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I N F O R M A T I O N A N D C O N T R O L 17, 145-160 (1970)
Syntax and Semantics" A Categorical View
DAVID B. BENSON
Department of Computer and Information Science, University of North Carolina, Chapel Hill, North Carolina 27514
A syntax is a category of strings and derivations between them. The semantic domain is a category of sets and functions. An interpretation is a cofunctor from the syntax to the semantics generated from a correspondence between produc- tions and certain functions. There is a Galois connection between congruences on derivations and classes of interpretations. The smallest congruence of interest, similarity, is shown to correspond to the class of all interpretations. By considering certain subclasses of interpretations and the corresponding congruences, three different versions of "context-sensitive" are explicated.
1. INTRODUCTION
T h o m p s o n (1966) proposed a notion of semantics for formal languages generated by a semithue system in which the interpretation of a derivation is a function. Implications of essentially these semantic notions for context-free languages, in particular the implications for compiler design, were found by Knuth (1968). These semantic notions bear strong resemblance to the seman- tic methods for logical calculi (cf. Cohn, 1965). I n the theory of models of logical calculi, the interpretations are to relational systems, and certain collections of interpretations play an important role. In programming languages, and formal language theory the semantics is an action, or function, on certain sets. For example, the retrieval of information or the call and sequential processing of a subroutine is such an action. Notions of truth are not central here. T h e classes of interpretations are, however, useful. Hotz (1966) (also see Schnorr, 1969) introduced the idea of using category theory to study the derivational structure imposed on a formal language by a rewriting system generating the language. T h e categorical formulation is refreshingly clear in its control of the details. A derivationat structure, here called a syntax, is a certain type of category generated by a semithue system. T h e derivational richness of a syntax depends on the intent of the study, as
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146 BENSON
described in the next paragraph. The semantic domain is a category of sets
and functions. The interpretations of a syntax are confunctors from the syntax
to the semantic domain, generated by an association of functions with each
production of the semithue system. The author has noted the apparent
relatedness of Lawvere's "theories" (cf. Eilenberg and Wright, 1967).
In somewhat greater detail, the syntax category has as objects strings of
letters drawn from some fixed alphabet and, if one chooses, only those strings
derivable from the axiom of the semithue system. The morphisms are the
derivations of one string from another. If the derivations are purely syntactic,
or free, then it has been generally recognized for some time that inessential
distinctions are made. One may say that the derivational structure is too rich.
Hotz (1966) and Griffiths (1968) solved this by considering a certain relation
between free derivations called similarty, which is shown to be an equivalence
relation under more general conditions than are needed here. In fact, the
relation is a congruence for the composition of derivations. The equivalence
classes are again called derivations by Hotz. The semantic theory shows this
is entirely suitable by relating derivational congruences and classes of inter-
pretations in analogy to the methods in algebraic logic. The relation of
similarity is shown to correspond to the class of all interpretations.
Even sparser derivational systems can be studied, by taking larger equiv-
alence classes corresponding to smaller classes of interpretations. The
application given here of the theory is the explication of three differing notions
of context sensitive systems. Each variety is defined without semantics in the
literature. We consider increasingly restricted subclasses of interpretations to
obtain each of the three.
2. NOTATION AND TERMS
A category is a collection of objects and a collection of morphisms. Mor-
phisms with domain a and codomain b are written a --->b Unless labeled: for
example, x : a --+ b. The set of morphisms from a to b is denoted by (a, b)
There is at least one morphism in (a, a) for every object a, the identity on a.
If x e (a, b) and y ~ (b, c) then the composition y x is a member of (a, c).
Composition is associative. A functor from one category to another is a pair
of functions: The object function maps objects to objects and the morphism
function maps morphisms to morphisms, preserving identities and morphism
composition. See MacLane and Birkhoff (1967), Mitchell (1965), Freyd (1964)
or Cohn (1965) for exact definitions.
Lower case Greek letters denote strings over the alphabet A. The excep-
tions are q~, denoting the empty set, and ~, denoting class membership. The
148 BENSON
and
C = ( / ~ 0 _ ~'0 ,..., /'~n--1-/"~.--1 , "/r0-- P0 , ' " , ~'r'm--1 - P m - 1 ) -
DEFINITION 2.3. x~ = (d, r, c), a derivation from i~Ov to/~¢v, is called the
(~, v)-extension of xl = ((00 ,..., 0.), q, (~r0 _ go,..., %-1 - P~-l)), a derivation
from 0 to ¢, iff d = (l~Oov,..., I~O,v), r = q, and c -~ (l~rro _ pov,..., tzrr~_~ _ pn_iv).
