic for rea­soning about relative complementarity relations. The deduction systems we shall de­velop will be :i Rasiowa-Sikorski style sequence formalism based on the use of signed formulae [RS63], and a Gentzen-style sequent calculus complete for theories obtained from the former system. The approach which we adopt follows closely the one used by [Kon97 a, Kon97b] to develop a logic for reasoning about parameterised similarity relations. Different systems, based on sufficiency (as opposed to modal) operators and relative semantics have been presented by fDOOO].

2 The language

The language£:, is parameterised by a fixed nonempty set ATTR. The reason for fix­ing that set is that in a real-life information system the set of objects can change quite fast, whereas the set of attributes used for characterising these objects remains as a rule constant throughout the lifetime of a current system version.

The alphabet of£ is the disjoint union of the following sets:

CONA = {a : a E ATTR} A set of constants representing individual attributes, one constant a for each single at­tribute a.

VARSA

{0}

VARO

VARSO

{-,u,n}

{•,V,A}

{(),[]}

A set of variables representing nonempty sets of attributes. A constant representing the empty set of attributes. A set of variables representing individual objects. A set of variables representing sets of objects. Symbols for set-theoretic operations on sets of attributes. Symbols for set-theoretic operations on sets of objects. Symbols for modalities.

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

276 Journal of Applied Non-Classical Logics. Volume 10- No. 3-4/2000

There are two kinds of expressions:

1. Terms, representing sets of attributes.

2. Formulae, representing sets of objects .

The set TERM of terms is the smallest set satisfying the following conditions:

(i) VARSA U CONA U {0} <;:;:TERM,

(ii) If A, BE TERM, then -A, AU B, An BE TERM.

The set FORM of formulae is the smallest set satisfying the following conditions:

(i). VARO U VARSO <;:;: FORM.

(ii). IfF, G E FORM, then •F, F V G, F 1\ G E FORM.

(iii). If A E TERM and F E FORM, then [A]F, (A)F E FORM.

We shall also use the derived operator--+ which serves as an abbreviation for

(5) •FVG.

3 Semantics of£

A complementarity frame, or just frame, is a pair

(6) F = < OBJ, {comp(A)}A\;ATTR >,

where OBJ is a nonempty set of objects, ATTR a fixed nonempty set of attributes, and {camp( A)} ACATTR is a family of relations on OBJ which satisfy Cl, C2, C3, 4, (2) on page 2. For each A <;:;: ATTR, we call comp(A) the complementarity rela­tion corresponding to (the attributes in) A. Except for (2), we do not put any restric­tions on the assignment A I-t camp( A). A model of£ is a pair M = (F, v), where F = (OBJ, { comp(A)} A\;ATTR) is a frame as in (6), and vis a (multi-sorted) valuation such that

• v(A) <;:;: ATTR for A E VARSA,

• v(O) <;:;: OBJ for 0 E VARSO,

• v(x) E 0BJ for x E VARO.

The valuation v is extended in a natural way to an interpretation TM of terms and an interpretation 'PM of formulae by interpreting 0 as 0, each constant a E CONA as the corresponding attribute a E ATTR (or, rather, as the singleton set containing a), the symbols-, U, nasset-theoretical operations on sets of attributes, •, V, 1\ as set­theoretical operations on sets of objects, and (A), [A] as the possibility and necessity modalities corresponding to the accessibility relation camp( TM (A)).

More precisely, the interpretation of terms in M is a function TM : TERM --+ 2ATTR defined inductively as follows:

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multimodal logic 277

(i). TM (a) = {a} for any a E ATTR, where a is the unique constant in CONA rep­resenting a,

(ii). TM(O) = 0,

(iii). TM(A) = v(A) for A E VARSA,

(iv). For any A, B E TERM,

TM(-A) = ATTR \ TM(A),

TM(AUB) = TM(A) UrM(B),

TM(A n B)= TM(A) n TM(B).

The interpretation of fm mulae is a function rpM : FORM --+ 20BJ such that for all x E VARO, 0 E VARSO, F, G E FORM, and A E TERM,

'PM(x) = {v(x)},

'PM(O) = v(O),

'PM(-..,F) = OBJ \ i.f'M(F),

'PM(F v G)= 'PM(F) u 'PM(G),

'PM(F A G)= 'PM(F) n 'PM(G),

'PM([A]F) = {o E OBJ: (Vo' E OBJ)((o,o') E comp(rM(A)) --to' E 'PM(A))}

'PM((A)F) = {o E 0BJ: (:3o' E 'PM(A))(o, o') E comp(rM(A))}.

A formula F E FORM is said to be true in a model M, written I= M F, iff rpM (F) = OBJ, i.e. iff it evaluates to the whole object-universe of this model. A formula F is called valid iff I= M F for every model M.

It can be easily seen that, in general, neither the formula F nor the formula -..,p need hold in a given mC1del. Consequently, due to our notion of satisfaction, -.., does not directly correspond to non-satisfaction, which should be kept in mind for things to come.

4 Signed formulae

In the development of our first deduction system, we shall use signed formulae, which are a well-known tool for expressing satisfaction and non-satisfaction of formulae. This type of "internalising" semantical notions has also been used successfully on the object level, see e.g. [I:la98]. The signs are dropped when we pass to the Gentzen deduction system, and the system we obtain is a sequent calculus for the original lan­guage. Syntactically, signed formulae are simply formulae in FORM preceded by one of the signed formula constructors T, N. More formally, the set SFORM of signed for­mulae is

SFORM = {T(F) :FE FORM} U {N(F): FE FORM},

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

278 Journal of Applied Non-Classical Logics. Volume 10- No. 3-4/2000

where T stands for "true", or "satisfied", and N for "false", or "not satisfied". The interpretation of signed formulae is a function (J"M : SFORM -+ {tt, ff},

where tt denotes truth and ff falsity, defined by

(J"M(T(F))={ tt iffi=M F, ( )) { tt iffnoni=M F, ff otherwise, (J"M(N F = ff otherwise

For a signed formula G, we say that G is true in M, and write I= M G, iff(]" M (G) = tt. We say that G is valid- and write I= G -if and only if I= M G for any model M.

The deduction system we are going to develop will be based on formulae express­ing the membership relation, namely,

(7) def X E F =X-+ F,

Special instances of such formulae are x E y and

(8) x cp(A) y ~r x E (A)y.

Here x, y E VARO, FE FORM, A E TERM. Since

FM F-+ G iff <t'M(F) ~ <t'M(G),

it can be easily seen that formulae of type (7) indeed represent the membership rela­tion with x E x representing equality, and formulae of type (8) the complementarity relation. For the corresponding signed formulae we have

(9) FM T(x E F) iffv(x) E <J'M(F),

(10) FM N(x E F) iffv(x) tf. <J'M(F)

(11) FM T(x cp(A) y) iff (v(x), v(y)) E comp(rM(A)),

(12) FM N(x cp(A) y) iff (v(x), v(y)) tf. comp(rM(A)),

(13) FM T(x E y) iffv(x) = v(y),

(14) FM N(x E y) iffv(x) =f. v(y).

