An Evaluation Semantics for DATR Theories 
Bill Keller 
School of Cognitive and Computing Sciences 
The University of Sussex, Brighton, UK 
billk@cogs, susx. ac. uk 
Abstract 
This paper describes an operational se- 
mantics for DATR theories. The seman- 
tics is presented as a set of inference rules 
that axiomatises the evaluation relation- 
ship for DATR expressions. The infer- 
ence rules provide a clear picture of the 
way in which DATR works, and should 
lead to a better understanding of the 
mathematical and computational prop- 
erties of the language. 
1 Introduction 
DATR was originally introduced by Evans and 
Gazdar (1989a; 1989b) as a simple, non- 
monotonic language for representing lexical inher- 
itance hierarchies. A DATR hierarchy is defined 
by means of path-value specifications. Inheritance 
of values permits appropriate generalizations to 
be captured and redundancy in the description of 
data to be avoided. A simple default mechanism 
provides for concise descriptions while allowing for 
particular exceptions to inherited information to 
be stated in a natural way. 
Currently, DATR is the most widely-used lexical 
knowledge representation language in the natural 
language processing community. The formalism 
has been applied to a wide variety of problems, 
including inflectional and derivational morphol- 
ogy (Gazdar, 1992; Kilbury, 1992; Corbett and 
Fraser, 1993), lexical semantics (Kilgariff, 1993), 
morphonology (Cahill, 1993), prosody (Gibbon 
and Bleiching, 1991) and speech (Andry et al., 
1992). In more recent work, DATR has been used 
to provide a concise, inheritance-based encoding 
of Lexiealized Tree Adjoining Grammar (Evans 
et al., 1995). There are around a dozen different 
implementations of DATR in existence and large- 
scale DATR lexicons have been designed for use in 
a number of natural language processing applica- 
tions (Cahill and Evans, 1990; Andry et al., 1992; 
Cahill, 1994). A comprehensive, informal intro- 
duction to DATR and its application to the design 
of natural language lexicons can tbund in (Evans 
and Gazdar, 1996). 
The original publications on DATR sought to 
provide the language with (1) a formal theory 
of inference (Evans and Gazdar, 1989a) and (2) 
a model-theoretic semantics (Evans and Gazdar, 
1989b). Unfortunately, the definitions set out in 
these papers are not general enough to cover all 
of the constructs available in the DATR language. 
In particular, they fail to provide a full and cor- 
rect treatment of DATR's notion of 'global inher- 
itance', or the widely-used 'evaluable path' con- 
struct. A denotational semantics for DATR that 
covers all of the major constructs has been pre- 
sented in (Keller, 1995). However, it still remains 
to provide a suitably general, formal theory of in- 
ference for DATR, and it is this objective that is 
addressed in the present paper. 
2 Syntax of DATR Theories 
Let NODE and ATOM be finite sets of symbols. E1- 
eInents of NODE are called nodes and denoted by 
N. Elements of ATOM are called atoms and de- 
noted by a. Elements of ATOM* are called values 
and denoted by a, /3, 7- The set DESC of DATR 
value descriptors (or simply descriptors) is built 
up from the nodes and atoms as shown below. \[n 
the following, sequences of descriptors in DESC* 
are denoted ¢, ~/J. 
1. a C DESC for any a E ATOM 
2. For any N C NODE and 66 C DESC*: 
N: (¢) e DES(; 
"N: (66)" E m.~sc 
t'(66)" E DESC 
"N" E DESC 
Elements of DESC are either atomic descriptors 
(1) or else inheritance descriptors (2). Inheritance 
descriptors are fllrther distinguished as being local 
(unquoted) or global (quoted). For ¢ C PESO* a 
sequence of descriptors, an expression (¢) is called 
a path descriptor. When each element of ¢ is 
atomic, then (¢) is called a path, and denoted P. 
For N a node, P a path and ¢ a (possibly 
empty) sequence of value descriptors, an equation 
of the form N : P == 66 is called a (definitional) 
646 
sentence. InforInally, N : P .... ¢ specifies a 
prolmrty of the node N, nalnely that the value 
of the path P is given by the sequence of value 
descriptors ¢. A DATR theory "Y is a finite se, t 
of definitional sentences subject to the fbllowing 
re(luirelnent of functionality: 
if N : 1' == ¢ 6 T & N : P =-- '(; 6 T th(!n ~/) -~ ~t/J 
t)Smctionality thus correst)onds to a semantic re- 
quirement that node/path pairs are associated 
with (at most) one value. 
