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<?xml version="1.0" standalone="yes"?> <Paper uid="P03-1047"> <Title>Bridging the Gap Between Underspecification Formalisms: Minimal Recursion Semantics as Dominance Constraints</Title> <Section position="3" start_page="0" end_page="0" type="metho"> <SectionTitle> 2 Minimal Recursion Semantics </SectionTitle> <Paragraph position="0"> We define a simplified version of Minimal Recursion Semantics and discuss differences to the original definitions presented in (Copestake et al., 1999).</Paragraph> <Paragraph position="1"> MRS is a description language for formulas of first order object languages with generalized quantifiers. Underspecified representations in MRS consist of elementary predications and handle constraints.</Paragraph> <Paragraph position="2"> Roughly, elementary predications are object language formulas with &quot;holes&quot; into which other formulas can be plugged; handle constraints restrict the way these formulas can be plugged into each other.</Paragraph> <Paragraph position="3"> More formally, MRSs are formulas over the follow- null ing vocabulary: 1. Variables. An infinite set of variables ranged over by h. Variables are also called handles.</Paragraph> <Paragraph position="4"> 2. Constants. An infinite set of constants ranged over by x,y,z. Constants are the individual variables of the object language.</Paragraph> <Paragraph position="5"> 3. Function symbols.</Paragraph> <Paragraph position="6"> (a) A set of function symbols written as P.</Paragraph> <Paragraph position="7"> (b) A set of quantifier symbols ranged over by Q (such as every and some). Pairs Qx are further function symbols (the variable binders of x in the object language).</Paragraph> <Paragraph position="8"> 4. The symbol [?] for the outscopes relation.</Paragraph> <Paragraph position="9"> Formulas of MRS have three kinds of literals, the first two are called elementary predications (EPs) and the third handle constraints: 1. h:P(x1,...,xn,h1,...,hm) where n,m [?] 0 2. h:Qx(h1,h2) 3. h1 [?] h2 Label positions are to the left of colons ':' and argument positions to the right. Let M be a set of literals. The label set lab(M) contains those handles of M that occur in label but not in argument position. The argument handle set arg(M) contains the handles of M that occur in argument but not in label position. Definition 1 (MRS). An MRS is finite set M of MRS-literals such that: M1 Every handle occurs at most once in label and at most once in argument position in M.</Paragraph> <Paragraph position="10"> M2 Handle constraints h1 [?] h2 in M always relate argument handles h1 to labels h2 of M.</Paragraph> <Paragraph position="11"> M3 For every constant (individual variable) x in argument position in M there is a unique literal of the form h:Qx(h1,h2) in M.</Paragraph> <Paragraph position="12"> We call an MRS compact if it additionally satisfies: M4 Every handle of M occurs exactly once in an elementary predication of M.</Paragraph> <Paragraph position="13"> We say that a handle h immediately outscopes a handle hprime in an MRS M iff there is an EP E in M such that h occurs in label and hprime in argument position of E. The outscopes relation is the reflexive, transitive closure of the immediate outscopes relation.</Paragraph> <Paragraph position="15"> An example MRS for the scopally ambiguous sentence &quot;Every student reads a book&quot; is given in Fig. 1. We often represent MRSs by directed graphs whose nodes are the handles of the MRS. Elementary predications are represented by solid edges and handle constraints by dotted lines. Note that we make the relation between bound variables and their binders explicit by dotted lines (as from everyx to readx,y); redundant &quot;binding-edges&quot; that are subsumed by sequences of other edges are omitted however (from everyx to studentx for instance).</Paragraph> <Paragraph position="16"> A solution for an underspecified MRS is called a configuration, or scope-resolved MRS.</Paragraph> <Paragraph position="17"> Definition 2 (Configuration). An MRS M is a configuration if it satisfies the following conditions.</Paragraph> <Paragraph position="18"> C1 The graph of M is a tree of solid edges: handles don't properly outscope themselves or occur in different argument positions and all handles are pairwise connected by elementary predications.</Paragraph> <Paragraph position="19"> C2 If two EPs h:P(...,x,...) and h0 :Qx(h1,h2) belong to M, then h0 outscopes h in M (so that the binding edge from h0 to h is redundant).</Paragraph> <Paragraph position="20"> We call M a configuration for another MRS Mprime if there exists some substitution s : arg(Mprime)mapsto-lab(Mprime) which states how to identify argument handles of Mprime with labels of Mprime, so that:</Paragraph> <Paragraph position="22"> The value s(E) is obtained by substituting all argument handles in E, leaving all others unchanged.