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<?xml version="1.0" standalone="yes"?> <Paper uid="W94-0204"> <Title>E DEFAULT FINITE STATE MACHINES AND FINITE STATE PHONOLOGY</Title> <Section position="4" start_page="0" end_page="33" type="metho"> <SectionTitle> NOTATIONAL PRELIMINARIES </SectionTitle> <Paragraph position="0"> We assume an alphabet L, with a reserved symbol 0 ~ PS for insertions and deletions. A replacement over PS is a pair of the form I = (1,1') where (1) ! E PS and (2) I I E PS or i I = 0; ReplacementsPS is the set of replacements over PS. US-stringsPS is the set of strings over the set PS2 U \[PS x {0}\] of replacements.</Paragraph> <Paragraph position="1"> 2The elsewhere condition is built into an implementation (due to Karttnnen) of the TWOL rule compiler; see (Dalrymple et al., 1993), pp. 28-32. But on this' approach, default reasoning and the elsewhere condition are not employed at a level of computation that is theoretically modeled; this reasoning is simply a convenient feature of the code that translates rules into finite state automata.</Paragraph> <Paragraph position="2"> We use roman letters to denote themselves: for instance, T denotes the letter I. Boldface letters denote constant replacements: for instance, I is the pair (l,l). Moreover, C/ is the empty string over L~, and ~ is the empty string over the PS replacements. When the name of a subset of/2 (e.g.</Paragraph> <Paragraph position="3"> C) is written in boldface, (e.g. C), the set of identity pairings is intended (e.g., C = {l:l/l E C}). We use ordinary italics as variables over let~rs, and boldface italics as variables over replacements and strings of replacements. Ordinarily, we will use I for replacements and z, 7t for strings of replacements. Finally, we use ' I:I&quot; for the pair (l,l'). Where z E US-stringsPS, U-String(a,.) is the underlying projection of z, and S-String(z) is its surface projection. That is, if z = (z,z'), then</Paragraph> <Paragraph position="5"/> </Section> <Section position="5" start_page="33" end_page="34" type="metho"> <SectionTitle> RULE NOTATION AND EXAMPLES </SectionTitle> <Paragraph position="0"> The rules with which we are concerned are like the rewrite rules of generative phonology; they are general, context-conditioned replacements. That is, a rule allows a replacement if (1) the replacement belongs to a certain type, and (2) the surrounding context meets certain constraints.</Paragraph> <Paragraph position="1"> If we represent the contextual constraints extensionally, as sets of strings, a rule will consist of three things: a replacement type, and two sets of US-Strings. Thus, we can think of a rule as a triple (X, Y, F), where X and Y are sets of US-strings.</Paragraph> <Paragraph position="2"> Imagine that we are given a replacement instance l in a context (z, y), where z and y are US-strings.</Paragraph> <Paragraph position="3"> This contextualized replacement (~, l, y) satisfies the rule ifzEX, yE Y, andIEF.</Paragraph> <Paragraph position="4"> For linguistic and computational purposes, the sets that figure in rules must somehow be finitely represented. The KIMMO tradition uses regular sets, which can of course be' represented by regular expressions, for this purpose. We have not been able to convince ourselves that regular sets are needed in phonological applications, a InaThe issue here is whether there are any linguistically plausible or well-motivated applications of the Kleene star in stating phonological rules. For instance, take the English rule that replaces &quot;e by 0 after a morpheme boundary preceded by one or more consonants preceded by a vowel.&quot; You could represent the context in question with the regular expression VC*C; but you could equally well use VC I VCC \] VCCC \] VCCCC.The only way to distinguish the two rule formulations is by considering strings that violate the phonotactic constraints of English; but as far as we can see, there are no intuitions about the results of applying English rules to underlying strings like typppppe+ed. We do not question the usefulness stead, we make the stronger assumption that contexts can be encoded by finite sets of strings. A string satisfies such a context when its left (or its right) matches one of the strings in this set.</Paragraph> <Paragraph position="5"> (Note that satisfaction is not the same as membership; infinitely many strings can satisfy a finite set of strings.) Assuming a finite alphabet, all replacement types will be finite sets. With these assumptions, a rule can be finitely encoded as a pair {(X, Y~, F), where the sets X and Y are finite, and F is s replacement type.