Diss. ETH No. 9328
A Framework for
Syntactic and Morphological Analysis and its Application in a Text-to-Speech System A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZÜRICH
for the
degree
of
Doctor of Technical Sciences
presented by THOMAS RUSSI
Dipl. El.-Ing. ETH 13, 1960 of Andermatt, Switzerland
born December Citizen
aecepted
on
the recommendation of
Prof. Dr. W. Prof. Dr. A.
Guggenbühl, Kündig,
1990
*-*
corresponds ras+st
to
a
surface
z:z
or
_._
Otherwise,
s.
rast, fliess+st
«-+
_¦;_*,
t:t.
:0>
16
Chapter
The
epenthesis rule describes the
?-+
widmA+st
The deleted
*-*
(realized
an
e
tense)
in the surface and
endings.
widmest
feature
the type of stem is encoded form5 with special symbols which are (and phonetic)
morphological
into the citation
and past
arbeitA+st *-* arbeitest, wartA+st «-» wartest, leidet, hiessC+st *-* hiessest ebnA+st *-* ebnest,
Examples: leidA+t
insertion of
(present
form between verb stems
Formalisms
2.
as
null
indicating
symbols)
in the surface form.
In the
previous example, the *-+ Operator was used to define that an morpheme boundary in the surface form if and if of the restriction holds. Although this Operator is by context one only far the most frequently used, there are two other Operators which can be used as well. The Operators have the same meaning as in Koskenniemi e can
be inserted at the
[Kos83b,
p 37
ff]:
context restriction:
a :
b
The lexical character
—?
o
LC
RC
matches the surface character
it is in the context of LC and RC. The
pair
.
when
only
a:b cannot appear in
any other context. surface coercion:
a :
b«— LC
In the context LC and
surface character b and
combined rule: This is
a
a :
b
*-*
RC, a lexical nothing eise.
LC
character
a
matches
o
in that context.
In the next
give
a
section, we relate two-level ruies with finite procedural interpretation for two-level ruies.
5The encoding
mic) string
of
morphological
basically
features into the lexical
is somewhat awkward and introduces
eral modifications have been
add
an
co¬
matches the
in the context LC and RC and that a:b
only
only pair allowed
a
RC
It states that the lexical character
surface character b
and
only
combination of the context restriction and surface
ercion ruies.
is the
RC
(graphemic
redundancy
automata
and
phone¬
in lexical entries. Sev¬
proposed ([Eme88], [Bea88b], [Bea88a], [Tro90]),
additional mechanism to the two-level ruies to
access
which
lexical features.
17
Two-Level Formalism
2.1.
Ruies and Finite Automata
2.1.3
regulär expressions to state in declarative manner strings pairs consisting of a lexical and a surface symbol. There are two basic approaches to processing regulär expressions. One possibiUty is to have them processed directly by an interpreter. This approach is pursued by Bear [Beei86], who implemented em extended version of the two-level model. In Bear's system, ruies are directly inter¬ preted as constraints on pairings of surface strings and lexical strings. The second approach is to apply a well-known theorem of automata theory, which says that, for every regulär expression r, a deterministic automaton can be constructed which accepts the language L(r) (see, for example, Hopcroft [HU79, p 28 ffj). The compiling of two-level ruies into finite automata was put forward by Koskenniemi [Kos83b] and is pursued in this project as well. The description of such a Compiler does not lie within the scope of this dissertation (see, for example, KartunTwo-level ruies
use
the set of
nen
of
[KKK87]). However,
and transition
•
The
graphs
operational
we
introduce the definitions of finite automata
for the
following
reasons:
semantics revealed
by
the finite automaton nota¬
tion is contrasted to the declarative notation of the two-level rule. This leads to
a
better
understanding
of the
procedural interpre¬
tation. •
The transition network formalism described in Section 2.3 is based
the concept of finite automata.
on
following sections we shall strive to use the same symbols to same things. We adopt the notation of Hopcroft [HU79] as possible. Unless it is stated otherwise, the reader may assume
In the
denote the far
as
that:
1.
Q
is the set of states of
the
symbols
2. E is
3. 6 is 4. F is
an
a
a
an
q and p, with
automaton, qo is the initial state, and or
without
input aiphabet; symbols
state transition funetion. set of final states.
a
subscripts,
and b
are
are
states.
input symbols.
