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<?xml version="1.0" standalone="yes"?> <Paper uid="P93-1026"> <Title>A COMPLETE AND RECURSIVE FEATURE THEORY*</Title> <Section position="4" start_page="194" end_page="194" type="metho"> <SectionTitle> 3 The Axioms </SectionTitle> <Paragraph position="0"> The first axiom scheme says that features are func- null The third and final axiom scheme will say that certain &quot;consistent feature descriptions&quot; are satisfiable. For its formulation we need the important notion of a solved clause.</Paragraph> <Paragraph position="1"> An exclusion constraint is an additional atomic formula of the form zfI (&quot;f undefined on x&quot;) taken to be equivalent to -~3y (xfy) (for some variable y # z).</Paragraph> <Paragraph position="2"> A solved clause is a possibly empty conjunction C/ of atomic formulae of the form xfy, Ax and xf~ such that the following conditions are satisfied: 1. no atomic formula occurs twice in C/ 2. ifAxEeandBxEC/,thenA=B 3. ifxfyEeandxfzEC/,theny=z 4. if xfy E C/, then xfT q~ C/.</Paragraph> <Paragraph position="4"> As in the example, a solved clause can always be seen as the graph whose nodes are the variables appearing in the clause and whose arcs are given by the feature constraints xfy. The constraints Ax, xfT appear as labels of the node x.</Paragraph> <Paragraph position="5"> A variable x is constrained in a solved clause C/ if C/ contains a constraint of the form Ax, xfy or xfT. We use CV(C/) to denote the set of all variables that are constrained in C/. The variables in V(C/) -CV(C/) are called the parameters of a solved clause C/. In the graph representation of a solved clause the parameters appear as leaves that are not not labeled with a sort or a feature exclusion. The parameters of the solved clause in Figure 2 are y and z.</Paragraph> <Paragraph position="6"> We can now state the third axiom scheme. It says that the constrained variables of a solved clause have solutions for all values of the parameters: (Ax3) ~/qxC/ (for every solved clause C/ and X = cv(C/)).</Paragraph> <Paragraph position="7"> The theory FT is the set of all sentences that can be obtained as instances of the axiom schemes (Axl), (Ax2) and (Ax3). The theory FTo is the set of all sentences that can be obtained as instances of the first two axiom schemes.</Paragraph> <Paragraph position="8"> As the main result of this paper we will show that FT is a complete and decidable theory.</Paragraph> <Paragraph position="9"> By using an adaption of the proof of Theorem 8.3 in \[15\] one can show that FTo is undecidable.</Paragraph> </Section> <Section position="5" start_page="194" end_page="194" type="metho"> <SectionTitle> 4 Outline of the Completeness Proof </SectionTitle> <Paragraph position="0"> The completeness of FT will be shown by exhibiting a simplification algorithm for FT. The following lemma gives the overall structure of the algorithm, which is the same as in Maher's \[12\] completeness proof for the theory of constructor trees.</Paragraph> <Paragraph position="1"> Lemma 4.1 Suppose there exists a set of so-called prime formulae such that: 1. every sort constraint Ax, every feature constraint xfy, and every equation x - y such that = ~ y is a prime formula 2. T is a prime formula, and there is no other closed prime formula 3. for every two prime formulae fl and fl' one can compute a formula 6 that is either prime or .1_ and satisfies flAi'MFT6 and )2(6)C_V(flAff) 4. for every prime formula fl and every variable x one can compute a prime formula i' such that 3xi MFT fl' and Y(t') C_ Y(3xfl) 5. if i, ill,''' ,fin are prime formulae, then</Paragraph> <Paragraph position="3"> 6. for every two prime formulae fl, fl I and every variable x one can compute a Boolean combination 6 of prime formulae such that 3~(fl^-,C/) I~FT 6 and Vff) C VO=(fl^-~l')).