{ "paper_id": "P01-1011", "header": { "generated_with": "S2ORC 1.0.0", "date_generated": "2023-01-19T09:29:51.053856Z" }, "title": "Underspecified Beta Reduction", "authors": [ { "first": "Manuel", "middle": [], "last": "Bodirsky", "suffix": "", "affiliation": { "laboratory": "Programming Systems Lab Saarland University", "institution": "", "location": { "postCode": "D-66041", "settlement": "Saarbr\u00fccken", "country": "Germany" } }, "email": "" }, { "first": "Katrin", "middle": [], "last": "Erk", "suffix": "", "affiliation": { "laboratory": "Programming Systems Lab Saarland University", "institution": "", "location": { "postCode": "D-66041", "settlement": "Saarbr\u00fccken", "country": "Germany" } }, "email": "" }, { "first": "Joachim", "middle": [], "last": "Niehren", "suffix": "", "affiliation": { "laboratory": "Programming Systems Lab Saarland University", "institution": "", "location": { "postCode": "D-66041", "settlement": "Saarbr\u00fccken", "country": "Germany" } }, "email": "" }, { "first": "Alexander", "middle": [], "last": "Koller", "suffix": "", "affiliation": { "laboratory": "", "institution": "Saarland University", "location": { "postCode": "D-66041", "settlement": "Saarbr\u00fccken", "country": "Germany" } }, "email": "koller@coli.uni-sb.de" } ], "year": "", "venue": null, "identifiers": {}, "abstract": "For ambiguous sentences, traditional semantics construction produces large numbers of higher-order formulas, which must then be-reduced individually. Underspecified versions can produce compact descriptions of all readings, but it is not known how to perform-reduction on these descriptions. We show how to do this using-reduction constraints in the constraint language for \u00a1-structures (CLLS).", "pdf_parse": { "paper_id": "P01-1011", "_pdf_hash": "", "abstract": [ { "text": "For ambiguous sentences, traditional semantics construction produces large numbers of higher-order formulas, which must then be-reduced individually. Underspecified versions can produce compact descriptions of all readings, but it is not known how to perform-reduction on these descriptions. We show how to do this using-reduction constraints in the constraint language for \u00a1-structures (CLLS).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Abstract", "sec_num": null } ], "body_text": [ { "text": "Traditional approaches to semantics construction (Montague, 1974; Cooper, 1983 ) employ formulas of higher-order logic to derive semantic representations compositionally; then -reduction is applied to simplify these representations. When the input sentence is ambiguous, these approaches require all readings to be enumerated andreduced individually. For large numbers of readings, this is both inefficient and unelegant.", "cite_spans": [ { "start": 49, "end": 65, "text": "(Montague, 1974;", "ref_id": "BIBREF10" }, { "start": 66, "end": 78, "text": "Cooper, 1983", "ref_id": "BIBREF2" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Existing underspecification approaches (Reyle, 1993; van Deemter and Peters, 1996; Pinkal, 1996; Bos, 1996) provide a partial solution to this problem. They delay the enumeration of the readings and represent them all at once in a single, compact description. An underspecification formalism that is particularly well suited for describing higher-order formulas is the Constraint Language for Lambda Structures, CLLS (Egg et al., 2001; . CLLS descriptions can be derived compositionally and have been used to deal with a rich class of linguistic phenomena .", "cite_spans": [ { "start": 39, "end": 52, "text": "(Reyle, 1993;", "ref_id": "BIBREF13" }, { "start": 53, "end": 82, "text": "van Deemter and Peters, 1996;", "ref_id": "BIBREF14" }, { "start": 83, "end": 96, "text": "Pinkal, 1996;", "ref_id": "BIBREF11" }, { "start": 97, "end": 107, "text": "Bos, 1996)", "ref_id": "BIBREF1" }, { "start": 417, "end": 435, "text": "(Egg et al., 2001;", "ref_id": "BIBREF3" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "They are based on dominance constraints (Marcus et al., 1983; Rambow et al., 1995) and extend them with parallelism (Erk and Niehren, 2000) and binding constraints.", "cite_spans": [ { "start": 40, "end": 61, "text": "(Marcus et al., 1983;", "ref_id": "BIBREF9" }, { "start": 62, "end": 82, "text": "Rambow et al., 1995)", "ref_id": "BIBREF12" }, { "start": 116, "end": 139, "text": "(Erk and Niehren, 2000)", "ref_id": "BIBREF4" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "However, lifting -reduction to an operation on underspecified descriptions is not trivial, and to our knowledge it is not known how this can be done. Such an operation -which we will call underspecified -reduction -would essentiallyreduce all described formulas at once by deriving a description of the reduced formulas. In this paper, we show how underspecified -reductions can be performed in the framework of CLLS.