Chapter
18
5. w,
_
and
z are
(consisting
of
strings zero
symbols;
of input
(DFA)
consists of
set of transitions from state to state that
a
symbols. We formedly define
a
DFA
Definition 2.1 A deterministic
(Q,E,6,q0,F)
as
finite
is
a
finite
set
of states,
(2)
E
ts a
finite
set
of input symbols,
(3)
6 is
-
a
finite set of
occur on
input
follows:
automaton
(DFA)
M is
a
5-tuple
where
(1) Q
a
denotes the empty string
symbols).
A deterministic finite automaton states and
.
Formalisms
2.
possibly partial funetion,
-
mapping from Qx_
to
Q, called
state
transition
(4)
0, by recursively traversing the network A. The transition from state s2 to .3 recognizes the strings {cm}, m > 0, by traversing the network C. The network grammar Gl recognizes the language: transition from state si to state s2
L(M3)
{xe {a,b,c}* |
=
x
=
anbncm and n,m >
0}
An RTN grammar not only speeifies the set of strings a language encompasses, but also assigns to each string a constituent strueture tree.
Figure
2.4 shows
a
constituent strueture tree for the
string aaabbbcc.
Constituent strueture trees represent three kinds of information
syntactic
strueture of
1. The hierarchical nance
relation).
a
on
the
string:
grouping
of the
string
into constituents
(domi¬
34
Chapter
y\\ a
b
C
i
A
c
a 2.4:
b
Formalisms
k
yy a
Figure
A
2.
c
b Constituent strueture tree
for the string
aaabbbcc
The UTN Formalism
2.3.
2. The
grammatical type
3. The
left-to-right
35
of each constituent.
order of the constituents
(precedence relation).
advantages over DFAs. Com¬ monly oecurring subpattems expressed as named networks, and be modular networks. In addition, RTNs into can large grammeirs split reflect the recursive strueture of language in a natural way. RTNs have certain obvious notational can
be
RTNs are equivalent to context-free grammars in their generative ca¬ pacity. For example, the network grammar of Figure 2.3 cem be mapped into a strongly equivalent context-free grammeir G (Vr, Vn,S,P), =
where
(1)
the terminal
(2)
the nonterminal
(3)
the start
(4)
the grammeir ruies
aiphabet
symbol
is
Vt
aiphabet is 5 €
are
=
{_,&,_},
is Vn
=
{S,A,C},
Vn and
R=
^
S^AC
C^Cc
A—taAb
C
—*
c
A->ab
limitations in
specifying the syntax of natural languages. First, there are linguistic phenomena which exceed the generative capacity of context-free grammars. For example, crossserial ordering of subordinate clauses in Swiss German6 can be formally stated as the string ambncmdn, which cannot be expressed by RTNs or type-2 ruies. Second, other frequent linguistic phenomena, for ex¬ ample, case-gender-number agreement between determiners and nouns in German, can be expressed as RTNs (or context-free ruies) only by introdueing a large number of transitions (or ruies). This obscures the However, RTNs have
severe
real nature of agreement. 6An example of cross-serial dependency in Swiss German is the subordinate clause ...
Jan säit, dass
This is n
an
dative
d 'Chind
em
a
Hans
es
Huus händ welle laa
hälfe
aastriche
aecusative NPs followed
by by m aecusative-demanding verbs and n dative-demanding detailed description including a proof, see Shieber [Shi87].
NPs,
verbs. For
mer
instance of the pattern
followed
NP™NPjV™ VJ1,
m
36
Chapter
overcome
two
Formalisms
Unification-Based Transition Networks
2.3.2
To
2.
the limitations of
RTNs,
we
have extended the concept in
important respects:
1. Terminal
and
nonterminal
(atomic) symbols,
symbols pairs
but name-term
are or
no
longer
monadic
feature structures.
2. In addition to the linear
precedence and immediate dominance topology of the networks, additional con¬ terminals and constituents can be speeified by equations.
relations encoded in the straints between
using
unification
These extensions
considerably increase the generative power of the formalism, which now includes indexed and fully context-sensitive gram¬ mars, without changing the simplicity and declarativeness of RTNs. We have we
developed
two variants of the UTN formalism. The variant
describe first is based
logic complex as
on
the notation of terms of first-order
described in Section 2.2.2. feature structures
as
predicate
The second variant is based
on
described in Section 2.2.3.
