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<Paper uid="P04-1032">
  <Title>Minimal Recursion Semantics as Dominance Constraints: Translation, Evaluation, and Analysis</Title>
  <Section position="3" start_page="2" end_page="4" type="metho">
    <SectionTitle>
2 Minimal Recursion Semantics
</SectionTitle>
    <Paragraph position="0"> This section presents a definition of Minimal Recursion Semantics (MRS) (Copestake et al., 2004) including EP-conjunctions with a merging semantics. Full MRS with qeq-semantics, top handles, and event variables will be discussed in the last paragraph. null MRS Syntax. MRS constraints are conjunctive formulas over the following vocabulary:  1. An infinite set of variables ranged over by h. Variables are also called handles.</Paragraph>
    <Paragraph position="1"> 2. An infinite set of constants x,y,z denoting indivual variables of the object language.</Paragraph>
    <Paragraph position="2"> 3. A set of function symbols ranged over by P, and a set of quantifier symbols ranged over by Q. Pairs Q x are further function symbols.</Paragraph>
    <Paragraph position="3"> 4. The binary predicate symbol '= q '.</Paragraph>
    <Paragraph position="4"> MRS constraints have three kinds of literals, two kinds of elementary predications (EPs) in the first two lines and handle constraints in the third line: 1. h : P(x  In EPs, label positions are on the left of ':' and argument positions on the right. Let M be a set of literals. The label set lab(M) contains all handles of M that occur in label but not in argument position, and the argument handle set arg(M) contains all handles of M that occur in argument but not in label position. Definition 1 (MRS constraints). An MRS constraint (MRS for short) is a finite set M of MRSliterals such that: M1 every handle occurs at most once in argument position in M,  M2 handle constraints h = q h prime always relate argument handles h to labels h prime , and M3 for every constant (individual variable) x in argument position in M there is a unique literal of the form h : Q</Paragraph>
    <Paragraph position="6"> ) in M.</Paragraph>
    <Paragraph position="7"> We say that an MRS M is compact if every handle h in M is either a label or an argument handle. Compactness simplifies the following proofs, but it is no serious restriction in practice.</Paragraph>
    <Paragraph position="8"> We usually represent MRSs as directed graphs: the nodes of the graph are the handles of the MRS, EPs are represented as solid lines, and handle constraints are represented as dotted lines. For instance, the following MRS is represented by the graph on the left of Fig. 1.</Paragraph>
    <Paragraph position="9">  Note that the relation between bound variables and their binders is made explicit by binding edges drawn as dotted lines (cf. C2 below); transitively redundand binding edges (e. g., from some</Paragraph>
    <Paragraph position="11"> however are omited.</Paragraph>
    <Paragraph position="12"> MRS Semantics. Readings of underspecified representations correspond to configurations of MRS constraints. Intuitively, a configuration is an MRS where all handle constraints have been resolved by plugging the &amp;quot;tree fragments&amp;quot; into each other. Let M be an MRS and h,h prime be handles in M.We say that h immediately outscopes h prime in M if there is an EP in M with label h and argument handle h prime , and we say that h outscopes h prime in M if the pair (h,h prime ) belongs to the reflexive transitive closure of the immediate outscope relation of M. Definition 2 (MRS configurations). An MRS M is a configuration if it satisfies conditions C1 and C2: C1 The graph of M is a tree of solid edges: (i) all handles are labels i. e., arg(M)=/0 and M contains no handle constraints, (ii) handles don't properly outscope themselve, and (iii) all handles are pairwise connected by EPs in M.</Paragraph>
    <Paragraph position="14"> : P(...,x,...) belong to M, then h outscopes h prime in Mi.e., binding edges in the graph of M are transitively redundant. We say that a configuration M is configuration of an MRS M prime if there exists a partial substitution s :</Paragraph>
    <Paragraph position="16"> ) that states how to identify labels with argument handles of M</Paragraph>
    <Paragraph position="18"> ) in M.</Paragraph>
    <Paragraph position="19"> The value s(E) is obtained by substituting all labels in dom(s) in E while leaving all other handels unchanged.</Paragraph>
    <Paragraph position="20"> The MRS on the left of Fig. 1, for instance, has two configurations given to the right. EP-conjunctions. Definitions 1 and 2 generalize the idealized definition of MRS of Niehren and Thater (2003) by EP-conjunctions with a merging semantics. An MRS M contains an EP-conjunction if it contains different EPs with the same label h.The intuition is that EP-conjunctions are interpreted by object language conjunctions.</Paragraph>
    <Paragraph position="21">  joined and their arguments are merged into a set. The MRS does not have configurations since the argument handles of the merged EPs cannot jointly outscope the node P  .</Paragraph>