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Our approach extends the work presented in (Bodirsky et al., 2001) , which defines -reduction constraints and shows how to obtain a complete solution procedure by reducing them to parallelism constraints in CLLS. The problem with this previous work is that it is often necessary to perform local disambiguations. Here we add a new mechanism which, for a large class of descriptions, permits us to perform underspecified -reduction steps without disambiguating, and is still complete for the general problem.", "cite_spans": [ { "start": 43, "end": 66, "text": "(Bodirsky et al., 2001)", "ref_id": "BIBREF0" } ], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Plan. We start with a few examples to show what underspecified -reduction should do, and why it is not trivial. We then introduce CLLS and -reduction constraints. In the core of the paper we present a procedure for underspecified -reduction and apply it to illustrative examples.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "In this section, we show what underspecifiedreduction should do, and why the task is nontrivial. Consider first the ambiguous sentence Every student didn't pay attention. In first-order logic, the two readings can be represented as", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Examples", "sec_num": "2" }, { "text": "EQUATION", "cite_spans": [], "ref_spans": [], "eq_spans": [ { "start": 0, "end": 8, "text": "EQUATION", "ref_id": "EQREF", "raw_str": "\u00a2 \u00a3\u00a4 \u00a6 \u00a5 \u00a7\u00a8\u00a2 \u00a9 \u00a4 \u00a2 \u00a4 \u00a4 \u00a3\u00a4 \u00a6 \u00a5 \u00a2 \u00a4 \u00a4 \u00a4 ! \" $ # % ' & % ' ( % ' ) % 1 0 % 2 # % ' 3 % 5 4 % 7 6 \u00a7\u00a8\u00a2 \u00a9 8 \u00a4 \u00a2 \u00a3\u00a4 \u00a6 \u00a5 \u00a2 \u00a4 \u00a4 9 \u00a4 \u00a4 ! \" @ 3 A C B A C & A ( A C 6 A C ) A D 0 \u00a7\u00a8\u00a2 \u00a9 \u00a4 \u00a2 \u00a4 \u00a4 \u00a4 E ! \" @ (", "eq_num": "Figure 1" } ], "section": "Examples", "sec_num": "2" }, { "text": ": Underspecified -reduction steps for 'Every student did not pay attention'", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Examples", "sec_num": "2" }, { "text": "\u00a2 \u00a2 F H G I \u00a9 \u00a3\u00a4 \u00a6 \u00a5 \u00a2 \u00a4 \u00a4 9 \u00a4 ! \" P & Q R & Figure 2: Description of 'Every student did not pay attention' \u00a7 ' S U T \u00a9 8 S ! T \u00a4 \u00a4 9 S W V V ! T X \u00a7 ' S U T \u00a9 8 S \u00a4 \u00a4 9 S W V V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Examples", "sec_num": "2" }, { "text": "A classical compositional semantics construction first derives these two readings in the form of two HOL-formulas:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Examples", "sec_num": "2" }, { "text": "T 8 F H G I \u00a9 V \u00a1 S Y T ! \u00a4 a \u00a4 S 1 V ! T T 8 F H G b \u00a9 8 V \u00a1 S Y T \u00a4 a \u00a4 S 1 V V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Examples", "sec_num": "2" }, { "text": "where F H G I is an abbreviation for the term", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Examples", "sec_num": "2" }, { "text": "F G I d c \u00a1 f e g \u00a1 ' h T X \u00a7 ' S U T e S h S 1 V V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Examples", "sec_num": "2" }, { "text": "An underspecified description of both readings is shown in Figure 2 . For now, notice that the graph has all the symbols of the two HOL formulas as node labels, that variable binding is indicated by dashed arrows, and that there are dotted lines indicating an \"outscopes\" relation; we will fill in the details in Section 3. Now we want to reduce the description in Figure 2 as far as possible. The first -reduction step, with the redex at Q R & is straightforward. Even though the description is underspecified, the reducing part is a completely known \u00a1 -term. The result is shown on the left-hand side of Figure 1 . Here we have just one redex, starting at % & , which binds a single variable. The next reduction step is less obvious: The ! operator could either belong to the context (the part between or to the argument (below % 5 4 ). Still, it is not difficult to give a correct description of the result: it is shown in the middle of Fig. 1 . For the final step, which takes us to the rightmost description, the redex starts at A D B . Note that now the ! might be part of the body or part of the context of this redex. The