</Paragraph> <Paragraph position="6"> Rule encodings, rule applicability and satisfaction are illustrated by the rule examples given below. The ideas are further formalized in the next section.</Paragraph> <Paragraph position="7"> Language: Let PS = {a,b,...,z,+,#,', i} Declare the following subsets of PS:</Paragraph> <Paragraph position="9"> pheme boundary and after a constant USconsonant, i.e. after (l,i), where ! E C.</Paragraph> <Paragraph position="10"> Example 3 Rule encoding: (({sh}, {i ^(#, O) / I E Csib}), {(+,e)}) Rule notation: + --~ e / sh_Csib #:0 Rule description: Keplace + with e after sh and before a suffix in Csib.</Paragraph> <Paragraph position="11"> Example rule applications: 1. The rule encoded in Example 1 is satisfied by (+,0) in the context (cat, s) because (1) for some , cat = z^e, (2) for some y, s = c ^y, and (3) (+,0) e {{+,0)}.</Paragraph> <Paragraph position="12"> of regular expressions in many computational applications, but are not convinced that they are needed in a linguistic setting. We would be interested to see a well motivated case in which the Kleene star is linguistically indispensable in formulating a two-level phonological rule. Such a case would create problems for the approach that we adopt here.</Paragraph> <Paragraph position="13"> 2. The rule encoded in Example 2 is not satisfied by (y,i) in the context (spot + :t, +:0 hess) because there is no s such that spot + :t = ~e ~l, where I E C.</Paragraph> <Paragraph position="14"> 3. The rule encoded in Example 3 is not satisfied by (+, 0) in the context (ash, s #:0). In fact, the context is satisfied: (1) sh = m-sh for some :e and (2) s #:0 E Csib ~y for some It. (3.1) Moreover, the underlying symbol of the replacement (namely, +) matches the argument of the ~ule's replacement function. Under these circumstances, we will say that the rule is applicable. But the rule is not satisfied, because (3.2) the surface symbol of the replacement (namely, 0) does not match the value of the rule's replacement function (namely, e): thus, (+,0) ~\[ {(+,e)}.</Paragraph> </Section> <Section position="6" start_page="34" end_page="35" type="metho"> <SectionTitle> INDEXED STRINGS AND RULES </SectionTitle> <Paragraph position="0"> We now restate the above ideas in the form of formal definitions.</Paragraph> <Paragraph position="1"> Definition 1. Context type.</Paragraph> <Paragraph position="2"> A context type is a pair C = (X, Y), where X and Y are sets of US-Strings.</Paragraph> <Paragraph position="3"> Definition 2. Indexed US-strings.</Paragraph> <Paragraph position="4"> An indexed US-String over PS is a triple (as, l,y), where a,y E US.stringsr and I E Replacementsr.</Paragraph> <Paragraph position="5"> An indexed US-string is a presentation of a nonempty US-string that divides the string into three components: (1) a replacement occurring in the string, (2) the material to the left of that replacement, and (3) the material to the right of it. Where (as, I, y) is an indexed string, we call as the left context of the string, I/the right context of the string, and I the designated replacement of the string.</Paragraph> <Paragraph position="6"> A rule licenses certain sorts of replacements in designated sorts of environments, or context types. For instance, we may be interested in the environment after a consonant and before a morpheme boundary. Here, the phrase &quot;after a consonant&quot; amounts to saying that the string before the replacement must end in a consonant, and the phrase &quot;before a morpheme boundary&quot; says that the string after the replacement must begin in a morpheme bound'ary. Thus, we can think of a context type as a pair of constraints, one on the US-string to the left of the replacement, and the other on the US-string to its right. If we identify such constraints with the set of strings that satisfy them, a context type is then a pair of sets of USstrings; and an indexed string satisfies a context type in case its left and right context belong to the corresponding types.</Paragraph> <Paragraph position="7"> Definition 3. Replacement types.</Paragraph> <Paragraph position="8"> A replacement type over PS is a partial function F from PS U {0} to PS U {0}. (Thus, a replacement type is a certain set of replacements.) Dora(F) is the domain of F.</Paragraph> <Paragraph position="9"> Definition 4. Rules.