On a connection between algebra, logic and linguistics

D ^IAGRAMMES

J. L

On a connection between algebra, logic and linguistics

Diagrammes, tome 22 (1989), p. 59-75

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D I A G R A M M E S V O L U M E 2 2 , 1 9 8 9 A C T E S D E S J O U R N E E S E . L . I . T .

( U N I V . P A R I S 7 , 2"7 JUIIM-2 J U I L L E T 1 9 8 8 )

ON A C O N N E C T I O N B E T W E E N A L G E B R A , LOGIC AND L I N G U I S T I C S

J . L A M B E K , M C G I L L U N I V E R S I T Y

Une grammaire indépendante du contexte (avec flèches renversées) con- siste en dérivations Ai . . . An —» An+ i ; on appelle les Ai types. O n peut con- sidérer u n e telle grammaire comme u n calcul de séquences à la Gentzen, mais sans ses trois règles structurelles: p e r m u t a t i o n , contraction et a t t é n u a t i o n . (Les trois règles sont permises dans la logique intuitioniste, les deux premières dans la "logique pertinente", et la première dans la "logique linéaire" de Gi- rard.)

L'idée d ' u n e multicatégorie trouve son origine dans la m é t h o d e des ap- plications bi-linéaires de Bourbaki. Formellement, une multicatégorie est une g r a m m a i r e indépendante d u contexte avec une relation d'équivalence appropriée entre dérivations (séquences). Elle peut être caractérisée pax son langage interne. Si on postule des équations appropriées correspondantes aux trois règles structurelles, on obtient une théorie algébrique (multisortale).

Une grammaire catégorielle est une grammaire indépendante d u con- texte avec u n e opération binaire / ( = sur) entre types; la n o t a t i o n logique est <= ( = si). D a n s une multicatégorie, A/B est le Hom(B,A) interne; dans le cas spécial, A 4= B est l'exponentiation AB. Le langage interne d'une multicatégorie avec / est u n e version unilatérale du À-calcul; avec les règles structurelles de Gentzen, elle devient le ÀJFf-calcul usuel de Church. Pour une g r a m m a i r e catégorielle (de l'anglais, p a r exemple) le passage d u premier au second est essentiellement ce que fait la sémantique de Montague.

It is claimed t h a t logic is t o linguistics as universal algebra is t o mul- tilinear algebra, the common t h r e a d being Gentzen's notion of "sequent", which goes u n d e r différent names in the four fields: déduction, dérivation, opération a n d multilinear form.

1. Contextfree grammars as multicategories.

A contextfree dérivation has t h e form / : A\ . . . An —> A, where Ai a n d A are syntactic types (also called "grammatical catégories"). T h e arrow hère

follows t h e convention in categorial grammar, in generative g r a m m a r t h e arrow is reversed.

Among t h e dérivations there is the reflexive law

a n d a rule for constructing new dérivations from old:

/ : A - » A g:TAA->B g<f>: TAA -> B '

called the eut rule, which represents substitution of / into g. Hère capital Greek letters dénote strings of types, e.g. A = Ai . . . An, n > 0. T h e n o t a t i o n g < f > for t h e new dérivation is somewhat ambiguous, inasmuch as it does n o t show where / h a s been substituted; b u t we shall use it just the same, as t h e place of s u b s t i t u t i o n will usually be clear from the context a n d because a more précise n o t a t i o n would appear to be r a t h e r cumbersome.

Among t h e syntactic types will be s for " s t a t e m e n t " a n d others t o b e discussed in Section 6 below. It is sometimes convenient to include English words among t h e types, think of them as one-element types. An analysis would yield two distinct dérivations

expérience teaches spiders that time /lies —* s,

interpreting t h e sentence to the left of the arrow in two différent ways. To see this compare t h e passive sentences spiders that time flies are taught by expérience a n d spiders are taught by expérience that time flies. On the o t h e r h a n d , there seems t o be no need for distinguishing the dérivations

John ( likes Jane) —> s a n d

(John likes) Jane —> s, assuming t h a t b o t h dérivations are possible.

Anyway, equality of dérivations is something t h a t should be considered.