3 Inference in DATR 
The probh'm of constructing an explicit theory 
of infhrence for DATR was originally addressed in 
(Evans and Gazdar, 1989a). In this work, an at- 
tempt is made to set out a logic of DATR state- 
merits. Consider for eXaml)le the following rule 
of in%rence, adapted from (Evans and Gazdar, 
1989a). 
NI:\['I--=N~:P'2, N~ : P'2 == ~ 
Ni : l'l ='-~'~ 
The prelnises are detinitional sentences which can 
be read: "the value of path 1'~ at node Nj ix (in- 
herited fl'om) the value of path P'2 at; N2" and 
"the vahle of path 1~,2 at, node N2 is el", respec- 
tively. Given the premise, s, the rule lieenees the 
conclusion "the value of path \['l at node Nj is 
(t". Thus, the rule captures a logical relationship 
between DATR sentences. For a given DATR the- 
ory T, rules of this kind lllay /)e used to deduce 
additional sentences as theorems of '\]-. 
In contrast, the system of inR;renee described 
ill this pal)er characterizes a relationship between 
DATR expressions (i.e. sequences of descriptors) 
and the vahles they may be used to cOlnlmte. 
As an example, consider the following (simpliiied) 
rule of the operational semantics: 
if N1 : l'~ == 4) G T the, n 
Ni : Pt ~ rt 
The rule is applieatfle just in case the theory T 
contains a detinitional sentence N, : t~l - - eft. It 
states that if the sequence of value descril)tors (/) 
on the right of the sentence evaluates to (-->) the 
sequence of atoms tt, then it may be concluded 
that the node/1)ath pair NI : I~ also evaluates 
to a. Rules of this kind may be used to provide, 
an inductive detinition of an evaluation relation 
between DATR expresskms and their values. 
Both approaches to inference in DATR aim to 
provide a system of deductioi~ that makes it possi- 
ble to (teterlnine formally, for a given DATR theory 
7~, what; follows fl'om the stateulellts in 7. The 
primary interest lies in deducing statements about 
the vahles associated with particular node/path 
pairs defined within the theory. UnRn'tunately, 
the proof rules described in (Ewms and Gazdar, 
1989a) are not su\[\[iciently general to support all of 
the required inferenees, and it is not obvious that 
the approach can be extended appropriately to 
(:over all of the available DATR constructs. A par- 
tieuiar t)rot)hnn (:on(:erns DATR's notion of non- 
local or global inheritance. The value (~xi)resse(l t)y 
a global inheritan(:e descriptor (lep(,nds on more 
than just the proi)ertie.s of the nodes sl)eeified by 
{;he definitional sentences of a theory. In fact, 
it only makes sense to talk about the value of 
a global descriptor relative to a given context of 
evaluation, or 91obal context. Because the proof 
rules of (Evans and C, azdar, 1989a) just talk about 
DATR sentences, which do not make explicit reii'~r- 
ence to contexl., it is not possible to give. a satisfac- 
tory a(:comlt of the global inheritance mechanism. 
The evaluation semantics described ill the fol- 
lowing sections provides a perspicuous treatment 
of both local and global inheritance in DATR. The, 
rules eaptme the essential details of the t)roeess of 
evaluating DATR expressions, and for this reason 
silouhl prove, of use to the language imt)lementer. 
4 Local Inheritance 
As a point of departure, this section provides 
rules of inference for a restricted variant of DATR 
which lacks both global inheritance and tilt de- 
fault me(:hmfism. This varianl; will be refl'ared to 
as DATRL. The syntax of DATRI, is as given in 
see|ion 2, except of course that the three forms 
of global inheritance descriptor are omitted. An 
exami>le of a simph; DATR~, theory is shown next. 
Noun: (cat) .... nouu 
@un) :_ 
Dog: (cat) == Noun 
(root) =-- dog 
(sing) .... (root) 
(plur) == (root) Noun: (sufl) 
In this and all subsequent examples, a nun> 
her of standard abbreviatory devices are adopted. 
Sets of definitional sentences with the same node 
on the left-hand side are groupe.d together and 
the node h;ft implicit in all but the tirst given 
sentence. Als% a definitional sent;en(:e such its 
Dog: (c.at) --=== Noun: (,'at), where the path on 
the right is identical to that on the left, is written 
more succinctly as Dog : (cat) --= Noun. Sim- 
ilarly, nodes oil the right of a sentence a.re sup- 
pressed whe.n identical to the node on the left. 
