Bernard Bel

PATTERN GRAMMARS IN FORMAL REPRESENTATIONS OF MUSICAL STRUCTURES

Keywords: pattern languages, formal grammars, stochastic automata, music analysis

This paper introduces several formal models of pattern representation in music. Polyvalent multimodal grammars describe partially overlapping sound events as found in polyphonic structures. Bol Processor grammars are characterizations of sequential events in terms of substring repetitions, homomorphisms, etc. Parsing techniques, stochastic production and recent developments of BP grammars are briefly described.

La musique est le suprême mystère des sciences de l'homme, celui contre lequel elles butent et qui garde la clé de leur progrès. (Lévi-Strauss 1964:27) Music representation is a vast subject covering two separate topics: sound models and musical structures. Although in the field of computer-aided composition much work has been devoted to new sound generation techniques, many studies of structural models are still limited to the analysis of Western tonal music. Beyond this domain (particularly in ethnomusicology) there has been a tendency to borrow concepts from linguistics on the basis of assumed parallels between language and traditional musical systems. Feld's provocative paper (1974), however, introduced to ethnomusicologists an epistemological dilemma which resulted in the virtual rejection of purely abstract speculations that Feld termed 'the hollow shell of formalism': Only Blacking (...) and Lindblom and Sundberg (...) have dealt explicitly with basic theoretical issues... The rest of the literature ignores issues like the empirical comparison of models, a metatheory of music, evaluation procedures, and the relation of the models to the phenomena they supposedly explain. (Feld 1974:210).

'Precomputer' studies have been limited to descriptions of 'frozen' data, thereby ignoring models that would produce musical scores:

1 Bernard Bel is a computer scientist with a background in electronics. Since 1979 he has been collaborating with anthropologists, musicologists and musicians on a scientific study of North Indian melodic and rhythmic systems. In 1981 he built a sophisticated real-time Melodic Movement Analyser (MMA) with the aid of which he developed automatic transcription and analysis of raga music (in collaboration with Dr. Wim van der Meer). In 1986 he joined GRTC, an AI laboratory of the Centre National de la Recherche Scientifique in Marseille. At present he is working on algorithms for automatic rule generation derived from hypotheses on training methods in traditional tabla (in collaboration with ethnomusicologist Dr. Jim Kippen). He is also involved in research on computer music at the Laboratoire Musique et Informatique de Marseille (MIM).

Our theory of music is...based on structural considerations; it reflects the importance of structure by concerning itself not with the composition of pieces but with assigning structures to already existing pieces. (Jackendoff & Lerdahl 1982:85) In fact, computers make it possible not only to check analytical models against large data sample sets but also to adjust grammars with a view to improving the machine's ability to compose music or to assist a human composer by suggesting channels of 'musical thinking' unknown to him/her. This is -broadly speaking -the 'expert system' approach (Kippen & Bel 1989c) 2 . Several teams are working on Western tonal music along these lines, trying to setup validations of the structural models proposed by Ruwet and Jackendoff & Lerdahl (Camilleri 1989).

Research in analysis and composition of music by means of computers implies the choice of adequate representational models of music. This paper deals with issues relevant to both approaches. Two models of musical patterns imbedded in formal grammars are introduced: Polymodal Multivalent Grammars (PMG) and Bol Processor (BP) grammars. PMGs have been proposed by Vecchione (1978ff) as a general syntactic model of music which lends itself to a paradigmatic analysis based on resemblance, and may therefore be used as an aid to music composition3 . BP grammars have been designed by [START_REF] Kippen | forthcoming 'The identification and modelling of a percussion "language", and the emergence of musical concepts in a machine-learning experimental set-up[END_REF] for an experimental study of improvisation/evaluation schemata in North Indian tabla drumming. The Bol Processor was first implemented in assembly language on the Apple IIc for fieldwork with traditional musicians 4 . Recent developments of this model in the light of contemporary music theory and practice are briefly introduced in §2.55 .

A detailed description of the BP's modus operandi is found in Kippen & Bel (1989a). The problem of transferring knowledge from informants to machines using automatic rule generation -the present focus of the author's project -is discussed in Kippen & Bel (1989b,c,d).

Pattern grammars, languages, and musical structures

Definitions and notations

It is assumed that elementary musical events ('atoms') may be transcribed with the aid of a finite set of symbols, namely the alphabet Vt of terminal symbols. For example, many pieces of tabla music belonging to the qa'ida repertoire may be transliterated with Vt = {dhin, tin, dhee, tee, dha, ta, ghi, ki, ge, ke, ka, na, ne, ra, -}, in which the hyphen denotes a silence6 . Each symbol is a quasi-onomatopoeic mnemonic representation of a musical event (stroke or bol7 ). In the same way, any finite set of elementary sound events (e.g. notes) may be viewed as a terminal alphabet for music.

A monodic sequence is a string on Vt. The set of finite strings on alphabet Vt is notated Vt*. The length of string s is |s|. For instance, |dhee na ge na| = 4. The nil string is λ, hence |λ| = 0. The set on non-nil finite strings on Vt is Vt + . In set notation, Vt + = Vt* -{λ}. The number of elements in a set S (the cardinal of S) is notated card(S). ∅ denotes the empty set, hence card(∅) = 0.

To describe languages we may need an enumerable set of variables or non-terminal symbols Vn such that Vn ∩ Vt = ∅. Throughout this paper, variables will be represented with upper-case characters sometimes followed with numbers denoting paradigmatic variants or metric durations, e.g. X2, TA14, etc... ta is notated as mir(dha), and teenakena as mir(dheenagena). Homomorphisms are suitable for describing any structural relationship that is built on transformations of the terminal symbols, e.g. transposition, octave shift, etc. in tonal music.

Pattern grammars

Leaving aside its parallels with language, music may be seen as an algorithmic (non-random) signal sequence. Since automatic (i.e. automata-generated) sequences lie somewhere between periodicity and chaos, yet closer to periodicity (Allouche 1987:264), a workable hypothesis is one that envisages the description of all kinds of musical events as generated by automata, or, equivalently, formal grammars.

Yet both the atomistic theory of music -"la mélodie n'est autre chose qu'une succession de sons déterminés" (Vincent d'Indy ) -and the classical 'scheme' model -"ABA is a Lied pattern" -are one-level conceptualizations of music. The atomistic conception, which was followed by Schönberg, Webern, etc. 8 , resulted in attempts to proclaim set theory as the foundation of tonal (cf. Babbitt) and atonal (cf. Forte) music. As put by Roads (1984:10):

With this approach, abstract properties -such as invariance relationships under formal operationsmay be studied, while properties which are not amenable to such an approach are often ignored (e.g. timbral and spatial relationships, articulation factors, clusters and clouds of sound, dramatic and expressive qualities, and so on). [START_REF] Ruwet | Roads, C. 'An overview of music representations[END_REF]Ruwet ( ,1972:100-34) :100-34) and Nattiez (1976:239-78) -both involved in surface, Schenkerian and prosodic analysis -and later Jackendoff & Lerdahl (1983) 9 , introduced linguistic/semiotic descriptions in which several layers of music constituents may be assigned to a single (monodic or homophonic) sequence.

Unfortunately, these sequential models have been unable to account for the structural organization of overlapping constituents as found in polyphonic musical structures:

At the present stage of development of the theory, we are treating all music as essentially homophonic (...) For the more contrapuntal varieties of tonal music, where this condition does not obtain, our theory is inadequate. We consider an extension of the theory to account for polyphonic music to be of great importance. However, we will not attempt to treat such music here except by approximation. (Lerdahl & Jackendoff 1983:37) To achieve this, it is necessary to disconnect the set of musical constituents from the set of the time-span intervals which support them 10 . The set of constituents is always structured as a tree (a hierarchy of partitions), whereas the set of time-span intervals is structured as an joinsemilattice (Birkhoff 1984:22) which may or may not be tree-shaped. Musical structure is therefore determined by a morphism between these two structures: a bijective morphism defines a sequential structure whereas injective morphisms define non-sequential structures [START_REF] Vecchione | Pour une théorie générale des grammaires musicales formelles[END_REF](Vecchione :139ff, 1985:207ff):207ff).