If this is done, a contextfree grammar becomes what has been called a mul- ticategory [L1969]. However, the rules of equality are a bit awkwaxd to s t a t e directly, so I now [L1989] prefer to do this with the help of the internai lan- guage: we adjoin countably many variables of each type, say xt- is one such variable of type A,-, t h e n fx^x . . . xⁿ is to be regarded as a term of type A

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whenever / : Ai . . . An —• A is a dérivation. If also g : Ai . . . An —> A, we shall say that / = g provided we can prove fx\ . . . xn = gx\... xn from the usual rules of equality, including the rule of substitution of tenus of a certain type for variables of that type.

While we are hère mainly interested in applications to linguistics, it should be pointed out that dérivations / : A —> A hâve appeared in multi- linear algebra under a différent name: multilinear forms.

2. The rôle of Gentzen's structural rules.

The reader will hâve noticed the similarity between dérivations in lin- guistics and déductions in logic, called "sequents" by Gentzen, who imposed three structural rules on déductions in intuitionistic logic, which axe absent in linguistics:

/ : TAABA -> C f{ : TBAAA -+ C f : TAAA -> C

/c : TAA - C

/ : TA -> C

P : TAA -> C

interchange,

contraction,

thinning.

Again our notation f%,fc,ff ignores the place where the interchange, con- traction and thinning occurs. The capital letters now dénote "formulas";

recall the popular slogan: "formulas as types".

While logicians axe not usually interested in the question when two déductions are equal, this is sometimes important. For example, one ought to distinguish between the two déductions A A A —> A obtained by putting J3 = A i n A A i ? — • A and A A B —> B respectively.

Going over to the internai language of a "Gentzen multicategory" [see Szabo 1978], one may think of the variable x,- of type A,- as an assumption that A, holds. If / : Ai . . . An —> A is a déduction à la Gentzen, one may think of / x i . . . xn as a proof of A from the assumptions x,(i = 1 , . . . , n) à la Prawitz. Note that the saine formula may be assumed several times in one proof.

Gentzen's structural rules may now be viewed as définitions of new dé- ductions, when we write x, y, z,... for variables of types A, B, C , . . . respec- tively, and u = Ui . . . um, v, w,... for strings of vaxiables of types T, A , A , . . .

respectively:

f*uywxv = fuxwyv, fcuxv = fuxxv, fluxv = fuv.

Actually, a Gentzen multicategory may be considered to be a many- sorted algebraic theory, in which the formulas axe usually called sorts and the déductions are called opérations.

To adopt a neutral terminology, let us use Gentzen's word sequent for / : Ai . . . A„ —• A. We may then summarize our observations by saying that sequents appear in linguistics as dérivations, in linear algebra as forms, in logic as déductions and in uni versai algebra as opérations. Curiously, in linguistics, logic and algebra, people hâve considered variants in which some of Gentzen's structural rules axe permitted while others axe forbidden.

In linguistics, the three structural rules should be absent from syntax, at least the way I see it, but présent in semantics, as conceived by Curry [1958] and Montague [1974]. However some linguists, notably van Benthem [1983, 1988], hâve advocated that the interchange rule should be admitted when one discusses languages with free word order. Steedman [1988] and Szabolcsi [1989] seem to forbid thinning and to allow contraction.

In intuitionistic logic, ail three rules axe permitted, in "relevance logic"

only interchange and contraction and, in Girard's linear logic as originally formulated [1987], only interchange. However, one can conceive of a non- symmetric lineax logic in which ail structural rules axe forbidden [see Girard 1989].

In algebra, I am only aware of multilineax algebra, in which ail three rules are forbidden, and universal algebra, in which ail three axe allowed.

I haven't corne across any other variants, unless one wishes to consider as an algebraic theory Church's A-calculus [1941], which forbids thinning, as opposed to his Àif-calculus, which allows it.

3. Type forming opérations.

Let us now look at opération on types, formulas or sorts as studied in linguistics, logic and algebra respectively.

We are primarily concerned with binaxy and nullary opérations, pre- sented in their most common notation in algebra and linguistics, as opposed to logic, where they axe called connectives and occur in a rival notation. In

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this connection, one may also consider "quantifiers" with respect to variable types.

algebra: ® / \ x + / 1 0 Ux \lx

logic: <= =» A V T ± Vx 3 x

We shall henceforth refer to thèse type forming opérations as "connectives".

