<?xml version="1.0" standalone="yes"?>
<Paper uid="E91-1016">
  <Title>A LOGICAL APPROACH TO !ARABIC PHONOLOGY</Title>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
LOGICAL FRAMEWORK
</SectionTitle>
    <Paragraph position="0"> Interval based tense logics are calculi of temporal reasoning in which propositions are assigned truth values over extended periods of time s. Three operators F (future), P (past) and O (overlaps) are introduced: F~b means &amp;quot;q~ will be the case (at least once)&amp;quot;, P~b means &amp;quot;~b was the case (at least once)&amp;quot; and O~b means &amp;quot;~b is the case at some overlapping interval (at least once)&amp;quot;. O corresponds to what phonologists call 'association'. Typically sentences are true at some intervals and not at others. (This is obviously the case, for example, if ~b encodes the proposition &amp;quot;the sun is shining&amp;quot;.) Blackburn (1989) has explored the effects of adding of a new type of symbol, called nominals, to tense logic. Unlike ordinary propositions, nominals are only ever true once. In a sense, a nominal is a 'name' (or a 'temporal indexical') for that unique period of time at which it is true. Certain observations about time can only be expressed in a theory which employs nominals. For example, i ~ -~Fi picks out precisely the irreflexive time flows, whereas no formula containing only propositional variables can do this. Nominals have been employed in the analysis of temporal reference in linguistic semantics. The present paper illustrates an application of nominals to a very different domain, namely phonology. In addition to F, P and O, we shall employ the modality O to represent phonological  sponds closely to the use of the term in feature logics. However. we treat dominance as a relation rather than as a collection of partial functions, for two reasons. First. in phonological structures, it is the nodes and not the arcs which are labelled. Second, there can be multiple arcs emanating from a node.</Paragraph>
    <Paragraph position="1"> - 89 -Syntax. Let X = {p, q, r,...} be the propositional variables and let N = {i, j, k,...} be the nominals of L. Then L is the smallest set such that all of the following hold, where C/, ~P E L.</Paragraph>
    <Paragraph position="2"> T,_k E L X,N C L &lt;&gt;~b,FC/,PC/,OC/ E L ~b v C/,-~C/ E L We define ---., ,---, and A as usual. We also define the duals of our modal operators in the usual fashion: G~b - ~F~C/ (~b is always going to be the case), H~b = -~P--,C/ (~b always has been the case), C~b ~. ~O--,q~ (~b holds at all overlapping intervals) and El~b ~ ~O~b (~b is true at all 'daughter' intervals). Two additional defined operators will play an important role in what follows: Me = P~b V O~b V F~b, and its dual LC/ ~_ ~M~C/. It follows from the semi-linear time semantics adopted below that M~b means '~b holds at some time' and L~b means '~b holds at all times'. We will often abbreviate &lt;&gt;(p A ~b) using the expression (p)~b and abbreviate a sequence of such applications (Pl)'&amp;quot; (p,,)~b using the expression {pl&amp;quot;&amp;quot; &amp;quot;p,)C/. We adopt a similar practice for the dual forms: \[p\]$ is shorthand for r3(p :... C/) and \[Pl &amp;quot; &amp;quot;P,,\]$ is shorthand for \[Pa\]'&amp;quot; \[P,,\]C/. We also write &lt;)n (or t3&amp;quot;) to stand for a length n sequence of 0s (or 13s). Semantics. Let T be a set of intervals (which we will think of as nodes), and let ~, &lt; and e be binary relations on T.</Paragraph>
    <Paragraph position="3"> As &lt; models temporal precedence, it must be irreflexive and transitive, o models temporal overlap (phonological association), and so it is reflexive and symmetric. &lt; and o interact as follows: (i) they are disjoint, (ii) for any tl, t2, t3, t4 G T, tl &lt; t2 o t3 &lt; t4 implies tl &lt; t4 (that is, precedence is transitive through overlap), and (iii) for any t~, t2 E T, tl &lt; t2 orhot2 orb &gt; t2 (thatis, our conception of time is semi-linear). Note that the triple (T, &lt;, o) is what temporal logicians call an interval structure.</Paragraph>