</Paragraph> <Paragraph position="23"> The MRS in Fig. 1 has precisely two configurations displayed in Fig. 2 which correspond to the two readings of the sentence. In this paper, we present an algorithm that enumerates the configurations of MRSs efficiently.</Paragraph> <Paragraph position="24"> parts from standard MRS in some respects. First, we assume that different EPs must be labeled with different handles, and that labels cannot be identified. In standard MRS, however, conjunctions are encoded by labeling different EPs with the same handle. These EP-conjunctions can be replaced in a preprocessing step introducing additional EPs that make conjunctions explicit.</Paragraph> <Paragraph position="25"> Second, our outscope constraints are slightly less restrictive than the original &quot;qeq-constraints.&quot; A handle h is qeq to a handle hprime in an MRS M, h =q hprime, if either h = hprime or a quantifier h:Qx(h1,h2) occurs in M and h2 is qeq to hprime in M. Thus, h =q hprime implies h [?]hprime, but not the other way round. We believe that the additional strength of qeq-constraints is not needed in practice for modeling scope. Recent work in semantic construction for HPSG (Copestake et al., 2001) supports our conjecture: the examples discussed there are compatible with our simplification. Third, we depart in some minor details: we use sets instead of multi-sets and omit top-handles which are useful only during semantics construction.</Paragraph> </Section> <Section position="4" start_page="0" end_page="0" type="metho"> <SectionTitle> 3 Dominance Constraints </SectionTitle> <Paragraph position="0"> Dominance constraints are a general framework for describing trees, and thus syntax trees of logical formulas. Dominance constraints are the core language underlying CLLS (Egg et al., 2001) which adds parallelism and binding constraints.</Paragraph> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 3.1 Syntax and Semantics </SectionTitle> <Paragraph position="0"> We assume a possibly infinite signature S of function symbols with fixed arities and an infinite set Var of variables ranged over by X,Y,Z. We write f,g for function symbols and ar(f) for the arity of f .</Paragraph> <Paragraph position="1"> A dominance constraint ph is a conjunction of dominance, inequality, and labeling literals of the following forms where ar(f) = n:</Paragraph> <Paragraph position="3"> Dominance constraints are interpreted over finite constructor trees, i.e. ground terms constructed from the function symbols in S. We identify ground terms with trees that are rooted, ranked, edge-ordered and labeled. A solution for a dominance constraint consists of a tree t and a variable assignment a that maps variables to nodes of t such that all constraints are satisfied: a labeling literal X : f(X1,...,Xn) is satisfied iff the node a(X) is labeled with f and has daughters a(X1),...,a(Xn) in this order; a dominance literal Xtriangleleft[?]Y is satisfied iff a(X) is an ancestor of a(Y) in t; and an inequality literal X negationslash=Y is satisfied iff a(X) and a(Y) are distinct nodes.</Paragraph> <Paragraph position="4"> Note that solutions may contain additional material. The tree f(a,b), for instance, satisfies the constraint Y :a[?]Z :b.</Paragraph> </Section> <Section position="2" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 3.2 Normality and Weak Normality </SectionTitle> <Paragraph position="0"> The satisfiability problem of arbitrary dominance constraints is NP-complete (Koller et al., 2001) in general. However, Althaus et al. (2003) identify a natural fragment of so called normal dominance constraints, which have a polynomial time satisfiability problem. Bodirsky et al. (2003) generalize this notion to weakly normal dominance constraints.</Paragraph> <Paragraph position="1"> We call a variable a hole of ph if it occurs in argu- null ment position in ph and a root of ph otherwise.</Paragraph> <Paragraph position="2"> Definition 3. A dominance constraint ph is normal (and compact) if it satisfies the following conditions. N1 (a) each variable of ph occurs at most once in the labeling literals of ph.</Paragraph> <Paragraph position="3"> (b) each variable of ph occurs at least once in the labeling literals of ph.</Paragraph> <Paragraph position="4"> N2 for distinct roots X and Y of ph, X negationslash=Y is in ph. N3 (a) if X triangleleft[?] Y occurs in ph, Y is a root in ph. (b) if X triangleleft[?] Y occurs in ph, X is a hole in ph. A dominance constraint is weakly normal if it satisfies all above properties except for N1(b) and N3(b). The idea behind (weak) normality is that the constraint graph (see below) of a dominance constraint consists of solid fragments which are connected by dominance constraints; these fragments may not properly overlap in solutions.