The idea of dissociating musical constituents from their time-span intervals comes in response to Xenakis ' (1963, 1972:57) suggestion to separate out-time structures, the structure of time and the resulting in-time structure, i.e. the Cartesian product of the former two.

Consider for instance the 9th measure (2nd theme of the first movement) in J.S. Bach's fifth Brandenburg Concerto in D major:

Fl. V. Pr. A1 C B1 B2 A2
mes. 9 (Vecchione 1984:137) The set of constituents may be structured as a tree 9 a model recognising four components: metrical structure, grouping structure, time-span reduction and Schenkerian prolongational reduction -the latter was criticised and expanded by Deliège . 10 This hypothesis has not been considered by all researchers in the field. See for instance Chemillier (1986Chemillier ( :2, 1987,2:380ff),2:380ff). Interestingly, Xenakis introduced three structures in his early work (1963:190-1,200): in-time, out-time structures and structure of time, the former being a mapping between the latter two. Yet in 1972 he did not deal any more with the structure of time.

∅

Note that if the universal lower bound '∅' is not represented, the structure becomes a joinsemilattice with universal upper bound I(S), the minimal elements of which are arranged in a strict sequence: I(A11)=I(B21), I(A12)=I(B111)=I(B22), I(A13)=..., I(C2).

The weakness of this representation lies in the fact that some of the time-span intervals have been chopped into smaller intervals, e.g. I(B11), I(B12) that do not necessarily support meaningful musical constituents. The same problem arises in set representations, e.g. [START_REF] Chemillier | Toward a theory of formal musical languages[END_REF] who developed a formal language for polyphonic music based on trace languages and partially commutative free monoïds [START_REF] Mazurkiewicz | Traces, histories, graphs: instances of a process monoïd[END_REF][START_REF] Chemillier | Monoïde libre et musique: deuxième partie[END_REF].

Another aspect of the present research, therefore, is an attempt to deal with time as a topological entity: considering non-Euclidian metric durations is necessary when dealing with contemporary music, most extra-European systems, and European music prior to the 15th Century. Wiggins & al. (1988) pursued this line of thinking in a logic representation of musical constituents, defining an ordered set of 'times' and a linearly ordered commutative group 'duration'. Another fruitful approach is based on a single set of open intervals (of rational numbers) carrying two binary relations: inclusion and precedence (Van Benthem 1983:58-79).

In the linguistic/semiotic model, applying a rule in a formal grammar amounts to a step in string concatenation. In the description of polyphony, however, thirteen distinct operations are possible (Van Benthem 1984:70, Vecchione 1984:150ff):

X α -α -β1 β1 β3 -β3 β4 -β4 γ -γ δ -δ β2 -β2 ( ) X Y Y X Y X Y X Y X Y X Y X Y X Y X Y X Y X Y X Y X Y
Comparisons of time-span intervals may be worked out by calculus or with reference to the fact that that the ear may perceive event X as starting or ending before, simultaneously, or after event Y.

If I(X) and I(Y) are the time-span intervals supporting X and Y respectively, operations α, β1, ..., δ yield a constituent X∪Y, the time-span interval of which is the smallest interval covering both I(X) and I(Y), e.g.

β3 δ I( I( (X,Y)) (X,Y)) I(X) I(Y) I(Y) I(X)
Note that γ is a strict sequence and both α and -α strict superimpositions. The latter two are identical.

If a binary-tree structure is assigned to constituents A1, A2, B1, B2 and C in the Bach example, e.g.

S A1 C B1 B2 A2
then S is the result of the following operation:

S = β1(γ(A1,C),-β3(B1,γ(B2,A2)))
Obviously this description does not retain the entire topological information: the combination of A1 and B1 might as well have been β4(A1,B1) or β2(A1,B1).

If more information must be made explicit, then alternative formulae, built on all the supposingly 'meaningful' binary trees -in terms of musical analysis -may reduce ambiguity:

S = β1(γ(A1,C),-β3(B1,γ(B2,A2))) = -β4(α(β4(A1,B1),γ(B2,A2)),C) = .
.. (up to 5! = 120 formulae). Vecchione (1984:185ff) implicitly used time operators as predicates (i.e. Boolean operators) in his syntactic model ('Ω-syntax'), actually using the same notation for operators and their corresponding predicates. In this way, the first expression of S may be written:

S -> β1(X,Y) X -> γ(A1,C) Y -> -β3(B1,Z) Z -> γ(B2,A2)
Rules in this format may contain more than one predicate, thereby allowing hierarchical descriptions more general than binary trees, e.g.

S A1 B1 X C B2 A2
resulting in a more compact (and less ambiguous) representation:

S -> β4(A1,B1) -β1(A1,X) γ(A1,C) -β3(B1,X) -β4(B1,C) -β4(X,C) X -> γ(B2,A2)
where the right side of the first rule is a conjunction of predicates 11 .

In many cases only part of the information on time-span intervals is musically significant. If E = {A1,A2,B1,B2,C}, P(E), the set of all subsets of E, is a complete lattice which may be represented as a graph in which each node indicates a subset. Edges marking the inclusion relation X ⊃ Y may be labeled with the predicate which is verified on (X,Y), in fact always α, β1, or β3. The graph may then be complemented until it is complete. Other temporal relations: β2, β3, β4, γ, δ (and their symmetric counterparts) are visible in the complete graph.

Consider for instance the subgraph concerning A1, A2 and B1 in the example:

{A1,A2,B1} {A1,A2} {A2,B1} {A1,B1} A1 A2 B1 α α β1 β3 β3 β3 β3 α β3 β1 β3 α β3 β3 β1 β3 β3 β4 β4 β4 β3
Bold edges are those belonging to the lattice.

If only part of the information about time is known (i.e. significant), it may be completed with the aid of a transitivity table: given f(X,Y) and g(Y,Z), the table yields f*g(X,Z). In many cases the result is a disjunction of predicates; in some cases it is even unknown (hence it is a disjunction of the thirteen predicates). Part of the 13x13 square table is:

-β2 β2 β4 γ δ -β2 β2 β4 γ δ -β2 -β1 ∨ -β2 ∨ β4 ∨ γ ∨ δ δ δ β2 ∨ β3 ∨ β4 ∨ γ ∨ δ δ δ δ δ δ δ δ δ δ δ β4 ∨ γ ∨ δ β2 ∨ β3 ∨ β4 β2 ∨ β3 ∨ β4 α ∨ β1 ∨ β2 ... β2 β2 ∨ β3 ∨ β4 ∨ γ ∨ δ -β1 ∨ -β2 ∨ β4 ∨ γ ∨ δ -β1 ∨ -β2 ∨ β4 -β1 ∨ -β2 ∨ β4 α ∨ β1 ∨ -β1 ∨ β2 ∨ -β2 ∨ β3 ∨ -β3 ∨ β4 ∨ -β4 (unknown) * f * g | f(X,Y) ∧ g(Y,Z) -> f*g (X,Z)
11 These examples of tree structures do not claim any musical relevance.

For example, β2(X,Y) ∧ γ(Y,Z) -> [β2 ∨ β3 ∨ β4] (X,Z). The graph with incomplete information about time may be computed as a constraint network (Allen & Kautz 1985:255ff). The model makes no assumptions on relations that have deliberately been left unknown. Rather it is a enumeration of all structures consistent with the input data.

Unambiguous predicate representation requires n(n-1)/2 predicates in a rule containing n variables. The following matrix lay out is self-explanatory:

A1 C X B1 A1 B1 X C α β4 -β1 γ -β4 α α α -β3 -β4 β1 β3 -β4 -γ β4 β4 S -> α β4 -β1 γ -β4 α -β3 -β4 β1 β3 α -β4 -γ β4 β4 α [A1,B1,X,C]
or equivalently:

α β4 -β1 γ -β4 α -β3 -β4 β1 β3 α -β4 -γ β4 β4 α S -> T [A1,B1,X,C] with T =
Vecchione's general format for context-sensitive time-predicate rules (which he calls 'Ωsyntax rules') is the following (1985:276):

θq [A J B] -> θr [C K D]
in which θq, θr are matrixes of time predicates, and A, B, C, D possible contexts. A, B, C, D, J, K are strings of variables corresponding to 'significant' units 12 . In general, |K| ≥ |J| to ensure decidability. If A=C and B=D the rule is context-sensitive in the common sense. The following subclass of rules

θq [C J D] -> θq [E J F]
may be called change of context. Depending on whether changes occur in the time-predicate matrix, in the medium part of the list ('J') or in its right and left contexts, eight modes of transformation can be defined and related to musical transformations 13 (Vecchione 1985:277ff).