They axe usually introduced in terms of the set [V, A] of ail sequents / : T —»

A. To do full justice to the quantifiers, one must declaxe the free variables and should write / : T-> A or / G [I\ A\x, where A' is a set of variables which includes ail the variables occurring freely in Y and A. Such free variables are of course subject to the usual law of substitution, to which I plan to return on another occasion. In principle, ail connectives are introduced by means of certain biunique correspondences, namely as follows:

[T(A®B)A,C]^[TABA,C], [T,A/B]^[TB,A], [T,B\A]^[BT,A], [T,AxB]^[T,A]x[T,B],

[r(A + B)A, C] S [rAA, C] x [VBA, C], [TIA,C)^[TA,C],

[ r , i ] - w , [roA,c]^{*}, [T,l[A(X))^[T,A(X))

,

x

[r\jA(X)A,C]^[rA(X)A,C}

.

X

On the right side, x dénotes the caxtesian product of sets and {*} is a typical one-element set.

If the reader is puzzled by the absence of / , ® and <= in the usual logical Systems, linear logic excepted, this cornes from the fact that, in the présence of Gentzen's three structural rules, 1 = 1, A®B = AxB and B\A = A/B.

[See L1989].

More precisely, the connectives axe introduced by specifying a particulax sequent and providing additional information to ensure that it induces a biunique correspondent as required:

(1) mAB:AB -+A®B;

if / : TABA - • C then 3!/§ : T(A ® B)A - • C such that /Sumxyu = fuxyv.

(2) eA B : (A/B)B -> A;

if / : TB -> A then 3!/* : T -» A/JB such that /*euy = / « y .

(2') e'AB : B ( B \ A ) - A;

if f :BT -> A then 3 ! /+ : T -> £ \ A such that e'yf+u = fyu.

(3) PAB : A x B -+ A, qAB :AxB-*B;

if / : T -> A and g : T -*• 5 then 3! < / , y >: T -* A x B such that P(< / , £ > « ) = / « , q(< />0 > «) = 0U-

(4) kAB : A -> A + B, lAB : B -» A + B;

if / : TAA -H- C and s : TBA - • C then 3![/,y] : T(A + B)A -* C such that

[/,y]ukxu = /uxu, [f,g]u £yv = ytxyu.

( 5 ) i : - I ;

if / : TA -> C then 3 ! / # : VIA - • C such that / * u i u = / u v .

(6) or : r -+ 1, 3!flf : T - 1.

(7) DFA : TOA - C;

3!^ : TOA -> C.

(8) * x : I l x A ( X ) ^ A ( X ) ;

if V( X ) : T$A(X) then 3!y>A : T - • Ux A(X) such that

*x<PAu =x <f{X)u.

(9) /ex : i l ( X ) * U x A(X);

if y>(X) : r A ( X ) A * C then 3!v?v : T ]\x A{X)A - • B such that

¥?^VU(KXZ)V = X <p(X)uzv.

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Note that, in principle, VCÏAB^AB etc should carry subscripts; but, in practice, we hâve usually omitted them. We hâve also been négligent in fail- ing to declaxe free variables other than variable types in the above équation.

This becomes more important when Gentzen's structural rules axe admitted.

It should be pointed out that, in ail cases considered, the existence and uniqueness of the sequent with a certain property can be expressed by means of équations which do not contain free variables, except variable types:

1) / § < m > = / , (g < m >)§ = g.

[2) e<r>=f,(e<g >)* = g.

;2') e' < / t > = / , (c' < g >)t = g.

[3) P < / , $ » = / , q < / , $ > = 0 , < p < ^ > , q < A > = h.

;4) [/,y] < k > = / , [f,g] < i>= g,[h<k>,h< £>} = h.

;5) / # < i > = / , (g<i>)*=g.

6) or = g.

8) 7TX < ¥>A > = x <p{X\ (TT^X < g >)A = g.

;9) c^v < /cx > = x <p(X), (g < KX > )v = g.

As regards (2), the notation for e^jsty in [LS 1986] was t'y (read t of y) and the usual notation for f*u is Xy^sfuy. The two équations in (2) then t a i e the more familiar form:

^yeBfuy cy =u,y fuy, \yeB(gu cy) =u grz,

where, for once, we hâve declaxed the variables. Moreover, in the first of thèse équations, we may substitute any term b of type B for y. In paxticulax, if b contains no free variables other than such as appeax in the string u, we obtain:

KtBfuy 'b=u fub.