Indeed, for (9),

<J'M(x E F)= <J'M('x V F)= (OBJ- {v(x)}) U <J'M(F),

whence FM T(x E F) iff <t'M(x E F)= OBJ iffv(x) E <t'M(F). Furthermore, (10) - ( 12) are just special cases of (9).

The formulae above provide the classical dichotomy of satisfaction vs non­satisfaction, not expressible directly in our original language, and are very useful for developing the deduction system. This is expressed in the following result from [Kon97a] (see also [Gab81]):

Lemma 1. For any formula F E FORM:

(i) T(F) is valid if!T(x E F) is valid, where x E VARO is any variable not occur­ring in F,

(ii) N(F) is valid whenever for any model M there is a variable x E VARO such thatN(x E F) is true in M.

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multi modal logic 279

5 Components and transformation to normal form

In order to cope with the modalities corresponding to various sets of attributes, we use the "component" approach described in [Kon97 a, Kon97b] for the case of the similarity logic: We replace the terms appearing in signed formulae by unions of certain special terms called "components" which evaluate to a disjoint cover of ATTR in any model.

For an arbitrary finite sequence n of signed formulae, let

CoNA(n) ={a E CONA: a occurs inn},

and

VARSA(n) = {A E VARSA :A occurs inn}

be the set of all constants in CON A and all variables in VARSA that appear in the terms ofn.

Suppose that

where n, m ?: 0. The sequence n is said to be nondegenerate iff m + n > 0, i.e. iff it contains symbols from VARSA U CONA 1.

Suppose that n is nondegenerate, and

for any term A. We denote

SCOMP(n) ={Q~1 n ... n Q~ : i1, ... , im E {+,-} },

COMP(n) ={aj n S: S E SCOMP(n), 1 'S j 'S n} U

{--a1 n ... n-ann S: S E ScoMr(n), 1-:; j-:; n}.

The elements of COMP(n) are called components for n, and those of ScoMr(n) subcomponents. Note that the notion of component is defined for nondegenerate se­quences only. By a positive component we mean any component of the form aj n S. Components will be always denoted by a suitably indexed C, and subcomponents by S. For nonpositi ve components we shall often use the shorthand notation of the form -an S, which should be read as -a1 n ... n-ann S, where a= {a1 , ... , an}·

The components and subcomponents have the following important properties, see [Kon97b] for a proof:

Lemma 2. For any nondegenerate sequence n of signed formulae and any model M,

1. The sets { TM (C)} CECOMP(n) form a disjoint cover of ATTR, i.e.

1The reader should note that 0 belongs neither to CONA nor to VARSA, whence any sequence whose terms contain no symbol for an attribute or set~ of attributes other than 0 is considered as degenerate.

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

280 Journal of Applied Non-Classical Logics. Volume 10- No. 3-4/2000

(a) TM(C) n TM(C') = 0forany C, C' E COMP(O) such thatC # C';

(b) UcECOMP(n) TM(C) = ATTR.

2. The sets { TM ( S)} 5 E SCOMP(n) also form a disjoint cover of ATTR.

3. For any term A occurring inn, either

(i) A is semantically equivalent to 0, or

(ii) there exists a unique subset { C1, ... , Ck} ofCOMP(O) such that the term C1 u ... u Ck is semantically equivalent to A in all models.

It is easy to see that every term A appearing in a degenerate sequence n is seman­tically equivalent to either 0 or -0.

Thus, for any term A in any sequence n, by thenormalform n(A) of A (with respect to 0) we shall understand either the sequence 0 or -0, if A is semantically equivalent to one of the above, or else

( ) def

n A = Ct U ... U Ck

where { C1 , ... , Ck} is a set of components with union semantically equivalent to A defined in Lemma 2.

The sequence obtained from any sequence n (degenerate or not) by replacing ev­ery term inn by its normal form with respect ton is denoted by n(O) and called the normal form ofO. The algorithm for obtaining n(O) is just a simple modification of the standard algorithm of transformation into complete disjunctive normal form.

By Lemma 2(3), the interpretation of n(O) in any model coincides with the inter­pretation of n. Hence, our deduction system will only be concerned with sequences in normal form.

6 DRS: a deduction system for signed formulae in Rasiowa- Sikorski style

The type of deduction system we are going to develop was introduced in [RS63], and will be henceforth called an R-S system. It consists of decomposition rules for se­quences of signed formulae, and of axiomatic sequences to be defined below. Us­ing a more common terminology, the decomposition rules are the "inference rules" of the system, whereas the axiomatic sequences represent its "axioms", or rather axiom schemes. An R-S system is dual to a tableau system of the type used in [Bet59] or [Fit83]. One basic difference is that, in contrast to a refutation based tableau system, we try to show the validity of a formula or sequence of formulae. We construct a de­composition tree with vertices labelled by sequences of formulae whose branches ter­minate "correctly" only if we encounter a simple, axiomatic sequence of formulae -like T(F), N(F), which is guaranteed to be valid; a sequence of formulae then is valid iff it has a finite decomposition tree all of whose branches end in axiomatic sequences. The other difference is that in the tableau system an application of a rule to a vertex v takes into consideration the labels of all the ancestors of v, whereas in an R-S system

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multimodal logic 281

the vertex explicitly inherits all the necessary information from its ancestor. In apply­ing a rule to a vertex we need not consider anything but the label of the vertex.

A sequence 0 = G 1 , G2 , ... , G n of signed formulae is called true in a model M, written FM 0, iff FM G; for some 1 ::; i ::; n. 0 is called valid iff FM 0 for every model M. A decomposition rule is either a pair 01, 02 or a triple 01, 02, 03 of sequences of formulae in SFORM, usually written as

respectively. 01 is called the conclusion of the rule, and 02 (02' 03) its premise (premises). A rule is said to be sound provided its conclusion is valid iff all its premises are valid.

Thus, a decomposition rule is sound iff it leads from valid sequences to valid se­quences both "downwards" and "upwards". Such a "two-way" notion of soundness, crucial for R-S systems, is a stronger than the usual one, and amounts to invertibility of rules in the common terminology. To underline this fact, we separate the premises from the conclusion in the decomposition rules by a double line instead of the usual single one.

It should be stressed that in our deduction system, the decomposition of a sequence of formulae will be preceded by transforming it to normal form; in other words, all decomposition rules will be applied to sequences in normal form only.

A sequence 0 of formulae is called axiomatic if it contains one of the formulae ( 15) - (17) or pairs of formulae (18), (19) given below:

(15)

(16)

(17)

(18)

(19)

N(x cp(O) y),

N(x cp(A) x),

T(x Ex),

T(F), N(F),

N(x cp(aj n S) y), N(x' cp(aj n S') y'),

where S, S' are two different subcomponents in SCOMP(O).