</Paragraph> <Paragraph position="4"> Then one can compute for every formula C/ a Boolean combination ~ of prime formulae such that C/ MET ~ and V(O C_ V(C/).</Paragraph> <Paragraph position="5"> Proof. Suppose a set of prime formulae as required exists. Let C/ be a formula. We show by induction on the structure of C/ how to compute a Boolean combination df of prime formulae such that C/ MET 6 and V(O C_ V(C/).</Paragraph> <Paragraph position="6"> If C/ is an atomic formula Ax, xfy or x - y, then C/ is either a prime formula, or C/ is a trivial equation z - z, in which case it is equivalent to the prime formula T.</Paragraph> <Paragraph position="7"> If C/ is -~C/, C/ ^ C/' or C/ V C/', then the claim follows immediately with the induction hypothesis.</Paragraph> <Paragraph position="8"> It remains to show the claim for C/ = 3=C/. By the induction hypothesis we know that we can compute a Boolean combination df of prime formulae such that</Paragraph> </Section> <Section position="6" start_page="194" end_page="196" type="metho"> <SectionTitle> 6 MFT ~) and V(6) C_ V(C/). Now ~ can be trans- </SectionTitle> <Paragraph position="0"> formed to a disjunctive normal form where prime formulae play the role of atomic formulae; that is, 6 is equivalent to 6'1 V... V C/,, where every &quot;clause&quot; qi is a conjunction of prime and negated prime formulae. Hence 3=C/ 14 3=(o-~ v... v ,..) I=13=o-~ v... v 3=o-., where all three formulae have exactly the same free variables. It remains to show that one can compute for every clause ~r a Boolean combination 6 of prime formulae such that =1=o- MET 6 and Y(6) C_ V(3xa).</Paragraph> <Paragraph position="1"> We distinguish the following cases.</Paragraph> <Paragraph position="2"> (i) a = fl for some basic formula i. Then the claim follows by assumption (4). Oi) o&quot; i^&quot; ~ , = Ai=I ti n > 0. Then the claim follows with assumptions (5) and (6). n T n Oil) tr = Ai=I -~ii, n > 0. Then a MET AA/=I -~fli and the claim follows with case (it) since T is a prime formula by assumption (2).</Paragraph> <Paragraph position="3"> (iv) ~ =ill ^...^tk ^-,ill ^... h t', k > 1, n ___ 0. Then we know by assumption (3) that either fll A...A flk MFT .L or fll A ... A flk MET fl for some prime formula ft. In the former case we choose 8 = -,T, and in the latter case the claim follows with case (i) or (ii). \[\] Note that, provided a set of prime formulae with the required properties exists, the preceding lemma yields the completeness of FT since every closed formula can be simplified to T or -~T (since T is the only closed prime formula).</Paragraph> <Paragraph position="4"> In the following we will establish a set of prime formula as required.</Paragraph> </Section> <Section position="7" start_page="196" end_page="197" type="metho"> <SectionTitle> 5 Solved Formulae </SectionTitle> <Paragraph position="0"> In this section we introduce a solved form for conjunctions of atomic formulae.</Paragraph> <Paragraph position="1"> A basic formula is either 3- or a possibly empty conjunction of atomic formulae of the form Ax, xfy, and x - y. Note that T is a basic formula since T is the empty conjunction.</Paragraph> <Paragraph position="2"> Every basic formula C/ ~ 3- has a unique decomposition C/ = CN ACG into a possibly empty conjunction CN of equations &quot;x -- y&quot; and a possibly empty conjunction CG of sort constraints &quot;Ax&quot; and feature constraints &quot;xfy&quot;. We call CN the normalizer and and C/G the graph of C/.</Paragraph> <Paragraph position="3"> We say that a basic formula C/ binds x to y if x - y E C/ and x occurs only once in C/. Here it is important to note that we consider equations as directed, that is, assume that x - y is different from y ~ x ifx ~ y. We say that C/ eliminatesx ifC/ binds x to some variable y.</Paragraph> <Paragraph position="4"> A solved formula is a basic formula 7 ~ 3- such that the following conditions are satisfied: 1. an equation x - y appears in 7 if and only if 7 eliminates x 2. the graph of 7 is a solved clause.