A similax, but distinct, notation could be adopted for (2'). But is it worth the trouble to introduce an analogous notation for (1)?

As regards (6) and (7), I am not entirely happy with the way they hâve been stated.

4. Freelv gênerated categorial grammars.

Let us concentrate on the applications to linguistics and multilineax al- gebra, so we can forget about Gentzen's structural rules. The question now

arises: given a contextfree g r a m m a r Q ( t h a t is, a multicategory), w h a t h a p - p e n s when we freely adjoin some or ail of t h e connectives ®, / , \ , x , . . . ? More precisely, we wish t o construct the left adjoint F to t h e forgetful functor U from contextfree grammaxs with connectives to contextfree grammaxs with- o u t . T h u s F{Ç) is the contextfree g r a m m a r with (g),/, etc. freely generated by Q. We hope t h a t the multifunctor G -» UF(Ç) will t u r n out t o be full a n d faithful.

T h e m a i n problem we face is t o calculate t h e set [T, A] of ail sequents r —> A in UF(G). T h e problem splits into two p a r t s :

(I) F i n d ail sequents / : T -> A.

(II) Given sequents / , g : T -> A, décide when / = g.

T h e solution t o Problem I will show t h a t Q -> UF(Q) is full, while the solution to P r o b l e m II, sometimes called the "cohérence problem", should imply t h a t it is faithful.

I hâve looked at Problem I repeatedly, using Gentzen's m e t h o d of eut élimination, first ignoring équations between sequents [L1958] a n d later bringing t h e m in [L1969]. I intend t o look at it once more, exploiting t h e internai language t o get a b e t t e r insight into w h a t is going on. W h a t this m e a n s , essentially, is t h a t one replaces Gentzen style déductions by Prawitz style proofs.

My a t t a c k s on Problem II [L1968, 1969] contained w h a t I still believe were some good ideas, b u t also contained some mistakes. More successful a t t a c k s , sometimes for related Systems, were carried out by m a n y people; in particulax Mine [1977] and Szabo [1978] m a d e use of normalization and t h e Church-Rosser property, which I believe to be the n a t u r a l m e t h o d . To this also I hope to r e t u r n on another occasion.

Hère we shall merely illustrate how Problem I is to be solved, by con- fining a t t e n t i o n t o one connective only, and also give a hint for attacking P r o b l e m IL Suppose we hâve a sequent 0 —> C in UF(G) containing at most t h e connective / , it must be of one of t h e following forms:

(i) it is already a sequent in G]

(ii) it is of t h e form 1 ^ : A —> A;

(iii) it is obtained by a eut:

/ : A - > A g:TAA->C g< f>: TAA -> C

££

(iv) it is of the form (2a):

(v) it is of the form (2b):

eAB : (A/B)B - A;

/ : eB -» A / * : 0 ^ A / B '

Given 0 and C, it is easy to find ail sequents 0 —• C (if any) of the form (i), (ii) and (iv). If C = A / B , we may reduce the problem of finding ail sequents 0 —• C of the form (v) to that of finding ail sequents QB —• A.

The difficulty is with the eut (iii): suppose we hâve decomposed 0 as TAA, how do we know which A to try? The way out, following an idea of Gentzen, is to replace (iv) (that is (2b)) by the following more powerful rule:

(iv') / : A -> B g: TAA

g[f] : T(A/B)AA -> C '

where g[f] = g < e < / > > is given by

g[f]utwv = guetfwv,

t being a variable of type A/B. While (iv') was constructed with the help of a eut, it is a haxmless rule, inasmuch as the premises contain nothing which does not already occur in the conclusion.

One byproduct of replacing (iv) by (iv') is that we may dispense with (ii). For, unless (ii) is already an arrow in Ç/, hence a spécial case of (i), it is of the form IA/B : A/B —» A/B, in which case it may easily be proved, with the help of the internai language, that

IA/B=(U[1B]Y-

Thus IA/B c a n be constructed without using (ii), provided 1,4 and 1# can be so constructed, which we may assume by an appropriate inductional as- sumption.