Lemma 3. The signed formulae (15)- (19) are valid.

Proof The validity of ( 15)- (17) follows from C1, C2 and (9), (10), and, clearly, the pair (18) is also valid.

For the proof of (19), consider a arbitrary model M. Since S, S' are two different subcomponents, then, by Lemma 2, the sets TM (S) and TM (S') are disjoint. Now, TM (aj) = { v(aj)} is a .>ingleton, and it follows that either TM(aj) n TM (S) = 0 or TM(a.i) n TM(S') = 0. In other words, either TM(a.i n S) = 0 or TM(aj n S') = 0. Sinceby4camp(0) = 0,eithercamp(TM(ajnS)) = 0orcamp(TM(ajnS')) = 0. In the first case, we obviously have ( v(x), v(y)) fl. camp( TM(a.i n S)), and in the second case ( v ( x'), v (y')) fl. camp( TM ( a.i n S')). Thus, by ( 10), one of the formulae in the pair must be true in M. D

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

282 Journal of Applied Non-Classical Logics. Volume 10- No. 3-4/2000

As any sequence of formulae containing a valid formula is valid, we have

Corollary 4. Every axiomatic sequence is valid.

To define the decomposition rules, we need the notion of an indecomposable signed formula or a sequence of such formulae. Intuitively, this an elementary formula (se­quence) which cannot be broken into any simpler formulae using the decomposition rules.

A formula G E SFORM is said to be indecomposable iff it has one of the following forms:

(i) T(x E 0) or N(x E 0), where x E VARO, 0 E VARSO U VARO,

(ii) T(x cp(C) y) or N(x cp(C) y), where x, y E VARO, and Cis 0, -0, or

(a) C is any component, if ATTR is infinite;

(b) C is any positive component, if ATTR is finite.

Otherwise a formula is said to be decomposable. Note that (ii) is justified by the fact that in decomposing sequences into normal form we split the unions of components into individual components which are considered as "atomic"; clearly, 0 is also "inde­composable", and -0 can be encountered only in the normal form of a degenerate for­mula, which has no components at all; thus, we cannot split -0 into anything smaller. The condition (b) in (ii) follows from the fact that in case of a finite ATTR we can elim­inate all negative components of the form -a n S by replacing them with a union of positive components of the form Ua'ECONA-a (a' n S).

A sequence n of signed formulae is called indecomposable iff all its elements are indecomposable. The decomposition rules for the language of signed formulae are given in Tables 1 - 32 .

In all DRS rules, x, y, z E VARO, F, G E FoRM, !:1' and !:1" are sequences of signed formulae, with the prerequisite that !:1' is indecomposable, A, B E TERM, Cis a component, and S a subcomponent. Since the decomposition process is applied to sequences in normal form only, the only terms encountered in the actually decomposed formulae will be unions of components; thus we need no rules for term constructors other than the union.

The rules (T) and (N), which are sound by Lemma 1, allow us to manage the awk­ward problem of "neither F nor •F holds" by reducing the original formulae (in re­peating steps in case of (N)) to dichotomous formulae of the form T(x E F), N(x E F). This reduction is analogous to that used in first-order logics in case of quantifiers: The semantic conditions for T(F), N( F) to be true involve implicit quantification over all objects, universal in case ofT( F), and existential in case of N( F).

The rules for finite ATTR are justified by the fact that, in case of a finite ATTR, each negative component of the form -anS can be replaced by a finite union of components of the form a' n S over a' E CONA \a.

From (9)- (11) and Lemma 1 we immediately obtain

2 Tables are placed at the end of the paper.

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multimodal logic 283

Lemma 5. The decomposition rules in the (DRS) system are sound.

The general idea of the deduction system is to break down the formulae occurring in a sequence of signed formulae into some elementary, indecomposable parts, whose validity determines the validity of the original sequence. In what follows, the five rules marked ( *) will be called expansion rules, and all others replacement rules. Roughly speaking, replacement rules replace the original formula they act upon by simpler for­mulae, whereas expansion rules add new formulae to the sequence, e.g. close it under some symmetry or transitivity law.

It can be easily seen that

• Replacement rules are applicable to decomposable sequences only.

• To any decomposable sequence of formulae we can apply at most one replace­ment rule; namely, the unique rule applicable to the leftmost decomposable for­mula in this sequence.

• Expansion rules can also be applied to indecomposable sequences.

Note that there may be several expansion rules applicable to a given sequence. To avoid ambiguity, we shall give a method of choosing an appropriate rule to be applied, as well as a variable in the quantifier-like rules.

7 Decomposition trees for sequences of formulae and completeness of the DRS system

In this Section we shall describe the mechanism of using the DRS system introduced in the preceding section to prove the validity of sequences of signed formulae, and show the completeness of the system.

The validity proofs in DRS consist in constructing decomposition trees for se­quences of formulae, using the decomposition rules in the way described below. We shall prove that a sequence is valid iff its decomposition tree is finite and all its branches end in axiomatic sequences only; this amounts to the completeness of the system.

In order to make the decomposition tree unique (i.e. to avoid ambiguity), we need one more notion. Let n be a sequence of signed formulae; a decomposition rule R in DRS is called correctly applicable to n if both the following conditions are satisfied:

(i) R is a rule applicable to n such that its application augments n by some new formula,

(ii) There is no rule with this property that can be applied to a formula or a subse­quence of formulae inn lying to the left of the formula or formulae, to which R can be applied3 .

3 If~ is the relation of "lying to the left", then we assume that for all ( F1 , ... , Fk), ( G1, ... , Gl), we have (Ft, ... , Fk) < (G1, ... , Gl), iff, for some 1 ~ m ~ min(k, 1), Fr = Gr for r = 1, ... , m-1 and Fm ~ Gm, where the latter relation is understood in the natural way.

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

284 Journal of Applied Non-Classical Logics. Volume 10 - No. 3-4/2000

Since there is at most one replacement rule applicable to any sequence of formulae, and (ii) uniquely defines the expansion rule which is correctly applicable to n, we obtain

Lemma 6. At most one decomposition rule is correctly applicable to a any given se­quence n of signed formulae.

Hence, we can talk about the unique ruleR correctly applicable to a given sequence D; this is of a fundamental importance for defining the notion a decomposition tree. To assure an unambiguous choice of variables in the decomposition process, we suppose for the rest of this paper that V ARO is a well-ordered set with respect to some ordering <.

By a decomposition tree DT(D) for a sequence n of signed formulae we mean a maximal binary tree with vertices labelled by sequences of signed formulae defined inductively as follows:

(i) The root of DT(D) is labelled by n(D), i.e. the normal form of n.