    <Paragraph position="4"> The remaining relation ~ encodes the hierarchical organization of phonological structures. As a phonological unit overlaps all of its constituents (cf. Hayes 1990:44), we demand that the transitive closure of 8 be contained within o. Furthermore, phonological structures are never cyclic and so we require that for any h , . . . , t, ~ T, if h tt~ ..... tn-l~tn then it is not the case that t,,btl. By a phonological frame F we mean a quadruple (T, &lt;, o, B}:of the type just described.</Paragraph>
    <Paragraph position="5"> It merely remains to link L with such structures. A valuation V is a function (X t3 N) --* 2 T that obeys three constraints. First, it must assign a singleton set to each nominal. Second, for each t E T, there is an i 6 N such that V(i) = {t}. Third, fib,t2 ~ V(p) wherep ~ X then tl o tu ---* t~ = t2. In short, valuations are functions which ensure nominals act as: names, where all intervals are named, and valuations capture the idea that phonological 'tiers' are linearly ordered. A model for L is a pair (F, V).</Paragraph>
    <Paragraph position="6"> Satisfaction. Let M = (F, V):, t E T, a ~ X O N. Then:</Paragraph>
    <Paragraph position="8"> M~, FC/ iffSt' : t &lt; t' and M ~,, 4~ M~,P~biffgt':t'&lt;tandM~,,~b If 9*( ~, C/ then we say that ~b is true in M at :t. Note that under this semantics, M really does mean 'at some time' and L means 'at all times' (by virtue of semi-linearity). Validities. If (F, '12) ~t ~b for all frames F, for all valuations V on F, and all t E T, then we say ff is a validity. The following are some examples of validities. The first group concerns our intervalic structure.</Paragraph>
    <Paragraph position="9">  (TI) i ---, --,Fi. Precedence is irreflexive.</Paragraph>
    <Paragraph position="10"> (T2) ~b ~ O~b. Overlap is reflexive.</Paragraph>
    <Paragraph position="11"> (T3) ~ --~ C0C/. Overlap is symmetric.</Paragraph>
    <Paragraph position="12"> (T4) Fi---,-~Oi. Pi---,-~Oi.</Paragraph>
    <Paragraph position="13"> Precedence and overlap are disjoint.</Paragraph>
    <Paragraph position="14"> (TS) FOFqS --. FC/.</Paragraph>
    <Paragraph position="15"> Precedence is transitive through overlap.</Paragraph>
    <Paragraph position="16"> (T6) F~AFC/--~ F(C/^FC/)vF(~bAFC/)vF{C/AO~b).</Paragraph>
    <Paragraph position="17">  Time is semi-linear 4 .</Paragraph>
    <Paragraph position="18"> The next two validities concern the dominance relation and its interaction with the interval structure. (D1) 0'*C/ ~ O~b. The transitive closure of dominance is included in the overlap relation.</Paragraph>
    <Paragraph position="19"> (D2) i ---* -~O'* i. Dominance is acyclic.</Paragraph>
    <Paragraph position="20"> The next group of validities reflect the constraints we have placed on valuations.</Paragraph>
    <Paragraph position="21">  (FORCE) Mi.</Paragraph>
    <Paragraph position="22"> Each nominal names at least one interval.</Paragraph>
    <Paragraph position="23"> (NOM) i A M(i A C/) --. C/.</Paragraph>
    <Paragraph position="24"> Each nominal names at most one interval.</Paragraph>
    <Paragraph position="25"> (PLIN) p A O(p ^ C/) -.-* ~.</Paragraph>
    <Paragraph position="26">  Phonological tiers are linearly ordered. Proof Theory. It is straightforward using techniques discussed in (Gargov et al. 1987, 1989, Blackburn 1990) to provide a proof theory and obtain decidability results. At present we are investigating efficient proof methods for this logic and hope to implement a theorem prover.</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
EXPRESSING PHONOLOGICAL CONSTRAINTS
</SectionTitle>
    <Paragraph position="0"> Feature Matrices. L can be used for describing feature matrices. For example, consider the matrix below.</Paragraph>