More importantly, if we use (iv') in place of (iv), any sequent constructed with eut is equal to one without eut. This is Gentzen's eut élimination

theorem in the absence of structural rules, although Gentzen himself did not bother about "equality".

Limitations of space do not permit me to include the proof of the eut élimination theorem hère. On the face of it, the démonstration that a sequent constructed with eut equals one constructed without makes use of both équa- tions e < / * > = / and (e < g >)* = g. When translated into A-notation however, it appeaxs that, while Axç^(/?(x) cx = (f(x) is used, the équation

A^XG A ( / ^CZ) = / is not used in full generality, but only a conséquence of it, namely the implication:

if tp(x) =x ip(x) then AxGA(p(x) = \XGAHX)'

Anyway, both the proof of eut élimination and the proof that IA/B =

( 1 A [1B])* make use of the following technique, which seems to be useful for attacking Problem II: to compaxe two axrows f,g : T —> A/B, compare the expressions efuy and eguy in the internai language. Détails of thèse arguments will be found in the proceedings of the fortheoming conférence

"Kleene '90" in Bulgaria.

5. Some linguistic applications.

Although I now believe that production grammaxs axe better suited to natural languages then categorial ones, I am willing to ask the question: of what possible significance axe the connectives ® , / , \ , x etc. in linguistics?

While we only discussed one of the connectives in some détail hère, it appeaxs that they may be freely added to a contextfree grammar without loss in generality. More precisely, when we look at the contextfree grammar G and the enhanced grammar UF{G) as multicategories, the multifunctor G —*

UF(G) is a full embedding. In particulax, it is a conservative extension.

For the purpose of illustration, let G be the contextfree component of the small fragment of English grammar discussed in "Grammar as mathematics"

[L1989b]. In this contextfree grammar one may, for example, generate the pseudo-sentence

(t) John may hâve Perf be Paxt corne often.

Hère Perf ( = perfect paxticiple of) and Paxt ( = présent participle of) axe certain grammatical terms, which I hâve called "inflectors". Of course Perf be should be converted to been and Paxt co7ne to coming, but the rules permitting this are not contextfree.

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One can of course handle strings like (f) in a categorial g r a m m a r . Let u s introduce t h e following types, except for subscripts the same as in [L1959]:

Si = statement in présent tense,

n3 = n a m e ( = third person n o u n phrase), i = infinitive,

p = présent paxticiple, q = past participle.

If t h e différent words in (j) axe assigned types as follows, (f) is easily calcu- lated to be of type Sj :

John may hâve Perf be P a r t corne often n3 ( n3\ s i ) / i i / q q / i i / p p / i i i \ i To convert (f) i n t o an actual sentence, t h e production g r a m m a r of [L1989b] requires two non-contextfree rules. As it is customaxy to reverse axrows in categorial grammaxs, thèse two rules will take t h e forbidden form:

coming —* Paxt corne, been —• Perf be.

B u t ail is not lost. Let us incorporate the tensor product ® to mimic j u x t a - position, t h e n thèse rules take the permitted form:

coming —» P a r t ® corne, been —> P e r f ® 6e.

We may infer t h a t coming has type ( p / i ) ® i —» p a n d 6een has type ( q / i ) ® ( i / p ) —> q / p . We need the more complex types to the left of the axrows, for example, to check t h a t

coming often ( p / i ) ® i i \ i

has type p , as is shown in the following Gentzen-style argument:

( P / J ) i

(p/i)i(i\i) -» p

((p/i)®i)(i\i)-p-

In a similax manner we find that has and is hâve types ( ( n3\ s i ) / i ) 0 (i/q) -> ( n3\ s i ) / q and

( ( n3\S l) / i ) ® ( i / p ) - ( n3\S l) / p respectively, but being does not hâve type

(p/i) ® ( i / p ) - p / p ,

plausible as this might be, since the following is not an English sentence:

* John is being coming.

The connectives ®, / and \ hâve been used before; in fact, the above account closely follows [L1959], With a bit of imagination, it is not difficult to find linguistic applications for some of the other connectives. For example, if

ni = first person noun phrase, 112 = plural noun phrase,

then hâve in they hâve corne will hâve the following type:

( ( n A s O / i ) ® (i/q) -+ ( n i \ s ) / q

and the same with ni replaced by 112. Ignoring the more elaborate type on the left of the arrow, for the purpose of illustration only, we may infer that hâve has type

(i/q) x ( ( n i \ s i ) / q ) x ( ( n2\ s i ) / q ) , which is isomorphic to

(i/q) x (((m + n

)\s)/q).