</Paragraph> <Paragraph position="5"> Note that a solved clause not containing exclusion constraints is a solved formula, and that a solved formula not containing equations is a solved clause. The letter 7 will always denote a solved formula. We will see that every basic formula is equivalent in FT0 to either 3- or a solved formula.</Paragraph> <Paragraph position="6"> Figure 3 shows the so-called basic simplification rules. With C/\[x ~-- y\] we denote the formula that is obtained from C/ by replacing every occurrence of x with y. We say that a formula C/ simplifies to a formula C/ by a simplification rule p if ~ is an instance of p. We say that a basic formula C/ simplifies to a basic formula C/ if either C/ = C/ or C/ simplifies to C/ in finitely many steps each licensed by one of basic simplification rules in Figure 3.</Paragraph> <Paragraph position="7"> Note that the basic simplification rules (1) and (2) correspond to the first and second axiom scheme, respectively. Thus they are equivalence transformation with respect to FTo. The remaining three simplification rules are equivalence transformations in general. Proposition 5.1 The basic simplification rules are terminating and perform equivalence transformations with respect to FT0. Moreover, a basic formula C/ ~ 3_ is solved if and only if no basic simplification rule applies to it.</Paragraph> <Paragraph position="8"> Proposition 5.2 Let C/ be a formula built from atomic formulae with conjunction. Then one can compute a formula 6 that is either solved or 3_ such that C/ ~FTo 6 and r(6) C_ l;(C/).</Paragraph> <Paragraph position="9"> In the quantifier elimination proofs to come it will be convenient to use so-called path constraints, which provide a flexible syntax for atomic formulae closed under conjunction and existential quantification. We start by defining the denotation of a path. The interpretations fit, g~ of two features f, g in a structure .4 are binary relations on the universe 1&quot;41 of .4; hence their composition fA o g.a is again a binary relation on 1-41 satisfying a(f A o gA)b C/=:C/, 3c ~ 1&quot;41: af Ac A cfAb for all a, b E 1&quot;41. Consequently we define the denotation p~t of a path p = fl &quot;'&quot; .In in a structure .4 as the composition (fl...fn) A :---- f:o...ofn A, where the empty path ~ is taken to denote the identity relation. If.4 is a model of the theory FTo, then every paths denotes a unary partial function on the universe of .4. Given an element a E \[.41, p~t is thus either undefined on a or leads from a to exactly one b ~ 1.41.</Paragraph> <Paragraph position="10"> Let p, q be paths, x, y be variables, and A be a sort. Then path constraints are defined as follows: .4, a ~ zpy :C/:~ o~(x) pA a(y) .4, a ~ xp.~yq :C/:=~ 3a E 1.41: degt(x)pa aAa(y)q A a .4, a~Azp :~=~3ael.41: a(z)p'4a^aeA &quot;~.</Paragraph> <Paragraph position="11"> Note that path constraints xpy generalize feature constraints x fy.</Paragraph> <Paragraph position="12"> A proper path constraint is a path constraint of the form &quot;Axp&quot; or &quot;xp ~. yq&quot;. Every path constraint can be expressed with the already existing formulae, as can be seen from the following equivalences:</Paragraph> <Paragraph position="14"> The closure \[3`\] of a solved formula 3` is the closure of the atomic formulae occurring in 7 with respect to the following deduction rules: x-y xpy yfz xpz yqz Ay xpy xEx xey zpf z xp I Yq Axp Recall that we assume that equations x - y are directed, that is, are ordered pairs of variables. Hence, xey E \[71 and yex ~ \[71 if x - y E 7.</Paragraph> <Paragraph position="15"> The closure of a solved clause 6 is defined analogously. null Proposition 5.3 Let 7 be a solved formula. Then: I. if ~v E \[7\], then 7 ~ ~r 2. xeyE\[7\] iff x=yorx--yE7 3. xfy E \[7\] iff zfy E 3` or 3z: z &quot;-- z E 7 and zfy E 7 4. xpfy e \[7\] iff 3z: xpz e \[7\] and zfy e 3` 5. if p 7 PS e and xpy, xpz E \[3`\], then y = z 5. it is decidable whether a path constraint is in \[3'\]. 