12 Derived from a meaningful segmentation.

In grammars, unary or n-ary predicates may also be used to express relationships other than those on time-span intervals: location in time and space, modes of significance, voice or instrument processing, etc. Grammars containing several classes of predicates (taking their arguments in several sets of values) have been called Polymodal Multivalent Grammars (PMG) (Vecchione 1985:257ff). Time predicates are equivalent to the thirteen primitive relations in Allen & Kautz's model of naive temporal reasoning (1985), while [START_REF] Oppo | Per una teoria generale del languaggio musicale[END_REF] considered only seven relations given the fact that, for any time predicate ω, ω(X,Y) =ω(Y,X). In a PMG it is necessary to use the thirteen values to obtain a canonic rule format that is meaningful for computing rule resemblances [START_REF] Bel | The "OC" project[END_REF].

Formulae using time operators and conjunctive expressions of time predicates may be matched or evaluated using tree unification as implemented in the Prolog language [START_REF] Colmerauer | Equations and inequations on finite and infinite trees[END_REF]. The Prolog II-III implementation of generation/parsing techniques for PMGs is under study at the MIM laboratory [START_REF] Risch | Prolog II et les grammaires temporelles[END_REF].

PMGs belong to a class of formal grammars which we call pattern grammars:

A pattern grammar is an ordered 6-tuple (Vt, Vn, A, S, F, P) such that:

Vt is the terminal alphabet, Vn a finite set of variables, Vt ∩ Vn = ∅ and S ∈ Vn;

A is any enumerable set; P is a set of predicates, i.e. applications of (Vn ∪ A) n to {true, false} where n is a positive integer;

F is a finite number of triplets (p,q,R) such that p,q ∈ (Vn ∪ Vt) + and R a conjunctive expression using predicates in P whose arguments are variables occurring in q. Triplets of F can be notated: p -> q where R . In many cases, A is the set of positive integers; consequently, predicates may accept numeric arguments, see for instance §2.1.

In the field of pattern recognition, [START_REF] Evans | Grammatical Inference in Pattern Analysis[END_REF] first proposed an effective procedure for inferring pattern grammars from examples. This author has achieved a similar goal using topdown parsing and postponed unification in a Prolog II program [START_REF] Bel | Grammaires de génération et de reconnaissance de phrases rythmiques[END_REF].

The pattern grammar concept may be further elaborated depending on its actual implementation (rewriting systems, logic, etc.) Contextual information may be conveyed in a sophisticated way, e.g. feature-grammars or metamorphosis grammars [START_REF] Colmerauer | Equations and inequations on finite and infinite trees[END_REF]. Metamorphosis grammars are easier to parse than context-sensitive grammars [START_REF] Véronis | Bottom-up parser generation in Prolog[END_REF]). An example of logic representation implicitly conveying contextual information may be found in Wiggins & al. (1988).

We shall see ( §2.1) that Bol Processor grammars are a combination of pattern grammars and a sequential model derived from pattern languages. The following ( §1.3-1.4) is a formal description of the sequential model which may be skipped by readers unfamiliar with formal language theory.

Pattern languages

If H denotes the set of all λ-free homomorphisms (with respect to concatenation) of (Vn ∪ Vt)* to itself, an element of H which is the identity when restricted to Vt is called a substitution. If the image set of a substitution is Vt*, the substitution is terminal. A substitution which is a bijection of Vn to itself when restricted to Vn is called a renaming of variables.

A pattern is any element of (Vn ∪ Vt)*. If p is a pattern and s a substitution, then s(p) is a derivation of p. A pattern containing no variable is a terminal derivation or a sentence. Two patterns, p and q, are equivalent (i.e. p ≈ q) if and only if there exists a renaming of variables r such that p = r(q). Another binary relation (notated p ≤ q) is defined as follows: p is less general than (or more specific than) q iff for some substitution s, p = s(q). Since a substitution is a λ-free homomorphism, p ≤ q => |p| ≥ |q|.

The language generated by a pattern p is the set of terminal derivations of p, namely L(p) = {s ∈ Vt + : s ≤ p}. It can be proved that: ≤ is transitive p ≤ q => L(q) ⊇ L(p) p ≈ q <=> p ≤ q and q ≤ p However, the question of finding an effective procedure to decide whether L(q) ⊇ L(p) given arbitrary patterns p and q appears to be open (Angluin 1980:49-52).

The mirror mapping (see 1.1) can be extended to the set of patterns with the following definition:

∀p,q ∈ (Vn ∪ Vt)*, p = mir(q) <=> ∀s ∈ Vt + , s ≤ q => mir(s) ≤ p A transcription of a few variations of a qa'ida is given below 14 : Tabulation has been used to denote beats; each beat comprises six strokes (bols) 15 .

Conventionally, the first item is the theme of the qa'ida whereas [2], [3] and [4] are called variations. The problem may be defined as the description of the whole set of possible variations of this theme. One obvious feature of this sample sequence is that some variations are 14 Qa'ida is an Urdu word meaning 'rules' or 'system', and is derived from the Arabic word qava'id meaning 'grammar'. Tabla players often refer to the concept of grammaticality to denote correct arrangements of 'words' built on a limited vocabulary of bols. 15 In the onomatopoeic system used by North Indian drummers, Vt is practically a prefix code mapping strokes/sounds to the English lower-case alphabet. In other words, any string of Vt* (like dheenedheenagena) can be identified deterministically from left to right as a unique sequence of bols (dhee/ne/dhee/na/ge/na). Therefore strings of Vt* are usually represented without spaces.

of the same duration as the theme (16 beats) while others are double (32 beats). Doubling the length is an improvisation technique commonly used by tabla players of the Lucknow style (Kippen 1988:166-67). These sets of variations may be roughly described with the following patterns, where X, Y, Z are variables: Single variations: Let L be the language representing all acceptable variations of this qa'ida, L1 the subset of single variations, and L2 the subset of double variations: L = L1 ∪ L2. L(p1) and L(p2) are the languages generated by p1 and p2 respectively. We will agree (on the basis of much larger sample sets) that L(p1) and L(p2) contain non trivial subsets of L1 and L2. An elaborated description of L1 and L2 has been published in Bel (1987b).

Ideally, the search is for descriptive patterns of L1 and L2. A pattern d is descriptive of a language L if L(d) ⊇ L and for every pattern q such that L(q) ⊇ L, L(q) is not a proper subset of L(d).

L is always a finite set: in this qa'ida, for example, all acceptable strings of Vt* have lengths shorter than or equal to 32 x 6 = 192 symbols, therefore card(L) ≤ (1+card(Vt)) 192 . Yet, all we know about L is a set of positive and negative instances (correct and incorrect variations) called the presentation. The subsets of positive and negative instances are notated S + and S -.

A pattern p matches a presentation S = S + ∪ Siff p is descriptive of S + and S -∩ L(p) = ∅. A pattern p is a tight fit of S if L(p) = S + . Evidently, any tight fit of S is descriptive of S + and matches S.

There is an effective procedure to infer a descriptive pattern from a set of positive instances S + , but the problem appears to be NP-complete in most cases (Angluin 1980:54-55). In addition, the class of pattern languages is not closed under any of the basic set operations: union, complement, and intersection. Therefore it is not possible to construct systematically a pattern description of a language. In order to achieve such an analytical task it is necessary to define more suitable classes of languages.

Restricted pattern languages (RPL)

If p is a pattern, a subclass of L(p) may be defined with the aid of restrictions on acceptable derivations of p. For example, we may define the subclass of strings in L(p) with specified length n: L n (p) = {s ∈ Vt + : s ≤ p and |s| = n}. In the example above, evidently |X| = |Y| = 24 and |Z| = 12. In addition, we can write that Y = mir(X).

