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<Paper uid="P93-1027">
  <Title>ON THE DECIDABILITY OF FUNCTIONAL UNCERTAINTY*</Title>
  <Section position="8" start_page="205" end_page="207" type="concl">
    <SectionTitle>
6 Kaplan and Maxwell's Method
</SectionTitle>
    <Paragraph position="0"> We are now able to compare our method with the one used by Kaplan and Maxwell. In our method, the non-deterministic addition of path relation and the evaluation of these relations are done at different times. The evaluation of the introduced constraints c~ - fl and o~ :C/ fl are done after (PathRel) in the first phase of the algorithm, whereas the evaluation of the divergence constraints is done in a separate second phase.</Paragraph>
    <Paragraph position="1"> In Kaplan and Maxwell's algorithm all these steps are combined into one single rule. Roughly, they substitute a clause {xL~y, xL2z, } O C/ nondeterministicly by one of the following clauses: ~  Recall that {XLly, xL2z} is expressed in our syntax by the clause 3' = {xay, o~ ~ L1, x~z, j~ ~ L2}, which is the example we have used on page 2. The first three cases correspond exactly to the result of the 2This is not the way their algorithm was originally described in \[5\] as they use a slightly different syntax. Furthermore, they don't use non-deterministic rules, but use a single rule that produces a disjunction. However, the way we describe their method seems to be more appropriate in comparing both approaches.</Paragraph>
    <Paragraph position="2"> derivations that have been described for 72, 73 and 3'4. By and large, the last case is achieved if we first add c~ \[I ~ to 3' and then turn over to the second phase as described in the last section.</Paragraph>
    <Paragraph position="3"> The problem with Kaplan/Maxwell's algorithm is that one has to introduce a new variable u in the last case, since there is no other possibility to express divergence. If their rule system is applied to a cyc!ic description, it will not terminate as the last part introduces new variables. Hence it cannot be used for an algorithm in case of cyclic descriptions.</Paragraph>
    <Paragraph position="4"> The delaying of the evaluation of divergence constraint may not only be useful when applied to cyclic feature descriptions. As Kaplan and Maxwell pointed out, it is in general useful to postpone the consistency check for functional uncertainty. With the algorithm we have described it is also possible to delay single parts of the evaluation of constraints containing functional uncertainty.</Paragraph>
    <Paragraph position="5"> Appendix Proof of Lemma 4.2. The first claim is easy to prove. For the second claim let {L1,...,Ln} C P(~+) be the set of regular languages used in C/ and let .Ai = (Q.4~, i.4~, cr a~, Fin.4~) be finite, deterministic automatons such that .A i recognizes Li. For each .Ai we define dec(.Ai) to be the set dee(A/) = {L~ \]p,q E QJt,}, whereL~ = {w E 2 &amp;quot;+ I a~,(p,w) = q}. It is easy to show that dec(.Ai) is a set of regular languages that contains Li and is closed under decomposition.</Paragraph>
    <Paragraph position="6"> Hence, the set A0 = \[.Jinx dec (Ai) contains each Li and is closed under decomposition. Let A = fi (A0) be the least set that contains A0 and is closed under intersection. Then A is finite and e-closed, since it contains each Li.</Paragraph>
    <Paragraph position="7"> We will prove that A is also closed under decomposition. Given some L E A and a word w = wlw2 E L, we have to find an appropriate decomposition P, S in A. Since each L in A can be written as a finite m L intersection L = Nk=l i~ where Lik is in A0, we know that w = wlw2 is in Li~ for 1..m. As A0 is closed under decomposition, there are languages Pi~ and Si~ for k = 1..m with wl E Pi~, w2 E Si~ and Pik'Sik C Li~. Let P = M~n=l Pik and S = s,~.</Paragraph>
    <Paragraph position="8"> Clearly, wl 6 P, w2 6 S and P.S C L. Furthermore, P, S 6 A as A is closed under intersection. This implies that P, S is an appropriate decomposition for  sound and globally X U 12-preserving. If A is closed under decomposition, then (LangDec^) is X U 12sound and globally X U IJ-preserving. The (Pre) rule is X-sound and X-preserving. All other rules are X U 13-sound and X U 13-preserving.</Paragraph>
    <Paragraph position="9">  Next we will prove some syntactic properties of the clauses derivable by the rule system. For the rest of the paper we will call clauses that are derivable from prime clauses admissible.</Paragraph>