    <Paragraph position="1"> \[ PHON (Kay, pats, Blackle) \] A possible description of this matrix is: (PHON)(Kay A Fi) A (PHON)(pats A i A F j) A (PHON)(Blackie A j). This representation of sequences (cf. Rounds &amp; Manaster-Ramer 1987) enables the expression of partial ordering constraints which are widely required in phonological descriptions. Note  that all instances of the following variant of the NOM schema are valid. E and E' are strings of modal operators from {&lt;&gt;, F, P,O}.</Paragraph>
    <Paragraph position="2"> (NOME) Ei A E'(i ^ C/) .--, E(i A C/) A E'i.</Paragraph>
    <Paragraph position="4"> This schema expresses a familiar equivalence on feature matrices. For example: \[,,,,o, \[ \[\] That is, nominals may be used in the representation of reentrancy (Bird 1991), Sort Lattices. Node labels in phonologists' diagrams (e.g. see example (1)) can be thought of as classifications. For example, we can think of p E X as denoting a certain class of nodes in a phonological structure (the mora nodes). Moras may be further classified into onset moras and coda moras, which are written as Po and/~c respectively. The relationship between /~, Po and Pc can then be expressed using the following formulas: L(p .-. po v pc) L(po A pc .-- i) Such constraints are Boolean constraints. For example, a simple Boolean lattice validating the two formulas concerning moras above is ( {p, #o, #c, _L} ; po I-1 pc = .L, po LJ pc = # ). This is depicted as a diagram as follows: ~o G  Each element of X appears as a node in the diagram. The join (U) of two sorts p and q is the unique sort found by following lines upwards from p and q until they first connect, and conversely for the meet (I-1). For convenience, constraints on node classifications will be depicted using lattice diagrams of the above form. Trading on the fact that L contains propositional calculus, Boolean constraints can be uniformly expressed in L as follows: (i) p I-I q = r becomes L(p A q ~ r) (ii) p U q = r becomes L(p V q .-. r) Appropriateness Constraints. As we shall see, the hierarchical prosodic structures of phonology are highly constrained. For example, syllables dominate moras and not the other way around, and so a structure where there is a mora dominating a syllable is ill-formed. Following (Bird 1990), we express these restrictions on dominance by augmenting the sort lattice with a binary' 'appropriateness' relation A, represented graphically using arrows. We can express in L the constraints captured by such appropriateness graphs. For example, L(p --, -~Oq) expresses the fact that a node with sort p cannot dominate a node with sort q. We can also use L to express stronger C/o~traints. For example, L(p -. Oq) expresses the fact that a node of sort p must dominate at least one node of sort q. In short, the O operator allows us to express graphical constraints.</Paragraph>
    <Paragraph position="5"> A THEORY OF PHONOLOGY A phonological theory is a collection of generalizations expressed in a language of the above logic. We choose as our language X = tr, O'h, ell, ao, trc, (rho, ~rhc, alo, ale, p, /to, Pc, 7r, Xc, xv, b, d, h, J, k, n, r, s, t, w, ?, a, u, I. The nine tr symbols are for the classification of syllables into heavy vs. light and open vs. closed and their various cross classifications. We have already been introduced to p, Po and pc, for moras, onsets and codas respectively. The remaining symbols are classifications of segments (x), firstly into vowels Orv) vs. consonants (xc), and then into the individual segments themselves (in boldface). This classification is depicted in (2) below.</Paragraph>
    <Paragraph position="6"> Syllable Structure. Phonological representations for sta, tat, taat and ast are given in (1) 5.</Paragraph>
    <Paragraph position="7">  (1) a. o b. o c. o d. o</Paragraph>
    <Paragraph position="9"> sta tat tat ast We can describe these pictures using formulas from L. For example, (lc) is described by the formula:</Paragraph>
