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<Paper uid="P92-1010">
  <Title>Reasoning with Descriptions of Trees *</Title>
  <Section position="4" start_page="74" end_page="75" type="metho">
    <SectionTitle>
3 Language
</SectionTitle>
    <Paragraph position="0"> Our language is built up from the symbols: K -- non-empty countable set of names, 1 r -- a distinguished element of K, the root &lt;1, ~+, ,~*, --&lt; -- two place predicates, parent, proper domination, domination, and left-of respectively, -- equality predicate, A, V, -~ -- usual logical connectives (,), \[, \] -- usual grouping symbols  Our atomic formulae are t ,~ u, t C/+ u, t &lt;* u, t -&lt; u, and t ~ u, where t, u * K are terms. Literals are atomic formulae or their negations. Well-formedformulae are generated from atoms and the logical connectives in the usual fashion.</Paragraph>
    <Paragraph position="1"> We use t, u, v to denote terms and C/, C/ to denote wffs. R denotes any of the five predicates.</Paragraph>
    <Section position="1" start_page="74" end_page="75" type="sub_section">
      <SectionTitle>
3.1 Models
</SectionTitle>
      <Paragraph position="0"> Quasi-trees as formal structures are in a sense a reduced form of the quasi-trees viewed as sets of relationships. They incorporate a canonical sub-set of those relationships from which the remaining relationships can be deduced.</Paragraph>
      <Paragraph position="1"> Definition 1 A model is a tuple (H,I, 7),79,.A,PS), where: H is a non-empty universe, iT. is a partial function from K to Lt (specifying the node referred to by each name), 7 9, .4, 79, and PS are binary relations over It (assigned to % ,a +, ,a*, and -4 respectively). Let T( denote 27(r).</Paragraph>
      <Paragraph position="3"> to description theory.</Paragraph>
      <Paragraph position="5"> And meeting the additional condition: for every x,z * U the set B=z = {Y I (x,Y),(Y,Z) * 79} is finite, ie: the length of path from any node to any other is finite. 2 A quasi-tree is consistent iff</Paragraph>
      <Paragraph position="7"> At least one normal, consistent quasi-tree (that consisting of only a root node) satisfies all of these conditions simultaneously. Thus they are consistent. It is not hard to exhibit a model for each condition in which that condition fails while all of the others hold. Thus the conditions are independent of each other.</Paragraph>
      <Paragraph position="8"> Trees are distinguished from (ordinary) quasi-trees by the fact that 79 is the reflexive, transitive closure of P, and the fact that the relations 79, 79, ,4, PS are maximal in the sense that they cannot be consistently extended.</Paragraph>
      <Paragraph position="9"> Definition 3 A consistent, normal quasi-tree M is a tree iff Tel 79M = (7~M)*, TC2 for all pairs (x, y) * U M X l~ M, exactly one of the following is true:</Paragraph>
      <Paragraph position="11"> Note that TC1 implies that .A M -- (79M)+ as well.</Paragraph>
      <Paragraph position="12"> It is easy to verify that a quasi-tree meets these conditions iff (H M, 79M) is the graph of a tree as commonly defined (Aho, Hopcroft &amp; Ullman, 1974).</Paragraph>
    </Section>
    <Section position="2" start_page="75" end_page="75" type="sub_section">
      <SectionTitle>
3.2 Satisfaction
</SectionTitle>
      <Paragraph position="0"> The semantics of the language in terms of the models is defined by the satisfaction relation between models and formulae.</Paragraph>
      <Paragraph position="2"> In addition we require that ZM(k) be defined for all k occurring in the formula.</Paragraph>
      <Paragraph position="3"> It is easy to verify that for all quasi-trees M (3t, u, R)\[M ~ t R u,-~t R u\] ==~ M inconsistent. If 2: M is surjective then the converse holds as well. It is also not hard to see that if T is a tree</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="75" end_page="76" type="metho">
    <SectionTitle>
4 Characterization
</SectionTitle>