</Paragraph> <Paragraph position="10"> A rule is a pair 7~ = (C, F), where C is a context type and/' is a replacement type.</Paragraph> <Paragraph position="11"> Definition 5. Rule applicability.</Paragraph> <Paragraph position="12"> A rule ((X, Y), F) is applicable to an indexed string (se, (i,l'), y) if and only if as E X, y ~ Y, and F(l) is defined, i.e., i E Dom(F).</Paragraph> <Paragraph position="13"> Definition 6. Rule satisfaction.</Paragraph> <Paragraph position="14"> An indexed string (as, i, y) satisfies a rule (C, F) if and only if as E X, y E Y, and F (l) = l deg. The above definitions do not assume that the contexts are finitely encodable. But, as we said, we axe assuming as a working hypotheses that phonological contexts are finitely encodable; this idea was incorporated in the method of rule encoding that we presented above. We now make this idea explicit by defining the notion of a finitely encod- null able rule.</Paragraph> <Paragraph position="15"> Definition 7. LeftExp( X ), RightExp( X ) LeftExp(X) = {z^,/, E X} Right~xp(X) = {(r)^z/ (r) ~ X} Definition 8. Finite encodability A subset X of US-strings j: is left-encoded by a set U in case X = LeftExp(U), and is right-encoded by 17 in case X = RightExp(V). (It is easy to get confused about the usage of &quot;left&quot; and &quot;right&quot; here; in left encoding, the left of the encoded string is arbitrary, and the right must match the encoding set. We have chosen our terminology so that a left context type will be left-encoded and a right context type will be right-encoded.) A context type C = (X, Y) is encoded by a pair (U, V) of sets in case U left-encodes X and V right-encodes Y.</Paragraph> <Paragraph position="16"> A rule ~ = (C, F) is finitely encoded by a rule encoding structure ((U, V),g) in case (U,V) encodes C, g = F, and ff and V are finite.</Paragraph> <Paragraph position="17"> In the following material, we will not only confine our attention to finitely encodable rules, but will refer to rules by their encodings; when the notation ((X, Y), F~ appears below, it should be read as a rule encoding, not as a rule. Thus, for instance, the indexed string (cat, +:0, s I satisfies the rule (encoded by) (({~}, {~}), {(+, 0)}), even though cat C/ {e}.</Paragraph> </Section> <Section position="7" start_page="35" end_page="36" type="metho"> <SectionTitle> SPECIFICITY OF CONTEXT TYPES AND RULES </SectionTitle> <Paragraph position="0"> We have a good intuitive grasp of when one context type is more specific than another. For instance, the context type preceded by a back vowel is more specific than the type preceded by a vowel; the context type followed by an obstruent is neither more nor less specific than the type followed by a voiced consonant; the context type preceded by a vowel is neither more nor less specific than the type followed by a vowel.</Paragraph> <Paragraph position="1"> Since we have identified context types with pairs of sets of strings, we have a very natural way of defining specificity relations such as &quot;more specific than&quot;, &quot;equivalent&quot;, and &quot;more specific than or equivalent&quot;: we simply use the subset relation.</Paragraph> <Paragraph position="2"> Definition 9. C < C'.</Paragraph> <Paragraph position="3"> Let C = (X1, Y1) and C' = (X2, Y2} be context types. C < C' if and only if X~ C_ X~ and Yt C_ Y~.</Paragraph> <Paragraph position="4"> Definition 10. C _= C ~.</Paragraph> <Paragraph position="5"> C =_ C' if and only if C < C' and C' _< C.</Paragraph> <Paragraph position="6"> Definition 11. C < C ~.</Paragraph> <Paragraph position="7"> C < C' if and only if C < C' and C I ~ C.</Paragraph> <Paragraph position="8"> It is not in general true that if LeflEzp(X) C LeflEzp(JO, then X C Y; for instance, LeftExp({aa, ba}) C_ PSeflExp({a}), but {aa, ba} {a}. However, we can easily determine the specificity relations of two contexts from their finite encodings: Lemma 1. LeflExp(X) C_ LeflEzp(Y) iff for all z E X there is a y E Y such that for some z, ffi = z Ay. Similarly, RightExp(X) C RightExp(Y) iff for all z E X there is a y E Y such that for some Proof of the lemma is immediate from the definitions. It follows from the lemma that there is a tractable algorithm for testing specificity relations on finitely encodable contexts: Lemma 2. Let C be finitely encoded by {X1, X2) and C' be finitely encoded by {YI, Y2). Then there is an algorithm for testing whether C < C ~ that is no more complex than O(m x n x k), where m = max(I Xxl, \[.X21), n = max(I Yll, I Y zl), and k is the length of the longest string in Y1 U Y2. Proof. Test whether for each zl E X1 there is a Yl E Yl that matches the end of zl. Then perform a similar test on X2 and Y~.