</Paragraph> <Paragraph position="5"> Note that Definition 3 always imposes compactness, meaning that the heigth of solid fragments is at most one. As for MRS, this is not a serious restriction, since more general weakly normal dominance constraints can be compactified, provided that dominance links relate either roots or holes with roots. Dominance Graphs. We often represent dominance constraints as graphs. A dominance graph is the directed graph (V,triangleleft[?]unionmultitriangleleft). The graph of a weakly normal constraint ph is defined as follows: The nodes of the graph of ph are the variables of ph. A labeling literal X : f(X1,...,Xn) of ph contributes tree edges (X,Xi) [?] triangleleft for 1 [?] i [?] n that we draw as X Xi; we freely omit the label f and the edge order in the graph. A dominance literal Xtriangleleft[?]Y contributes a dominance edge (X,Y) [?] triangleleft[?] that we draw as X Y . Inequality literals in ph are also omitted in the graph. f a gFor example, the constraint graph on the right represents the dominance</Paragraph> <Paragraph position="7"> A dominance graph is weakly normal or a wnd-graph if it does not contain any forbidden subgraphs: Dominance graphs of a weakly normal dominance constraints are clearly weakly normal.</Paragraph> <Paragraph position="8"> Solved Forms and Configurations. The main difference between MRS and dominance constraints lies in their notion of interpretation: solutions versus configurations.</Paragraph> <Paragraph position="9"> Every satisfiable dominance constraint has infinitely many solutions. Algorithms for dominance constraints therefore do not enumerate solutions but solved forms. We say that a dominance constraint is in solved form iff its graph is in solved form. A wnd-graph Ph is in solved form iff Ph is a forest. The solved forms of Ph are solved forms Phprime that are more specific than Ph, i.e. Ph and Phprime differ only in their dominance edges and the reachability relation of Ph extends the reachability of Phprime. A minimal solved form of Ph is a solved form of Ph that is minimal with respect to specificity.</Paragraph> <Paragraph position="10"> The notion of configurations from MRS applies to dominance constraints as well. Here, a configuration is a dominance constraint whose graph is a tree without dominance edges. A configuration of a constraint ph is a configuration that solves ph in the obvious sense. Simple solved forms are tree-shaped solved forms where every hole has exactly one out-going dominance edge.</Paragraph> <Paragraph position="12"> one configuration, and for every configuration there is exactly one solved form that it configures.</Paragraph> <Paragraph position="13"> Unfortunately, Lemma 1 does not extend to minimal as opposed to simple solved forms: there are minimal solved forms without configurations. The constraint on the right of Fig. 3, for instance, has no configuration: the hole of L1 would have to be filled twice while the right hole of L2 cannot be filled.</Paragraph> </Section> </Section> <Section position="5" start_page="0" end_page="0" type="metho"> <SectionTitle> 4 Representing MRSs </SectionTitle> <Paragraph position="0"> We next map (compact) MRSs to weakly normal dominance constraints so that configurations are preserved. Note that this translation is based on a non-standard semantics for dominance constraints, namely configurations. We address this problem in the following sections.</Paragraph> <Paragraph position="1"> The translation of an MRS M to a dominance constraint phM is quite trivial. The variables of phM are the handles of M and its literal set is:</Paragraph> <Paragraph position="3"> [?]{hnegationslash=hprime |h,hprime in distinct label positions of M} Compact MRSs M are clearly translated into (compact) weakly normal dominance constraints. Labels of M become roots in phM while argument handles become holes. Weak root-to-root dominance literals are needed to encode variable binding condition C2 of MRS. It could be formulated equivalently through lambda binding constraints of CLLS (but this is not necessary here in the absence of parallelism).</Paragraph> <Paragraph position="4"> Proposition 1. The translation of a compact MRS M into a weakly normal dominance constraint phM preserves configurations.</Paragraph> <Paragraph position="5"> This weak correctness property follows straight-forwardly from the analogy in the definitions.