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<Paper uid="E91-1036">
  <Title>CLASSICAL LOGICS FOR ATTRIBUTE-VALUE LANGUAGES</Title>
  <Section position="3" start_page="0" end_page="0" type="metho">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> This paper describes a classical logic for attribute-value (or feature description) languages which are used in unification grammar to describe a certain kind of linguistic object commonly called attribute-value structure (or fcz~ture structure). From a logical point of view an attribute-vMue structure like e.g. tile following (in matrix notation)</Paragraph>
  </Section>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
PRED 'PROMISE'
TENSE PAST
</SectionTitle>
    <Paragraph position="0"> suBJ Pl :i) 'JOliN'\] XCOMP \[ SUBJ m \] PRED 'COME'  can be regarded as a graphical representation of a minimal model of a satisfiable feature description. If we assume that the attributes (in the example: PRED, TENSE, SUB J, XCOMP) are unary partial function symbols and the values (a, 'PROMISE', PAST, 'JOIIN', 'COME') are constants then the given feature structure represents graphically e.g. the minimal model of the following description:</Paragraph>
    <Paragraph position="2"> I Note that the terms arc h)rnlcd without using brackets. (Since all function symbols are unary, the introduction of brackets would So, in the following attribute-value languages are regarded &amp; quantifier-free sublanguages of classical first order language~ with equality whose (nonlogical) symbols are given by a set o&amp;quot; unary partial function symbols (attributes) and a set of constants (atomic and complex values). The logical vocabulary includes all propositional connectives; negation is interpreted (:lassically. 2 For quantifier-free attribute-value languages L we give an axiomatic or IIilbert type system lldegv which simply results from an ordinary first order system (with partial function symbols), if its language were restricted to the vocabulary of L. According to requirements of tile applications, axioms for the constantconsistency, constant/complex-consistency and acyclicity can be added to force these properties for the feature structures (models).</Paragraph>
    <Paragraph position="3"> For deciding consistency (or satisfiability) of a feature description, we assume .first, that the conjunction of the formulas ill,the feature dc'scription is converted to disjunctive normal form. Since a formula in disjunctive normal form is consis.</Paragraph>
    <Paragraph position="4"> tent, ill&amp;quot; at least one of its disjuncts is consistent, we only need all algorithm for.deciding consistency of finite sets of literals (atomic formulas or negated atomic formulas) S. In contrast to the reduction algorithms which normalize a set S accord.</Paragraph>
    <Paragraph position="5"> ing to a complexity norm in a sequence of norm decreasing rewrite steps 3 wc use a restricted deductive closure algorithm for deciding the consistency of sets of literMs. 4 The restriction results from the fact that it is sufficient for deciding the consistency of S to consider proofs of equations from ,.q with a certain subterm property. For tile closure construction only those equations are derived from S whose terms are subterms of the terms occurring in the formulas of S. This guarantees that the construction terminates with a finite set of literals. The adequacy of this subterm property restriction, which was already shown for the number theoretic calculus K in \[Kreisel/Tait 61\] by \[Statman 74\], is a necessary condition for the development of more efficient Cut-free Gentzen type systems for attributenot improve tile readability essentially.) Therefore we write e.g. PRED SUBJa instead of PRED(SUBJ(a)).</Paragraph>
    <Paragraph position="6"> 2For intuitionistic negation cf. e.g. \[Dawar/Vijay-Shanker 90\] and \[Langholm 89\].</Paragraph>