DEFINITION 2.4. T h e length zero derivation ((0), ( ) , ( ) ) is called the
0-identity derivation.
DEFINITION 2.5. T h e length one derivation ((~, fi), (~ ~ fl), (A_ h)) is
called the (a--* fl) derivation.
3. SYNTAX
Three categories, called syntactic categories, are defined. Each of these
categories demonstrates the free derivational structure G-induced on A +. T h e
complete category, F, is a G-induced relational algebra in categorical clothing.
T h e objects of F are all the strings of A+ and the set of morphisms (0, ~b) is
the set of derivations from 0 to ~b. F is a category since (i) composition of
derivations, which is associative, is the morphism composition, and (ii) for
each object 0 the 0-identity derivation is the categorical identity for 0 under
composition of derivations.
For each ~ --* fi a production of P, the length one (~ --~ fi) derivation has
domain ~, codomain fi, and production application ~ --~ ft. This morphism is
purposely confused with the production a--~ fi so that the productions are
morphisms of F.
For each x : 0 --~ ¢, a morphism o f F , and for each pair of objects/~, v, there
is a unique morphism of F, i~Ov --~ ~bv, which is the (/~, v)-extension of x. In
an extension of x, the only action is from 0 to ¢ with/~ and v remaining un-
changed throughout the derivation ~Ov --~ I ~ v.
Morphisms are called derivations when the linguistic structure is to be
emphasized. T h e linguistic interest in F is as a convenient method of defining
the following subcategory.
T h e syntax, A, is the full subcategory of F such that (0, ~b) are morphisms
of A just in case (a, 0) is not empty in F. T h e objects of A are recovered from
the identities remaining in A, and are the strings in A* derivable from the
axiom, a.
If we divide the alphabet A into disjoint alphabets I and E, the internal and
SYNTAX AND SEMANTICS 149
external alphabets respectively, we define the external syntax, E, as the full
subcategory of A such that (0, ¢) are morphisms of E just in case there is an
o) E E + such that (~b, ~o) is not empty in A. T h e strings in E + which are also
objects of E form the language of the grammar G, in the usual sense. T h e
term "external" is used to avoid conflict with the use of the term "terminal" in category theory. T h e external syntax is not considered in the sequel. T h e following facts are immediate. If G is monogenic then A is ordered. G is loop free if and only if for each object 0 of A, (0, 0) contains only the iden-
tity. If P is a binary relation from A + -- E + to A +, then any morphism of A
with codomain a member of E + is epic. A does not, in general, have terminal objects.
4. SEMANTICS AND INTERPRETATIONS
T h e semantic domain is taken to be some fixed category of sets and func- tions, S, in which cartesian products are available. Cartesian product is considered to be an associative operation. With this, the extension of func-
tions is defined as follows: If W, X, Y, and Z are sets and f : X - ~ Y any
function from X to Y, then t : W × X × Z -~ W × Y × Z is an extension o f f preserving W and Z iff t acts on Y as does f and acts as the identity on W and Z. That is, t is an extension o f f if the following three diagrams commute, where the unlabeled arrows are the projections.
W × X × Z 2 ~ W × Y × Z
X Y , Y
W X X X Z ~ W X Y X Z
W
w × x x z ~ w × Y x Z
Z
SYNTAX AND SEMANTICS 151
I($) = N , the set of natural numbers; I ( # ) --- {1}, the set whose sole member
is the natural number 1; I(@) = A, a postulated identity for the cartesian
product operation. One could avoid using A, but it simplifies the presentation
of the example. The interpretation of the production set is given by
I($ -+ $@$) : N 2 --~ N = + :N 2 ~ N,
the ordinary addition of natural numbers;
I($ - - # ) : {1} --~ N,
the injection of 1 into the set of natural numbers.