There are other ways to formalize specifications with the aid of rewriting rules: we notate p -> q, the fact that every acceptable derivation of q is an acceptable derivation of p, i.e. q ≤ p. If S is the set of all substitutions, p -> q denotes the subset Sp->q such that ∀s ∈ Sp->q, s(p) = q. Since the problem of deciding whether p ≤ q given arbitrary patterns p and q is NPcomplete (Angluin 1980:52) Sets of substitutions for rules 10 and 14 may be represented as S p10->q10 = {..., ZD ZE/ZG ZH, ...} and S p14->q14 = {..., ZC/dhin--,...}, where '...' indicates the (denumerable) possible substitutions of all strings of (Vt ∪ Vn) + except the specified ones (ZD ZE and ZC). We find that S p10->q10 ∩ S p14->q14 = {..., ZD ZE/ZG ZH, ... , ZC/dhin--,...}.

The result of all intersections is:

S o = {P192/C Z D A E Z D B, A/tagetirakitadhin--dhagena D,
B/tagetirakitadhin--dhagenadhatigegenakadheenedheenagena, C/dhin--dhagenadha--dhagenadhatigegenakadheenedheenagena A, D/dhatigegenakateeneteenakena, Z/ZC ZD ZE ZF, ZC/dhin--, ZD ZE/ZG ZH, ZF/dha--, ZG ZH/dhagenadha--} where every derivation is explicit. Therefore S o contains a unique substitution that yields the sentence.

If we select R o = {1,2,3,4,5,8,10,13,14,15,21} we get S p10->q10 = {..., ZD ZE/ZG ZH, ...}, S p15->q15 = {..., ZD/dhin--,...}, and S p20->q20 = {..., ZE/dha--,...}. Let us call S 15,20 = S p15- >q15 ∩ S p20->q20 = {..., ZD/dhin--,..., ZE/dha--,...}. For any s ∈ S 15,20 , s(ZD ZE) = s(ZD) s(ZE) = dhin--dha--. Therefore ZD ZE may never be substituted to ZG ZH, which contradicts S p10->q10 . Consequently, S o = ∅ and the derivation is aborted.

Remark: we may also define S o as a maximal (hence most specific) consistent conjunctive expression using predicates (p i = q i ) such that (p i -> q i ) ∈ R .

Theorem:

The class of finite languages coincides exactly with that of RPLs.

Proof:

Each subset of the (finite) set of rules in a RPL yields at most one terminal derivation. Therefore the set of terminal derivations is finite. Conversely, for any finite language there exists a non-imbedding rightlinear grammar G that generates exactly L. Let G = (Vt, Vn, S, R) where S is the starting symbol and R a finite set of rules in format A -> a or A -> aB such that A,B ∈ Vn, A≠B, and a ∈ Vt. It is easy to prove that G generates exactly the RPL L R (p) such that p = S, which completes the proof.

Corollary:

The class of RPLs is recursive, and it is closed under union, catenation and intersection. We can define an effective procedure for constructing L R (p) = L R1 (p1)∪L R2 (p2) as follows:

Assume that L R1 (p1) and L R2 (p2) are defined with: p1 = S1 and R1 = {S1 -> q1, ...} p2 = S2 and R2 = {S2 -> q2, ...} in which we renamed variables in R2 so that no variable occurs in both R1 and R2 . We construct L R (p) defined with: p = S and R = {S -> S1, S -> S2} ∪ R1 ∪ R2.

The set L R (p) of terminal derivations of S is the union of the sets of terminal derivations of S1 and S2, therefore L R (p) = L R1 (p1)∪L R2 (p2). In addition, since R1 and R2 contain only well-formed rules, R contains also well-formed rules. Consequently, L R (p) is a RPL.

RPLs form a class of formal languages that is closed under union, and there is even a procedure for building a language from its subsets. In addition, the class is recursive and therefore membership tests can be computed. These properties are useful for building representational models that lend themselves to descriptive generalization.

Bol Processor grammars

Pattern rules

Since stroke sequences in North Indian drumming are homophonic, then in any pattern xyz the time predicates γ(x,y), γ(y,z) and δ(x,z) are implicit. Therefore the only predicates that need to be made explicit are those denoting repetitions (equal) and mirror substitutions (mirror).

For example the rewriting rules defining P96 and P192 in §1.4 may be represented with the following pattern grammar: P96 -> X A Y B where mirror(X,Y) and length(X,24) P192 -> C Z1 D A E Z2 D B where equal(Z2,Z1) and length(Z1,12) and mirror(C,E)

In this example the meaning of predicates mirror, length and equal is self-explanatory.

It is clear that if two variables linked with an equal/mirror predicate appear in the conclusion of a rule, the leftmost variable denotes a sequence of strokes that has been played first and served as a reference. For example, using the following set of rules:

A -> B C where equal(B,C) B -> dhagena the terminal derivation dhagenadhagena is such that the first dhagena has served as a reference for the second. Thus we may notate this sentence:

(= dhagena) (: dhagena)
with the meaning that the parenthesis with an equal sign denotes the reference, and that with a colon is the copy. In fact, in the machine implementation, the content of the second parenthesis is merely a pointer to the reference:

We use the same convention in rules, e.g.:

A -> (= B) (: B)
As to notating mirrors, we use an asterisk to indicate that the following sequence between brackets is unvoiced. For example, given the mirror mapping of §1.1, in the following grammar: (2) The same variable may appear twice in a rule premise (if not linked with repetition markers), e.g. A A -> q is correct but not (=A) (:A) -> q.

(3) Mirrors shall be represented with an asterisk.

Using these conventions, the BP grammar generating the RPL language defined under §1.3 is:

( 9,10,11,15,16,17,20,21 and 23 have been deleted. This representation is more compact although it generates the same language. The syntactic graph of the sentence generated in §1.4 is now: In BP representations, a grammar is generally layered in several transformational 'subgrammars'. The term 'transformational' is used here as it is defined in formal language theory (Bel 1987a:356), not linguistics. In a subgrammar there is more than one starting symbol, and a derivation sequence consists of transforming a string of starting symbols P1 to a string of terminal symbols P2. Symbols that are terminal to a subgrammar may become the starting symbols of the next subgrammar(s) to be applied, and so on (Kippen & Bel 1989a:3).

1) P192 -> (=C) (=Z) (=D) A * (:C) (:Z) (:D) B (2) A -> tagetirakitadhin--dhagena D (3) B -> tagetirakitadhin--dhagenadhatigegenakadheenedheenagena (4) C -> dhin--dhagenadha--dhagenadhatigegenakadheenedheenagena A (5) D -> dhatigegenakateeneteenakena (6) Z -> tirakitatirakitatirakita (7) Z -> Z6 Z6 (8) Z -> Z3 Z3 Z3 Z3 (12) Z3 Z3 -> dhagenadhin-- (13) Z3 Z3 -> dhagenadha-- (14) Z3 -> dhin-- (18) Z3 -> dha-- (22) Z6 -> dheenedheenedheene (24) Z6 -> dha-dha-dha- in which rules

The membership test in Bol Processor grammars

BP grammars may be viewed as declarative statements on the languages they generate. Nevertheless, a membership test is a recognition procedure by which a given string is checked against the grammar. Since the membership test algorithm implemented in the BP is bottom-up deterministic, emphasis is put on a procedural description. Discriminating compositional and analytical processes has been one of the major reasons why the BP could be used for modelling part of the musical 'behaviour' of traditional musicians 16 .

A membership test should provide an explanation of its failures by pointing to a deficiency of the grammar, i.e. an easy step by step trace. For these reasons the test implemented in the BP is bottom-up deterministic. Informally, the algorithm is the following:

(1) If G is a transformational grammar, its dual is obtained by swapping the premise and conclusion in every rule.

(2) If G 1 , G 2 , ... G n are the subgrammars of the language, a membership test is the result of the context-sensitive canonic rightmost derivation of the sentence by G n , ... G 2 , G 1 .

(3) If the membership test has yielded S, the starting symbol, the input sentence is assessed 'correct'. It is 'incorrect' in any other case.