Similaxly, a modal verb such as may has type n \ ( s1/ i ) , where n = ni + n2 + n3. However, we must distinguish between hâve of type ((ni + n 2 ) \ s i ) / q and has of type ( n3\ s i ) / q .

A case can also be made for assigning to the word only the polymorphic type f ] [x( X / X ) , where X is a variable type, although some provision has to

70

be m a d e to ensure t h a t only modifies only expressions which axe stressecL We may also try to ascribe to the word and the polymorphic t y p e nA-(-X"\JC)/JÏ";

b u t there is a difficulty with this: when and is placed between any two noun phrases, whether of type n i , n2 or n3, the resulting expression always has t y p e n2.

T h e r e seems to b e no d o u b t t h a t contextfree g r a m m a r s can be mean- ingfully enhanced by introducing opérations on types such as / , \ , ®, x a n d + , a n d perhaps also Ylx a n <^ others. More controversial is t h e claim implicit in many articles on categorial grammar, including eaxly papers of the présent a u t h o r , t h a t the entire grammar of English can be squeezed into t h e dictionary by assigning appropriate types to t h e words of the language.

In view of the décision procédure provided by eut élimination, this would imply t h a t the set of English sentences is not only recursively enumerable, which almost no one dénies, b u t even recursive.

6. Postscript.

At a récent workshop and conférence on categorial g r a m m a r , the a u t h o r was surprised by the number of people seriously engaged in research in this axea. In a paxallel course of lectures by Moortgat a n d Oehrle, t h e above décision procédure was discussed in great détail. Following van Benthem [1983, 1988], they associated A-expressions to dérivations, particularly in connection with eut élimination, although appaxently their A-calculus was t h e usual one a n d neither t h e lineax variant discussed hère nor the variant p e r m i t t i n g interchange by van Benthem.

Not ail the speakers at t h e conférence, however, were aware of t h e exis- tence of a décision procédure. T h u s , one speaker proposed the rule

(X/(Y/Z)) (Y/W) -> X/(W/Z)

on empirical grounds, although it is in fact provable. A n o t h e r speaker pro- posed the rule

((Z/Y)/X) (Y/X) -> Z/X,

which is not provable, in fact, not valid syntactically, a l t h o u g h it holds of course in the semantic calculus one obtains when Gentzen's s t r u c t u r a l rules are a d m i t t e d .

Several people suggested applications for some of t h e not so traditional connectives. T h u s Oehrle h a d an interesting use for x , allowing one to ac- count for a phonological component in addition to the syntactic one. Morrill

[1989] also proposed uses for + and even Girard's [1987] "why n o t " . Actually, Giraxd's linear logic also allows a négation and consequently the De Morgan d u a l s of ® and "why n o t " , b u t thèse seem to hâve no obvious applications in linguistics.

T h e r e is however a n o t h e r kind of duality, in view of the fact t h a t the d u a l of a production g r a m m a r is also a production g r a m m a r . T h u s one might consider the connectives corresponding to "over" a n d "under" in t h e dual g r a m m a r . They could be viewed as two kinds of s u b t r a c t i o n , say "less" and

"from", even t h o u g h the connective corresponding t o ® is still ®. As fax as I know, n o one h a s explored applications of thèse last mentioned connectives.

Corrections added in proof.

To Section 1, p e n u l t i m a t e paxagraph, add:

More generally, if ai is a n y terni of type A,-, / a i . . . an is a t e r m of type A,-.

W e stipulate t h a t l ^ x = x and t h a t g < f > u i . . . umwx ... wnvi ... Vk = gui . . . umfw1 . . . wnv1 ...vk.

In Section 3 , (2), replace / * e by e / * .

To Section 4, add: Alternatively, we could hâve constructed 1.4/B = eAB-

To Section 5, last p a r a g r a p h , add:

T h e above claim amounts to saying t h a t the g r a m m a r of English has the form F{G), where G has only sequents of the form 1 ^ .

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Mathematics Department McGill University

Montréal, Québec H3A 2K6.