To
explain the two variants of the UTN formalism, we use the gram¬ (see Figure 2.5), a transition network grammar consisting of four networks for simple Germern sentences. Network 5, the top-level net¬ work, speeifies an (infinite) set of sentences consisting of a noun phrase (NP) and a verb phrase (VP). The NP consists of mar
•
G2
(optional) determiner, zero, one or more eidjectives and e.g., der sternenübersäte Himmel (the star-spangled sky), an
Herbert,
a
noun,
or
•
a
proper name, e.g.,
•
a
pronoun, e.g.,
•
a recursively defined noun phrase followed by a prepositional phrase (PP), e.g., umweltfreundliche Autos mit niedrigem Ben¬ zinverbrauch (non-polluting cars with low petrol consumption).
er
(he),
or
or
2.3.
The UTN Formalism
37
>QPrePfrQ NP»Q
Figure
2.5:
Transition network grammar G2
for simple German
sen¬
tences
The VP consists of lowed
by
an
em
NP. A VP
intransitive verb
can
also have
a
or
of
a
transitive verb fol¬
number of PPs attached to it.
This grammar recognizes sentences such as Die berühmte Astronomin beobachtet den sternenübersäten Himmel im Observatorium mit dem Ra¬
dioteleskop (The famous woman astronomer observes the star-spangled sky in the observatory with the radio telescope). Appendix C contains the code for this example grammeir. There is one version of this gram¬ mar based on name-term pairs and a second version based on feature structures.
UTN and First-Order Terms
The first variant of the UTN formalism is based
•
sets of name-term
pairs
to
on
the notation of
represent terminals and constituents
and •
unification
equations
to
speeify
constraints that must be satisfied
between terminals and constituents.
38
Chapter
Terminals and constituents name-term
represented
pair>
»(»{"("
::=
an
(unordered)
symbol and is (a symbol prefixed by "?") or an optional in peurentheses (infix notation). is a
as
set of
pairs
aBj
S
and
w
6
FIRST^)}
The REACHABILITY relation 3. holds between two B if there is
a
in the
dominated
string
Definition 3.5 LetR A £ N and B any sentential
symbols
A and
derivation from A to B such that B is the first element
_
by A.
More
formally:
(_V, S, M,S)
=
(N U E).
be
a
B is reachable
network, 4- Sa in
recursive transition
from A, Ä&B
A
form.
The FIRST relation is
a
subset of the REACHABILITY relation.
The REACHABILITY relation closure of the left-corner
can
also be defined
relation, i.e.,
as
the transitive
all
tuples consisting of the sym¬ bol of the left-hand side emd the first symbol of the right-hand side of all grammeir mies in em e-free contex-free grammar. For example, in grammar G2 (see Figure 2.5), the REACHABILITY relation includes the following tuples: {(S, NP), (S, det), (NP, noun),...}. The above relations
can
be
precomputed
and stored for
a
specific
At parse time, this information can be used to effieiently the rule invocation strategy. In the next two sections, we describe
grammeir.
guide four top-down relations.
and four
These
bottom-up strategies which make use of eight strategies have been implemented in our
these chart
parser.
3.3.1
Top-Down Strategies
Top-down parsing can be viewed as finding a derivation for an input string. Beginning with the start symbol, nonterminal symbols are re¬ placed step by step by the right-hand sides of the corresponding ruies until the string consists of terminal symbols only. Top-down parsing can also be regarded as construeting a parse tree for the input string starting from the root and creating the tree in preorder. There
several top-down strategies (see, for example, [AU72], but quite inefficient top-down algorithms based on general [ASU86]), recursive or backtracking algorithms and nonrecursive algorithms based on LL(k) tables, where k indicates the number of lookahead symbols. are
64
Chapter
Between the
which
general (exponential) algorithms based
be used to parse context-sensitive
can
(ünear) LL(k)-algorit_ms, which are, subset of the context-free languages, there is
ficient
Algorithms
3.
on
backtracking,
languages,
however,
and the ef¬
restricted to
a
class of
quite efficient general context-free languages. These algo¬ rithms, sometimes also called tabular parsing methods [AU72], belong to the family of chart parsing algorithms, which are especially well-suited to parse natural languages.