6 Feature Trees and Feature Graphs In this section we establish three models of FT consisting of either feature trees or feature graphs. Since we will show that FT is a complete theory, all three models are in fact elementarily equivalent. A tree domain is a nonempty set D _C FEA* of paths that is prefix-closed, that is, if pq E D, then p E D. Note that every tree domain contains the empty path.</Paragraph> <Paragraph position="16"> A feature tree is a partial function a: FEA* --+ SOR whose domain is a tree domain. The paths in the domain of a feature tree represent the nodes of the tree; the empty path represents its root. We use D~ to denote the domain of a feature tree ~. A feature tree is called finite \[infinite I if its domain is finite \[infinite\]. The letters a and 7. will always denote feature trees.</Paragraph> <Paragraph position="17"> The subtree pa of a feature tree a at a path</Paragraph> <Paragraph position="19"> A feature tree a is called a subtree of a feature tree 7- if ~r is a subtree of 7- at some path p E Dr, and a direct subtree if p = f for some feature f.</Paragraph> <Paragraph position="20"> A feature tree a is called rational if (1) cr has only finitely many subtrees and (2) a is finitely branching (i.e., for every p E D~, the set {pf E D~ \[ f E FEA} is finite). Note that for every rational feature tree a there exist finitely many features fl,...,In such that Do C_ {fl,..-,fn}*.</Paragraph> <Paragraph position="21"> The feature tree structure'T is the SOR~FEAstructure defined as follows: * the universe of 7- is the set of all feature trees</Paragraph> <Paragraph position="23"> subtree of a at f).</Paragraph> <Paragraph position="24"> The rational feature tree structure 7~ is the sub-structure of T consisting only of the rational feature trees.</Paragraph> <Paragraph position="25"> Theorem 6.1 The feature tree structures T and 7~ are models of the theory FT.</Paragraph> <Paragraph position="26"> A feature pregraph is a pair (x, 7) consisting of a variable x (called the root) and a solved clause 7 not containing exclusion constraints such that, for every variable y occurring in 7, there exists a path p satisfying xpy E \[7\]- If one deletes the exclusion constraints in Figure 2, one obtains the graphical representation of a feature pregraph whose root is x. A feature pregraph (x, 7) is called a subpregraph of a feature pregraph (y,~) if 7 _C 6 and x -- y or x E \]2(~). Note that a feature pregraph has only finitely many subpregraphs.</Paragraph> <Paragraph position="27"> We say that two feature pregraphs are equivalent if they are equal up to consistent variable renaming. For instance, (x, xfy A ygx) and (u, ufx A xgu) are equivalent feature pregraphs.</Paragraph> <Paragraph position="28"> A feature graph is an element of the quotient of the set of all feature pregraphs with respect to equivalence as defined above. We use (x, 7) to denote the feature graph obtained as the equivalence class of the feature pregraph (x, 7).</Paragraph> <Paragraph position="29"> In contrast to feature trees, not every node of a feature graph must carry a sort.</Paragraph> <Paragraph position="30"> The feature graph structure ~ is the SOR FEA-structure defined as follows: * the universe of ~ is the set of all feature graphs</Paragraph> <Paragraph position="32"> Theorem 6.2 The feature graph structure ~ is a model of the theory FT.</Paragraph> <Paragraph position="33"> Let ~&quot; be the structure whose domain consists of all feature pregraphs and that is otherwise defined analogous to G. Note that G is in fact the quotient of jc with respect to equivalence of feature pregraphs. Proposition 6.3 The feature pregraph structure yr is a model of FTo but not of FT.