end result is precisely a description of the two readings as first-order formulas.", "cite_spans": [], "ref_spans": [ { "start": 59, "end": 67, "text": "Figure 2", "ref_id": null }, { "start": 606, "end": 614, "text": "Figure 1", "ref_id": null }, { "start": 940, "end": 946, "text": "Fig. 1", "ref_id": null } ], "eq_spans": [], "section": "Examples", "sec_num": "2" }, { "text": "\" d # and % 7 & ) \u00a2 \u00a3\u00a4 \u00a6 \u00a5 i \u00a2 p \u00a4 q ! i \u00a2 p q ! Q % r s", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Examples", "sec_num": "2" }, { "text": "So far, the problem does not look too difficult. Twice, we did not know what exactly the parts of the redex were, but it was still easy to derive correct descriptions of the reducts. But this is not always the case. Consider Figure 3 , an abstract but simple example. In the left description, there are two possible positions for the ! : above Q or below % . Proceeding na\u00efvely as above, we arrive at the right-hand description in Fig. 3 . But this description is also satisfied by the term", "cite_spans": [], "ref_spans": [ { "start": 225, "end": 233, "text": "Figure 3", "ref_id": "FIGREF0" }, { "start": 431, "end": 437, "text": "Fig. 3", "ref_id": "FIGREF0" } ], "eq_spans": [], "section": "Examples", "sec_num": "2" }, { "text": "i T ! T t p T q V V V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Examples", "sec_num": "2" }, { "text": ", which cannot be obtained by reducing any of the terms described on the left-hand side. More generally, the na\u00efve \"graph rewriting\" approach is unsound; the resulting descriptions can have too many readings. Similar problems arise in (more complicated) examples from semantics, such as the coordination in Fig. 8 .", "cite_spans": [], "ref_spans": [ { "start": 307, "end": 313, "text": "Fig. 8", "ref_id": null } ], "eq_spans": [], "section": "Examples", "sec_num": "2" }, { "text": "The underspecified -reduction operation we propose here does not rewrite descriptions. Instead, we describe the result of the step using a \" -reduction constraint\" that ensures that the reduced terms are captured correctly. Then we use a saturation calculus to make the description more explicit.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Examples", "sec_num": "2" }, { "text": "In this section, we briefly recall the definition of the constraint language for \u00a1 -structures (CLLS). A more thorough and complete introduction can be found in (Egg et al., 2001) .", "cite_spans": [ { "start": 161, "end": 179, "text": "(Egg et al., 2001)", "ref_id": "BIBREF3" } ], "ref_spans": [], "eq_spans": [], "section": "Tree descriptions in CLLS", "sec_num": "3" }, { "text": "We ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Tree descriptions in CLLS", "sec_num": "3" }, { "text": "The idea behind \u00a1 -structures is that a \u00a1 -term can be considered as a pair of a tree which represents the structure of the term and a binding function encoding variable binding. We assume u contains symbols \u00a4 E (arity 0, for variables), \u00a3\u00a4 g \u00a5 (arity 1, for abstraction), \u00a2 (arity 2, for application), and analogous labels for the logical connectives. ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lambda structures", "sec_num": "3.1" }, { "text": "Definition 1. A \u00a1 -structure h is a pair T w \u00a1 V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lambda structures", "sec_num": "3.1" }, { "text": "Now we define the constraint language for \u00a1 structures (CLLS) to talk about these relations. Q w % w A are variables that will denote nodes of a", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Basic constraints", "sec_num": "3.2" }, { "text": "\u00a1 -structure. | m I m c Q v w % } a Q z c % } a Q { % } | t t | y } Q m i TQ # w y y y 2 w Q o V T t \u00a4 T i V C c V } \u00a1 TQ V c % } \u00a1 W # TQ R & V c d v Q # w y y y w Q o A constraint |", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Basic constraints", "sec_num": "3.2" }, { "text": "is a conjunction of literals (for dominance, labeling, etc). We use the abbreviations ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Basic constraints", "sec_num": "3.2" }, { "text": "Q v x % for Q v x w % Q z c % and Q c % for Q v w % % v w Q . The \u00a1 -binding literal \u00a1 TQ V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Basic constraints", "sec_num": "3.2" }, { "text": "\u00a3\u00a4 \u00a6 \u00a5 \u00a4 E \u00a4 Q Q # Q 3 Figure 4: The constraint graph of \u00a1 # TQ V c g v Q # w Q R 3 Q v x w Q # Q v x w Q R 3", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Basic