</Paragraph> </Section> <Section position="6" start_page="0" end_page="0" type="metho"> <SectionTitle> 5 Constraint Solving </SectionTitle> <Paragraph position="0"> We recall an algorithm from (Bodirsky et al., 2003) that efficiently enumerates all minimal solved forms of wnd-graphs or constraints. All results of this section are proved there.</Paragraph> <Paragraph position="1"> The algorithm can be used to enumerate configurations for a large subclass of MRSs, as we will see in Section 6. But equally importantly, this algorithm provides a powerful proof method for reasoning about solved forms and configurations on which all our results rely.</Paragraph> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 5.1 Weak Connectedness </SectionTitle> <Paragraph position="0"> Two nodes X and Y of a wnd-graph Ph = (V,E) are weakly connected if there is an undirected path from X to Y in (V,E). We call Ph weakly connected if all its nodes are weakly connected. A weakly connected component (wcc) of Ph is a maximal weakly connected subgraph of Ph. The wccs of Ph = (V,E) form proper partitions of V and E.</Paragraph> <Paragraph position="1"> Proposition 2. The graph of a solved form of a weakly connected wnd-graph is a tree.</Paragraph> </Section> <Section position="2" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 5.2 Freeness </SectionTitle> <Paragraph position="0"> The enumeration algorithm is based on the notion of freeness.</Paragraph> <Paragraph position="1"> Definition 4. A node X of a wnd-graph Ph is called free in Ph if there exists a solved form of Ph whose graph is a tree with root X.</Paragraph> <Paragraph position="2"> A weakly connected wnd-graph without free nodes is unsolvable. Otherwise, it has a solved form whose graph is a tree (Prop. 2) and the root of this tree is free in Ph.</Paragraph> <Paragraph position="3"> Given a set of nodes Vprime [?]V , we write Ph|Vprime for the restriction of Ph to nodes in Vprime and edges in VprimexVprime. The following lemma characterizes freeness: Lemma 2. A wnd-graph Ph with free node X satisfies the freeness conditions: F1 node X has indegree zero in graph Ph, and F2 no distinct children Y and Yprime of X in Ph that are linked to X by immediate dominance edges are weakly connected in the remainder Ph|V\{X}.</Paragraph> </Section> <Section position="3" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 5.3 Algorithm </SectionTitle> <Paragraph position="0"> The algorithm for enumerating the minimal solved forms of a wnd-graph (or equivalently constraint) is given in Fig. 4. We illustrate the algorithm for the problematic wnd-graph Ph in Fig. 3. The graph of Ph is weakly connected, so that we can call solve(Ph).</Paragraph> <Paragraph position="1"> This procedure guesses topmost fragments in solved forms of Ph (which always exist by Prop. 2).</Paragraph> <Paragraph position="2"> The only candidates are L1 or L2 since L3 and L4 have incoming dominance edges, which violates F1. Let us choose the fragment L2 to be topmost.</Paragraph> <Paragraph position="3"> The graph which remains when removing L2 is still weakly connected. It has a single minimal solved form computed by a recursive call of the solver, where L1 dominates L3 and L4. The solved form of the restricted graph is then put below the left hole of L2, since it is connected to this hole. As a result, we obtain the solved form on the right of Fig. 3.</Paragraph> </Section> </Section> <Section position="7" start_page="0" end_page="0" type="metho"> <SectionTitle> 6 Full Translation </SectionTitle> <Paragraph position="0"> Next, we explain how to encode a large class of MRSs into wnd-constraints such that configurations correspond precisely to minimal solved forms. The result of the translation will indeed be normal.</Paragraph> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 6.1 Problems and Examples </SectionTitle> <Paragraph position="0"> The naive representation of MRSs as weakly normal dominance constraints is only correct in a weak sense. The encoding fails in that some MRSs which have no configurations are mapped to solvable wndconstraints. For instance, this holds for the MRS on the right in Fig 3.</Paragraph> <Paragraph position="1"> We cannot even hope to translate arbitrary MRSs correctly into wnd-constraints: the configurability problem of MRSs is NP-complete, while satisfiability of wnd-constraints can be solved in polynomial time. Instead, we introduce the sublanguages of MRS-nets and equivalent wnd-nets, and show that they can be intertranslated in quadratic time.</Paragraph> <Paragraph position="2"> solved-form(Ph) [?] Let Ph1,...,Phk be the wccs of Ph = (V,E) Let (Vi,Ei) be the result of solve(Phi) return (V,[?]ki=1Ei) solve(Ph) [?] precond: Ph = (V,triangleleftunionmultitriangleleft[?]) is weakly connected choose a node X satisfying (F1) and (F2) in Ph else fail Let Y1,...,Yn be all nodes s.t. X triangleleftYi Let Ph1,...,Phk be the weakly connected components of Ph|V[?]