algorithms
In the
starting
which
can
following,
a
parse
we
present four top-down rule invocation strategies,
with the most
simple but leeist efficient one, which is a pure topThe three strategies use the FIRST and FOLLOW other strategy. relations to pmne search paths which do not lead to a parse. down
Strategy
Tl
(top-down)
Strategy
Tl is the
simplest top-down
rule
invocation strategy discussed here. After initializing the chart with an inactive edge [i,i + l,Cj,Cj,e], for eeich input word a< of category Ci,
edge [1,1, S,e, X] is added to the chart for each transition (qs,X,p). hypotheses predict that the input string will be parsed as a constituent of type S. The top-down parser proeeeds as follows: For every pedr of active and inactive edges, the fundamental rule is apphed. In addition, each time an active edge "seeking" as next symbol a nonterminal X is eidded to the chart, an empty active edge of category X is eidded to the chart at the vertex where the active edge ends (unless it is already in the chart). The fundamental mle and the prediction of new, empty active edges are apphed until no more edges a new
active
These initial
can
be added to the chart.
If the chart contains
one or more
inactive
edges of type 5 (i.e., [l,n,5,a, e]) that span the entire chart, the input string has been recognized. Otherwise, the string does not belong to the language defined by the grammar. Figure 3.7 shows a simplified version of the recognition algorithm. In
implementation, an edge can be extended to contain a parse tree, thereby turning the recognizer into a parser. In addition, an effi¬ cient indexing scheme can be used instead of a simple hst to maintain the set of active and inactive edges. Furthermore, a second data stmc¬ ture called an agenda can be used to störe the tuples of active and inactive edges to which the fundamental rule is to be apphed. Depend¬ ing on whether the agenda is implemented as a Stack, queue or sorted list, the algorithm behaves as a depth-first, breadth-first or heuristic an
3.3.
Chart
Algorithm:
65
Parsing
3
Input: A recursive transition network (RTN) -1-203 ...an with a< 6 input string w =
Output: An inactive edge [1, n Method: Initialize the set of
steps (2)
and
(3)
until
1,5, a, e]
+
an
active
edge
to set I
fedlure
input string, add an edge edges I. For each transition and p 6 Qs of Ms, add an edge of
[i,j, A,a,B] is eidded to set /, (qß,X,p) of Mb, a new active edge ej
=
(unless be
=
tive
Figure
this
edge is already
an
inactive
in set
Top-down
chart
I).
For each
=
3.7:
an
S
edge. [i,j,A,a, e] edge efc [k,i,B,ß, A] of set _" and for each (6*(QB,ßÄ),X,p), add a new edge [k,j,B,ßA,X]
3. Let et-
and
of the
every transition
[_/,_•',5,e,__]
(N, S, M, S)
by performing step (1). Repeat edges can be added to the set I.
[i,i + l,Ai,Ai,e] to the set (qs, X,p) with X € (S U N) [1,1,5, €,__"] toset/. add, for
=
I
edges
no new
1. For every terminal et,-
2. Whenever
or
R
parsing algorithm
ac¬
transition to set L
66
Chapter
search
3.
Algorithms
algorithm.
Strategy T2 (top-down with selectivity) Grammars for natural languages tend to have a large branching faetor, as, for a nonterminal A, there are frequently several ruies which expand A. It is often possible to restrict the number of alternatives if it is known which set of terminals can
derive the first nonterminal of the
network).
transition
This is
right-hand
side of
exactly the information
a
a
rule
(or
predictive top-
down parser uses to select one of a set of alternative ruies [ASU86]. Each time the parser enters a transition network of category A, each
edge [.,.,__,e,.B] is tested to see whether it cem derive the input symbol a,- by examining whether a< e FIRST(B). Therefore, step (2) two of the top-down algorithm of Figure 3.7 is modified as follows:
active
[i,j, A, a, B] is added to set /, eidd, for every transition (qi,,X,p) of Mb, a new active edge [j,j,B,e,X] to set I if aj £ FIRST(B). Whenever
an
active
edge
e
=
Remark:
This strategy is similar to
ever, it is
more
predictive LL(k) parsing.
How¬
general because it parses all context-free grammars. It to Kay's "directed top-down" scheme [Kay82], a directed
corresponds top-down strategy that uses the FIRST relation to test whether the next input symbol is in the FIRST set of the active edge each time an empty active edge is created.
Strategy T3 (top-down with lookahead) The use of the FIRST relation significantly reduces the number of useless active edges. The applieation of the FOLLOW relation can be used in a similar way to reduee the number of useless inactive edges. This is important, since inactive edges not only use storage but may also trigger new active edges. Each time an inactive edge [i,j,A,a,e] is added to the chart, it is tested whether the next input symbol aj (to the right) is in the FOLLOW set of the nonterminal A. Therefore, step (3) of the top-down algorithm (see Figure 3.7) is modified as follows: Let e,-
edge
=
ej_
[i,j,A,a,e] be [k,i,B,ß,A],
=
an
if
inactive
edge.