The concept of context-sensitive canonic (leftmost) derivation was defined by Hart (1980:82) for strictly context-sensitive grammars, i.e. grammars in which all rule premises contain no more than one variable. This author has extended this definition to all length-increasing grammars:

Context-sensitive canonic rightmost derivation: definition

The formal definition of the context-sensitive rightmost definition adapted from [START_REF] Hart | Derivation Structures for Strictly Context-Sensitive Grammars[END_REF] and used in the membership test of LIN grammars is the following:

Let G be a length-increasing grammar. The derivation in G:

W 0 => W 1 => ... => W n is context-sensitive rightmost iff: ∀i ∈ [0, n-1], W i = X i L i C i R i Y i W i+1 = X i L i D i R i Y i when applying rule f i : L i C i R i -> L i D i R i ,
and at least one of the two following conditions is satisfied:

(C1) |C i+1 R i+1 Y i+1 | > |Y i | (C2) |L i+1 C i+1 R i+1 Y i+1 | > |R i Y i |
In Hart's definition (1980:82)

, |C i | = |C i+1 | = 1, so that C1 may be written |R i+1 Y i+1 | ≥ |Y i |.
The diagram and commentary below illustrate conditions C1 and C2:

Yi+1 Ri+1 Ci+1 Li+1 Xi Li Di Xi+1 Xi+1 Li+1 Ci+1 Ri+1 Yi+1 Yi Ri Ri Yi Ci Li Xi C1 C2
Suppose that neither conditions C1 nor C2 are satisfied. Since C2 is not true, rule f i+1 could have been applied before f i as L i+1 C i+1 R i+1 would be a substring of R i Y i . Besides, since C1 is not true, applying rule f i+1 would only modify Y i without changing the context R i . In such a case the order of application of f i and f i+1 might have been inverted. This change in the order would have been justified since all symbols rewritten by f i+1 are to the right of those rewritten by f i . Now, an explanation of how the inference engine of the BP handles ambiguity in bottom-up parsing will be given. Suppose that for a working string W i there are two candidate rules:

f i L i C i R 1 -> L i D i R i f' i L' i C' i R' i -> L' i D' i R' i where X i L i C i R i Y i = X' i L' i C' i R' i Y' i = W i
The selection criterion is the following: f i will have priority over f' i if one of the following conditions (in this order) is satisfied:

(D1) |X i L i C i | > |X' i L' i C' i | (D2) |X i L i C i R i | > |X' i L' i C' i R' i | (D3) |L i C i R i | > |L' i C' i R' i | (D4) i > i'
It can be proved (Bel 1987a:8) that once D1 has been considered, D2 is no longer relevant and ambiguity may therefore be handled by D3 and D4. D3 makes a decision on the basis of the length of the conclusions of the two rules, and D4 is a final arbitrary decision that takes into account the order in which the rules appear in the grammar.

To augment efficiency, D3 is not considered by the inference engine of the BP; it is therefore the task of the analyst to take into account the following partial ordering of rules:

'Chunk' rule:

In a BP grammar, the conclusion of rule f i may not be a substring of the conclusion of f j such that j<i.

There are a few more restrictions on rule formats imposed by the membership test procedure (Bel 1987a-b). One is that right contexts may only contain symbols of an alphabet external to the grammar.

Templates

Sentences can only be parsed if they have been entered into the editor complete with structural information, i.e. brackets, asterisks, etc. Yet in view of the fact that the Bol Processor has to perform membership tests on large amounts of data comprising examples to which it is not always easy to assign a structure, this limitation is held to be unacceptable. In response to this, the inference engine of the Bol Processor is able to generate templates from a grammar, i.e. a list of possible structures in which each dot represents a terminal symbol of one unit. Templates are enumerated and stored in the grammar file, for instance: In analysis the Bol Processor takes a new sentence and superimposes it on each template in strict order. A membership test is performed each time the sentence matches a template, so allowing for any structural ambiguity to be assessed. An example of template matching may be found in Kippen & Bel (1989c: appendix).

Stochastic model

The only realistic method for testing a grammar with an expert musician is to instruct the machine to produce one randomly chosen sentence at a time. If the sentence is assessed correct, the procedure is invoked again and another sentence is generated. Generally, the grammar is considered to be satisfactory if all sentences generated within a few sessions have been accepted by the expert.

Since the correctness of a grammar can never be fully assessed -indeed, like musicians themselves, machines may be allowed casual mistakes -it is important to enable the stochastic production process to generate sentences from a wide and representative subset of the language. This can be achieved by weighting the decisions of the inference engine.

Formal grammars offer the possibility of defining consistent probabilistic models that apply to a very wide class of languages. We developed such a model as a response to the need to inhibit rules in context-free grammars.

This model is derived from probabilistic grammars/automata as defined by [START_REF] Booth | Applying Probability Measures to Abstract Languages[END_REF], the difference being that a weight -within the range [0,255] -rather than a probability is attached to every context-free rule. The rule probability is computed from weights as follows: if the weight is zero then the probability is zero; if the weight is positive then the inference engine calculates the sum of weights of all candidate rules, and the rule probability is the ratio of its own weight to the sum. Candidate rules are those whose premise is a substring of the work string. Consider, for example, the set of rules

[1] <100> V3 -> dhagena [2] <100> V3 -> dhatrkt [3] <50> V3 -> dha-- [4] <5> V3 -> dhati-
in which the sum of the weights is 100+100+50+5 = 255. The probability of choosing rule [4] in the derivation of a string containing V3 is therefore 5/255 = 0.0196. Using weights instead of probabilities has the advantage that is does not presuppose the sum of coefficients of all candidate rules to be 1.

Weights (and their associated probabilities) are used in modus ponens to direct the Bol Processor's production along paths more likely to be followed by musicians. In some contextfree grammars -those that fulfil the consistency condition expressed by Booth & Thompson (1973:442) -they may be used for computing a probabilistic sentence function, i.e. a coefficient representing the likelihood of occurrence of each sentence in the language. Grammars that are constructed in a systematic way (or generated from sample sentences, see Kippen & Bel 1989b:205) are good examples of consistent probabilistic grammars.

Another remarkable feature of consistent grammars is that rule probabilities can be inferred from a set of sentences (Maryanski & Booth 1977:525). Given a correct grammar and a subset of the language that this grammar generates (for instance a sample sequence taken from a performance of an expert musician), rule weights are inferred as follows: let all weights be reset to zero; then analyse every sentence and increment by one unit the weights of all rules used in the derivation. (The algorithm described here is more general than the one devised by Maryanski and Booth, since the latter requires the choice of a sample set in which all rules have been used.)

Evidently, rules that are never used in the analysis of a sample set remain with weight (and therefore probability) zero, which inhibits their use in modus ponens. Those rules may be scrutinized to see whether they are incorrect or whether they point to fragments of the language that have not yet been explored. To test this, their weights are set to a high value so that the Bol Processor is forced to generate sentences that either have never been assessed or at least are not being considered by the informant at the time.

An example of weight inference based on the qa'ida introduced in §1.2 may be found in Kippen & Bel (1989a).

Using weighted rules resulted in a marked improvement in the quality of the generated music. This went a long way towards solving the problem of musical credibility encountered in earlier experiments, a problem that arose from the complete randomness of the generative process.

Recent developments of the Bol Processor

These developments have taken place as the result of collaboration with Western composers, more specifically musicians working with digital synthesizers. Many have felt a strong need to implement general abstract models of musical structures. A new version of the Bol Processor has been implemented in C language on the Macintosh. 'BP2' accepts polyphonic structures in a format loosely related to Diener's T-trees (1988). Unlike T-trees, however, where each individual terminal (e.g. a note or a stroke) is viewed as an object (in an object-oriented programming environment), BP2 remains based on strings. Structural units of overlapping events may be represented as sets or trees. Several λ-free homomorphisms may be defined on the same alphabet. Wildcards (local variables that may be instantiated on Vn ∪ Vt ) can be used in the left argument of context-sensitive rules to represent patterns. BP2 is also able to handle metagrammars, i.e. grammars that generate grammars.

A MIDI interface is under study so that the sentences generated may be immediately checked in the way they actually sound.