{X,Y1,...,Yn} Let (Wj,E j) be the result of solve(Phj), and Xj [?]Wj its root</Paragraph> <Paragraph position="4"/> </Section> <Section position="2" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 6.2 Dominance and MRS-Nets </SectionTitle> <Paragraph position="0"> A hypernormal path (Althaus et al., 2003) in a wnd-graph is a sequence of adjacent edges that does not traverse two outgoing dominance edges of some hole X in sequence, i.e. a wnd-graph without situations Y1 X Y2.</Paragraph> <Paragraph position="1"> A dominance net Ph is a weakly normal dominance constraint whose fragments all satisfy one of the three schemas in Fig. 5. MRS-nets can be defined analogously. This means that all roots of Ph are labeled in Ph, and that all fragments X : f(X1,...,Xn) of Ph satisfy one of the following three conditions: strong. n [?] 0 and for all Y [?]{X1,...,Xn} there exists a unique Z such that Y triangleleft[?] Z in Ph, and there exists no Z such that X triangleleft[?] Z in Ph.</Paragraph> <Paragraph position="2"> weak. n [?] 1 and for all Y [?]{X1,...,Xn[?]1,X} there exists a unique Z such that Y triangleleft[?] Z in Ph, and there exists no Z such that Xn triangleleft[?] Z in Ph.</Paragraph> <Paragraph position="3"> island. n = 1 and all variables in {Y |X1 triangleleft[?] Y} are connected by a hypernormal path in the graph of the restricted constraint Ph|V[?]{X1}, and there exists no Z such that X triangleleft[?] Z in Ph.</Paragraph> <Paragraph position="4"> The requirement of hypernormal connections in islands replaces the notion of chain-connectedness in (Koller et al., 2003), which fails to apply to dominance constraints with weak dominance edges.</Paragraph> <Paragraph position="5"> For ease of presentation, we restrict ourselves to a simple version of island fragments. In general, we should allow for island fragments with n > 1.</Paragraph> </Section> <Section position="3" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 6.3 Normalizing Dominance Nets </SectionTitle> <Paragraph position="0"> Dominance nets are wnd-constraints. We next translate dominance nets Ph to normal dominance constraints Phprime so that Ph has a configuration iff Phprime is satisfiable. The trick is to normalize weak dominance edges. The normalization norm(Ph) of a weakly normal dominance constraint Ph is obtained by converting all root-to-root dominance literals X triangleleft[?] Y as follows: null X triangleleft[?] Y = Xn triangleleft[?] Y if X roots a fragment of Ph that satisfies schema weak of net fragments. If Ph is a dominance net then norm(Ph) is indeed a normal dominance net.</Paragraph> <Paragraph position="1"> Theorem 2. The configurations of a weakly connected dominance net Ph correspond bijectively to the minimal solved forms of its normalization norm(Ph).</Paragraph> <Paragraph position="2"> For illustration, consider the problematic wndconstraint Ph on the left of Fig. 3. Ph has two minimal solved forms with top-most fragments L1 and L2 respectively. The former can be configured, in contrast to the later which is drawn on the right of Fig. 3.</Paragraph> <Paragraph position="3"> Normalizing Ph has an interesting consequence: norm(Ph) has (in contrast to Ph) a single minimal solved form with L1 on top. Indeed, norm(Ph) cannot be satisfied while placing L2 topmost. Our algorithm detects this correctly: the normalization of fragment L2 is not free in norm(Ph) since it violates property F2.</Paragraph> <Paragraph position="4"> The proof of Theorem 2 captures the rest of this section. We show in a first step (Prop. 3) that the configurations are preserved when normalizing weakly connected and satisfiable nets. In the second step, we show that minimal solved forms of normalized nets, and thus of norm(Ph), can always be configured (Prop. 4).</Paragraph> <Paragraph position="5"> Corollary 1. Configurability of weakly connected MRS-nets can be decided in polynomial time; configurations of weakly connected MRS-nets can be enumerated in quadratic time per configuration.</Paragraph> </Section> <Section position="4" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 6.4 Correctness Proof </SectionTitle> <Paragraph position="0"> Most importantly, nets can be recursively decomposed into nets as long as they have configurations: Lemma 3. If a dominance net Ph has a configuration whose top-most fragment is X : f(X1,...,Xn), then the restriction Ph|V[?]{X,X1,...,Xn} is a dominance net.</Paragraph> <Paragraph position="1"> Note that the restriction of the problematic net Ph by L2 on the left in Fig. 3 is not a net. This does not contradict the lemma, as Ph does not have a configuration with top-most fragment L2.