The DATRL theory defines the propertie~s of two 
nodes, Noun and Dog. The detinitional sen- 
te, iices specify values for node/path l)airs, where 
the st)eeitication is either direct (a particular value 
is e×hitfited), or indirect (the wflue is obtained 
by local inheritance), l%r e×alnpte, the value of 
the node/path pair Noun : {eat} is specitied di- 
rectly as noun. Ill contrast, the node/path pair 
Dog : (cat} obtains its value indirectly, by local 
647 
Val~tes : 
a~a Val 
Definitions : 
Sequences : 
Evaluable Paths : 
if N : (a) == ¢ 6 T then 
Def N: (a) ~ 
Seq 
¢ ===¢ a N : (a) =:::V p Sub 
N: (¢) ==~ fl 
Figure 1: Evaluation Semantics for DATRL 
inheritance from the value of Noun : (cat). Thus 
Dog : <cat) also has the value noun. The value of 
Dog : (plur) is specified indirectly by a sequence 
of descriptors Dog: (root) Noun: (suiT). Intu- 
itively, the required value is obtained by concate- 
nating the values of the descriptors Dog : (root) 
and Noun : (surf}, yielding dog s. 
We wish to provide an inductive definition of 
an evaluation relation (denoted ~) between se- 
quences of DATR descriptors in DESC* and se- 
quences of atoms (i.e. values) in ATOM*. We write 
to mean that the sequences of descriptors ¢ eval- 
uates to the sequence of atoms a. With respect 
to the DATR/ theory above we should expect 
that Dog : (cat) ~ noun and that Dog : 
(root) Noun: (surf) ~ dog s, amongst other 
things. 
The formal definition of ==v for DATRL is pro- 
vided by just four rules of inference, as shown in 
figure 1. The rule for Values states simply that 
a sequence of atoms evaluates to itself. Another 
way of thinking about this is that atom sequences 
are basic, and thus cannot be evaluated further. 
The rule for Definitions was briefly discussed in 
the previous section. It permits inferences to be 
made about the values associated with node/path 
pairs, provided that the theory T contains the ap- 
propriate definitional sentences. The third rule 
deals with the evaluation of sequences of descrip- 
tors, by breaking them up into shorter sequences. 
Given that the values of the sequences ¢ and ¢ 
are known, then the value of ¢¢ can be obtained 
simply by concatenation. Note that this rule in- 
troduces some non-determinism, since in general 
there is more than one way to break up a se- 
quence of value descriptors. However, whichever 
way the sequence is broken up, the result (i.e. 
value obtained) should be the same. The follow- 
ing proof serves to illustrate the use of the rules 
Val, Def and Seq. It establishes formally that the 
node/path pair Dog : (plur) does indeed evaluate 
to dog s given the DATRL theory above. 
dog==~dog Val s ~ s Val Dcf 
Dog: <root) :=~ dog Noun: <suiT) ~ s Def Seq 
Dog: <root) Noun: (surf) ==~ dog s Dcf 
Dog: (plur) ~ dog s 
The final rule of figure 1 deals with DATR's evalu- 
able path construct. Consider a value descriptor 
of the form A: (B: 0). To determine the value of 
the descriptor it is first necessary to establish what 
path is specified by the path descriptor (B : 0)- 
This involves evaluating the descriptor B : 0 and 
then 'plugging in' the resultant value a to obtain 
the path (a). The required value is then obtained 
by evaluating A : (a). The rule for Evaluable 
Paths provides a general statement of this pro- 
cess: if a sequence of value descriptors ¢ evaluates 
to a and N: (a) evaluates to/3, then N: (¢) also 
evaluates to ~. 
5 Global Inheritance 
DATR's local inheritance mechanism provides for 
a simple kind of data abstraction. Thus, in the 
DATRL theory above, information about the plu- 
ral suffix is stated once and for all at the abstract 
Noun node. It is then available to any instance 
of Noun such as Dog via local inheritance. On 
the other hand, information about the formation 
of singular and plural forms of dog must still be 
located at the Dog node, even though the pro- 
cesses involved are entirely regular. To overcome 
this problem, DATR provides a second form of in- 
heritance: global inheritance. This section pro- 
vides an evaluation semantics for a default-free 
variant of DATR with both local and global in- 
heritance (DATRG). A simple DATRG theory is 
shown below. 
Noun: <cat) :: noun 
<sum == <sing) 
=- "<root)" 
<pint) == "(root)" <surf) 
Dog : (cat) == Noun 
<root) == dog 
(sing) == Noun 
(plur) == Noun 
The new theory is equivalent to that given pre- 
viously in the sense that it associates exactly the 
same values with node/path pairs. However, in 
the DATRa theory global inheritance is used to 
capture the relevant generalizations about the sin- 
gular and plural forms of nouns in English. Thus, 
the sent~ence Noun : <sing) == "<root)" states 
that the singular form of any noun is identical to 
its root (whatever that may be). The sentence 
Noun: (plur) == "(root)" (surf) states that the 
plural is obtained by attaching the (plural) suffix 
to the root. 