    <Paragraph position="0"> We now show that this formalization is complete in the sense that a consistent quasi-tree as defined characterizes the set of trees it describes. Recall that the quasi-tree describes the set of all trees which satisfy every literal formula which is satisfied by the quasi-tree. It characterizes that set if every literal formula which is satisfied by every tree in the set is also satisfied by the quasi-tree. The property of satisfying every formula which is satisfied by the quasi-tree is captured formally by the notion of subsumption, which we define initially as a relationship between quasi-trees.</Paragraph>
    <Paragraph position="1"> Definition 5 Subsumption. Suppose M</Paragraph>
    <Paragraph position="3"> tent quasi-trees, then M subsumes M z (M ~ M I) iff there is a function h : lA M ~ 14 M' such that:</Paragraph>
    <Paragraph position="5"> We now claim that any quasi-tree Q is subsumed by a quasi-tree M iff it is described by M.</Paragraph>
    <Paragraph position="6"> Lemma 1 If M and Q are normal, consistent quasi-trees and 3 M is surjective, then M E Q iff for all formulae C/, M ~ C/ ~ Q ~ C/.</Paragraph>
    <Paragraph position="7"> The proof in the forward direction is an easy induction on the structure of C/ and does not depend either on normality or surjectiveness of I M. The opposite direction follows from the fact that, since Z M is surjective, there is a model M' in which/~M' is the set of equivalence classes wrt ~ in the domain of Z M, such that M E M~ E Q-The next lemma allows us, in many cases, to assume that a given quasi-tree is normal.</Paragraph>
    <Paragraph position="8"> Lemma 2 For every consistent quasi-tree M, there is a normal, consistent quasi-tree M ~ such that M E M~, and for all normal, consistent quasi-tree M', M E M&amp;quot; ::C/. M ~ E M'.</Paragraph>
    <Paragraph position="9"> The lemma is witnessed by the quotient of M with respect to S M, where sM = { (x, y) I (x, y), (y, x) e vM}.</Paragraph>
    <Paragraph position="10"> We can now state the central claim of this section, that every consistent quasi-tree characterizes the set of trees which it subsumes.</Paragraph>
    <Paragraph position="11"> Proposition 1 Suppose M is a consistent quasitree. For all literals C/ M ~ C/ C/~ (VT, tree)\[M E T ::~ T ~ C/\] The proof follows from two lemmas. The first establishes that the set of quasi-trees subsumed by some quasi-tree M is in fact characterized by it. The second extends the result to trees. Their proofs are in (Rogers &amp; Vijay-Shanker, 1992).</Paragraph>
    <Paragraph position="12"> Lemma 3 If M is a consistent quasi-tree and C/ a literal then (3Q, consistent quasi-tree)\[M E_ Q and Q ~ -~C/\] Lemma 4 If M is a consistent quasi-tree, then there exists a tree T such that M E T.</Paragraph>
    <Paragraph position="14"> C/V -~(3Q, consistent q-t)\[M E Q and Q ~ -~C/\] (==~ by lemma 4, C/= since T is a quasi-tree) (::~ by lemma 3, C/=: by lemma 1) O</Paragraph>
  </Section>
  <Section position="6" start_page="76" end_page="77" type="metho">
    <SectionTitle>
5 Semantic Tableau
</SectionTitle>
    <Paragraph position="0"> Semantic tableau as introduced by Beth (Beth, 1959; Fitting, 1990) are used to prove validity by means of refutation. We are interested in satisfiability rather than validity. Given E we wish to build a model of E if one exists. Thus we are interested in the cases where the tableau succeeds in constructing a model.</Paragraph>
    <Paragraph position="1"> The distinction between these uses of semantic tableau is important, since our mechanism is not suitable for refutational proofs. In particular, it cannot express &amp;quot;some model fails to satisfy C/&amp;quot; except as &amp;quot;some model satisfies -C/&amp;quot;. Since our logic is non-classical the first is a strictly weaker condition than the second.</Paragraph>
    <Paragraph position="2"> Definition 6 Semantic Tableau: A branch is a set, S, of formulae.</Paragraph>
    <Paragraph position="3"> A configuration is a collection, {S1,...,S~}, of branches.</Paragraph>
    <Paragraph position="4"> A tableau is a sequence, (C1,..., Cnl, of configurations where each Ci+~ is a result of the application of an inference rule to Ci.</Paragraph>