    <Paragraph position="10"> Proposition A.2 Every admissible clause is basic.</Paragraph>
    <Paragraph position="11"> Ira -~ 13, o~ -- \[3 or c~ (I 13 is contained in some admissible clause C/, then there is a variable z such that zc~y and zflz is in C/.</Paragraph>
    <Paragraph position="12"> Note that (by this proposition) (Pre) (resp. (Eq)) can always be applied if a constraint c~ 4 \[3 (resp. -/3) is contained in some admissible clause. The next lemma will show that different applications of (Pre) or (Eq) will not interact. This means the application of one of these rule to some prefix or path equality constraint will not change any other prefix or path equality constraint contained in the same clause. This is a direct consequence of the way (PathP~el) was defined.</Paragraph>
    <Paragraph position="13"> Lemma A.3 Given two admissible clauses 7, 7' with 7 ---~r 7' and r different from (PathRel). Then c~ &amp;quot;- 13 E 7' (resp. ~ 4 13 E 7 I) implies ~ -- 13 E 7 (resp. a :C/ \[3 E 7). Furthermore, if a.13 is contained in 7', then either a.fl or a -~ 13 is contained in 7. Note that this lemma implies that new path equality or prefix constraints are only introduced by (PathRel). We can derive from this lemma some syntactic properties of admissible clauses which are needed for proving completeness and quasitermination. null Lemma A.4 If C/ is an admissible clause, then  1. If c~ :&lt; 13 is contained in C/, then there is no other  prefix or equality constraint in C/ involving 13.</Paragraph>
    <Paragraph position="14"> Furthermore, neither 13.\[3~ nor 13~.\[3 is contained in C/.</Paragraph>
    <Paragraph position="15"> e. ira.13 fi 13' is in C/, then either 13' equals a or C/ contains a constraint of form afi t3', a - 13' or :~ ~'. The first property will guarantee that concatenation does not occur in prefix or equality constraints and that the length of path concatenation is restricted to 2. The second property ensures that a constraint c~.13 fi 13' is always reducible.</Paragraph>
    <Paragraph position="16"> Theorem A.5 For every finite A the rule system 7~a is quasi-terminating.</Paragraph>
    <Paragraph position="17"> Proof. The rule system produces only finitely many different clauses since the rules introduce no additional variables or sort symbols and the set of used languages is finite. Additionally, the length of concatenation is restricted to 2. \[\] Lemma A.6 There are no infinite derivations using only finitely many instances of (Pre).</Paragraph>
    <Paragraph position="18"> Since the rule system is quasi-terminating, the completeness proof consists of two parts. In the first part we will proof that pre-solved clauses are just the irreducible clauses. In the second part we will show that one finds for each solution Vx of a prime clause C/ a pre-solved e-derivative 7 such that Vx is also a solution of 7.</Paragraph>
    <Paragraph position="19"> Theorem A.7 (Completeness I) Given an admissible clause C/ ~ _1_ such that C/ is not in pre-solved form. If A is e-closed and closed under decomposition, then C/ is T~A-reducible.</Paragraph>
    <Paragraph position="20"> Theorem A.8 (Completeness II) For every prime clause C/ and for every A that is e-closed, closed under decomposition and intersection we have IC/\] _c U b\] z 7 E pre-solved (C/,R^) where pre-solved(C/,R^) is the set of pre-solved (C/, R A )-derivat ives.</Paragraph>
    <Paragraph position="21"> Proof (Sketch) We have to show, that for each prime clause C/ and each Vx, V~,Z with (Vx, V~) ~z C/ there is a pre-solved (C/, T~A)-derivative 7 such that Vx E ~7\] z. We will do this by controlling derivation using the valuation (Vx, VT~). The control will guarantee finiteness of derivations and will maintain the first completeness property, namely that the irreducible clauses are exactly the pre-solved clauses. We allow only those instances of the non-deterministic rules (PathRel) and (LangDecA), which preserve exactly the valuation (Vx, V~). That means if (Vx,V~) ~z C/ and C/ --~r 7 for one of these rules, then (Va', V~) ~z 7 must hold. Note that the control depends only on VT,. E.g. for the clause C/ = {xc~y, a ~ L1, x13z, 13~ L2} and arbitray Z, Vx this means that if VT,(a) = f, V~,(13) = g and (Vx, VT,) ~z C/, the rule (PathRel) can transform C/ only into {a h 13} U C/.</Paragraph>