</Paragraph> <Paragraph position="9"> DFSM'S A DFSM's transitions are labelled with finitely encodable rules rather than with pairs of symbols. Moreover, nondeterminism is restricted so that in case of conflicting transitions, a maximally specific transition must be selected. The critical definition is that of minimal satisfaction of an arc by an indezed path, where an indexed path represents a DFSM derivation, by recording the state transitions and replacements that are traversed in processing a US-String.</Paragraph> <Paragraph position="10"> Definition 12. Arcs.</Paragraph> <Paragraph position="11"> An arc over a set S of states and alphabet PS is a triple A = (s, sl,~), where s,s I E S and 7~ is a rule over/:.</Paragraph> <Paragraph position="12"> Definition 13. DFSMs.</Paragraph> <Paragraph position="13"> A DFSM on ~: is a structure .hd = {S,i,T,.A}, where S is a finite set of states, i E S is the initial state, T C S is the set of terminal states, and .,4 is a set of arcs over S on PS.</Paragraph> <Paragraph position="14"> Definition 14. Paths.</Paragraph> <Paragraph position="15"> A path ~&quot; or ~r(s0, an) over .M from state so to state sn is a string s011stll...lnsn, where for all m, 0 _< m _< n, sm is a state of .h4 and lm E US-strings~c.</Paragraph> <Paragraph position="16"> Remark I: n >_ 0, so that the simplest possible path has the form s, where s is a state. Remark & we use the notations ~r and ~r(s, s ~) alternatively for the same path; the second notation provides a way of referring to the beginning and end states of the path.</Paragraph> <Paragraph position="17"> Definition 15. Recovery of strings from paths. Let lr = solzszll...lnsn. Then String(~') = Definition 16. Indezed paths.</Paragraph> <Paragraph position="18"> An indexed path over .Ad is a triple (%1, 7r') where 7r, 7c' are paths, and l,n E US-stringsPS. (Tr, 1, or') is an indexing of path a if and only if o&quot; --&quot; C/r ~l ~lr I.</Paragraph> <Paragraph position="19"> Definition 17. Applicability of an arc to an indezed path.</Paragraph> <Paragraph position="20"> An are (u,u',7~) is applicable to an indexed path {lr(s, t), 1, ~'~(s ~, t')} if and only if t = u and the rule 7~ is applicable to the indexed string (String0r), 1, String(~')). Definition 18. Satisfaction of an arc by an in. dezed path.</Paragraph> <Paragraph position="21"> (~'(s, t), 1, r~(s ~, t~)) satisfies an are {u, u ~, ~) if and only if t -- u, s ~ = u ~, and the indexed string {String(~r), 1, String(~'~)) satisfies the rule Definition 19. Minimal satisfaction of an arc by an indezed path.</Paragraph> <Paragraph position="22"> Ca',l, z&quot;) minimally satisfies an arc A = (s, s', 7~) of.M if and only if (a', 1, lr') satisfies A and there is no state s&quot; and arc A' = i s, s&quot;, ~') of Ad such that A' = (s, s','g') is applicable to (a',l, a&quot;) and ~' < g.</Paragraph> <Paragraph position="23"> As we said, the above definition is the crucial component of the definition of DFSM's. According to this definition, to see whether a DFSM derivation is correct, you must check that each state transition represents a maximally specific rule application. This means that at each stage the DFSM does not provide another arc with a competing replacement and a more specific context. (&quot;Competing&quot; means that the underlying symbols of the replacement match; a replacement competes even if the surface symbols does not match the letter in the US-String being tested.) 4 Definition 20. Indezed path acceptance by a DFSM.</Paragraph> <Paragraph position="24"> M = (8, i,T,.A) accepts an indexed path (Tr, l,z &quot;~) if and only if there is an arc A I = (s, s I, g~) of .M that is minimally satisfied by (,~, I, 7r').</Paragraph> <Paragraph position="25"> Definition 21. Path acceptance by a DFSM.</Paragraph> <Paragraph position="26"> = (8, i, T, ,4) accepts a path a'(s, s ~) if and only if .Ad accepts every indexing of ~', s = i, and s' G T.</Paragraph> <Paragraph position="27"> Definition 22. US-String acceptance by a DFSM. .Ad accepts z E US-stringsr if and only if there is a path ~r such that ,Ad accepts ~r, where z = String(Jr).</Paragraph> <Paragraph position="28"> Definition 23. Generation of SF from UF by a DFSM.