There are two distinct phrase markers for derivations from $ to # @ # @ # ,
but each derivation has the interpretation f : {1} × {1} × { 1 } - ~ N such
that f(1, 1, 1) = 3. Exercising the terminology, the interpretation of the
sentence # @ # @ # is f : {1} × {1} X {1}-+ N; the meaning of # @ # @
is the image o f f , {3}; the semantics of the interpretation I is in this case a
subcategory of S. The example suggests the resemblance of this formulation
to that of tree automata (cf. Thatcher, 1967).
- CONGRUENCES
Let ~ be an equivalence relation defined on each set (0, ¢) of morphisms.
If two morphisms do not share the same domain and the same codomain then
they are inequivalent. Suppose the equivalence, ~-~, is such that for all
'7, 0, ~b, ~o, w E @, 0), x, y E (0, ~b), z ~ (¢, ~o) it is the case that x ~-~y implies
xw ~ y w and z x ~ zy. Then ~ is called a congruence. The equivalence
classes into which each (0, ¢) is partitioned by the congruence ~-, are the
morphisms of the quotient category A/~-~ (Mitchell, 1965, p. 4).
Let q~ be the class of interpretations of A. For each congruence ~-~ define
S(--~) as the class of all I e q~ such that whenever x ~ y, then I(x) = I ( y ).
For each subclass S define similarity modulo ~, ~ ( ~ ) , by : x ~ y(3) iff x
and y share domains and share codomains and for all I e 3, I(x) = I ( y ). The
correspondences
form a Galois connection (cf. Cohn, 1965, p. 44) between congruence
relations and classes of interpretations, the pertinent facts being recorded in
the following proposition.
152 BENSON
PROPOSITION 5.1.
classes o f qL T h e n
Z l 2 &
L e t H 1 a n d ~-~ be congruences on A, 31 a n d S 2 sub-
implies .~-,7(t~-,'l)~_ .~-~7(,,-'~-'2),
implies ~( ~ ) D_ ~.~( $2) ,
"~"1 C ,'~-'(~.,7(.'~-'1)),
S~ __CZ(~(Z~)). I
Intuitively, two derivations are congruent if they possess the same structure, therefore imposing the same collection of possible meanings on the derived sentence. T h e Galois connection results in a semantic definition of structure. The structural description of a derivation is the equivalence class to which it belongs, and by the Galois connection this is a function of the particular subclass of interpretations considered acceptable by some external criteria. T h e uninterpreted notion of structural description arises from the con- gruence relation of similarity, defined by Griffiths (1968). Let ~-~ denote the similarity relation for the rest of the paper. In this section we show that ~,~ = ~.~(~b). That is, two derivations are similar if and only if they are iden- tically interpreted by each possible interpretation. This result can be con- sidered an additional reason for studying the X-categories of G. Hotz. T h e free X-categories are the quotient categories of complete syntaxes modulo similarity. If ~ ( ~ ) is a proper subclass of q~ for some congruence ~ , the corresponding quotient category is a nonfree X-category. T o begin, the definition of uninterpreted, or syntactic, similarity is required. Similarity is the least congruence relation on A such that if
x : O - ~ ¢ , y : O ' ~ ¢ '
are derivations of A with extensions
Xo: 0 0 ' - + ¢0',
x l : 0 ¢ ' ~ ~¢',
Yo : 00' ~ 0~',
yl : ¢0' -+ ¢¢',
and u = y l x o , v = x l y o , then u is similar to v, u ~-~ v. T o motivate this definition, a brief description of Griffiths' development is given. A derivation can be interchanged into another by switching a pair of production applications when the applications do not interact. From the
154 BENSON
and I(0') = A s. Every square and triangle commutes by the properties of
extensions and the above categorical fact. I n particular, I(u) = I(v). Since
the generating relation for similarity is a subset of ~-~(¢), it follows from the definition of similarity as the least congruence than ~-~ _C ~-~((b). | T o show that u N v ( ¢ ) implies u ~ v requires developing an inter- pretation, Q, which recovers derivations up to similarity. Since Q ~ # , the desired result will follow from showing that a canonical derivation similar to
u can be obtained from Q(u). Q m a y be thought of as the best G6del n u m -
bering of derivations it is possible to obtain as an interpretation. T h e G6del
n u m b e r i n g is in the following generalized arithmetic, G(A, P), over the
alphabet A and productions P.
(i) A C_ G(A, P).