    <Paragraph position="11"> It is possible to use formulas from L to describe ill-formed syllable structures. We shall rule these out by stating in L our empirical generalizations. We begin by specifying (i) the relationship between the sorts (i.e. the set X) using a sort lattice and (fi) how the sorts interact with dominance using an appropriateness relation. We then express in L the constraints graphically represented in the appropriateness graph in (2).</Paragraph>
    <Paragraph position="12"> (2) T aLo~l~degho~b k . t ? w a u i ..L The arrows may be glossed as follows: (i) all syllables must dominate an onset more~ (ii) heavy syllables must dominate a coda mora and (iii) all moras must dominate a segmenL The fact that potential arrows are absent also encodes constraints. For example: (i) syllables, moras and segments alike cannot dominate syllables and (ii) light syllables do not have coda moras. Constraints concerning the number of nodes of sort p that a node of sort q can dominate, and constraints concerning temporal organization cannot be stated in this graphical style. Nevertheless, they can be expressed in L as follows.</Paragraph>
    <Paragraph position="13"> (AI) L((po)/p --~ \[po\]~b). Onsets are unique.</Paragraph>
    <Paragraph position="14"> SSeC/ (Bird 1990) for arguments justifying this view of syllable structure. Moras are traditionally employed in the representation of syllable weight: a syllable with two moras is heavy, and a syllable with one mora is light.</Paragraph>
    <Paragraph position="15">  In a heavy syllable, the onset cannot end with a consonant. null We can express the constraint that two syllables cannot share a mora as follows.</Paragraph>
    <Paragraph position="16"> (A3) L(i Acr A (p)j --~ -~M(&amp;quot;~i A (#)j)).</Paragraph>
    <Paragraph position="17"> Two syllables cannot share a mora.</Paragraph>
    <Paragraph position="18"> An interesting alternative is to add an operator &lt;&gt;-1 that looks backwards along the dominance relationS.: The constraint that two syllables cannot share a mora could then be written L(# A ((r)-l~b ---, \[cr\]-l~b). There are further phonological phenomena which suggest that this may be an interesting extension of L to explore. For example, the requirement that all moras and segments must be linked to the hierarchical structure (prosodic licensing) may be expressed thus: L((# v 7r) ~ O-IT).</Paragraph>
    <Paragraph position="19"> Partiality. Crucially for the analysis of Arabic, it is possible to have a formula which describes more than one diagram. Consider the formula M(a A (#, x)(tA Fi)) A M(trA (th ~r)(aA i)), which may be glossed 'there is a syllable which dominates a t, and a syllable which dominates an a, and the t is before the a'. :This formula describes the three diagrams in (3) equally well:  (3) a. o b. o e. o o I II l.t It It Itit A II II t a t a ; t a  If a level of hierarchical structure higher than the syllable was employed, then it would not be necessary to use the M operator and we could write: (tr, #, x).(t A Fi) A (a, #, 7r)(a A i).</Paragraph>
  </Section>
  <Section position="6" start_page="0" end_page="0" type="metho">
    <SectionTitle>
ARABIC VERB MORPHOLOGY
</SectionTitle>
    <Paragraph position="0"> In the Semitic languages,: individual morphemes are often not manifested as contigl!ous strings of segments. A morphologically complex form must be expressed as the * intercalation of its component morphemes. An example of this phenomena is illustrated in Figure 1 for the perfective active r. Consider the form kattab in the second row. Its particular arrangement of four consonants and two e That is, O-1 is to O as P is to F.</Paragraph>
    <Paragraph position="1"> 7 Note that these are uninflected forms. Some forms are actually non-existent (for semantic reasons); these are i~dicated by a dash. However, this is unimportant since the present interest is in phonological structure and in potential forms.</Paragraph>