    <Paragraph position="22"> We call a configuration merging if it contains EPconjunctions, and merging-free otherwise. Merging configurations are needed to solve EP-conjuctions such as {h : P</Paragraph>
    <Paragraph position="24"> }. Unfortunately, they can also solve MRSs without EP-conjunctions, such as the MRS in Fig. 3. The unique configuration of this  must be identified with the only available argument handle. The admission of merging configurations may thus have important consequences for the solution space of arbitrary MRSs.</Paragraph>
    <Paragraph position="25"> Standard MRS. Standard MRS requires three further extensions: (i) qeq-semantics, (ii) tophandles, and (iii) event variables. These extensions are less relevant for our comparision.</Paragraph>
    <Paragraph position="26"> The qeq-semantics restricts the interpretation of handle constraints beyond dominance. Let M be an  in M. Every qeq-configuration is a configuration as defined above, but not necessarily vice versa. The qeq-restriction is relevant in theory but will turn out unproblematic in practice (see SS6). Standard MRS requires the existence of top handles in all MRS constraints. This condition doesn't matter for MRSs with connected graphs (see (Bodirsky et al., 2004) for the proof idea). MRSs with unconnected graphs clearly do not play any role in practical underspecified semantics.</Paragraph>
    <Paragraph position="27"> Finally, MRSs permit events variables e,e prime as a second form of constants. They are treated equally to individual variables except that they cannot be bound by quantifiers.</Paragraph>
  </Section>
  <Section position="4" start_page="4" end_page="4" type="metho">
    <SectionTitle>
3 Dominance Constraints
</SectionTitle>
    <Paragraph position="0"> Dominance constraints are a general framework for describing trees. For scope underspecification, they are used to describe the syntax trees of object language formulas. Dominance constraints are the core language underlying CLLS (Egg et al., 2001) which adds parallelism and binding constraints.</Paragraph>
    <Paragraph position="1"> Syntax and semantics. We assume a possibly infinite signature S = {f,g,...} of function symbols with fixed arities (written ar(f)) and an infinite set of variables ranged over by X,Y,Z.</Paragraph>
    <Paragraph position="2"> A dominance constraint ph is a conjunction of dominance, inequality, and labeling literals of the following form, where ar(f)=n:</Paragraph>
    <Paragraph position="4"> 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, edgeordered and labeled. A solution for a dominance constraint ph consists of a tree t and an assignment a that maps the variables in ph to nodes of t such that all constraints are satisfied: labeling lit-</Paragraph>
    <Paragraph position="6"> ) are satisfied iff a(X) is labeled with f and its daughters are a(X</Paragraph>
    <Paragraph position="8"> nodes.</Paragraph>
    <Paragraph position="9"> Solved forms. Satisfiable dominance constraints have infinitely many solutions. Constraint solvers for dominance constraints therefore do not enumerate solutions but solved forms i. e., &amp;quot;tree shaped&amp;quot; constraints. To this end, we consider (weakly) normal dominance constraints (Bodirsky et al., 2004). 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="10"> Definition 3. A dominance constraint ph is normal if it satisfies the following conditions.</Paragraph>
    <Paragraph position="11"> N1 (a) each variable of ph occurs at most once in the labeling literals of ph.</Paragraph>
    <Paragraph position="12"> (b) each variable of ph occurs at least once in the labeling literals of ph.</Paragraph>
    <Paragraph position="13"> N2 for distinct roots X and Y of ph, X negationslash=Y is in ph.</Paragraph>
    <Paragraph position="15"> We call ph weakly normal if it satisfies the above properties except for N1 (b) and N3 (b).</Paragraph>
    <Paragraph position="16"> Note that Definition 3 imposes compactness: the height of tree fragments is always one. This is not  its two solved forms (right).</Paragraph>
    <Paragraph position="17"> a serious restriction, as weakly normal dominance constraints can be compactified, provided that dominance links relate either roots or holes with roots. Weakly normal dominance constraints ph can be represented by dominance graphs. The dominance graph of ph is a directed graph G =(V,E</Paragraph>
    <Paragraph position="19"> fined as follows. The nodes of G are the variables of ph. Labeling literals X : f(X  . Inequality literals are not represented in the graph. In pictures, labeling literals are drawn with solid lines and dominance edges with dotted lines.</Paragraph>
    <Paragraph position="20"> We say that a constraint ph is in solved form if its graph is in solved form. A graph G is in solved form iff it is a forest. The solved forms of G are solved forms G prime which are more specific than Gi.e., they differ only in their dominance edges and the reachability relation of G extends the reachability of G</Paragraph>
    <Paragraph position="22"> minimal solved form is a solved form which is minimal with respect to specificity. Simple solved forms are solved forms where every hole has exactly one outgoing dominance edge. Fig. 4 shows as a concrete example the translation of the MRS description in Fig. 1 together with its two minimal solved forms. Both solved forms are simple.</Paragraph>