    <Paragraph position="7"> aCf. e.g. \[Kreisel/Tait 61\], \[Knuth/Bendix 70\], and applied to attrlhute-value languages \[Johnson 88\], \[Beierle/Plntat 88\], \[Smolka 89\].</Paragraph>
    <Paragraph position="8"> 4Since we allow cyclicity, unrestricted deductive closure algorithms (cf. e.g. \[Kasper/Rounds 86\] and \[Kasper/nounds OO\]) cannot be applied.</Paragraph>
    <Paragraph position="9"> - 204 value languages) Moreover, this closure construction is the direct prooI.</Paragraph>
    <Paragraph position="10"> theoretic correlate of the congruence closure algorithm (cf.</Paragraph>
    <Paragraph position="11"> \[Nelaon/Oppen 80\]), if it were used for testing satisffability of finite sets of literals in HOt,. As it is shown there, the congruence closure algorithm can bc used to test consistency if the terms of the equations are represented as labeled graphs and the equations as a relation on the nodes of that graph.</Paragraph>
    <Paragraph position="12"> O~ the basis of the algorithm for deciding satlsfiability of finite sets o |formulas we then show the completeness and decidability of//~t,.</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
2 Attribute-Value Languages
</SectionTitle>
    <Paragraph position="0"> In this section we define the type of lauguagc wc want to con- null sider i~nd introduce some additional notation. 2.t Syntax 2.1. DEFINITION. A quantifier-free attribute-value language (L.:%~) consists of the Jogical connective~ +- (false), ~ (negation), :) (implication), the equality symbol ,~ and the parentheses (,). The nonlogical vocabulary is given by a finite set of constants C and a finite set of unary partial Junction symbols r; (C/nr~ =~).</Paragraph>
    <Paragraph position="1"> 2.2. DEFINITION. The class of terms (7&amp;quot;) of L is recursively  defined as follows: each constant is a term; if f is a function symbol and r is a term, then fr is a term.</Paragraph>
    <Paragraph position="2">  2.3. DEFINITION. The set of atomic formulas: of L is !n ~ &amp;quot;~ I r,, r~+7,} u {+-}.</Paragraph>
    <Paragraph position="3"> 2.4. DEFINITION. The formulas of L are the atomic formulas 4nd, whenever ~ and ~b are formulas, then so are (+ ~b) and ~.5. DEFINITION. If ~ is a well-formed expressio n (term or  formula), then a\[r~/r~\] is used to designate an expression obtained from a by replacing some (possibly all or none) occurrC/nces Of r~ in ~ by r~.</Paragraph>
    <Paragraph position="4"> We assume that the connectives V (disjunction), ~:(conjunction) and ~ (equivalence) are introduced by their usual definitions, Furthermore, we write sometimes ri ~ rz ;instead of -,, ~'~ ~ r2 and drop the parentheses according tolthe usual conventions, e</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2~2 Semantics
</SectionTitle>
      <Paragraph position="0"> A model|or L consists of a nonempty universclt anti an inter.</Paragraph>
      <Paragraph position="1"> pre~a|ion function 9. Since not every term denotes an element In M if the function symbols are interi)reted as unary partial functions, we generalize the partiality of the denotation by asstltl~l~Ig that ~) itself is a partiM function. Thus in general not tCf. also \[Statmml 77\].</Paragraph>
      <Paragraph position="2"> sWe drop the outermost brackets, assume that the connectives h~ty e the precedence ,~&gt; &amp; &gt; v &gt;:), _-- and are left associative. all of the constants and function symbols are interpreted by ~).</Paragraph>
      <Paragraph position="3"> Redundancies which result from the fact that non-interpreted function symbols and function symbols interpreted as empty functions are then regarded as distinct are removed by requiring these partial funct~ions to be nonempty. Suppose \[X ,-, Y~(p) designates the set of all (partial) functions from X to Y~ then a model is defined as follows: 2.6. DEFINITION. A model for L is a pair M = (//, ~)), cpnsisting of a nonempty set U and an interpretation function</Paragraph>