To understand the way in which global inheri- 
tance works, it is necessary to introduce DATR's 
648 
notion of global contea't. Suppose that we wish 
to determine the value of Dog : (sing) in the ex- 
alnt)le DATRc; theory. Initially, the global context 
will be the pair (Dog, sing), bS"om tile theory, the 
value of Dog : (sing} is to be inherited (locally) 
fl'om Noun : (sing), which ill turn inherits its 
value (globally) from the quoted path "(root)". 
rio evaluate the quoted path, the global context is 
examined to find the current global node (this is 
Dog) and the vahle of "(root)" is then obtained 
by evaluating Dog : (root), which yields dog as 
required. 
More generally, the global context is used to fill 
in the missing node (t/ath) when a quoted path 
(node) is encountered. In addition, as a side ef- 
fect of evahlating a global inheritance descriptor 
the global context is updated. Thus, after encoun- 
tering the quoted path "(root}" in the preced- 
ing example, tile global context is changed from 
(Dog, sing) to (Dog, root). That is, the path 
component of the context is set to tile new global 
path root. 
Let T be a DATRa theory defined with respect 
to the set of nodes NODE and the set of atoms 
ATOM. The set (:ON'X' of (.qlobal) contexts of 7- is 
defined as the set of all pairs of the form (N, (t), for 
N G NODE and (.~ G ATOM*. Contexts are denoted 
t)y C. The evaluation relation ~ is now taken 
to be a mapping from elements of CeNT X \])ESC* 
to ATOM*. We write 
cF4)~ 
to mean that ¢ evaluates to fl in the global context 
C. 
To axiomatise the IleW evaluation relation, the, 
DATRc rules m'e modified to incorporate the 
global context parameter. For example, the rule 
for Evaluable Paths now becolnes: 
0 t- 4) ~, a C t- N : (,~) =-> fl Sub~ 
C P N : (¢) ==> fl 
Two sinfilar rules are required for sentences con- 
tMning quoted descriptors of the forms "N : {¢)" 
and "(qS)". Note that the context (7 plays no sl)e- 
cial role here, but is simply carried unchanged 
from premises to conclusion. The rules for Values, 
Definitio'ns and Sequences are modified in an en- 
tirely similar inanner. Finally, to capture tile way 
in which values are derived for quoted descriptors 
three entirely new rules are required, one for each 
of the quoted fi/rms. These rules are shown in 
figure 2. 
Consider for examt)le the Quoted Path rule. 
The premise states that N : (c~) evahmtes to fl 
in the glohal context (N, a). Given rills, the rule 
licences the conehlsion that the quoted descrip- 
tor' "(oe)" also evaluates to fl in any context with 
the same node component N. in other words, to 
ewduate a quoted path "(a)" in a context (N, (f), 
just evahmte the local descriptor N : (a) in the 
Quoted Node/Path : 
Quoted Path 
Quoted Node 
(N, .) ~- N: <,~> ~/J QUO 
C ~- "N : (c~)" ==~ ,{3 
(N, a) ~ N: <oe> ==:>/3 
Q~O 2 (U,~¢) ~ "<,~)" ~ fi 
(N,,,) ~- N: (.) =./~ O,,o:, 
(N', ,t) ~- "N" ==~ fl 
Figure 2: Evaluation of Quoted Descriptors 
updated global context (N,a). The rules (leal~ 
ing with global node/t)adl pairs, and global nodes 
work in a similar way. 
The following proof illustrates the use of tt{e 
Quoted Path rule (Qu%). It demonstrates that 
Dog : (sing) evaluates to dog, given the DATRo 
theory, aim when the initial global context is taken 
to be (Dog, sing). 
Val (Dog, root) t- dog =-=~ dog 
Def (Dog, root) \[- Dog: (root) ==~ ,log Q'ao2 
(Dog, sing) t- "(root)" ==~ dog Def 
(Dog, sing) F- Not,,,: (sing) :=~ dog Def 
(Dog, sing) F- Dog: (sing) ~ dog 
6 Path Extensions and Defaults 
In DATR, wflues may be associated with particu- 
lar node/path pairs either explicitly, in terms of 
local or global inheritance, or implicitly 'by de- 
fault'. The basic idea underlying DATR's default 
umchanism is as follows: any definitional sentence 
is applicable not only to the path specified on its 
left-hand side, but also for any rightward exten- 
sion of that path for which no more specitic def- 
initional sentence exists. Making use of defimlts, 
the DATRc: theory given above can be expressed 
more succinctly as shown next. 