</Paragraph> <Paragraph position="2"> Proof. First note that as X is free in Ph it cannot have incoming edges (condition F1). This means that the restriction deletes only dominance edges that depart from nodes in {X,X1,...,Xn}. Other fragments thus only lose ingoing dominance edges by normality condition N3. Such deletions preserve the validity of the schemas weak and strong.</Paragraph> <Paragraph position="3"> The island schema is more problematic. We have to show that the hypernormal connections in this schema can never be cut. So suppose that Y : f(Y1) is an island fragment with outgoing dominance edges Y1 triangleleft[?] Z1 and Y1 triangleleft[?] Z2, so that Z1 and Z2 are connected by some hypernormal path traversing the deleted fragment X : f(X1,...,Xn). We distinguish the three possible schemata for this fragment: strong: since X does not have incoming dominance edges, there is only a single non-trival kind of traversal, drawn in Fig. 6(a). But such traversals contradict the freeness of X according to F2.</Paragraph> <Paragraph position="4"> weak: there is one other way of traversing weak fragments, shown in Fig. 6(b). Let X triangleleft[?] Y be the weak dominance edge. The traversal proves that Y belongs to the weakly connected components of one of the Xi, so the Ph[?]Xn triangleleft[?] Y is unsatisfiable. This shows that the hole Xn cannot be identified with any root, i.e. Ph does not have any configuration in contrast to our assumption.</Paragraph> <Paragraph position="5"> island: free island fragments permit one single non-trivial form of traversals, depicted in Fig. 6(c). But such traversals are not hypernormal.</Paragraph> <Paragraph position="6"> Proposition 3. A configuration of a weakly connected dominance net Ph configures its normalization norm(Ph), and vice versa of course.</Paragraph> <Paragraph position="7"> Proof. Let C be a configuration of Ph. We show that it also configures norm(Ph). Let S be the simple solved form of Ph that is configured by C (Lemma 1), and Sprime be a minimal solved form of Ph which is more general than S.</Paragraph> <Paragraph position="8"> Let X : f(Y1,...,Yn) be the top-most fragment of the tree S. This fragment must also be the top-most fragment of Sprime, which is a tree since Ph is assumed to be weakly connected (Prop. 2). Sprime is constructed by our algorithm (Theorem 1), so that the evaluation of solve(Ph) must choose X as free root in Ph.</Paragraph> <Paragraph position="9"> Since Ph is a net, some literal X : f(Y1,...,Yn) must belong to Ph. Let Phprime = Ph|{X,Y1,...,Yn} be the restriction of Ph to the lower fragments. The weakly connected components of all Y1, ..., Yn[?]1 must be pairwise disjoint by F2 (which holds by Lemma 2 since X is free in Ph). The X-fragment of net Ph must satisfy one of three possible schemata of net fragments: weak fragments: there exists a unique weak dominance edge X triangleleft[?] Z in Ph and a unique hole Yn without outgoing dominance edges. The variable Z must be a root in Ph and thus be labeled. If Z is equal to X then Ph is unsatisfiable by normality condition N2, which is impossible. Hence, Z occurs in the restriction Phprime but not in the weakly connected components of any Y1, ..., Yn[?]1. Otherwise, the minimal solved form Sprime could not be configured since the hole Yn could not be identified with any root. Furthermore, the root of the Z-component must be identified with Yn in any configuration of Ph with root X. Hence, C satisfies Yn triangleleft[?] Z which is add by normalization.</Paragraph> <Paragraph position="10"> The restriction Phprime must be a dominance net by Lemma 3, and hence, all its weakly connected components are nets. For all 1 [?] i [?] n[?]1, the component of Yi in Phprime is configured by the subtree of C at node Yi, while the subtree of C at node Yn configures the component of Z in Phprime. The induction hypothesis yields that the normalizations of all these components are configured by the respective subconfigurations of C. Hence, norm(Ph) is configured by C. strong or island fragments are not altered by normalization, so we can recurse to the lower fragments (if there exist any).</Paragraph> <Paragraph position="11"> Proposition 4. Minimal solved forms of normal, weakly connected dominance nets have configurations. null Proof. By induction over the construction of minimal solved forms, we can show that all holes of minimal solved forms have a unique outgoing dominance edge at each hole. Furthermore, all minimal solved forms are trees since we assumed connectedness (Prop.2). Thus, all minimal solved forms are simple, so they have configurations (Lemma 1).</Paragraph> </Section> </Section> class="xml-element"></Paper>