S*(qs,ßA)
£
For each active
Fß (a final
State
3.3.
Chart
Parsing
67
reached) and aj £ FOLLOW(B), add an inactive edge [k, j,B,ßA,c] to set I; eise, if 6*(qß,ßA) & Fb (non-final state reached), add, for each transition (6*(qs,ßA),X,p), an active edge [k, j,B,ßA,X] to set I. is
Remark: This
with
a
strategy corresponds one-symbol lookahead.
Strategy
T4
most directed
gies
(top-down strategy
to the
algorithm
with lookahead and
is obtained
T2 and T3. This leads to
a
of
Earley [Ear72]
selectivity)
by combining algorithm
very efficient
FIRST and FOLLOW relations whenever
The
the features of strate¬
active
that
uses
the
inactive
edge is added to the chart. Steps (2) and (3) of the top-down algorithm of Figure 3.7 are replaced by steps (2) and (3) of strategies T2 and T3, respectively. Remark:
an
or an
Predictive
top-down parsing has been proposed by several ([Kay82], [Wir87]). Top-down parsing with lookahead is described by Earley [Ear72]. However, the combination of prediction and lookahead has never been studied. Based on our experiments (see Chapter 4), a most directed strategy, such as strategy T4, seems to outperform other strategies. researchers
3.3.2
Bottom-Up Strategies
Bottom-up strategies can be considered to construet a parse tree for an input string beginning at the leaves (bottom) and working up to the root (top). Shift-reduce algorithms [ASU86] are among the bestknown bottom-up strategies that reduee an input string to the start symbol by creating a right-most derivation in reverse. A subclass of the shift-reduce family often used to implement parsers for programming languages are the LR(k) algorithms, which are basically non-backtrack shift-reduce parsers whose shift and reduee actions are guided by an FA. Besides the general backtracking-based bottom-up algorithms capable of handling all context-sensitive languages and the special shift-reduce algorithms capable of handling only a subset of the context-free lan¬ guages (called LR languages), there are quite efficient algorithms to recognize general context-free languages. These algorithms belong to
Chapter
68
3.
Algorithms
parsing methods [AU72]. In the foUowing, we de¬ algorithm, a type of bottom-up rule invocation strategy. We start with the simplest but least efficient one and continue with improved versions. the class of tabular
scribe four variants of the left-corner
Algorithm:
4
Input: A recursive tremsition network RTN R _ia2a3... an with _,• £ S input string w
=
(_V,E,M,5)
and
an
=
Output: An inactive edge [1,»' + l,5,a,e] Method: Initialize the set of
steps (2) and
(3)
until
edges
no new
or
failure
by performing step (1). Repeat edges cein be added to the set I. /
input string, add an edge [*,»' + input items _,- and for of all Mb € Af, add a new active (qB,A,p)
1. For every terminal a,- of the
1, .li, Ai,e]
to the set of edges I. For all
each transition
edge [i, i, B, c, Ai]
to set i\
[i,j, A, a, e] is ewlded to set I, add, for every tremsition (qß,A,p) of Mb £ M, a new active edge [i,i,B,e,A] to set I (unless this edge is edready in set
2. Whenever
an
inactive
edge
e
=
[i,j,A,a,e] be an inactive edge. For each ac¬ [k,i,B,ß,A] of set / and for each transition (6*(QB,ßA),X,p), add a new edge [k,j,B,ßA,X] to set I.
3. Let e,tive
=
edge
Figure
et,
=
3.8:
Bottom-up chart parsing algorithm
Strategy Bl (left-corner) Before describing the left-corner algo¬ rithm, we introduce some terminology. The left corner of a rule is the leftmost symbol (terminal or nonterminal) on the right side. Similar, the left
corner
terminals
a
of
a
transition network is the set of terminals and
network
can
start with.
non-
We often refer to the transitive
using the term reachability relation parsing is to index each transi¬ tion network by its left corners. When a phrase is found, networks that have that phrase as their left corner eure tried in turn by looking for closure of the left-corner relation as
well.
The basic idea of left-corner
3.3.