Noun : (cat) ---: noun 
<~ing> -= "<root)" 
(plur) =:: "<root>" (still) 
(.,,ft-) --=. 
Dog : 0 == Noun 
(root) == dog 
Ilere, the relationship between the nodes Dog and 
Noun has effectively been collapsed into just a 
single statement Dog : 0 == Noun. This is p0s- 
sible because, the sentence now corresponds to a 
whole class of implicit definitional sentences, each 
of which is obtained by extending the paths found 
on the left- and right-hand sides ill the same way. 
Accordingly, the value of Dog : {cat) is specified 
implicitly as the value of Noun : (eat), and sim- 
ilarly for Dog : (sing) and Dog : (surf}. In con- 
trast, the specification Dog : {root} == Noun : 
649 
Ygllte8 : 
Val Ct-cx~a 
Definitions : if a is the longest prefix of a 7 
s.t. N : <a) -------- ¢ C T, then 
C~¢~fl D4 
Sequences : 
Seq C F ¢¢ ~, aft 
Evaluable Paths : 
Sub 1 C t- N: <¢> ===~ fl 
C \[- "N: (¢>" :::=~, fl Sub2 
Sub3 c ~- "<¢>" =% ,~ 
Quoted Descriptors : 
(N, ~) ~ N: <~> ~ 
C ~- "N : <a)" =:::~, fl 
(N, aT) I- N: (a) :=::V~ fl 
(N, ~.¢) ~ "<~>" ~ 
Q~o, 
Quo~ 
(N, aT) ~- N: ((~> ==~.y /3 
Q~tO 3 (N', a) ~- "N" =:::~, fl 
Path Extensions : 
C F N : (c~7) ~ fl Ezt 
C V N : <o~) ==:~.~ fl 
Figure 3: The Evaluation Semantics for DATR 
(root} does not follow 'by default' from the def- 
inition of Dog, even though it can be obtained 
by extending left and right paths in the required 
manner. The reason is that the theory already 
contains an explicit statement about the value of 
Dog: {root}. 
The evaluation relation is now defined as a map- 
ping from elements of CONT × DESC* X ATOM* 
(i.e. context/descriptor sequence/path extension 
triples) to ATOM*. We write: 
to nman that ¢ evaluates to a in context C given 
path extension 7. When 7 = e is the emi)ty path 
extension, we will continue to write C ~- ¢ ~ c,. 
A complete set of iifference rules for DATR is 
shown in figure 3. The rules for Values, Sequences 
and Evaluable Paths require only slight modi- 
fication as the path extension is simply passed 
through from premises to consequent. The rules 
for Quoted Descriptors are also much as hefore. 
Here however, the path extension 7 appears as 
part of the global context in the premise of each 
rule. This means/;hat when a global descriptor is 
encountered, any path extension present is treated 
'globally' rather than 'locally'. The main change 
in the Definitions rule lies in the conditions un- 
der which it is applicable. The amended rule just 
captures the 'most specific sentence wins' default 
mechanism. Finally, the new rule for Path Ezten- 
sions serves as a way of making any path exten- 
sion explicit. For example, if Dog : (eat} eval- 
uates to noun, then Dog : (} also evaluates to 
noun given the (explicit) path extension cat. 
An example proof showing thai; Dog : <plur> 
evaluates to dog s given the DATR theory pre- 
sented above is shown in figure 4. 
7 Conclusions 
The evaluation semantics presented in this paper 
constitutes the first fully worked out,, formal sys- 
tem of inference, for DATR theories. This fulfills 
one of the original objectives of the DATR pro- 
gramme, as set out in (Evans and Gazdar, 1989a; 
Evans and Gazdar, 1989b), to provide the lan- 
guage with an explMt theory of inference. The 
inference rules provides a clear picture of the way 
in which the different constructs of the language 
work, and should serve as a foundation for future 
investigations of the mathematical and computa- 
tional properties of DATR. Although the rules ab- 
stract away from particular impleInentational de- 
tails such as order of evaluation, they can be rea& 
ily understood in computational terms and may 
prove useful as a guide to the construction of prac- 
tical DATR interpreters. 
Acknowledgements 
The author wishes to thank Roger Evans, Gerald 
Gazdar and David Weir for suggestions and com- 
ments relating to this work. 
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Val (Dog, root) b dog ~ dog 
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