    <Paragraph position="5"> If s is an inference rule, (Ci\{S}) U {sl,..., s',} is the result of applying the rule to G iff z eG.</Paragraph>
    <Paragraph position="6"> A tableau for ~, where E is a set of formulae, is a tableau in which C1 = {E}.</Paragraph>
    <Paragraph position="7"> A branch is closed iff (9C/)\[{C/,--,C/} C 5'\]. A configuration is closed iff each of its branches is closed, and a tableau is closed iff it contains some closed configuration. A branch~ configuration, or tableau that is not closed is open.</Paragraph>
    <Section position="1" start_page="76" end_page="77" type="sub_section">
      <SectionTitle>
5.1 Inference Rules
</SectionTitle>
      <Paragraph position="0"> Our inference rules fall into three groups. The first two, figures 3 and 4, are standard rules for propositional semantic tableau extended with equality (Fitting, 1990). The third group, figure 5, embody the properties of quasi-trees.</Paragraph>
      <Paragraph position="1"> The --,,~ rule requires the introduction of a new name into the tableau. To simplify this, tableau are carried out in a language augmented with a countably infinite set of new names from which these are drawn in a systematic way.</Paragraph>
      <Paragraph position="2"> The following two lemmas establish the correctness of the inference rules in the sense that no rule increases the set of models of any branch nor eliminates all of the models of a satisfiable branch.</Paragraph>
      <Paragraph position="3"> Lemma 5 Suppose S' is derived from S in some tableau by some sequence of rule applications. Suppose M is a model, then: M~S'::~M~S.</Paragraph>
      <Paragraph position="4"> This follows nearly directly from the fact that all of our rules are non-strict, ie: the branch to which an inference rule is applied is a subset of every branch introduced by its application.</Paragraph>
      <Paragraph position="5"> Lemma 6 If S is a branch of some configuration of a tableau and ,S' is the set of branches resulting from applying some rule to S, then if there is a  consistent quasi-tree M such that M ~ S, then for some 5;~ E S' there is a consistent quasi-tree M' such that M' ~ S~.</Paragraph>
      <Paragraph position="6"> We sketch the proof. Suppose M ~ S. For all but --,,a it is straightforward to verify M also satisfies at least one of the S~. For ~,~, suppose M fails to satisfy either u ,~* t or -,t ,~* u. Then we claim some quasi-tree satisfies the third branch of the conclusion. This must map the new constant k to the witness for the rule. M has no such requirement, but since k does not occur in S, the value of 2: M(k) does not affect satisfaction of S. Thus we get an appropriate M' by modifying z M' to map k correctly.</Paragraph>
      <Paragraph position="7"> Corollary 1 If there is a closed tableau for C/ then no consistent quasi-tree satisfies C/.</Paragraph>
      <Paragraph position="8"> No consistent quasi-tree satisfies a closed set of formulae. The result then follows by induction on the length of the tableau.</Paragraph>
    </Section>
  </Section>
  <Section position="7" start_page="77" end_page="79" type="metho">
    <SectionTitle>
6 Constructing Models
</SectionTitle>
    <Paragraph position="0"> We now turn to the conditions for a branch to be sufficiently complete to fully specify a quasi-tree.</Paragraph>
    <Paragraph position="1"> In essence these just require that all formulae have been expanded to atoms, that all substitutions have been made and that the conditions in the definition of quasi-trees are met.</Paragraph>
    <Section position="1" start_page="77" end_page="78" type="sub_section">
      <SectionTitle>
6.1 Saturated Branches
</SectionTitle>
      <Paragraph position="0"> u -4 t E S or t ,~* u E S or u ,~* t E S.</Paragraph>
      <Paragraph position="1"> The next lemma (essentially Hintikka's lemma) establishes the correspondence between saturated branches and quasi-trees.</Paragraph>
      <Paragraph position="2"> Lemma 7 For every consistent downward saturated set of formulae S there is a consistent quasi-tree M such that M ~ S. For every finite consistent downward saturated set of formulae, there is a such a quasi-tree which is finite.</Paragraph>