</Paragraph> <Paragraph position="29"> .A4 generates a surface form z' from an underlying form z (where z and z' are strings over PS) if and only if there is a a E US-stringsPS such that .Ad accepts z, where U.String(v) = z and</Paragraph> <Paragraph position="31"/> </Section> <Section position="8" start_page="36" end_page="36" type="metho"> <SectionTitle> EXAMPLE: SPELLING RULES FOR ENGLISH STEM+SUFFIX COMBINATIONS </SectionTitle> <Paragraph position="0"> The following is an adaptation of the treatment in Antworth (1990) of English spelling rules, which 4This use of competition builds some directional bias into the definition of DFSM's, i.e., some preference for their use in generation. Even if we are using DFSM's for recognition, we will need to verify that the recognized string is generated from an underlying form by a derivatio~ that does not allow more specific competing derivations.</Paragraph> <Paragraph position="1"> in turn is taken from Karttunen and Wittenburg (1983).</Paragraph> <Paragraph position="2"> * .M = (S, i, T, A), where S = {i, s, t}. T = {t}. - Task of i: Begin and process left word boundary. null - Task of s: Process stem and suffixes.</Paragraph> <Paragraph position="3"> - Task oft: Quit, having processed right word boundary.</Paragraph> <Paragraph position="4"> * Remark: the small number of states is deceptive, since contexts are allowed on the arcs. An equivalent finite-state transducer would have many hundreds of states at least.</Paragraph> <Paragraph position="5"> * Remark: the relatively small number of arcs enumerated below is also deceptive, since two of these &quot;arcs,&quot; are 3 and arc 13, are actually schemes. In the following discussion we will speak loosely and refer to these schemes as arcs; this will simplify the discussion and should create no confusion.</Paragraph> <Paragraph position="6"> * Declare the foUowing subsets of PS: Ltr= {a, b, c, d, e, f, g, h,i,j, k, 1, m, n, o, p, q, r S, t~ U, V, W, X, y, Z)</Paragraph> <Paragraph position="8"> Where s,s' E 8, let A,,,, = {A/A G A and for some 7C/,A = (s, s', 'g)}. We present arcs by listing the rules associated with the arcs, for each appropriate pair (s, s') of states. We will give each arc a numerical label, and give a brief explanation of the purpose of the arc.</Paragraph> <Paragraph position="9"> I. The derivation that relates #kiss+s# to 0kisses0 proceeds as follows.</Paragraph> <Paragraph position="10"> 1. Begin in state i looking at #:0.</Paragraph> <Paragraph position="11"> 2. Follow arc 2 to s, recognizing k:k. (This is the only applicable arc.) 3. Follow arc 3 to s, recognizing i:i. (This is the only applicable arc.) 4. Follow arc 3 to s, recognizing s:s. (This is the only applicable arc.) 5. Follow arc 3 to s, recognizing s:s. (This is the only applicable arc.) 6. Follow arc 6 to s, recognizing +:e. (Arc 2 is also applicable here; but see the next illustration.) null 7. Follow arc 3 to s, recognizing s:s. (This is the only applicable arc.) 8. Follow arc 14 to f, recognizing #:0. (This is the only applicable arc.) II. No derivation relates #kiss+s# to 0kiss0s0. Any such derivation would have to proceed like the above derivation through Step 5. At the next step, the conditions for two arcs are met: arc 2 (replacing + with 0) and arc 6 (replacing + with e). Since the context of the latter ~llere, C + can be any string of no more than four consonants.</Paragraph> <Paragraph position="12"> arc is more specific, it must apply; there is no derivation from this point using arc 2.</Paragraph> <Paragraph position="13"> III. The derivation that relates #try+ing# to 0try0ing0 proceeds as follows.</Paragraph> <Paragraph position="14"> 1. Begin in state i looking at #:0.</Paragraph> <Paragraph position="15"> 2. Follow arc 2 to s, recognizing t:t. (This is the only applicable arc.) 3. Follow arc 3 to s, recognizing r:r. (This is the only applicable arc.) 4. Follow arc 8 to s, recognizing y:y. (There are three applicable arcs at this point: arc 3, arc 7, and arc 8. However, arcs 3 and 7 are illegal here, since their contexts are both less specific than arc 8's.) 5. Follow are 2 to s, recognizing +:0. (This is the only applicable arc.) 6. Follow arc 3 to s, recognizing i:i. (This is the only applicable arc.) 7. Follow arc 3 to s, recognizing n:n. (This is the only applicable arc.) 8. Follow arc 3 to s, recognizing g:g. (This is the only applicable arc.) 9. Follow arc 14 to f, recognizing #:0. (This is the only applicable arc.) IV. No derivation relates #try+ing# to 0tri0ing0. Any such derivation would have to proceed like the above derivation through Step 3. At the next step, arc 7 cannot be traversed, since arc 8 is also applicable and its context is more specific. Therefore, no arc is minimally satisfied and the derivation halts at this point.