Conclusion

Pattern grammars (which may be imbedded in production rules, logic descriptions, tree algebra, etc.) are not only powerful tools for music analysis. They also lend themselves to descriptions of creative/evaluation processes. In the phenomenological approach, a pattern is viewed more as a dynamic than static relation:

(La forme) n'est pas la somme des détails intégrés dans l'ensemble qui constitue l'oeuvre. Elle appartient au niveau des structures, c'est à dire des principes. Elle s'identifie avec le schème d'organisation qui suggère l'assemblage des parties en considération de leurs liaisons avec les lois propres du schème organisateur. La forme est liée aux principes de cohésion du système, à la réalisation d'un dynamisme particulier. (Vecchione 1984:11-12) Descriptions may be normative (i.e. based on some kind of rigorous reasoning) when dealing with highly constrained systems (e.g. improvisation in traditional music) or empirical (i.e. based on multi-criteria classification) when applied to the modelling of a compositional process.

The problem lies less in finding a 'universal' abstract representation of music, than in delineating forms of musical 'thinking' that may be operative in the design of tools for computer-aided music creation and performance. _______________________

  Tabla strokes are either open (voiced) or closed (unvoiced). Any sequence of unvoiced strokes may be seen as the closed (mirror) image of a sequence of open strokes through a mapping of Vt* to itself. This mapping mir is a λ-free homomorphism of Vt* to itself whose reduction to Vt is an idempotent mapping of Vt to itself, for instance:

  S -> (= D) * (: D) D -> dhagedheenagena the (only) terminal derivation is (= dhagedheenagena) * (: taketeenakena) and its internal representation: Using these conventions we can simplify the format of rewriting rules in RPLs: (1) If a variable appears several times in the RPL rewriting rule, it shall be represented with repetition markers. For instance, A -> B C B C B shall be represented A -> (=B) (=C) (:B) (:C) (:B).

  [1] (=................)(=........)(=........)*(................)(:........)(=........) [2] (=...............)(=................)*(................)(:................) [3] (=....................)(=........)*(........................)(=........)

  , we must restrict to well-formed rules :

	dhin--dhagena	dha--dhagena dhatigegenaka	dheenedheenagena C
	tagetirakita	dhin--dhagena	dhatigegenaka	teeneteenakena
	dhin--dhagena dha--dha--	dhatigegenaka	teeneteenakena	Z D
	tagetirakita	dhin--dhagena	dhatigegenaka	teeneteenakena	A
	tin--takena	ta--takena	tatikekenaka teeneteenakena	E
	taketirakita	tin--takena	tatikekenaka teeneteenakena
	dhin--dhagena	dha--dha--	dhatigegenaka	teeneteenakena	Z D
	tagetirakita	dhin--dhagena	dhatigegenaka	dheenedheenagena B
	which may be represented with the following context-sensitive syntactic graph, where 'equ' and
	'mir' indicate repetition and mirror structures:

Another approach, which will not be discussed here, is the discovery of a system for music composition or improvisation from musical scores[START_REF] Laske | KEITH: a rule system for making music-analytical discoveries[END_REF], Kippen & Bel 1989b).

Vecchione's model takes in consideration polyphonic time structures and other features like modes of space location, modes of significance, modes of voice/instrument processing, modes of performance, etc.

Initial work was part of a research scheme by the International Society for Traditional Arts Research (ISTAR, New Delhi) generously funded by the Ford Foundation and the National Centre for the Performing Arts (NCPA, Bombay). Subsequently, Kippen's work was also supported by the Leverhulme Trust in the UK.

An outcome of discussions in the 'OC' study group involving composers and scientists at Laboratoire Musique et Informatique de Marseille (MIM).

A system of transliterating tabla strokes to non-ambiguous symbolic representations has been proposed byKippen (1988:xvi-xxiii).

from the verb bolna, 'to speak', in North Indian languages; for this reason the machine was named 'Bol Processor'.

This corroboratesLaske's view (1984:168) on analysis versus composition.

Acknowledgements

I am indebted to Prof. Bernard Vecchione and Dr. Jim Kippen for their help in providing musical examples and the raison d'être of this paper. Their competence in music theory, practice, and in the history of musical analysis has been essential for the difficult task of linking formal models with musical concepts.

Definition:

Rule p -> q is well-formed iff p ∈ Vn + , q ∈ (Vt ∪ Vn) + , |q| ≥ |p|, and no variable occurs twice in p.

Theorem:

If rule p -> q is well-formed then L(p) ⊇ L(q).

Proof:

All we need to find is a substitution s such that q = s(p) that yields q ≤ p. A procedure for finding s is the following:

(1) substitute the |p|-1 leftmost variables in p with the |p|-1 leftmost symbols in q (2) substitute the rightmost variable in p with the string made of the |q|-|p|+1 rightmost symbols in q. Since no variable occurs twice in p, there are no restrictions on q that render substitutions impossible. This procedure may in fact be derived from Makanin's algorithm (see [START_REF] Makanin | The problem of solvability of equations in a free semi-group[END_REF][START_REF] Roussel | Programmation de l'algorithme de Makanin en Prolog II[END_REF]. For example, a substitution of XYZ yielding abcX is {X/a, Y/b, Z/cX}. Let L(p) be a pattern language and R a set of well-formed rewriting rules. Every rule (p -> q) defines a set of substitutions Sp->q. To construct a sentence of the restricted pattern language L R (p) we proceed as follows:

The selection of R o requires the comments:

(1) If a variable X does not appear in any premise of the selected set of rules, then it may be substituted to any q ∈ (Vt ∪ Vn) + . Consequently, S o is infinite and L R (p) = ∅ .

(2) There are subsets of R that yield S o = ∅, i.e. aborted derivations. For example, if a variable X appears as the premise of two distinct rules in R o , X -> q i and X -> q j , then S X->qi ∩ S X->qj = ∅, hence S o = ∅.

(3) In cases other than (1) and (2) every variable occurs in a pattern with a unique derivation, so that card(S o ) = 1.

The following example will clarify points (2) and (3). A RPL for double-length variations would be L R (p2) where: p2 = P192 and R is the following set of rules:


Christian Rossignol
email: christian.rossignol@lpl.univ-aix.fr

Introduction

The translation of measurement instruments, such as "questionnaires", "scales" or "classifications" raises different types of problems.

We will deal here with a certain number of translation issues, which appear when the texts, which have to be translated, are not only measurement tools. Some instruments are presented as measurement or classification tools, but aim equally at producing certain effects on their users or at modifying the reality they are designed to measure. Their objectives are sometimes manifold and not always explicit.

They develop what we call "textual strategies"; the function of textual strategies is to induce a particular type of representation or to bring the user or the respondent to favor a particular interpretation of the text or of the facts to which it refers.

We will present a few examples drawn from the World Health Organization's "International Classification of Impairments, Disabilities and Handicaps". WHO presents this text as a "hierarchical classification" and as a statistical and research tool.

A brief history of the "classification"

In the seventies, the World Health Organization which has already long circulated an International Classification of Diseases (ICD), solicited one of its consultants, Dr. Philip WOOD in order to investigate the possibilities of a classification of "consequences of disease" that will be compatible with the principles of the former.

Dr. WOOD defines himself as an "amateur taxonomist" and he acknowledges that working at this project showed him that "the difficulties met were not only the result of the nomenclature, but also that of the confusion there was concerning the basic concepts" and that a unique classification which would comply with the taxonomic principles of the classification of diseases was not possible.

According to him the "impairments" could be dealt with according to these principles, but trying to do so with the "handicaps" would lead to arbitrary compromise and contradictions.

Thus he suggests basing the classification on a different type of structure.

In May 1976, the publication of a "Classification of Impairments and Handicaps", complementary but distinct from the International Classification of Diseases, was approved by the World Health Assembly for experimentation purposes.

The text published four years later, in English, (which we will refer to as ICIDH-80) already contains a great number of modifications compared to WOOD's original project:

-the classification of "disabilities" has been modified following the model of that of "impairments" -functional disorders (i.e. which are not correlated with detectable organic impairments) which were originally placed in the classification of "disabilities", now belong to the classification of "impairments".

The French translation was to be published eight years later, in 1988, by the French WHO Collaborating Center. This translation, implemented by a team of a French medical research institute is considered as mediocre. It has been highly and rightly criticized for its omissions, additions, intentional modifications and errors.