    <Paragraph position="2">  vowels identifies it with the second conjugation. Certain forms have additional affixes which are underlined in the above table. In what follows, we make a number of observations about the patterning of consonants in the above forms, showing how these observations can be stated in L.</Paragraph>
    <Paragraph position="3"> Arabic Syl'lable Structure. It is now widely recognized amongst phonologlsts that an analysis of Arabic phonology must pay close attention to syllable structure s . From the range of syllabic structure possibilities we saw in (1), only the following three kinds are permitted in Arabic.</Paragraph>
    <Paragraph position="4">  (4) a. C b. C c.</Paragraph>
    <Paragraph position="5"> It. It It }.t It A N /11 t a: t a t a t  The following &amp;quot;generalizations can be made about Arabic syllable structutre.</Paragraph>
    <Paragraph position="6"> (A4) L(ac --* ~rh). Closed syllables are heavy.</Paragraph>
    <Paragraph position="7"> There is a maximum of one consonant per node.</Paragraph>
    <Paragraph position="8"> (A6) L((xv)~-. \[xv\]~).</Paragraph>
    <Paragraph position="9"> There is a maximum of one vowel per node.</Paragraph>
    <Paragraph position="10"> (AT) L0,~ ^ (~)~ -, \[~\]~).</Paragraph>
    <Paragraph position="11"> There is a maximum of one segment per coda.</Paragraph>
    <Paragraph position="12"> (AS) ~((~, ,~v)q, -, b,, ,,-v\],/,).</Paragraph>
    <Paragraph position="13"> There is a maximum of one vowel per syllable, SThe approadaes to Arabic phonology presented by Kay (1987) and Gibbon (1990)---while addressing important computational issues--fail to represent the hierarchical organization of phonological structures.</Paragraph>
    <Paragraph position="15"> Onsets must have a consonant and a vowel, in that order.</Paragraph>
    <Paragraph position="16"> There are certain phonological phenomena which appear to move us beyond the bounds of L: the need to specify defaults. Phonologists often employ default consonants and vowels, which appear when the consonant or vowel positions in syllable structures have not been filled. In Arabic, the default consonant is ? (the glottal stop) and the default vowel is a. The default consonant only appears word initially. There are two ways we can treat such defaults. First, we can regard them as instructions on how one ought to 'compute' with L. That is, we regard them as instructions to attempt to build certain preferred models. Alternatively, we could combine L with a default logic.</Paragraph>
  </Section>
  <Section position="7" start_page="0" end_page="0" type="metho">
    <SectionTitle>
MORPHOLOGICAL COMBINATION
</SectionTitle>
    <Paragraph position="0"> Consider the forms kattab and dahraJ. Both consist of two closed syllables. This observation is expressed below.</Paragraph>
    <Paragraph position="1"> (II) M(ac A i A Fj) A M(ac A j) A L(a --~ iV j) Similarly, the two consonantisms can be represented as follows. (Note that il, i2 and iz are introduced in the (KTB) lbrmula as labels of syllable nodes; these labels will be referred to in the subsequent discussion.)  To derive kattab, we simply form 0I) ^ (KTB). The final conjunct of (H) requires that there be only two syllables. Consequently, each syllable mentioned in (KTB) has to be identified with i or j. There are eight possibilities, which fall into three groups. In what follows, i ~ j is shorthand for L(i *--, j), i.e. L is rich enough to support a form of equational reasoning 9.</Paragraph>
    <Paragraph position="2"> (i) il ,~ i2 ~ is ~ i or il ~, i2 ,.~ is ~ j. This would require a syllable to dominate three distinct consonants. However, fxom (A1), (A2) and (A5), Arabic syllables contain a maximum of two consonants.</Paragraph>
    <Paragraph position="3"> (ii) i~,.~is~i,i2~j;i2~i3~i,i~j;i2~i, il i3 ,~ j; or is ~ i, il .~ i2 ,~ j. In all of these cases, we have the following reductio ad absurdum, for some k' E {kl, k2, ks}.</Paragraph>
    <Paragraph position="4"> M(~, ^ 0', ~C/)k' ^ Fj) ^ M(,, ^ j ^ (#, ~aFk')</Paragraph>
    <Paragraph position="6"> Fj). and ; o j, which expands to L(i -40 j).</Paragraph>