  </Section>
  <Section position="5" start_page="4" end_page="4" type="metho">
    <SectionTitle>
4 Translating Merging-Free MRS-Nets
</SectionTitle>
    <Paragraph position="0"> This section defines MRS-nets without EPconjunctions, and sketches their translation to normal dominance constraints. We define nets equally for MRSs and dominance constraints. The key semantic property of nets is that different notions of solutions coincide. In this section, we show that merging-free configurations coincides to minimal solved forms. SS5 generalizes the translation by adding EP-conjunctions and permitting merging semantics.</Paragraph>
    <Paragraph position="1"> Pre-translation. An MRS constraint M can be represented as a corresponding dominance con-</Paragraph>
    <Paragraph position="3"> is weakly normal, and the graph of M is the transitive reduction of the graph of ph</Paragraph>
    <Paragraph position="5"> Nets. A hypernormal path (Althaus et al., 2003) in a constraint graph is a path in the undirected graph that contains for every leaf X at most one incident dominance edge.</Paragraph>
    <Paragraph position="6"> Let ph be a weakly normal dominance constraint and let G be the constraint graph of ph. We say that ph is a dominance net if the transitive reduction G prime of G is a net. G prime is a net if every tree fragment F of G prime satisfies one of the following three conditions, illustrated in Fig. 5: Strong. Every hole of F has exactly one outgoing dominance edge, and there is no weak root-to-root dominance edge.</Paragraph>
    <Paragraph position="7"> Weak. Every hole except for the last one has exactly one outgoing dominance edge; the last hole has no outgoing dominance edge, and there is exactly one weak root-to-root dominance edge. Island. The fragment has one hole X, and all variables which are connected to X by dominance edges are connected by a hypernormal path in the graph where F has been removed.</Paragraph>
    <Paragraph position="8"> We say that an MRS M is an MRS-net if the pre-translation of its literals results in a dominance net</Paragraph>
    <Paragraph position="10"> Note that this notion of MRS-nets implies that MRS-nets cannot contain EP-conjunctions as otherwise the resulting dominance constraint would not be weakly normal. SS5 shows that EP-conjunctions can be resolved i. e., MRSs with EP-conjunctions can be mapped to corresponding MRSs without EPconjunctions. null If M is an MRS-net (without EP-conjunctions), then M can be translated into a corresponding dominance constraint ph by first pre-translating M into</Paragraph>
    <Paragraph position="12"> by replacing weak root-to-root dominance edges in weak fragments by dominance edges which start from the open last hole.</Paragraph>
    <Paragraph position="13"> Theorem 1 (Niehren and Thater, 2003). Let M be an MRS and ph M be the translation of M.IfM is a connected MRS-net, then the merging-free configurations of M bijectively correspond to the minimal solved forms of the ph</Paragraph>
    <Paragraph position="15"> The following section generalizes this result to MRS-nets with a merging semantics.</Paragraph>
  </Section>
  <Section position="6" start_page="4" end_page="4" type="metho">
    <SectionTitle>
5 Merging and EP-Conjunctions
</SectionTitle>
    <Paragraph position="0"> We now show that if an MRS is a net, then all its configurations are merging-free, which in particular means that the translation can be applied to the more general version of MRS with a merging semantics.</Paragraph>
    <Paragraph position="1"> Lemma 2 (Niehren and Thater, 2003). All minimal solved forms of a connected dominance net are simple.</Paragraph>
    <Paragraph position="2"> Lemma 3. If all solved forms of a normal dominance constraint are simple, then all of its solved forms are minimal.</Paragraph>
    <Paragraph position="3"> Theorem 2. The configurations of an MRS-net M are merging-free.</Paragraph>
    <Paragraph position="4"> Proof. Let M prime be a configuration of M and let s be the underlying substitution. We construct a solved form ph M prime as follows: the labeling literals of ph  cific than the graph of ph M because the graph of M prime satisfies all dominance requirements of the handle constraints in M, hence ph  The merging semantics of MRS is needed to solve EP-conjunctions. As we have seen, the merging semantics is not relevant for MRS constraints which are nets. This also verifies Niehren and Thater's (2003) assumption that EP-conjunctions are &amp;quot;syntactic sugar&amp;quot; which can be resolved in a pre-processing step: EP-conjunctions can be resolved by exhaustively applying the following rule which adds new literals to make the implicit conjunction explicit:  ' is a complex function symbol. If this rule is applied exhaustively to an MRS M, we obtain an MRS M prime without EPconjunctions. It should be intuitively clear that the configurations of M and M prime correspond; Therefore, the configurations of M also correspond to the minimal solved forms of the translation of M prime .</Paragraph>
  </Section>
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