      <Paragraph position="5"> The (partial) denotation function for terms ~ (~;C/\[T ~-*/at\] e) induced by 9 is defined as follows: 7 2.7. DEFINITION. For every ceC anti freT&amp;quot; (feFl),</Paragraph>
      <Paragraph position="7"> undefined otherwise.</Paragraph>
      <Paragraph position="8"> 2.8. DEFINITION. The satisfaction relation between models M and formulas ~b (~M ~b, read: M satisfies ~, M is a model of ~b, ~ is true in M) is defined recursively:</Paragraph>
      <Paragraph position="10"> A formula ~b is valid (\[= ~), iff ~b is true in all models. A formula ~b is satisfiable, iff it has at least one model. Given a set of formulas F, we say that M satisfies r (~ r), iff M satisfies each formula ~b in F. F is satisfiable, iff there is a model that satisfies each formula in F. ~ is logical consequenC/~ of F (F ~ C/), iff every model that satisfies F is a model of ~.</Paragraph>
    </Section>
  </Section>
  <Section position="6" start_page="0" end_page="13" type="metho">
    <SectionTitle>
3 The System Hdegv
</SectionTitle>
    <Paragraph position="0"> ? In this section we describe an axiomatic or Hilbert type system Hdegv for quantifier-free attribute-value languages L. We give a decision procedure for the saris|lability of finite sets of formulas and show the completeness and decidability of H~v on the b~mis of that procedure.</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.1 Axioms and Inference Rules
</SectionTitle>
      <Paragraph position="0"> If L is a fixed attribute-value language, then the system consiSts of a traditional axiomatic propositional calculus for L ud two additional equality axioms. For any formulas ~,~b,X , terms 71n the text following tile definition we drop the overllne.</Paragraph>
      <Paragraph position="1"> - 205 r, r', and every sequence of functors a (aeF;) of L the form,las under A1 - A4 are propositionalaxioms s and the formulas under El and E2 are equality axioms. deg The Modus Ponens (MP) is the only inlerence rule) deg</Paragraph>
      <Paragraph position="3"> A formula ff is derivable from a set of formulas F (I&amp;quot; b ~,), iff there is a finite sequence of formulas ff~...qL, such that ft, = q~ and every ~i is an axiom, one of the formulas in U or follows by MP from two previous formulas of the sequence, ff is a theorem (F ~), iff ~ is derivable from the empty set. A is derivable from F (r I- A), iff each formula of A is derivable from P. F and A are deductively equivalent (I&amp;quot; -U- A), iff r I- A and A F I'.</Paragraph>
      <Paragraph position="4"> The system is sound: n</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.2 Satisfiability
</SectionTitle>
      <Paragraph position="0"> We now prove 3.3. TIIEOREM. The satisfiability of a fi.ite set oJ formulas F is decidable.</Paragraph>
      <Paragraph position="1"> by providing a terminating procedure: First the conjunctio, of all formulas in F (denoted by A F) is converted into disjunctive normal form (DNF) using the well-known standard techniques. Then A F is equivalent with a DNF = (4,&amp;4~&amp;...&amp;C/k,) v (4~&amp;...&amp;4~,~) v ... v ~v-, ... v,k., where the conjuncts 4i (i = 1 .... n; j = 1 .... ki) are either atomic formulas or negations of atomic formulas, henceforth called iiterals. By the definition of the satisfiability we get: scf. e.g. \[Church 56\].</Paragraph>
      <Paragraph position="2">  consistency are to be guaranteed for a set Of atomic values V (V C_ C), for each a, beV (a # b) and leFt, axiomsof the form (i.) F a ~ b and (ii.) b fa ~ Ja have to be added (a finite set). I\[ also acyclicity has to be ensured, axioms of the form (iii.) bar ~ ~', with C/eFI + , veT, have to be added. Although this set is i,finite, we only need a finite subset for the satisfiability test and for deci,lal,illty (see below). II F'or the propositional calculus of. the sta,dard proofs, l&amp;quot;or axioms E1 and It,2 cf. \[Johnson 88\].</Paragraph>