Our analysis of the text in the nineties showed that:

-the changes brought about through the translation referred to political, economic or corporatist stakes -it was not an exact translation but rather a French adaptation implemented by translators whose work was largely under the influence of the lobbies of the concerned professional fields (medical, socio-medical and social) 1 .

In June 1997, WHO published a revised draft of its classification, titled "International Classification of Impairments, Activities and Participation. A manual of Dimensions of Disablement and Functioning. Beta-1 Draft for Field Trials" (which acronym is ICIDH-2).

We have implemented the French translation and the analysis of this draft, on the request of the French WHO Collaborating Center. The analysis of this draft showed that the main translation difficulties were different from those expected by WHO, listed in the "translation guidelines" that were circulated along with the new draft.

The real difficulties refer most often to a problem inherent to the construction of the classification and to the definition of the categories. These difficulties stem from ambiguities used as elements of textual strategies linked to political strategies, which are themselves referring to professional stakes.

It is the case in particular, when terms or expressions such as "disablement", "concept", "model", "norm", "unit of classification" are used either in the same sentence or in various places in the text with different meanings. Thus, either translating them by the same term makes the text incomprehensible, or translating them by different terms ruins the textual strategy and therefore alters the (political) functioning of the original text. Here are a few examples applying to: the choice of the title, the definition of the object of the classification, the definition of the "units of classification", the organization of the "categories".

The title

Usually, the title of a classification is intended to set forth what is classified and possibly the object of the classification.

In the present case, the conception of what is meant to be classified gives rise to endless debates between professionals with contradictory stakes, and to increasing ambiguous wordings.

As a result, the title has been changed at each phase of the revision process: Far from getting clearer along the revision process, the title becomes less and less precise and more and more enigmatic.

The same occurs with the object of the classification, which is not mentioned any longer in the title.

The object of the classification

The text states that its objective is:

« To provide a scientific basis for understanding and studying the functional states associated with health conditions » (Aims of ICIDH-2 p. 9)

The text states also that:

« The ICIDH-2 classification has been organized in a hierarchical scheme keeping in mind the standard taxonomic principles. […] Within each dimension or component, the categories have been created in a stem-branchleaf scheme, so that a fourth-level category shares the attributes of the higherlevel categories of which it is a member. » (Appendix 3 p.191)

The analysis however shows that in fact, the "categories" are not defined in terms of properties or attributes; the definitions are not definitions in comprehension and they are not organized according to a hierarchical structure. Thus the text leads the reader deliberately toward an erroneous interpretation.

If it were effectively a hierarchical classification, the expression "Disability and

Functioning" -as the label of the generic category -should be defined in terms of properties or attributes common to all the sub-categories. It is not the case.

The text does not provide an operational definition of what is meant to be classified. Instead of using such terms as "generic category" or "superordinate category", the authors use the metaphoric expression of "umbrella term".

The text also does not provide a formal definition of "functioning and disability", but tautological expressions defining these with synonymic terms. For instance: the categories of the first hierarchical level are defined with synonymic terms 2 and as "dimensions" of "functioning and disability", the latter being then defined as the "umbrella term" which encapsulates the whole of the "dimensions".

The question of what is the object, whose "dimensions" are covered by an "umbrella term"

remains unanswered. And one is lead to think that the authors of ICIDH-2 cannot or do not wish to answer the question.

However it must be said that if the text provided a clear definition of what constitutes the object of the classification and of what has to be classified, the question of the title and of its possible translations would then be solved. But our analyses lead us to hypothesize that the political functioning of this text requires that these questions not be answered.

In order to interpret and translate the expression "Functioning and Disability" and in the absence of definition, we have to refer to the contexts of its occurrences.

There are two possible options:

2 For instance: « Body Functions » are defined as the « functions of body systems »; « Activity » is defined as « the performance of a task or action by an individual »; « Participation » is defined as « an individual's involvement in ». These definitions are not definitions in terms of properties or attributes but in synonymic terms.

1°. one can either consider "Functioning and Disability" as an expression of common language and interpret it in relation to the contexts of its occurrences in the text itself, 2°. or one can interpret this expression in relation to the theoretical context it refer to3 . One must then consider the expression as the name of the generic category of a scientific classification.

1°) If we confine the study to the uses of the expression in the text itself, one observes that it is used to designate:

1°. the "functional states associated with health conditions" (p.7&11) As a result, the expression "Functioning and Disability" has no precise signification and the multiplicity of the usage categories4 of the terms, which compose it, will allow the translators and readers to interpret it freely.

2°) If we take into account the contextual indications that the text contains, one will interpret the expression "functioning and disability" in relation to the theoretical context which it is referred to, following the criterion of coherence:

The definition of the generic category of a hierarchical classification has to be intensional and must express the set of properties or attributes common to all the categories of the classification and only to them.

It should then be possible to reconstruct this definition by identifying the properties or attributes of the sub-categories and by extracting those that are common to all of them.

But this attempt fails and it appears impossible to identify a permanent core -even very general -of properties common to all categories. It even appears impossible to identify only one property or attribute common to all categories and only to them.

This failure is due notably to the fact that categories are not defined in terms of common properties or attributes, as the authors claim 5 , but on the basis of "resemblances"; what one could call a network of "family resemblances".

In fact, the type of analysis that WITTGENSTEIN applied to the notion of game and that Umberto ECO applied after him to the notion of "fascism" 6 may be used here to analyze the contents of "Functioning and Disability".

"Functioning and Disability" designates a series of "domains of functioning" which include the loss of a limb as well as the "quality of sleep", compliance with a cultural or a legal norm, "temperament" or "personality". These "domains" show at the most a certain "resemblance" that one does not really know on what it is founded.

Let us postulate a series of domains of functioning:

1 2 3 4 a b c b c d c d e d e f
"Domain 1 is characterized by the features "abc", domain 2 by "bcd" and so on. Two is similar to One , having two features in common. Three is similar to Two and Four is similar to Three for the same reason. Three is also similar to One (they have feature "c" in common).

The strangest case is Four, obviously similar to Three and to Two, but without any common feature with One. However, because of the uninterrupted series of decreasing similarities from One to Four, a family likeness remains between Four and One through to a kind of illusory (illusive?) similarity" 7 .

"Functioning and disability" is thus an expression, which is likely to adapt to any situation, for even if one removes one or several features, it will always be possible to acknowledge a "functional state associated with health conditions".

But if these expressions are not defined within a scientific or technical context, can they however be considered as expressions of common language? We do not think so.

Common language uses expressions such as "this disability made him eligible for a pension", "he is under a disability", "he has a health condition", "he seems severely impaired". "his appearance is a great handicap" "he was greatly handicapped by his accent".. Even an uneducated observer is able to interpret at least the general meaning of these syntagms, if not their precise signification. But with such expressions as "functioning and disability", "process of functioning and disability", "functional states associated with health conditions", etc. the common reader will find it difficult to give them a precise meaning. He may well suppose that they do have precise definitions in scientific or technical contexts, which are not for the common run of people. These expressions constitute a pseudo-scientific, pseudoordinary language.

One may question whether under the "family resemblance" which link the categories, there is a very general property that survives in each of the terms of the chain. And whether it is possible to construct, on the basis of this property, an object that is not identified to any of the "categories" studied but accounts for each of them.

There is indeed a property that does not describe any of the "categories" of ICIDH-2 but accounts for the existence of each of them. Our hypothesis is that each of the categories of this "classification" tries to identify properties, qualities, attributes, situations of persons that constitutes the present or potential clienteles of medical and medico-social professions.

We have elsewhere 8 hypothesized and showed that inasmuch as the categories define the clienteles and the professional fields, the expressions that designate these categories, their definitions or absence of definition and consequently their translations in different languages constitute political stakes and are subject to fierce debates.

The problem occurs also in regard to the "unit of classification".

The "Unit of classification"

A chapter of the introduction titled "Unit of classification" proposes to develop the following In the second sentence: « The unit of classification is therefore the "domain" of functioning (the area where functioning or disability occurs) », the expression « unit of classification »

refers to what constitutes the units of classification or Taxa 9 that is to say the categories of the classification.