</Paragraph> </Section> <Section position="9" start_page="36" end_page="39" type="metho"> <SectionTitle> COMPUTATIONAL COMPLEXITY </SectionTitle> <Paragraph position="0"> We now consider the complexity of using DFSM's to create one side of a US-string, given the other side as input. There are basically two tasks to be analyzed: * DFSM GENERATION: Given a DFSM, D, over an alphabet, PS, and an underlying form, u, does D generate a surface form, s, from u? * DFSM RECOGNITION: Given a DFSM, D, over an alphabet, PS, and a surface form, s, does D generate an underlying form, u, from s? These two tasks are related to the tasks of KIMMO GENERATION and KIMMO RECOGNITION, the various versions of which Barton et al. (1987) proved to be NP-complete or worse.</Paragraph> <Paragraph position="1"> Relationship to Kimmo The DFSM is not a generalization of KIMMO; it is an alternative architecture for two-level rules. KIMMO takes a programming approach; it provides a declarative rule formalism, which can be related to a very large FS automaton or to a system of parallel FSI automata. The automata are in general too unwieldy to be pictured or managed directly; they are manipulated using the rules. By integrating rules into the automata, the DFSM approach provides .a procedural formalism that is compact enough to be diagrammed and manipulated directly.</Paragraph> <Paragraph position="2"> DFSM rules are procedural; their meaning depends on the role that they play in an algorithm. In a DFSM with many states, the effect achieved by a rule (where a rule is a context-dependent replacement type) will in general depend on how the rule is attached to states. In practice, however, the proceduralism of the DFSM approach can be limited by allowing only a few states, which have a natural morphonemic interpretation. The English spelling example that we presented in the previous section illustrates the idea. There are only four states. Of these, two of them delimit word processing; one of them begins processing by traversing a left word boundary, the other terminates processing after traversing a final word boundary. Of the remaining two states, one processes the word; all of the rules concerning possible replacements are attached to arcs that loop from this state to itself. The other is a nonterminal state with no arcs leading from it. Inthe example, the only purpose of this state is to render certain insertions or deletions obligatory, by &quot;trapping&quot; all US-strings in which the operation is not performed in the required context.</Paragraph> <Paragraph position="3"> In cases of this kind, where the ways in which rules can be attached to arcs are very restricted, tile proceduralism of the DFSM formalism is limited. The uses of rules in such cases correspond roughly to two traditional types of phonological constructs: rules that allow certain replacements to occur, and constraints that make certain replacements obligatory.</Paragraph> <Paragraph position="4"> Although DFSM's are less declarative than KIMMO, we believe that it may be possible to interpret at least some DFSM's (those in which the roles that can~ be played by states are limited) using a nonmonotonic formalism that provides for prioritization of defaults, such as prioritized default logic; see (Brewka, 1993). In this way, DFSM's could be equated to declarative, axiomatic theories with a nonmonotonic consequence relation. But we have not carried out the details of this idea.</Paragraph> <Paragraph position="5"> Though it is desirable to constrain the number of states in a DFSM, there may be applications in which we may want more states than in the English example. For instance, one natural way to process vowel harmony would multiply states by creating a word-processing state for each vowel quality. Multiple modes of word-processing could also be used to handle cases (as in many Athabaskan languages) where different morphophonemic processes occur in different parts of the word.</Paragraph> <Paragraph position="6"> If they are desired, local translations of the four varieties of KIMMO rules deg into DFSM's are available, by using only one state plus a sink state.