Chart
phrases
Parsing
69
that span
remaining paths through the network. Roughly, in peirsing, the left comer of a transition network is recognized bottom-up and the remainder of the network is recognized top-down. Figure 3.8 shows the algorithm for left-corner parsing. A left-corner parser traverses the parse tree bottom-up and inorder. left-corner
Strategy strategies
(bottom-up
B2
with
top-down filter)
Bottom-up
often propose constituents that do not match higher-level constituents. This is a severe problem for grammars that have many common right factors. If, for example, the NP network has two paths
which derive det the
noun
and noun, this network is triggered twice on (the man), once on der and once on Mann.
der Mann
input string Bottom-up parsers are overproductive in edges that do not attach to phrases on the left. Directed bottom-up parsing avoids this problem by a teehnique that is the dual of predictive parsing. Directed bottom-up parsing is somewhat like running a top-down parser in parallel. Each time an inactive edge is added to the chart, it is tested whether there is an active edge at the start-vertex of the inactive edge which can be extended by the inactive edge. Step (2) of the bottom-up algorithm is modified
as
follows:
Whenever
an
[i,i,B,e,A]
edge [i, j, A, a, e] is added to set I, add, (g_j, A,p) of Mb £ M, a new active edge i" if there is em active edge [k,i,C,a,D]
inactive
for each transition to set
and D$tA.
(bottom-up with lookahead) Left-corner parsing optimized in another way by using a kind of lookahead similar to that of strategy T3. Each time an inactive edge is added to the chart, it is tested whether the next input symbol to the right of the inactive edge is in the FOLLOW set of that edge. Step (2) of the bottom-up algorithm is modified as follows: Strategy
can
B3
also be
edge e [i,j, A, a, e] is added to set I for every transition (qs,A,p) FOLLOW(A), add, a active new M, edge \j, k,B,e, A] to set I.
Whenever and aj £ of Mb £
an
inactive
=
Chapter
70
Strategy
head)
B4
3.
Algorithms
(bottom-up
The most efficient
with top-down filtering and looka¬ bottom-up algorithm is obtained by com¬
bining the top-down filter of strategy B2 and the lookahead of strategy B3. Step (2) of the algorithm of Figure 3.8 is modified in the following way:
edge e [i,j,A,a,e] is added to set I for add, FOLLOW(A), every transition (qu, A,p) a active new M, edge [i,i,B,e,Ä] to set I if there active edge [k, i, C, a, D] and DMA.
Whenever
an
inactive
=
and _j £ of Mb £ is
an
Remark: This strategy is similar to Tomita's extended version of the edgorithm [Tom86] which can be used to parse general context-free
LR
languages.
Computational Complexity
3.3.3
presented eight rule invocation strategies parsing. In this section, we discuss the computational complexity of chart parsing, i.e., its worst-case asymptotic time and space complexity. Time complexity is a measure for the number of elementary mechanical Operations executed as a funetion of the input. Space complexity is a measure of the memory that is required to störe intermediate results as a funetion of the size of the input. To indicate complexity, we use the 0-notation9. In order to analyze the complexity of chart parsing, we restate the algorithm in a form revealing the parallelism between context-free parsing emd matrix multiplication. This was originally shown by Martin et al. [MCP87]. Without loss of generedity, we assume that the grammeir is in Chomsky Normal Form [AU72]. Edges between vertex -i and vertex Vj consist of all possible combinations of edges from vertices u,- to vk and edges from vertices vk to Vj as created by the applieation of the fundamental rule of chart parsing. In the
previous sections,
we
within the framework of chart
9The use of the O-notation for upper bound wipes out constants from complexity For example, an algorithm with complexity 8n3 + 5n is O(n'). More formally, we say that a funetion / is "of order g" ox 0(g) iff there exists positive constants c and fc such that, for all n > fc, |/(n)| < c|_f(n)|. formulas.
3.3.
Chart
Parsing
71
chart(i,j) The chart
:=
Ui"
"{"
char>
symbol>
expr>
Appendix
Syntax
B
of UTN
Formalism
::=
_}
:: "(" -[} ")" =
::=
::=
":term"
| ":graph"
-(}
::=
"(" {} ")"
::=
::=
"(" ")"
"(" {} ")"
107
Appendix
108
I
Syntax of
B.
UTN Formalism
::=
I |
::=
"(" "cat" ")"
::=
"(" "call" ")"
::= "(" "jump" I
")"
::=
"(" "reply" ( ) ")"
pairs>
::=
feature>
::=