</Paragraph> </Section> <Section position="8" start_page="197" end_page="198" type="metho"> <SectionTitle> 7 Prime Formulae </SectionTitle> <Paragraph position="0"> We now define a class of prime formulae having the properties required by Lemma 4.1. The prime formulae will turn out to be solved forms for formulae built from atomic formulae with conjunction and existential quantification.</Paragraph> <Paragraph position="1"> A prime formula is a formula 3X7 such that 1. 7 is a solved formula 2. X has no variable in common with the normalizer of 3' 3. every x E X can be reached from a free variable, that is, there exists a path constraint ypx E \[7\] such that y ~t X.</Paragraph> <Paragraph position="2"> The letter/3 will always denote a prime formula. Note that T is the only closed prime formula, and that 3X 7 is a prime formula if 3x3X 7 is a prime formula. Moreover, every solved formula is a prime formula, and every quantifier-free prime formula is a solved formula.</Paragraph> <Paragraph position="3"> The definition of prime formulae certainly fulfills the requirements (1) and (2) of Lemma 4.1. The fulfillment of the requirements (3) and (4) will be shown in this section, and the fulfillment of the requirements (5) and (6) will be shown in the next section.</Paragraph> <Paragraph position="4"> Proposition 7.1 Let 3X 7 be a prime formula, .A be a model of FT, and ,4, a ~ 3X7. Then there exists one and only one X-update (~' of ~ such that A,a' ~7.</Paragraph> <Paragraph position="5"> The next proposition establishes that prime formulae are closed under existential quantification (prop null from atomic formulae with conjunction and existential quantification. Then one can compute a formula 6 that is either prime or I such that C/ ~FT 8 and Vff) _C V(C/).</Paragraph> <Paragraph position="6"> The closure of a prime formula 3X7 is defined as follows:</Paragraph> <Paragraph position="8"> The proper closure of a prime formula/3 is defined as follows:</Paragraph> <Paragraph position="10"> \[/3\], then/3 p ~ (and hence --,,~ p --,/3).</Paragraph> <Paragraph position="11"> We now know that the closure \[ill, taken as an infinite conjunction, is entailed by/3. We are going to show that, conversely,/3 is entailed by certain finite subsets of its closure \[/3\].</Paragraph> <Paragraph position="12"> An access function for a prime formula/3 = 3X 7 is a function that maps every x * 1)(7 ) - X to the rooted path xC/, and every x E X to a rooted path x'p such that x'px * \[7\] and x' ~ X. Note that every prime formula has at least one access function, and that the access function of a prime formula is injective on 1)(3') (follows from Proposition 5.3 (5)). The projection of a prime formula/3 = 3X7 with respect to an access function @ for/3 is the conjunction of the following proper path constraints: {Ax'p I Ax E 7, x'p = @x} U {='pf~y'q \[ xfy E 7, x'p = @x, y'q = @y}.</Paragraph> <Paragraph position="13"> Obviously, one can compute for every prime formula an access function and hence a projection. Furthermore, if )~ is a projection of a prime formula/3, then )~ taken as a set is a finite subset of the closure \[/3\]. Proposition 7.7 Let )~ be a projection of a prime formula/3. Then )t C \[/3\]* and )t ~=~FT /3&quot; As a consequence of this proposition one can compute for every prime formula an equivalent quantifier-free conjunction of proper path constraints. null We close this section with a few propositions stating interesting properties of closures of prime formulae. These propositions will not be used in the proofs to come.</Paragraph> <Paragraph position="14"> let )d be a projection of/3'. Then \]3 ~FT /3t \[#\]* _~ k'. Proposition 7.11 gives us a decision procedure for &quot;/3 ~FT /3&quot; since membership in \[/3\]* is decidable, k' is finite, and ,V can be computed from/3'.</Paragraph> </Section> class="xml-element"></Paper>