(ii) Let v, labelled by I;, be an end node of a branch B of the tree constructed up to now. Then,

(a) We terminate the branch Bat node v if either:

(a!) I; is an axiomatic sequence, or

(a2) I; is indecomposable and no expansion rule is correctly applicable to I;;

(b) Otherwise we expand the branch B beyond v by attaching to this node

(b 1) a single son labelled I; 1, if the unique rule correct! y applicable to I; is I;

of the form=, I;l

(b2) two sons labelled I;1 and I;2, respectively, if the unique rule correctly I;

applicable to I; is of the form I . I;l I;2

In a rule involving the choice of a new variable we choose the first variable with respect to the ordering::::; which does not appear in I;, see [RS63] for the details.

The decomposition tree starts with a single node, labelled by the normal form of n. The initial node is then expanded into a tree by means of the rules in DRS. In case (al) there is no sense to extend branch B any further, since we already have an axiomatic sequence at its end. In case (a2) no replacement rule is applicable to I;, and we can­not augment this sequence by applying an expansion rule, whence branch B cannot be extended beyond node v. Otherwise the branch is expanded by means of the unique correctly applicable rule; the conditions on the choice of variables in quantifier-like rules assure the uniqueness of the extension. Hence, DT(D) is uniquely determined by n(D).

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multimodal logic 285

It is easy to see that DT(Q) may be infinite- due to the (N, T E (A)) rules. Furthermore, a node of DT(Q) is terminal (a leaf), iff its label E is either a axiomatic sequence, or an indecomposable sequence closed under all the expansion rules4

.

In the sequel, we will refer to sequences of signed formulae labelling the terminal nodes of DT(Q) as the terminal sequences of n.

The notion of provability in our system is as follows: A sequence n of signed formulae is called provable in DRS, written as 1-DRS n,

iff its decomposition tree DT(Q) is finite and all its terminal sequences are axiomatic. From the equivalence character of the rules, it is evident that

Lemma 7. The system DRS is sound, i.e. every provable sequence n of signed formu­lae is valid.

The cornerstone of the converse result- the completeness theorem- is the fol­lowing crucial

Lemma 8. For any sequence n of signed formulae, each valid terminal sequence E in DT(Q) is axiomatic.

Proof Let n and E satisfy the assumptions of the Lemma, and assume that E is not ax­iomatic. Since Eisa terminal sequence of DT(Q), it follows thatE is indecomposable and closed under all expansion rules. Hence, by the definition of an indecomposable sequence, each element of E must have one of the following forms:

T(x E 0), N(x E 0), T(x cp(C) y), N(x cp(C) y),

where x, y E VARO, 0 E VARSO U VARO, and Cis either 0 or -0, or any component if ATTR is infinite, and any positive component if ATTR is finite, see the definition on page 10. Let

VARSA(r2) = {Ql,oo· ,Qm},

and set CONA(r2) =a. Then the terms C which can appear in E are as follows:

(i). c = 0.

(ii). C = -0. Since the normal form of any term A with respect to a non-degenerate sequence n is different from -0 by Lemma 2, this case is only possible if n is degenerate, i.e. if all the terms in Q are Boolean combinations of Os,

(iii). If C tf_ { 0, -0} and ATTR is infinite, then C can be either of the form aj n S or of the form -an S, where S E SCOMP(r2); if ATTR is finite, then C can only be of the form aj n s.

Recall that any subcomponentS E SCOMP(r2) is of the form

S = { Q~' n 00. n Q~ : i1, 00. , im E { +,-}}, 4 We say that E is closed under an expansion rule R if either R is not applicable to E or its application

cannot add any new formulae to that sequence.

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

286 Journal of Applied Non-Classical Logics. Volume 10- No. 3-4/2000

where A+= A, A- =-A for any term A. We are going to construct a counterexample to I:, i.e. a model in which I: is not

true. Let p ~ VARO x VARO be a binary relation defined by

p(x,y) ifftheformulaN(x E y) occurs in the sequence I:.

Then pis both symmetric and transitive, because I: is closed under the (sym E) and (t1·an E) rules. Hence the relation

p* = p u {(X' X) : X E VARO}

is an equivalence relation on VARO. As our counter-example we take a modified Herbrand-type model of the form

H = (F, v),

with

F = (OBJ, {comp(A)}Ac;AnR)

where

0BJ = VARO/ p*

is the set of all equivalence classes of p*.

The multi-sorted valuation v is defined as follows:

1. For all x E VARO,

v(x) = [x]p•,

where [x]p• is the equivalence class of p* which contains x. By definition of p, we have

(20) v(x) = v(y) iff the formula N(x E y) occurs in I:

for all x, y E VARO, x f. y.

2. For all 0 E VARSO,

v(O) = {v(x): N(x E 0) occurs iP I:}.

3. To define the valuation of the variables ranging over sets of attributes, recall that

CONA(D) =a= {a1, ... , an},

We consider two cases:

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multimodal logic 287

(a) ATTR is infinite: In this case A' = ATTR \ { a 1 , ... , an} is also infinite. As the set of sub­components SCOMP(Sl) is finite, there exists an injection

<I> : SCOMP(Sl) Y A'

with

<I>(S) =/ aj for all S E ScoMP(Sl) and all j, 1 ::; j ::; n.

By clause (19) on page 9, any sequence containing both N(x cp(aj n S)) and N(x' cp(aj n S') y'), where S # S', is axiomatic. Since, by our as­sumption, ~ is not axiomatic, it follows that for any 1 ::; j ::; n, there is at most one 3 E SCOMP(Sl) such that for some x, y E VARO the formula N(x cp(aj r, S) y) occurs in~- If such an Sexists for a given j, we de­note it by SJ and say that j is positive in ~- We put v( Q) = 0 for any Q E VARSA \ {Q,, ... , Qm} and

v(Qk) = {<I>(S): S E SCOMP(Sl) and Qt occurs inS} U{ aj : j positive in~ and Qt occurs in Sj}

(b) ATTR is finiTe:

In this case the only components that can occur in~ are of the form aj n S, where S E SCOMP(Sl). The definition of v restricted to VARSA is a simplification of the one given above; we put v( Q) = 0 for any Q E VARSA- {Q,, ... , Qm} and

v( Qk) = { aj : j positive in~ and Qt occurs in Sj}

Recall that by convention a = { a 1 , ... , an}, and -a = -a, n ... n -an. One can show that for the interpretation of terms TH induced by the valuation v

(i) 7

(a. n S) = { { aj} if j is ~ositive in~ and S = Sj H J 0 otherwrse.

(ii) If ATTR is infinite, then rH(-a n S) = {<I>(S)} =/ {aj} for any 1 s; j s; n.

Thus, TH evaluates all components occurring in~ to singletons or 0. This is the key property which helps us to define the family {camp( A)} A<;ATTR of complementarity relations on OBJ x OBJ, which is the last step needed to complete the definition of the model. Due to property C3, complementarity relations corresponding to individual at­tributes in ATTR generate the whole family { comp(A)} ACATTR; they can also be de­fined independently of each other, which solves the basic technical problem connected with modalities parameterised by arbitrary sets of attributes.