      <Paragraph position="3"> 3.4. LEMMA. Let A St v A Sav ... v A s&amp;quot; be a DNF d/A r consisting of conjunctions A Si of the literals in S i, then A r is satisfiable, iff at least one disjuncl A Si is satisfiablel We complete the proof of Theorem 3.3 by an algorithm that converts a finite set of literals S i into a deductively equivalent set of literals in normal form S i which is satisfiable iff it is not equM to {.L}.</Paragraph>
      <Paragraph position="4">  The normal form is constructed by closing S deductively by those equations whose terms are subterms of the terms occurring in S. For the construction we use the following derived rules:</Paragraph>
      <Paragraph position="6"> We get RI and R2 from E1 and E2 by the deduction theorem.</Paragraph>
      <Paragraph position="7"> R3 is derivable from R1 and R2, since we get from r ~ r' first r ~ r by R1 and then r' ~, z by R2.</Paragraph>
      <Paragraph position="8"> If Ts denotes the set of terms occurring in the formulas of S (Ts</Paragraph>
      <Paragraph position="10"> then the normal form is constructed according to the following inductive definition.</Paragraph>
      <Paragraph position="11"> 3.5. I)EFIN1TION. For a given set of literals S we define a sequence of sets Si (i &gt;_ O) by induction:</Paragraph>
      <Paragraph position="13"> oil tile basis of the subterm condition either with a finite.set of literals or with {l}. If each term of the equations in Si+, is a subterm of tile terms in Ts, no term of the equations in $~+1 can be longer than the longest term in Ts.</Paragraph>
      <Paragraph position="14"> EXAMPLE 1. Assume that L consists of the constants a, b, c, e and the function symbols f,g, h,m, n,p. Then, for the set of literals ga = ha, a .~ If a, ngffa ~ e the following sequence of sets is constructed. We represenL the equations of a set Si by tile system of sets of equivalent terms ind.ced hy S,. I.e.: If O is a set of terms under Si and</Paragraph>
      <Paragraph position="16"> r,r'rO, then r ~ r'cSi. Furthermore, we mark by an arrow that a set under Si is also induced (without modifications) by the equations in Si+l.</Paragraph>
      <Paragraph position="17">  3.6. DEFINITION. Let S, = S,; with t = min{i I S, = S,/~}. 3.7. LEMMA. For Sv holds: S -iF- Sv.</Paragraph>
      <Paragraph position="18"> PROOF. If Sv # {.I-}, then S and Su are deductively equiva null lent, since S is a subset of Sv and Sv only contains formulas derivable from S. For Sv = {.1_} the same holds for S~_t. Since S~_~ is inconsistent, S is deductively equivalent with {.1.}. Note that for each equation in Si (Si # {a_}) there is a proof from S with the anbterm property, as defined below. This follows from the subterm condition in the inductive construction. 3.8. DEFINITION. A proof of an equation from S has the subterm property, iff each term occurring in the equations of that proof is a subterm of the terms in Ts, i.e. an element of su~(7-s).</Paragraph>
      <Paragraph position="19"> So, if S is not trivially inconsistent (PS not in S), the construction terminates with {_1.}, since there exists a proof of an equation from S with the subterm property, whose negation is in $.</Paragraph>
      <Paragraph position="20">  The deductive closure construction restricted by the subterm property is a proof-theoretic simulation of the congruence closure algorithm (cf. \[Nelson/Oppen 80\]t3), if used for testing satisfiability of finite sets of literals in Hdegv. Strictly speaking, if i. the congruence closure algorithm is weakened for partial functions, ii. S is not trivially inconsistent (.1_ not in S), and iii. the failure in the induction step of 3.5. is overruled, tZCL also \[Gallier 87\].</Paragraph>