    <Paragraph position="22"> If V~, satisfies V~, (tr) 7~ V~, (13) for ~ different from fl with zcry E C/ and 213z E C/, we cannot add any prefix constraint using this control. Hence, (Pre) cannot be applied, which implies (by lemma A.6) that in this case there is no infinite controlled derivation. We will call such path valuations prefix-free with respect to C/. If V~, is not prefix-free, then (Pre) will be applied during the derivations. In this case we have to change the path valuation, since (Pre) is not P-preserving. If (Vx, V~) ~z C/ = {a k 13} U C/ and we apply (Pre) on cr -~ fl yielding 7, then the valuation VC/ with v (13) = and = for # will satisfy (Vx, pz % We will use for controlling the further derivations.</Paragraph>
    <Paragraph position="23"> If we change the path valuation in this way, there will again be only finite derivations. To see this, note that every time (Pre) is applied and the path valuation is changed, the valuation of one variable is shortened by a non-empty path. As the number of variables used in clauses does not increase, this shortening can only be done finitely many times. This implies, that (Pre) can only finitely often be applied under this control. Hence (by lemma A.6), there are again only finite controlled derivations. 1:3</Paragraph>
    <Section position="1" start_page="207" end_page="207" type="sub_section">
      <SectionTitle>
A.2 Consistency of Pre-Solved Clauses
</SectionTitle>
      <Paragraph position="0"> We will first do a minor redefinition of divergence.</Paragraph>
      <Paragraph position="1"> We say that two paths u, v are directly diverging (written u u0 v) if there are features f ~ g such that u E f/'* and v 6 g/'*. Then u n v holds if there are a possible empty prefix w and paths u', v' such that u = wu' and v = wC and u' n0 v'.</Paragraph>
      <Paragraph position="2"> We will reformulate the reduction of divergence constraints in order to avoid constraints of form a.fl fi fl'. Handling such constraints would make the termination proof somewhat complicated. For the reformulation we use a special property of pre-solved clauses, namely that a fi fl is in a pre-solved clause C/ iff zay and zflz is in C/. Hence, if a fi/? and ~ fi df is in C/, then a Ii df is also in C/. This implies, that we can write ep as fi(At) ~...~ fl(A,) t9 C/, where fl (A) is a syntactic sugar for fi(A) = {a fia' I a # a'Aa, a' 6 A}, As,...,An are disjoint sets of path variables and C/ does not contain divergence constraints. Note that for every Ai = {al,...,a,} there are variables x, Yt,...,yn such that {xatyt,...,x~,y,} C_ C/. Now given such that a constraint fi (A), we assume that a whole set of path variables A1 C A diverges with the same prefix ft. That means we can replace fl(At) C fl(A) by As = fl.A',O fi0(A~), where fl is new, A~ = {a~,..., a~} is a disjoint copy of A1 = {or1,...,an} and A - fi.A~ is an abbreviation for the clause {al - fl'a~,..., c~, - fl.a~}. fl 0(A) is defined similar to fl (A). Assuming additionally that the common prefix fl is maximal implies that fl fl a holds for a E (A-A1). If we also consider the effects of A1 = fl'A'l on the subterm agreements in C/ that involves variables of At, then we result in the following rule:  o f~#f~, for a#~' together with the rules (LangDech), (Join) and (Empty) completes the rule system 7~ degiv. (Reds) is needed as path variables always denote non-empty paths. We will view (Redz) and (Red2) as one single rule (Reduce).</Paragraph>
      <Paragraph position="3"> A clause ~ is said to be solved if (1) a.fl ~ L and ot~0 is not in ep; (2) a~L1 in ep and a~L~ in ep implies Lz = L2; (3) C/ does not contain constraints of form afl fl, a Ii0 fl, oL :&lt; fl, or a -&amp;quot; fl; and (4) for every {xay, z/~z} _C ~ with a C//? there are features f #g with {a~fLs,fl~gL2} _C C/. It is easy to see that every solved clause is consistent. Note that every solved clause is also prime.</Paragraph>
      <Paragraph position="4"> Lemma A.9 The rules (Reduce) = (Redt) + (Reds) and (Solv) are X-sound and globally Xpreserving. Furthermore, 7~ sdeglv is terminating. Lemma A.10 Let C/ be a pre-soived clause. If A is e-closed, closed under intersection and decomposition, then a (C/, TiSdeglv)-derivative different from 1 is irreducible if and only if it is solved.</Paragraph>
      <Paragraph position="5"> Finally we can combine both phases of the algorithm. Theorem A.11 Consistency of prime clauses is decidable. null</Paragraph>
    </Section>
  </Section>
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