We begin with defining the family {camp( a) }aEATTR of complementarity relations with respect to individual attributes. There are two cases:

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

288 Journal of Applied Non-Classical Logics. Volume 10- No. 3-4/2000

CASE 1: I; contains the term -0; this can happen only if n is degenerate. Hence in this case, for any formula of the form either T(x cp( C) y) or N(x cp(C) y) occurring in I:, we have either C = 0 or C = -0. We put

comp(a) = {(v(x), v(y)) : the formula N(x cp( -0) y) is in I:}

for each a E ATTR. CASE 2. I: does not contain -0; then, for any formula of the form T(x cp( C) y)

or N(x cp( C) y) occurring in I:, the term Cis either a component or 0. We have to consider two subcases: (i) ATTR is infinite. Then, for any a E ATTR, we define

{(v(x), v(y)) : N(x cp(aj n Sj) y) is in I:} if a= aj, 1 ~ j ~ n,

and j is positive in I:,

comp(a) = { ( v(x), v(y)) : N(x cp( -an S) y) is in I:} if a = <P(S), and

S E SCOMP(O),

0 otherwise.

The definition is correct, because <f> is one-to-one and <f>(S) :f:. aj for j = 1, ... , n. (ii) ATTR is finite. The definition is again a simplification of the definition for in­

finite ATTR: we put

{

{(v(x), v(y)) : N(x cp(aJ n Sj) y) is in I:} comp(a) =

0

Having defined the singleton-based relations, we set

(21)

and

(22)

for all 0 :f:. A ~ ATTR.

comp(0) = 0,

comp(A) = n comp(a) aEA

if a = aj, 1 ~ j ~ n, and j is positive in I:, otherwise.

We shall show that the family { comp(A)} A~ATTR defined above is a family of com­plementarity relations which satisfies conditions C 1 - C3 and ( 4 ). The degenerate Case 1 is clear, and thus, we will only show Case 2.

Conditions C 1 and C3 follow immediately from the definitions (21) and (22). Thus, all that is left to show is C2. Since the intersection of complementarity relations is again a complementarity relation, and by (22), it is enough to consider the case when A is a singleton, say, A = {a}.

For irreftexivity, assume that ([z], [z]) E comp(a) for some z E VARO. Then, there existx,y E VAROandA E TERMsuchthatv(x) = v(y) = [z]andN(xcp(A)y) E I:. Butv(x) = v(y) impliesN(x E y) E I:by(20), whencebyrule(tran E cp) we obtain

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multi modal logic 289

N(x cp(A) x) E :E. (16) now implies that :E is axiomatic, a contradiction. It follows, that comp( a) is irreftexive.

The fact that comp( a) is 3-transitive is proved in a similar way, based on rule (3-trancp).

Finally, symmetry follows directly from rule (sym cp): Since :E is closed under this rule, N(x cp(A) y) E :E implies N(y cp(A) x) E :E for any term A.

Thus, F = (OBJ, {comp(A)}ACATT&) is a complementarity frame as defined in (6), and H = (F, v) is model of our language.

The proof that ~ H :E is by considering each type of signed formula that can occur in :E, i.e.

T(x E y),N(x E y), T(x E O),N(x E 0), T(x cp(C) y),N(x cp(C) y),

where x, y E VARO, 0 E VARSA, Cis either a component or C E {0, -0}, and proving that it cannot be true in H.

For example, ifN(x E y) is in :E, then, by definition, we havev(x) = v(y), whence by (14) on page 6, ~MH N(x E y).

When proving that the T -type formulae are not true, we also make use of the fact that :E is not axiomatic. For example, if T(x E y) is in :E, then x -j:. y, because of (17), and, in addition, N(x E y) cannot occur in :E by (18). This yields v( v) -j:. v(y), whence ~H T(x E y) by (13).

The proofs for the cp-type formulae follow from the equalities giving v( C) for any component C that can occur in :E, as well as from the definition of the complementarity relations camp( a). The details of analogous proof for similarity logic can be found in [Kon97a].

In this way, we arrive at a contradiction, whence a valid terminal sequence :E must be axiomatic. 0

Now we can state the completeness theorem:

Theorem 9. Every valid sequence of signed formulae 0 E SFORM is provable, i.e. it has a .finite decomposition tree whose terminal sequences are all axiomatic.

Proof Suppose n is valid, and recall that the root of DT(O) is labelled by n(O), the normal form of n. If DT(O) is finite, then from the (two-way) soundness of the de­composition rules in DRS it follows that n(O) is valid iff all the terminal sequences of DT(O) are valid. However, by Lemma 8, the later holds iff all of these are axiomatic. Thus if n is valid and DT(O) is finite, then also n(O) is valid, and hence all the ter­minal sequences of DT(O) are axiomatic.

To complete the proof we have to prove that if DT(O) is infinite, then n cannot be valid. We achieve this by modifying the standard proof used in [RS63], p. 302, to suit the structure of our lanruage. The details are analogous to those in [Kon97a], so we shall give only the basic outline of the method.

Suppose DT(O) is infinite. As DT(O) is a binary tree, it follows from Konig's Lemma that it has an intinite branch B starting at the root. Let us denote by L\ the set of all indecomposable formulae which occur in the sequences labelling the nodes of

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

290 Journal of Applied Non-Classical Logics. Volume 10- No. 3-4/2000

B. Assume that~ is an axiomatic subsequence of.:::\. Since any subsequence of.:::\ consists of indecomposable formulae only, and each vertex in DT(D) inherits all the indecomposable formulae from its ancestors, there is some node v of B such that ~ is a subsequence of its label nv. It follows that nv is axiomatic as well, and therefore, B terminates at v. This contradicts the fact that B is infinite. Hence, .:::\ can contain neither of the three single formulae (15)- (17) nor any of the two pairs of formulae (18), (19) on page 9, whose presence in a sequence make it axiomatic.

Thus, reasoning exactly in the same way as in the proof of Lemma 8, we can build a model H such that &t H G for every G E .:::\. Indeed, the only assumptions used in building a counter-example H for a sequence ~ were that~ was indecomposable, and that it did not contain any axiomatic sequence. Both those assumptions hold for .:::\ as well; the only difference is that .:::\ is infinite, but this is irrelevant for the construction of H.

By definition of a decomposition tree, the top node of B which coincides with the root of DT(D) is labelled by n(D). Hence, arguing by induction on the complexity of a formula, the fact that no indecomposable formula in the labels of B is true in H implies that n(D) is not true in H. Thus, n cannot be valid. Details of the argument can be found in [Kon97a]. 0

8 A sequent calculus for complementarity

In this section, we will present a sequent calculus arising from the complete R-S proof system for sequences of signed formulae we have just described. The method of trans­forming an R-S system into a sequent calculus was described in [Kon97a], which the reader is invited to consult for any details omitted below.