In the third sentence: « It is important to note, therefore, that in ICIDH-2 persons are not the units of classification; that is, ICIDH-2 does not classify people », « unit of classification » designate the objects or « concrete occurrences » which are meant to be classified.

It appears that the categories (which are abstract objects, set of properties) and the empirical objects that have to be divided up into these categories according to their properties are here merged and confused under one and the same term. The type and the token, the concept and the occurrences of the concept are here mistaken one for the other.

To say that in ICIDH-2 the units of classification (or taxa) are not the persons is exact. It is even a truism since the units of classification are nothing else than collections of properties 10 .

But one cannot infer from that that ICIDH-2 does not classify the persons; on the contrary if these properties and attributes are those of the persons what could ICIDH-2 classify but persons?

9 In the taxonomic model to which the text refers, each hierarchical level of the classification is called taxon and each taxon is defined by a number of features (or « attributes» or properties) common to the species (or to the hyponymous categories), which constitutes it. 10 The text states that "definitions (of the taxa) are statements that set out the essential attributes (i.e. Qualities, properties or relationships) that constitute the concept or thing designated by the category." (Beta-2, p.193). It says also that ICIDH-2 is a classification of health characteristics of the persons in the context of their situation of individual life. We should deduce from this that the properties used to define the categories are attributes of the persons. This series of confusions not only shows that ICIDH-2 does not comply with taxonomic rules;

it is likely that it is an element of textual strategy focusing on eluding the true object of the classification which the political functioning of the text requires not to unveil.

Let us continue with the analysis of the categories and their organization.

Organization of the categories

The text then proposes a brief description of the organization of the information according to three dimensions named "functions and structures of the body" (B), "activities" (A) and "participation" (P). These dimensions are themselves divided into "domains of functioning".

A quick study of the "items" of each of the three "dimensions": "Body functions", "activities" and "participation" show that most categories of the "classification" of "Body functions" merely describe activities: motor, perceptual, nutritive, sexual, intellectual which are simply presented out of a context, in a « learned » language and often arbitrarily qualified as "functions".

But the observable facts they are supposed to categorize are nothing else than activities such as: chewing, swallowing, sucking, breathing, sleeping, talking, writing, calculating, looking, listening, interpreting, etc.

The same can be said of the categories of "participation", which merely describe activities: motors, perceptual, nutritive, sexual, intellectual, etc., often the same as the former, but grouped differently and presented in a social context and re-qualified as "Participation in...".

But the observable facts they list and categorize are no different from those classified in the two other "dimensions" and are nothing other than human activities: to move around, to prepare food, to feed oneself, to take care of oneself, to produce and interpret signs, to talk, to write, to learn, to work, to play, to have sexual intercourse, to do economic transactions, to pray, to vote, etc.

The distinction between the three dimensions: body functions/ activities/ participation is based on false evidence of common sense. Rhetoric procedures and artifacts support the likelihood of these distinctions whereas empirical facts, which constitute the categories occurrences, are generally the same for all three cases.

The same ordinary observable fact: an activity (X) can be placed:

-first in a category of "body functions" defined as such:

bXXX Function (X)

Function of (X) ("scientific" language)

-secondly in a category of "activities" defined as:

aYYY Activity of (X)

Actions and tasks performed by an individual for (X) (common language)

-thirdly in a category of "participation" defined as:

pZZZ Participation in (X)

Involvement in (X) (common language) as a function of the availability and accessibility to... We have showed elsewhere that these distinctions are not justified either from a theoretical point of view or from an empirical point of view.

A metaphorical language is used to justify the quasi-total overlap between dimensions.

We hypothesize that these three "parallel classifications" organize the same information but considered in relation to the needs of different professional sectors. A first version of the text stated that:

"For specific purposes (...) the orientation of the impairment, disability and handicap codes will be respectively, for medical, rehabilitation, and welfare services." ICIDH-80, p. 18.

But this mention disappeared in later versions.

The analysis can be carried on with the lower-level categories.

For the classification of "impairments", it is more complex 11 and cannot be presented here.

Let us say in short that while the text states as a principle that:

"Impairments have been conceptualized in congruence with biological knowledge at the level of tissues or cells and at the sub-cellular or molecular level" and that "the biological foundations of impairments have guided the classification", textual strategies are set up in order to avoid as much as possible to refer to biological knowledge 12 .

The analysis of the organization of "categories" will show that in spite of the principle stated in the introduction, the organization of categories does not follow the conceptual organization of existing biological knowledge. Rather it consists in following -or setting up -the "boundaries" of biomedical and medico-social specialties (and that is the reason why they represent political stakes and are object of bitter debate)

The analysis of the overlaps between categories show that they do not refer to overlaps between fields of knowledge but rather to overlaps between professional fields.

We may thus hypothesize that the categories of the WHO classification do not aim principally at the scientific organization of a field of knowledge but rather focus implicitly on the social organization of a professional field.

The formal model of the hierarchical classification and the scientific references along with it seem to be dedicated to induce the ordinary reader into the idea that the expressed normative proposals are scientifically founded and therefore convey a "universal value"; idea which the use of the term "international" will reinforce.

Conclusion

The translation of this type of text raises technical and ethical problems, which we will mention briefly.

-If one is attached to restitute as exactly as possible the content of each proposal, one is then led to implement what is considered a "good translation" with regard to the acknowledged criteria. But by doing so, one ruins a number of textual strategies and one elicits inconsistencies and contradictions, which are hidden in the original text by rhetorical procedures and artifacts. In other words one produces a relatively exact translation, but socially and politically the text no longer functions as intended.

-If on the contrary, one tries to protect the textual strategies and the procedures dedicated to induce the reader into a certain type of attitude or into a particular interpretation (and it 11 ROSSIGNOL, C., « Classification internationale des altérations du corps, invalidités et handicaps : modèles et enjeux du processus de révision » in Handicap, n°79-80, juillet -décembre 1998.

12 Ibid.

is this option that the French translators of ICIDH-80 and the Canadian translators of the later versions seem to have adopted), the translator then meets a number of technical difficulties related to the fact that these textual strategies often draw upon characteristics of form or usage of the source language which have no equivalent in the target language (such as homonyms, usage categories of a term which cannot be found in the target language, etc.). Frequently the translator cannot overcome these difficulties but with omissions, additions, intentional modifications, changes which will appear as translation errors, but which once analyzed, reveal to be meant to protect the social and political functioning of the text. It seems that this has been the case of the first French translation of this classification.

The translation of a measurement instrument such as questionnaire, nomenclature or classification should not raise particular problems as long as the question is to express in a different language a precisely defined content. Such a translation is always possible, even though it may sometimes require "painful" periphrasis.

Specific problems, which may not find any solution, occur when the so-called measurement instrument does not solely aim at measuring but also at producing a particular impact on its user or on the social reality it claim to measure.

Christian ROSSIGNOL & Catherine BARRAL

October 2000

Figure 1 .

 1 Figure 1. Structure of the « classification »

  1 1 ROSSIGNOL C. 1992. Classification internationale des Handicaps ? Présupposés et enjeux politiques d'un choix de traduction ; approche sociolinguistique et historique, Langage et société, N° 62 décembre 1992.

  --« International Classification of Impairments and Handicaps » in 1975, becomes: -« International Classification of Impairments, Disabilities, and Handicaps. A manual of classification relating to the consequences of disease » in 1980. It is then changed into: -« International Classification of Impairments, Activities and Participation. A manual of Dimensions of Disablement and Functioning » for the ICIDH-2 βeta-1 Draft, then into: -« International Classification of Functioning and Disability » for ICIDH-2 βeta-2 draft.

  rule: "A classification should be clear enough about what it classifies" (p.11) and the reader is thus expecting an answer to the question of what ICIDH-2 classifies. The following four lines compose the entire chapter. They are composed of three sentences linked together by groundless deductive inferences (using unjustifiably the term "therefore"); each of them referring to a different meaning of the expression "Unit of classification".

Finally

  and above all, one should note that the text states what the units of classification are not but does not state what they are. It says what ICIDH-2 does not classify but does not say what it classifies.

Appendix 3: « Taxonomic and terminological issues »

These categories of usage would be even more numerous if wee were to take into account previous versions.