      <Paragraph position="21"> then r ,.mr' is in Sv iff the nodes which represent the terms r and r' in the graph constructed for S are congruentfl t Moreover, for unary partial functions the algorithm is simpler, since the arity does not have to be controlled.</Paragraph>
      <Paragraph position="22"> 3.9. LEMMA. The set ol all equations in S~ is closed under subterm reflexivity, symmetry and transitivity.</Paragraph>
      <Paragraph position="23"> PROOF. For S~ = {.!_} trivial. If S~ # {.L}, then Sv is closed under subterm reflexivity and symmetry, since these properties are inherited from So to its successor sets. Sv is closed under transitivity, since we first get ra~SUB(Ts) from rl ~ r2, r~ ~ rsESu and then according to the construction also 7&amp;quot;1 ~ r2\[r2/rs\]~Sv+l = Sv, with r2\[ra/rs\] = rs. \[3  For the proof that the satisfiability of a finite set of fiterals is decidable we first show that a set of literals in normal form is satisfiable, iff the set is not equal to {.L}. For Sv = {.L} we get trivially: 3.10. LEMMA. Sv = {.1.} ~ &amp;quot;~3M(J=M Sv).</Paragraph>
      <Paragraph position="24"> Otherwise we can show the satisfiability of Sv by the construction of a canonical model that satisfies S~.</Paragraph>
      <Paragraph position="25"> Let Ev be the set of all (nonnegated) equations in Sv, TE~ the set of terms occurring in Ev and mEv the relation induced by E~ on T~ ({(r,r') \[ r ~ r'eE~}). Then, we choose as the universe of the canonical model M~ = (Uv,~v) the set of all equivalence classes of ~ on TE~, if T~ #- g. By Lemma 3.9 this set exists. If Sv contains no (unnegated) equation, we set Uv = {fl}, sittce the universe has to be nonempty.</Paragraph>
      <Paragraph position="26"> 3.11. DEFINITION. For a set of iiterals S~ in normal form, the canonical term model for Sv is given by the pair My - (Uv, ~lv}, consisting of the universe llv = {0} otherwise attd the interpretation function ~v, which is defined for cC/C, feFt and \[r\]d4v by: Is f \[c\] if ccT~ ~c(c)</Paragraph>
      <Paragraph position="28"> It follows from the definition that ~ is a partial function. Suppose further for ~)Ft(f) that \[rl\] = \[r2\] and that ~Ft(f)(\[rt\]) is defined. Then ~F, (f)(fn\]) = ~F~ (f)(fr2\]).</Paragraph>
      <Paragraph position="29"> For this, suppose ~F,(f)(\[rl\]) -- \[frq, with r'e\[rl\]. Since ~E~ is an equivalence relation we get r'e\[r~\] and thus</Paragraph>
      <Paragraph position="31"> For each term r in Tg~ it follows from tile definition of ~c and ~,: ~(r) = \[d.</Paragraph>
      <Paragraph position="32"> By the following lemma we show in addition that the domain of PSrv restricted to Ts~ is equal to TE~.</Paragraph>
      <Paragraph position="33"> 3.12. LEMMA. For each term r in Ts~: 11 ~ is defined for r, then ~,(r) = \[r\], with retd,.</Paragraph>
      <Paragraph position="34"> PROOF. (By induction on the length of r.) Suppose first that ~v is defined for r. For every coustant c it follows from the definition of ~)c that i~c(c) = \[c\], with c(7&amp;quot;E~. Assume for fr by inductive hypothesis ~v(r) = \[r\], with roTEs, then it follows from the definition of ~F~(f) that ~rt(f)(\[r\]) ----- \[fr~\]~ witlt frtcTF.~ and r'(\[r\]. Since r' is a subterm of \]r', wc first get r'eT-i;~ and by Lemma 3.9 fr' .~ fr',r' &amp;quot;~ r~S~. Because of fr(SUB(Ts), then also fr m \]r(Sv. So, fr must also be in Tg~ and hence c~, (f)(\[r\]) = \[fr\]. \[3 Next we show for the model My: 3.13. LEMMA. S~ # {,L} -.I=M~ S~.</Paragraph>
      <Paragraph position="35"> PROOF. (We prove I=~ @, for every C/, i, S~ hy induction oil the structure of @.) L is not element of S~. If 1 were in S~, we would get by the definition of S~ S~ = {a.} which contradicts our assumption. For @ =~ ,L, ~=MJ&amp;quot; PS holds trivially.</Paragraph>