The sequents will only involve ordinary, non-signed formulae of the language£ as described in Section 2; sets of such formulae will be denoted by r, .:::\ with suitable indices. A model M of£ is called a model off iff every formula in r is satisfied in Min the semantics given in Section 3.

By a sequent we mean a pair (r, .:::\)of finite sets r, .:::\ ~ FORM, usually written in the form r f-.:::\. In the sequent notation r, F f-.:::\, G, commas denote set-theoretical union, and individual formulae F, G E FORM are identified with the respective sin­gletons.

The sequent r f- .:::\ is called valid, written f= (r f- .:::\), iff

(23) FM r implies (3F E.:::\) FM F.

Thus, r f- .:::\ is valid if a model M which satisfies each formula in r also satisfies at least one formula in.:::\.

For any finite set of formulae Ax ~ FORM, and any F E FORM, we say that F is a semantic consequence of Ax, and write Ax f= F, iff F holds in every model of Ax. The set Ax may be treated as a set of axioms of some specific theory T; then Ax f= F iff F is a semantic consequence of the theory T. Evidently, Ax f= F iff the sequent Ax f- F is valid, which implies that the syntactic entailment f- in a valid sequent may be viewed as a formal counterpart of the semantic consequence relation.

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multi modal logic 291

In other words, a complete axiomatisation of the sequent calculus will automatically provide a deduction system complete for theories.

The sequent calculu~ (SC) which we shall develop consists of axioms, having the form of single valid sequents, and inference rules, leading from valid sequents to valid sequents. A deduction rule has one of the two forms

or

The sequent Swill be called the conclusion of the rule, and the sequent(s) S 1 (S 1 , S2) -its premise(s). A rule is called sound iff its conclusion is valid whenever all its premises are valid.

It turns out that there is a simple connection between validity of sequents and va­lidity of sequences of signed formulae. For any finite set r = { F1 , F2, ... , Fk} ~ FORM, let T(r) and N(r) be sequences of signed formulae obtained by preceding all elements of r with the operators T and N, respectively, so that

We now have

Lemma 10. A sequent r f- ~is valid whenever the sequence N(f), T(~) of signed formulae is valid.

The above fact suggests a straightforward method of developing a sequent calculus for our original formal language based on the DRS deduction system for sequences of signed formulae; we simply have to derive from the decomposition rules of DRS the corresponding inference rules of the sequent calculus, leading from valid sequents to valid sequents.

For any sequence 0 of signed formulae, let us denote

0+ ={FE FORM: T(F) is in 0}, 0- ={FE FORM: N(F) is in 0}.

Lemma 10 implies the f,Jllowing basic result:

Lemmall.

• For every axiomatic sequence 0 of DRS,

r,0-r~,0+

is a valid axiom of the sequent calculus.

0',II,0" • For every rule in DRS,

0', :E,0"

r, :E- f- ~, :E+ r,rr r ~,rr+

is a valid inference rule of the sequent calculus.

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

292 Journal of Applied Non-Classical Logics. Volume 10- No. 3-4/2000

S1', II, S1" • For every rule in DRS,

n', ~1, n" I n', ~2, n"

r, ~11- L)., ~t r, ~2 1- L)., I:t r,rr 1- L).,rr+

is a valid rule of the sequent calculus.

The resulting axioms and inference rules of the sequent calculus (SC) are given in Tables 4- 7 alongside the original axiomatic sequences and rules of DRS from which they were derived. Note that, in addition to the rules derived from DRS, we have also included thinning rules in the sequent calculus, which allow us simplify some of the derived rules. In their presence, we can avoid repeating certain formulae appearing in the premise(s) of the rule in its conclusion; this concerns especially the expansion rules.

That is exactly why some of the sequent calculus rules given in Tables 5-6 do not correspond to the general "recipe" for generating such rules given in Lemma 8.2. For example, the sequent rule for introducing (A) on the right hand side should formally be

r 1- L)., y E F, X E (A)F r 1- L)., X cp(A) y, X E (A)F r 1- L)., x E (A)F

but thanks to the weakening rules we could replace it by the simpler rule

r 1- L)., y E F r 1- L)., X cp(A) y r 1- L)., x E (A)F

In all axioms and rules given in Tables 4 - 7, we suppose that x, y, z E VARO, F, G E FORM, the S1's are sequences of signed formulae, r, L). are finite sub­sets of FORM, A, BE TERM, a= {al' ... 'an} <;;;; CONA, --a= -al n ... n-an. C is a component, and S a subcomponent. The components and subcomponents for a sequent r, L). are defined as components and subcomponents for the sequence of signed formulae N(f), T(L).) equivalent to that sequent.

In the rules marked by the symbols listed below, we also make the following stip­ulations:

*- F is not of the form z E G for any z E VARO and any G E FORM, x is a a new variable in VARO, i.e. x occurs neither above the double line in the DRS rule nor below the single line in the SC rule.

**- F is not of the form z E G for any z E VARO and any G E FORM, y is an arbitrary variable in VARO

***- z is a new variable, i.e. occurs neither above the double line in the DRS rule nor below the single line in the SC rule.

• - y is an arbitrary variable in VARO.

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multimodal logic 293

Before defining the notion of provability in our sequent calculus, we recall that the decomposition rules of DRS were tailored to sequences of signed formulae in normal form, and the decomposition tree of such a sequence n started with the normal form n(!J) of the sequence. Thus, our derived inference rules will be applicable to sequents in normal form, which i~ defined in analogy to the normal form of sequences of signed formulae; hence, like in the DRS system, the deduction mechanism of SC is applied to the normal form of a sequent instead of the original sequent.

More formally, the normal form of a sequent S of the form r f-- ~ is the sequent n(S) of the form n(f) f- n(~), where n(f), n(~) are obtained from r, ~by replac­ing every term A occurring in these sets by its normal form n(A), defined with respect to the sequence N(f), T(~) in the way described in Section 6.

We say that a sequent Sis provable, and write f-- sc S, iff n(S) can be derived from the axioms of the sequent calculus given in Table 4 by a finite number of applications of the inference rules given in Tables 5-7. As above, such a definition is justified by the fact that n(S) can be obtained from S by a simple algorithm, and Sis valid iff n(S) is.

Clearly, all SC rules are sound, and all axioms are valid; this follows from Lemmas 10 and 11. Hence,

Theorem 12. Any provable sequent is valid, i.e. f-- sc (r f-- ~) implies f= (f f-- ~).

Proof Suppose the sequentS is provable; then its normal form, n(S), can be derived in a finite number of steps 7rom the axioms of SC. The axioms are valid sequents, and all the derivation steps consist in applying the inference rules of SC which lead from valid sequents to valid sequents; it follows that n(S) is valid. Since n(S) is semantically equivalent to S, the latter sequent is valid as well. 0

Theorem 13. Anyvalidsequentisprovable, i.e. f= (f f-- ~) impliesf--sc (f f-- ~).