      <Paragraph position="3"> Again, we sketch the proof. Consider the set T(S) of terms occurring in a downward saturated set S.</Paragraph>
      <Paragraph position="4"> I-I6 and I-/7 assure that ~ is reflexive and substitutive. Sincet ~u,u~v E S=~t ~v E S, and u~u,u,~vE S~v~ u E Sby substitution of v for (the first occurrence of) u, it is transitive and symmetric as well. Thus ~ partitions T(S) into equivalence classes.</Paragraph>
      <Paragraph position="5"> Define the model H as follows:</Paragraph>
      <Paragraph position="7"> Since each of the conditions C1 through Cx2 corresponds directly to one of the saturation conditions, it is easy to verify that H satisfies Cq. It is equally easy to confirm that H is both consistent and normal. null  We claim that C/ E S =C/- H ~ C/. As is usual for versions of Hintikka's lemma, this is established by an induction on the structure of C/. Space prevents us from giving the details here.</Paragraph>
      <Paragraph position="8"> For the second part of the lemma, if the set of formulae is finite, then the set of terms (and hence the set of equivalence classes) is finite.</Paragraph>
    </Section>
    <Section position="2" start_page="78" end_page="79" type="sub_section">
      <SectionTitle>
6.2 Saturated Tableau
</SectionTitle>
      <Paragraph position="0"> Since all of our inference rules are non-strict, if a rule once applies to a branch it will always apply to a branch. Without some restriction on the application of rules, tableau for satisfiable sets of formulae will never terminate. What is required is a control strategy that guarantees that no rule applies to any tableau more than finitely often, but that will always find a rule to apply to any open branch that is not downward saturated.</Paragraph>
      <Paragraph position="1"> Definition 8 Let EQs be the reflexive, symmetric, transitive closure of { (t, u) l t ~ u e S}.</Paragraph>
      <Paragraph position="2"> An inference rule, I, applies to some branch S of a configuration C iff</Paragraph>
      <Paragraph position="4"> formulae Cj(a) E Si, there is no term t and pairs (ul, va), (u2, v2), . . . E EQs such that for each of the Cj, C/{t/a, ul/Vl,~2/v2,...} E S.</Paragraph>
      <Paragraph position="5"> (Where C/{t/a, Ul/Vl, U2/V2,...} denotes the result of uniformly substituting t for a, ul for vl, etc., in C/.) The last condition in effect requires all equality rules to be applied before any new constant is introduced. It prevents the introduction of a formula involving a new constant if an equivalent formula already exists or if it is possible to derive one using only the equality rules.</Paragraph>
      <Paragraph position="6"> We now argue that this definition of applies does not terminate any branch too soon.</Paragraph>
      <Paragraph position="7"> Lemma 8 If no inference rule applies to an open branch S of a configuration, then S is downward saturated.</Paragraph>
      <Paragraph position="8"> This follows directly from the fact that for each of H1 through H26, if the implication is false there is a corresponding inference rule which applies.  Proposition 2 (Termination) All tableau for finite sets of formulae can be extended to tableau in which no rule applies to the final configuration. This follows from the fact that the size of any tableau for finite sets of formulae has a finite upper bound. The proof is in (Rogers &amp; Vijay-Shanker, 1992).</Paragraph>
      <Paragraph position="9"> Proposition 3 (Soundness and Completeness) A saturated tableau for a finite set of formulae exists iff there is a consistent quasi-tree which satisfies E.</Paragraph>
      <Paragraph position="10"> Proof: The forward implication (soundness) follows from lemma 7. Completeness follows from the fact that if E is satisfiable there is no closed tableau for E (corollary 1), and thus, by proposition 2 and lemma 8, there must be a saturated tableau for E. \[\]</Paragraph>
    </Section>
  </Section>
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