(ii) a ~ A, r ~ P and g ~ G(A, P) implies (a, r, g) ~ G(A, P).
(iii) Finite sequences of m e m b e r s of G(A, P ) are m e m b e r s of G(A, P).
T h e required projections from G(A, P) are denoted as follows:
I f g = (a, r, g') ~ G(A, P) then Pog = a, p~g = r, and p~g = g'.
Define the interpretation Q on letters of the alphabet by, for a ~ A,
Q(a) = {a} u {(a, r, g)[ r ~ P & g ~ G(A, P)}. Q is defined on the production
r = ~ -+/3 where ~ = a l ' " d e , as Q(r) : Q(fl) ~ Q(o~) such that for each
g ~ Q(fl), Q(r)(g) = ( ( a l , r, g),..., (ak, r, g)). Q is then extended to A as in
Section 4.
I f u is a derivation from 0 to ~b define the G6dcl n u m b e r of u, Gd(u),
by Gd(u) = Q(u)(¢). Given Gd(u) = (gl ..... gm) the domain of u is recover-
able as a 1 "" am where ai ----gi i f g i is not a triple and ai = Pogi i f g i is a triple.
Recovery of the codomain of u from Gd(u) is implicit in the recovery of the
canonical derivation similar to u.
PROPOSITION 5.4. U and v are similar iff they have the same G6del number.
Proof. u ~-~ v implies Q(u) = Q(v), thus Gd(u) = Gd(v). Assume that
Gd(u) = Gd(v). I t suffices to assume, in addition, that v is canonical. Applica-
tion of the following procedure recovers v from Gd(u) in a finite n u m b e r of
iterations.
Assume Gd(u) = (gl ,..., gin), gi ~ G(A, P). F o r the largest i such that each
of gl ,-.., gl is a letter and therefore not a triple, no production was applied to the prefix y = gl "'" gi • I f i = m the construction is complete. I f i v~ m
consider gi+l ~ - ( a , r , g') where a ~ A, r ~ P, g' ~ G(A, P). I f r = o~-+/
SYNTAX AND SEMANTICS 155
the length of e is k, ~ is equal to the concatenate ofpogi+ 1 ,..., Pogi+~, each of Pdi+l ,...,Plgi+~ is equal to r, and g' = P~gi+l = "'" = P2gi+~, then the production r = ~ --~/3 is applied at this point in the canonical derivation v. T h e next finite sequence to which the reduction process applies is constructed by replacing, in (gl ,-.., g~), the subsequence (gi+l ..... gi+k) by the sequence g'. T h e procedure is iterated on the new sequence. In the above test for the applicability of r = ~--+ /3 if there is some j, 1 ~ j ~< k, such that c~j =/= Pogi+j or r ~ Pxg~+~ or g' ~ P~gi+j, the production r must be defered since some other production application occured first, overlapping the application under test at g~+j. In this case begin testing at position i + j. u is similar to some canonical derivation, say x. Hence Q(u) =- O(x) and Gd(u) --- Gd(x) ~ Gd(v). Since x is canonical by the above construction it is 7). T h u s u ~-~ 7). | From Propositions 5.3 and 5.4 we have
PROPOSITION 5.5. F o r u, 7) derivations of A , u ,-~ 7) iff 1z ~.~ 7.)((I)). | As corollaries note that if ~2 is a class of interpretations with Q E ~2 then ~(~2) = ~ and 3(,~(f2)) = ~. T h e congruence of equality on A has as its corresponding class of interpretations ¢, thus ,--~(Z(=)) = ~.
6. CONTEXT SENSITIVE SYSTEMS
There are at least three different notions of the structural descriptions of sentences generated by a context-sensitive grammar. T h e first is the purely syntactic view in which the idea of allowing the rewriting by a context-free production only in a fixed (and local) context is avoided by requiring only that the length of production antecedents be less than or equal to the length of the corresponding production consequents. In this sense, the semithue produc- tion ~ - + fi is type one context sensitive just in case the length of a is less than or equal to the length of ft. It is well-known that the above is weakly equiv- alent to the more stringent condition on productions. However the correspond- ing structural descriptions lack the intuitively desired strength. Let P be a set of type one context sensitive productions and (A, P, s) generate the syntax C. T h e class of interpretations of C is denoted by ~ and the structural des- criptions of derivations in C are the morphisms of C/~-~. I n the purely syntactic view no additional structure is imposed. C is called a type one con- text sensitive syntax. T h e second and third notions arise from considering the context sensitive
SYNTAX AND SEMANTICS 157
A type three context-sensitive syntax is a type two context-sensitive syntax, C, with class of admissible interpretations, g?, the largest subclass of 3 such that for each I ~ f2 and each production x = a --+ 13/7 _ 3 generating C there exists a function
f ( x , I ) : I(fi) ---* I(a).