Proof If a sequent r f-- ~is valid, then the corresponding sequence N(f), T(~) is also valid. By the completeness theorem for the DRS system, this means that this se­quence has a finite decomposition tree DT with only axiomatic sequences at its leaves. Based on this tree, it is straightforward to construct a proof of the normal form of the original sequent in the SC calculus. An outline is as follows: We start from the terminal nodes; these are labeled by axiomatic sequences, so the corresponding sequents can be derived from the axioms of the sequent calculus by means of the thinning rules. Now we go upwards in DT, replacing each downward application of a decomposition rule by an upward application of the corresponding sequent calculus rule. For a detailed proof in case of an analogous similarity logic, the reader is referred to [Kon97a]. 0

Since syntactic entailment f-- in a sequent corresponds to the semantic consequence re­lation, Theorem 13 means that the deduction system we have developed in this section is complete for theories:

Theorem 14. If Ax= {F1, F2, ... , Fn} isafinitesubsetofFORM, then,forany FE FORM, Ax f= F iff the sequent Ax f-- F is provable in SC.

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

294 Journal of Applied Non-Classical Logics. Volume 10- No. 3-4/2000

9 Summary

Each information systems gives rise to various types of information relations on the object set. In his paper, we have studied relations camp( A) of complementarity, pa­rameterised by the set of subsets of the attribute set in such a way that for any family A of nonempty subsets of ATTR,

camp (UA) = n comp(A). AEA

We have presented multimodallogics for these families of complementarity relations along with two sound and complete deduction systems. The first, a Rasiowa-Sikorski style calculus, involved the use of variables to model both the membership relation and modalities, and representation of the modality parameter A as a union of components. This enabled us to replace formulae containing modalities of the form [A], (A) by for­mulae with modalities [ C], (C) parameterised by atomic, disjoint sets. Thus, we could reduce reasoning about formulae with arbitrary modalities to reasoning about simple signed formulae of the form T( x cp( C) y), N ( x cp( C) y), where C is a component and x, y are individual variables, avoiding the usual problems arising in case of multiple accessibility relations. It should be noted that the DRS modality rules cannot be ex­pressed as Hilbert-style axioms within propositional modal logic: In the DRS rules, the use of variables to model quantification implicit in modalities relies heavily on the fact that these rules model semantic consequence on the validity level, whereas Hilbert axioms model consequence on the truth level.

The second proof system was a Gentzen sequent calculus which does not use signed formulas. It was obtained observing that a sequent r 1- .6. is valid whenever the se­quence N(f), T( .6.) of signed formulae is valid. This led to a straightforward lr:msla­tion of the RS calculus DRS to the sequent calculus SC, and soundness and complete­ness were obtained in analogy.

Acknowledgement

We should like to thank the anonymous referee for thoughtful and constructive re­marks.

References

[Bet59] E.W. Beth, The foundations of mathematics, North Holland, Amsterdam, 1959.

[Bia98] P. Blackburn, Internalizing labeled deduction, Tech. Report 102, Comput­erlinguistik, Universitat des Saarlandes, 1998.

[DGOOO] Ivo Dtintsch, Gunther Gediga, and Ewa Orlowska, Relational attribute sys­tems, Submitted for publication, 2000.

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multimodal logic 295

[D098] Stephane De1mi and Ewa Orlowska, Complementarity relations: reduc­tion of decision rules and informational representability, Rough sets in knowledge discovery, Vol. 1 (Lech Polkowski and Andrzej Skowron, eds.), Physica-Verlag, Heidelberg, 1998, pp. 99-106.

[DOOO] lvo Dlintsch ond Ewa Orlowska, Logics of complementarity in information systems, Mathematical Logic Quarterly 46 (2000), 267-288.

[Fit83] M. C. Fitting, Proof methods for modal and intutionisticlogics, Reidel, Dor­drecht, 1983.

[ Gab81] Dov Gab bay, An irrefiexivity lemma with applications to axiomatizations of conditions on tense frames, Aspects of Philosophical Logic (U we Mtinnich, ed.), D. Reidel Publishing Co., Dordrecht, 1981, pp. 67-89.

[Kon97 a] B. Konikowska, A logic for reasoning about similarity, Studia Logica 58 (1997), 185-226.

[Kon97b] Beata Konikc•wska, A logic for reasoning about similarity, Incomplete In­formation - Eough Set Analysis (Ewa Orlowska, ed.), Physica - Verlag, Heidelberg, 1997, pp. 462-491.

[Lip76] W. Lipski, Informational systems with incomplete information, Third Inter­national Colloquium on Automata, Languages and Programming (Univer­sity of Edinburgh) (S. Michaelson and R. Milner, eds.), Edinburgh Univer­sity Press, 1976, pp. 120-130.

[OP87] Ewa Orlowsk a and Zdzislaw Pawlak, Representation of nondeterministic information, Theoretical Computer Science 29 (1987), 27-39.

[Orl95] Ewa Orlowska, Information algebras, Proceedings of AMAST 95, Lecture Notes in Computer Science, vol. 639, Springer-Verlag, 1995.

[Paw73] Zdzislaw Pawlak, Mathematicalfoundations of information retrieval, ICS Research Report 101, Polish Academy of Sciences, 1973.

[RS63] H. Rasiowa and R. Sikorski, The mathematics of metamathematics, Polska Akademia N auk. Monografie matematyczne, vol. 41, PWN, Warsaw, 1963.

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

296 Journal of Applied Non-Classical Logics. Volume 10- No. 3-4/2000

Table 1: DRS: the decomposition rules for signed formulae I

(T)

(N)

(T E ...,F)

(N E ...,F)

(T E /\)

(N E A)

(T E V)

(N E V)

0', T(F), O"

0', T(x E F), 0" ' where F is not of the form z E G for any z E VARO, G E FORM, x E VARO, and x does not occur above the double line,

0', N(F), 0"

O',N(y E F),O",N(F)

where F is not of the form z E G for any z E V,\RO, G E FORM, andy is any variable in VARO,

0', T(x E ....,F),O"

O',N(x E F),O"

O',N(x E ....,F),O"

0', T(x E F), 0"

0', T(x E F 1\ G), 0"

0', T(x E F), 0" I 0', T(x E G), 0"

0', N(x E F 1\ G), 0"

O',N(x E F),N(x E G),O"

O',T(x E FvG),O"

0', T(x E F), T(x E G), 0"

O',N(x E F V G),O"

O',N(x E F),O" I O',N(x E G),O"

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multimodal logic 297

Table 2: DRS: the decomposition rules for signed formulae II

(T E (A})

(N E (A})

(T E [A])

(N E [A])

(Tcp(AUB))

(N cp(A u B))

(sym E)

(tran E)

(sym cp)

(tran E cp)