* The following correspondences provide translations, in polynomial time, to one or more DFSM arcs: Exclusion, u : s/ ~ LC__RC: an arc u s / LC--RC from the state to a sink state .... ..... Context Restriction, u : s ~ LC_-RC: a loop u --~ s / LC__RC, and an arc u --~ s / _ to a sink state.</Paragraph> <Paragraph position="7"> Surface Coercion, u : s ~ LC__RC: a loop u s / LC--RC, and for each surface character s t E PS, an arc u --~ s t / LC.--RC to a sink state.</Paragraph> <Paragraph position="8"> Composite, u : s C/~ LC._RC: all of the arcs mentioned in Context Restriction or Surface Coercion. : Extended DFSM's The differences between KIMMO and DFSM's prohibit the complexity analysis for the corresponding two KIMMO problems from naturally extending to an analysis of DFSM generation and recognition. In fact, we can define an extended DFSM (EDFSM), which drops the finite encodability requirement that KIMMO lacks, for which we have the following result: Theorem 1. EDFSM GENERATION is PSPACE-hard null Proof by reduction of REGULAR EXPRES-SION NON-UNIVERSALITY (see Figure 1). Given an alphabet E, and a regular expression, a C/ ~b, over E, we define an EDFSM over the alphabet, U {$}, where $ ~ E. We choose one non-empty string ceEL(a) of length n. The EDFSM first recognizes each character in a, completing the task at state n0: al al I(PS:PS)*--(PS:PS)* 7 From no, there are two arcs, which map to different states: eSproat (1992), p. 145.</Paragraph> <Paragraph position="10"> where the latter rule traverses to some state 81, with a being the expression which replaces each atom, b, in a by its constant replacement, b:b, and likewise for ~.</Paragraph> <Paragraph position="11"> From Sl, the EDFSM then recognizes o~ again, terminating at the only final state. We provide this EDFSM, along with the input ot2o~ to EDFSM GENERATION. This EDFSM can accept c~$ot if and only if, at state so, the context (~3&quot;, ~*) is not more specific than the context ((a + $), (a + 2)). So, we have:</Paragraph> <Paragraph position="13"> The Complexity of DFSM GENERATION Finite encodability foils the above proof technique, since one can no longer express arbitrary regular expressions over pairs in the contexts of rules. In fact, as we demonstrated above, there is a polynomial-time algorithm for comparing the specificities of finitely-encodable contexts. Finite encodability does not, however, restrict the complexity of DFSM's enough to make DFSM GEN-</Paragraph> </Section> <Section position="10" start_page="39" end_page="40" type="metho"> <SectionTitle> ERATION polynomial time: </SectionTitle> <Paragraph position="0"> Theorenl 2. I)I&quot;SM GENERATION is NI L complete.</Paragraph> <Paragraph position="1"> Proof DFSM GENERATION is obviously in NP. The proof of NP-hardness is a reduction of 3-SAT. Given an input formula, w, we construct a DFSM consisting of one state over an alphabet consisting of 0, 1, ~, one symbol, u~, for each variable in w, and one symbol, ej, for each conjunct in w. Let m be the number of variables in w, and n, the number of conjuncts. For each variable, ui, we add four loops:</Paragraph> <Paragraph position="3"> The first two choose an assignment for a variable, and the second two enforce that assignment's consistency. For each conjunct, Ijl V 1/2 V ljs, where the l's are literals, we also add three loops, one for each literal. The loops enforce a value of 1 on the symbol uj~ if lj~ is a positive literal, or 0, if it is negative. For example, for the conjunct ul V qua V u4, we add the following three rules:</Paragraph> <Paragraph position="5"> Thus, the input to DFSM GENERATION is the above DFSM plus an input string created by iterating the substring ul...umcj for each conjunct. The input string corresponding to the formula, ('~ul V u2 V u4) A (~u~ V us V'~u4) A (ul V u2 V us), would be ~ulu2usu4clulu2uau4e2ulu2uau4cs. The DFSM accepts this input string if and only if the input formula is satisfiable; and this translation is linear inm+n. D Compilation Of course, we should consider whether the complexity of DFSM GENERATION can be compiled out, leaving a polynomial-time machine which accepts input strings. This can be formalized as the separate problem: * FIXED-DFSM-GENERATION: For some DFSM, D, over alphabet, PS, given an underlying form, u, does D generate a surface form, s, from u? Whether or not FIXED DFSM GENERATION belongs to P remains an open problem. It is, of course, no more difficult than the general DFSM GENERATION problem, and thus no more difficult than NP-complete. The method used in tile proof given above, however, does not naturally extend to the case of FIXED DFSM GENERATION, since we cannot, with a fixed DFSM, know in advance .... .. ,.,.: how many variables to expect in a given input formula, without which we cannot use the same trick with the left context to preserve the consistency of variable assignment.