      <Paragraph position="36"> Suppose ~ = r ~ r', then r,r' are in T~, ~ is defined for r and r ~, and ~v(r) = \[r\], ~(r') = \[r'\]. Because of r r'(S~, it follows that \[r\] = \[r'\]. So ~v(r) = ~(r') and hence ~M,, 7&amp;quot; ,~, r t.</Paragraph>
      <Paragraph position="37"> Assume that @ is ~ (r ~ r'). If r .m r' were satisfied by M~, ~(r) would be equal to ~,,(r'). By Lemma 3.12 we would then get $~(r) = \[r\] and ~v(r') = \[r'\], with r, r'(Tg~. Since ~g, is an equivalence relation on 7&amp;quot;g~, r ~ r'C/Su would follow from \[r\] = Jr'l, and, contradicting the assumption, we would get S~ = {'L} by tile defipition of S~. n It can be easily shown that Mv is a unique (up to isomorphism) minimal model for Sv. :s Strictly speaking, if M is &amp; model for 16It can be verified very easily by using this fact that we need to add to a set of literals S only a finite number of axioms to ensure the =cycllcity. All axioms of the form ~&amp;quot; ~ ~ (C/~C/Ft, ~'e'T), with la'r~ _~ ISUB(T~)I, are e.g. more than enough, since from a consistent but cyclic set of literals S must follow an equation ar ~ ~ (aeFi + ,~'eT), with I~1 &lt; I~1, and I~1 _&lt; ISUB(TE)I holds by the construction of S~ homomorl~hic to My, then every minimal submodel of M tl, al, satisfies c~, is isomorphic to My.</Paragraph>
      <Paragraph position="38"> From the two leuinlata above it follows first that tile sails\]lability of sets of formulas in normal form is decidable: Since S, and S are deductively equivalent, we can establish by the following lemma that the satisfiability of arbitrary finite sets of literals S is decidable.</Paragraph>
      <Paragraph position="39"> 3.14. LEMMA. S~ # {_L} ~ 3M(~M S).</Paragraph>
      <Paragraph position="40"> PROOF. (--,) If Sv # {,L}, we know by Lemma 3.13 that My is a model for S~. Then, by the soundness Su i- S &amp;quot;--* VM(~M Sv --*~M S). Since S is derivable from Sv, it follows ~M, S and thus S~ # {.L} ---, :IM(~M S).</Paragraph>
      <Paragraph position="41"> (,-) If S~ = {.L}, then for each model M V=M S~. From the soundness we get S I- Sv --* VM(~M S &amp;quot;-*~M Sv). Since S=. is derivable from S, it follows VM(~M Sv &amp;quot;*~=M S) amd hence S~ = {.l_} -- VM(~M S). O</Paragraph>
    </Section>
    <Section position="3" start_page="0" end_page="13" type="sub_section">
      <SectionTitle>
3.3 Completeness and Decidability
</SectionTitle>
      <Paragraph position="0"> Using tile procedure for deciding satisfiability we can easily show the completeness and decidability of ltdegA v .</Paragraph>
      <Paragraph position="1"> 3.15. TIIEOREM. For euery finite set of formulas P, and\]or each formula ~: 1I F ~ q~, then r b ~.</Paragraph>
      <Paragraph position="2"> PROOF. By definition @ is a logical consequence of F, iff F O {N @} is unsatisfiable. Using the equivalences of Theorem 3.3, wc first get: r o {~ C/} + {A(r u {~ C/})}.</Paragraph>
      <Paragraph position="3"> S,,l,l,OSe, that A S' v,..vAs&amp;quot; is a DNFofA(Fu{~ @}), then</Paragraph>
      <Paragraph position="5"> and by tile decision procedure V= ru {~ ~b} ,-., s~ = {_L} A...A Sv n = {.L}. If r U {.-, @) is unsatisfiable, it follows that PS U {,,, @} -iF {2.}, since each S i is deductively equivalent with {.L}. From PS U {.~ @} k -L it follows by the deduction theorem first FI-,,.~D.L and thus Ft-,-, -L D ~. From I'F~ / D ~ and F I-~ -L by MP then r I- ~. 13 3.16. COROLLARY. For every finite set o\] \]ormulas F and each \]ormula ~, F ~&amp;quot; ~ is decidable.</Paragraph>
      <Paragraph position="6"> PROOF. By the completeness and soundness we know F I- @ .-. I' ~ ~. Since @ is a logical consequence of r, iff ~ r u {,., ~}, we can decide r I-- C/~ by tile procedure for deciding ~= FU{,., ~}.</Paragraph>
    </Section>
  </Section>
class="xml-element"></Paper>