    <Paragraph position="7"> (iii) il ~ i2 ~ i, i3 ~ j or ix ~ i, i2 ~ i3 ~ 3. It follows from the above default stipulations that two of the four consonants of (It) must be identical. By a similar process to that used in (ii) above, we can show that the coda consonant of i is identical to the onset consonant of j. The result is shown in (5a).</Paragraph>
    <Paragraph position="8"> (5) a. a a b. a a A A A A AVe\ A\A\ k ~v t gv b d ~ v h r n v J The case of (II) A (DHRJ) is depicted in (5b). The four consonants of (DHRJ) satisfy the requirements of the second conjugation template (II) without the need for reentrancy.</Paragraph>
  </Section>
  <Section position="8" start_page="0" end_page="0" type="metho">
    <SectionTitle>
OTHER PHENOMENA
</SectionTitle>
    <Paragraph position="0"> Consonant Doubling. In conjugations IX, XI, XII, XIV and QIV there is a non-geminate doubling of consonants.</Paragraph>
    <Paragraph position="1"> In the exceedingly rare XH, the second consonant (t) is doubled. In all the ;other cases, the final consonant is doubled. The most direct solution is to posit a lexical rule which fIecly applies to consonantisms, doubling their final consonant. For example, the rule would take the (KTB) form provided above and produce: (KTB') M((I~, ~r)AkAFkl))AM((#, 7r)AtAkl AFk2)) AM((#;rc) AbAk2AFk3))AM((#, rr) A bA k3)) It would be necessary to prevent this extended form from being used in conjugations 17 and V. since the patterns katbab and takatbab are unattested.</Paragraph>
    <Paragraph position="2"> The Reflexive Affix. Conjugations V, VI and VIII are marked by the.presence of a reflexive affix t. Rather than viewing these conjugations as basic (as Figure 1 implies), we can treat them as having been derived from conjugations H, IN and I respectively. As this affix always appears as the onset of a light syllable, we shall represent it thus: M(ct I A (/z, a')t). This morpheme will actually be expressed lexically as a function from conjugations to conjugations which attaches a syllable of the above form to the existing phonological material of a conjugation. The affix must be ordered relative to the other syllables. A constraint encoding the observation that all conjugations end in a closed syllable would prevent the affix from being a suffix. The fact that the affix is a light syllable correctly rules out ktattab (V) and ktaatab (VI), leaving only the attested forms for (V) and (VI). A constraint which prevented the first two syllables of a conjugation both being light (cf. McCarthy 1981:387), easily expressed in L, would rule out takatab (VIII) leaving only the attested form for (VIII).</Paragraph>
    <Paragraph position="3"> Extrametricality. Above we specified conjugation H as having a closed final syllable. Looking back at Figure 1, it would appear as if all conjugations end in a closed syllable.</Paragraph>
    <Paragraph position="4"> However, a study of the inflected forms reveals that this is not the case. For example, the third person plural of kattab is kattabu. From (Tll), the b must be syllabified with the vowel to its right. However, the first person plural - 93 is kattabna, and the b is syllabified with the vowel to its left. Similarly, the s of staktab is not part of the syllable tak. It is actually the coda of a previous syllable. In order to pronounce this form, ?i is prefixed, producing ?istaktab.</Paragraph>
    <Paragraph position="5"> Therefore, the conjugations are not merely sequences of complete syllable templates, but rather they are sequences bounded by unsyllabified (or extrametrical) consonants. The definition of lI should therefore be modified to be Mac A M(~C/ A (/~o)) A M(@ A (/~)) 10. This is intended to leave open the possibility for the final consonant to be syllabified with the second syllable or with the third syllable, while simultaneously requiring it to ultimately be syllabifled somewhere.</Paragraph>
  </Section>
class="xml-element"></Paper>