</Paragraph> <Paragraph position="6"> Even more interestingly, the technique used in the proof of PSPACFE-hardnees of EDFSM GENERATION does not naturally extend to fixed EDFSM's either; thus, whether or not FIXED DFSM GENERATION belongs to P is an open question as well s. Dropping finite encodability, of course, affects the compilation time of the problem immensely.</Paragraph> <Paragraph position="7"> Nulls The two proofs we have given remain valid if we switch alll of the underlying forms with their surface counterparts. Thus, without nulls, EDFSM RECOGNITION is PSPACE-hard, DFSM RECOGNTION is NP-complete, and, if FIXED DFSM GENERATION is in P, then we can presumably use the same compilation trick with the roles of underlying and surface strings reversed to show that FIXED DFSM RECOGNITION is in P as well. If nulls are permitted in surface realizations, however, DFSM RECOGNTION becomes much more difficult, even with finite encodability enforced: null Theorem 3. DFSM RECOGNTION with nulls is PSPACE-hard.</Paragraph> <Paragraph position="8"> Proof by reduction of CONTEXT-SENSITIVE LANGUAGE MEMBERSHIP (see Figure 2). Given a context-sensitive grammar and an input string of length m, we let the input surface form to the DFSM RECOGNTION problem be the same as the input string. We then design a DFSM with an alphabet equal to E U {$,!}, where ~ is the the set of non-terminals plus the set of terminals. The DFSM first copies each surface input symbol to the corresponding position in the underlying form, and then adds the pair $:0, completing the task in a state So.</Paragraph> <Paragraph position="9"> Having copied the string onto the underlying side of the pair, the remainder of the recognized underlying form will consist of rewritings of the string for each rule application, and will be paired with surface nulls at the end of the input string. Each rewriting will be separated by a $ symbol, and, as the string length changes, it will be padded by ! symbols. For each rule a ~ #, we add a cycle to the DFSM, emanating from state so, which first sit is quite unlikely, however, since the reduction can probably be made with a different PSPACEcomplete problem, from which the NP-completeness of FIXED EDFSM GENERATION would follow as a corollary.</Paragraph> <Paragraph position="10"> writes j copies of the ! symbol to the underlying form, where j = b - a, b = Ifll, and a = lal: ! -* 0 / ~:PS(PS:PS ...~ PS:PS)-- :.</Paragraph> <Paragraph position="11"> j >_ 0 since the rules are context-sensitive. The cycle then copies part of the most recent S-bounded string of symbols with a family of loops of the form: o&quot; --+ 0 / o':PS (PS: ...m+j PS:PS )-- (r l) for each o&quot; E ~. It then recognizes ~, and : writes a, with:</Paragraph> <Paragraph position="13"> o<, --+ 0/-It then copies the rest of the most recent gbounded string, using copy of the family of loops in (rl), and then adds a new $ with a rule that also ensures that this second loop has iterated the appropriate number of times by checking that the length has been preserved: $ -~ 0 / $:PS (PS:L...m L:L )&quot; (r2) The DFSM also has a loop emanating from so which adds more ! symbols: ! -..+ 0 / hPS ( PS:PS ...m PS:PS )-All of the rule-cycles will use this to copy previously-added ! symbols, as the string shrinks in size. The proper application of this loop is also ensured by the length-checking of (r2).</Paragraph> <Paragraph position="14"> Finally, we add one arc to the DFSM from So to the only final .state which checks that the final copy of the string contains only the distinguished symbol, S: $ -* 0 / ( hPS ...~-I hPS) S:PS $:PS__ L,.</Paragraph> <Paragraph position="15"> Thus, the DFSM recognises the surface form if and only if there is a series of rewritings from the input string to S using the rules of the grammar, and the translation is linear in the size of the input string times the number of rules. O Since there exist fixed context-sensitive grammars for which the acceptance problem is NP-hard 9, the NP-hardness of FIXED DFSM RECOGNITION with nulls follows as a corollary.</Paragraph> </Section> class="xml-element"></Paper>