(3-tran cp)

0', T(x E (A) F), 0" 0', T(y E F), 0", T(x E (A)F) I 0', T(x cp(A)y), 0", T(x E (A) F) '

where y is an arbitrary variable in VARO,

O',N(x E (A)F),O"

0', T(x E [A]-.F), 0"

0', Ttx E [A]F),O"

O',N(xcp(A)z),T(zEF),O"' where z E VARO, and z does not occur above the double line,

0', N(x E [A]F), O"

0', T(x E (A)•F), 0"

0', T(x cp(A U B) y), 0"

0', T(x cp(A) y), 0" I 0', T(x cp(B) y), 0"

0', N(r cp(A u B) y), 0"

O',N(xcp(A) y),N(xcp(B) y),O"

0', N(x E y), 0"

O',O",N(l· E y),N(y Ex)

0', Nl X E y), 0", N(y E F), 0"'

O',O",O"',N(x E y),N(y E F),N(x E F)

O',N(xcp(C) y),O"

0', 0", N(l cp(C) y), N(y cp(C) x)

0', N(x E y), 0", N(y cp(C) z), 0"'

0', 0", 0"', N(x E y), N(y cp(C) z), N(x cp(C) z)

0', l'"(xcp(C)y), 0", N(y cp(C) z), 0"', N(zcp(C)t), O""

0', 0", 0"', 0"", N(xcp( C)y), N(ycp( C)z ), N(zcp( C)t), N(xcp( C)t), ( *)

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

298 Journal of Applied Non-Classical Logics. Volume 10- No. 3-4/2000

Table 3: Additional DRS rules for finite ATTR

(T cp -a) 0', T(x cp( -an S) y), 0"

0', T(x cp(Ua'ECONA-a a' n S) y), 0"

(N cp -a) 0', N(x cp( -an S) y), n"

0', N(x cp(Ua'ECONA-a a' n S) y), 0 11

Table 4: Sequent axioms

I Axiomatic subsequence: DRS J Sequent axiom: SC

T(F), N(F) T(x Ex) N(x cp(O) y) N(x cp(A) x) N(x cp(aj n S) y), N(x' cp(aj n S') y'), where S # S' are subcomponents

~,FI-f,F

~ 1-- r, x Ex ~' x cp(O) y I- f ~,xcp(A)xl-f

~'X cp(aj n S) y, x1 cp(aj n S') y', I- r where S # S' are subcomponents

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multimodallogic 299

Table 5: SC rules I

I Decomposition rules· DRS Sequent calculus rules: SC

0', T(F),O"

0', T(x E F), 0" * r 1-- t::.., x E F

r 1-- t::..,F *

O',N(F),O" f, y E F I-!::.. O',N(y E F),O",N(F) * * r,F 1-- t::.. * *

0', T(x E •F),O" r, x E F 1-- t::.. 0', N(x E F), 0" f I- !::..,x E •F

0', N(x E •F), 0" r I- !::..,x E F 0', T(x E F), 0" f,xE•FI--6.

0', T(x E F 1\ G), 0" r 1-- t::.., x E F r 1-- t::.., x E G 0', T(x E F), 0" I 0', T(x E G), 0" r 1-- t::.., x E F A G

0', N(x E F 1\ G), O" r, x E F, x E G 1-- t::.. O',N(x E F),N(x E G),O" f,xEF/\GI--6.

O',T(xEFVG),O" r 1-- t::.., x E F, x E G 0', T(x E F), T(x E G), 0" f I- !::..,x E FV G

O',N(x E FvG),O" f,xEFI--6. f,xEGI--6. 0', N(x E F), 0" I 0', N(x E G), 0" f,xEFVGI--6.

0', T(x E (A)F),O"

0', T(y E F), 0", T(x E (A) F) I 0', T(x cp(A)y), 0", T(x E (A) F) fl--f::,.,y E F fl--/::,.,xcp(A)y

r 1-- t::.., x E (A)F

0', N(x E (A) F), 0" f I- /::,.,x E [A]·F 0', T(x E [A]•F),O" r, x E (A)F 1-- t:,.

0', T(x E [A]F),O" f,xcp(A)zl--!::..,zE F 0', N(x cp(A) z ), T(z E F), 0" * ** r I- !::..,x E [A]F * **

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

300 Journal of Applied Non-Classical Logics. Volume 10- No. 3-4/2000

Table 6: SC rules II

Decomposition rules· DRS Sequent calculus rules· SC ...

!!', N(x E [A]F), !!" r f- t.," E (A}~F !!', T(x E (A}~F), !!" f, .T E [AJF f- t.

!!', T(x cp(A u B) y), !!" r f- t.,xcp(A) y r f- t., X cp( B) y !!', T(x cp(A) y), !!" I!!', T(x cp(B) y), !!" r f- t., X ocp( A u R) y

!!', N(x cp(A u B) y), !!" r, X cp(A) y, X cp(B) y f- t. !!', N(x cp(A) y), N(x cp(H) y), o<~ f,xcp(AUB)yf- t.

!I', N(x E y), !!" r, y Ex r- t. !I',!!",N(x E y),N(y Ex) r, x c.: y r- t.

!!', N(x E y), !!", N(y E F), 0"' f,xEFf-t. n',n",!I"',N(x E y),N(y E F),N(x E F) r, x E y, Y E F r- t. •

!!', N(x cp(C) y), n" r' y cp( C) X f- t. !!', !!", N(x cp(C) y), N(y cp(C) x) r,xcp(C)yf- t.

!!', N(.t· E y), 0", N(y cp(C) z), !!"' fx cp(C) z f- t. !!', 0", !!"', N(x E y), N(y cp(C) z), N(x cp(C) z) l', x E y, y cl'( C) z f- t. •

!!', N(x cp(C) y), 0", N(y cp(C) z), !!"', N(z cp(C) t), ll""

!!', !!", O"', N(x cp(C) y), N(y cp(C) z ), N(z cp(C) t), N(x cp(C) t)

r,xcp(C)tf-t. • r, X cp(C) y, y cp(C) z, z cp(C) t f-t.

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014

A multimodallogic 301

Table 7: SC rules III

Rules for finite ATTR: DRS Rules for finite ATTR: SC

0', T(x cp( -an S) y), 0" r 1- ~'X cp(Ua'eCONA-a a' n S) y 0', T(x cp(Ua'FCONA-a a' n S) y), 0" r 1- ~'X cp( -an S) y

0', N(x cp( -an S) y), 0" r, X cp(Ua'eCONA-a a' n S) y 1- ~ 0', N(x cp(Ua'FCONA-a a' n S) y), O" r,xcp(-anS) y 1- ~

Thinning rules

fl-~ r 1- ~ r,FI-~' r 1- ~, F

Dow

nloa

ded

by [

Hac

ette

pe U

nive

rsity

] at

19:

10 1

9 D

ecem

ber

2014