s'2 and the f u n c t i o n s f (x, I ) must satisfy the properties:
(i) T h e commutivity of
i(7/38 ) if(x)> i(717l~)
1(/3) I(0e,/) > I(a)
where the projections are again unlabeled.
(ii) If x = a ~ fi/y _ 3 and y = a --~ flirt _ p are productions then for each I e ~ , f ( x , I) : 1(/3) --~ I(a) = f ( y , I) : 1(/3) --~ I(a).
T h e latter property corresponds to considering a ~/3/Y - 3 and a --+/3/~- _ p to be a single context-free production a - + / 3 restricted to the context 7 - 3 or to ~r _ p. Without this property the underlying p-marker cannot be obtained. T h e structural description of a derivation x in C is the ~(g2) equivalence class [x]a. T o each equivalence class corresponds a unique p-marker of the under- lying context-free syntax, as the following demonstrates. Suppose P is the set of productions which together with their contexts generate the type three context-sensitive syntax C with class of interpreta- tions f2. T h e n R = {a ~ / 3 I a ~ / 3 / 7 - 8 ~ P } is the set of context-free productions generating the context-free syntax, D, with p-markers D / ~. T h e function F : P -+ R with F ( a --,/3/7 - 8) = a --+/3 extends to the context- freeing functor F : C -+ D. For each I : C ~ S in f2 define Jz : D ~ S by
(i) For each object 0, .1,(0) = I(0) (ii) On morphisms by induction from the productions R, where for each a --~ fi E R, Ji(a --~/3) = f ( x , I ) for any x = a -~ fi/7 - 8 e P such that = - + / 3.
T h e diagram C F + D
S
158 BENSON
commutes for each I ~/2. Furthermore {Jr I I E/2} is equal to q)n, the entire
class of interpretations of D, since any K e q~o is extendible to some I ~/2. From the above association of a context-free system with the type three
context-sensitive syntax C, one notes that the p-marker of x, a derivation in
C, is the similarity class of F(x), IF(x)|.
PROPOSITION 6.1.
C ~ ~ D
C//2 ~ , D/,.~
commutes where F : C --+ D is the context-freeing functor for C, C//2 is the
quotient category of C modulo ~-~(/2),D/~-~ is the quotient category of D modulo
similarity, and G : C//2-+ D/~-~ is the structural description or p-marker
functor, the identity on objects and on morphisms, G([x]a ) ---- [F(x)].
Proof. G is well-defined since for any x, y e (0, ~b), if x ~ y(/2) then
for any I ~/2, Jl(F(x)) = I(x) = I(y) -~ J1(F(y)), and so F(x) ~ F(y).
The commutivity of the diagram is obtained directly from the defi- nition of G. |
PROPOSITION 6.2. G is faithful, i.e., injective on morphisms.
Proof. Suppose G([x]a) = G([y]a), or equivalently, F(x) ,-~ F(y). Since
for all K E q)D, K(F(x)) = K(F(y)), and since every I ~/2 is an extension of
some K ~ q~n, it is the case that for a l l / ~ / 2 , I(x) = I(y), so that x ~ y(/2). |
From the propositions above one has
PROPOSITION 6.3. I f X, y ~ (0, @) are derivations of C then x and y are
similar modulo/2 if and only if they have the same p-marker. That is, x ~.~ y(/2)
i f f F ( x ) , ~ F ( y ). |
Kuno (1967) defines trees augmented by quadruples of integers at each node as the structural descriptions of context sensitive derivations. T h e corresponding uninterpreted congruence relation here is called cs-similarity. Cs-similarity, __~, is defined to be the least congruence on C such that
(i) x ~-~ y implies x ~__y, and
160 BENSON
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