constraints", "sec_num": "3.2" }, { "text": "We draw constraints as graphs ( Fig. 4 ) in which nodes represent variables. Labels and solid lines indicate labeling literals, while dotted lines represent dominance. Dashed arrows indicate the binding relation; disjointness and inequality literals are not represented. The informal diagrams from Section 2 can thus be read as constraint graphs, which gives them a precise formal meaning.", "cite_spans": [], "ref_spans": [ { "start": 32, "end": 38, "text": "Fig. 4", "ref_id": null } ], "eq_spans": [], "section": "Basic constraints", "sec_num": "3.2" }, { "text": "Finally, we define segments of \u00a1 -structures and correspondences between segments. This allows us to define parallelism and -reduction constraints.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Segments and Correspondences", "sec_num": "3.3" }, { "text": "A segment is a contiguous part of a \u00a1 -structure that is delineated by several nodes of the structure. Intuitively, it is a tree from which some subtrees have been cut out, leaving behind holes. ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Segments and Correspondences", "sec_num": "3.3" }, { "text": "). A segment l of a \u00a1 - structure T w \u00a1 V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (Segments", "sec_num": null }, { "text": "T l V of nodes of l is T l V c d v Y } T l V v x w w and not q v x n for all s r s", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (Segments", "sec_num": null }, { "text": "To exempt the holes of the segment, we define ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (Segments", "sec_num": null }, { "text": "T l V c T l V ' \u00a6 T l V . If ' \u00a6 T l V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (Segments", "sec_num": null }, { "text": "i , n m i T C w y y y w 1 V n \u00a7 \u00a5 T V m i T \u00a5 T C V w y y y \u00a5 T 1 V V y", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (Segments", "sec_num": null }, { "text": "There is at most one correspondence function between any two given segments. The correspondence literal co", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (Segments", "sec_num": null }, { "text": "T \u00a3 @ w V TQ V c %", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (Segments", "sec_num": null }, { "text": "expresses that a correspondence function \u00a5 between the segments denoted by \u00a3 and exists, that Q and % denote nodes within these segment, and that these nodes are related by \u00a5 . Together, these constructs allow us to define parallelism, which was originally introduced for the analysis of ellipsis (Egg et al., 2001 ). The parallelism relation l holds iff there is a correspondence function between l and that satisfies some natural conditions on \u00a1 -binding which we cannot go into here. To model parallelism in the presence of global \u00a1 -binders relating multiple parallel segments, Bodirsky et al. (2001) generalize parallelism to group parallelism. Group parallelism", "cite_spans": [ { "start": 297, "end": 314, "text": "(Egg et al., 2001", "ref_id": "BIBREF3" }, { "start": 582, "end": 604, "text": "Bodirsky et al. (2001)", "ref_id": "BIBREF0" } ], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (Segments", "sec_num": null }, { "text": "T l # w y y y 2 w l 2 o V T # w y y y 2 w o V is entailed \u00a3\u00a4 \u00a6 \u00a5 x i \u00a2 \u00a3\u00a4 g \u00a5 \u00a2 \u00a4 \u00a4 q i \u00a2 \u00a4 q & 1 y & # 6 y # 3 4 ( 1 y ( Figure 5: i T T \u00a1 S y \u00aa \u00a9 T k S W V V T q V V \u00ac \u00ab i T \u00a9 T q V V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (Segments", "sec_num": null }, { "text": "by the conjunction o q b # l q q of ordinary parallelisms, but imposes slightly weaker restrictions on \u00a1 -binding. By way of example, consider the \u00a1 structure in Fig. 5 , where", "cite_spans": [], "ref_spans": [ { "start": 162, "end": 168, "text": "Fig. 5", "ref_id": null } ], "eq_spans": [], "section": "Definition 2 (Segments", "sec_num": null }, { "text": "T & # w 3 a 4 w ( V T 1 y & 1 y # w 1 y # 1 y 4 w 1 y 4 V holds. On the syntactic side, CLLS provides group parallelism literals T # w y y y 2 w o V T \u00a1 # w y y y \u00ae w \u00a1 o V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (Segments", "sec_num": null }, { "text": "to talk about (group) parallelism.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Definition 2 (Segments", "sec_num": null }, { "text": "Correspondences are also used in the definition of -reduction constraints (Bodirsky et al., 2001) . A -reduction constraint describes a singlereduction step between two \u00a1 -terms; it enforces correct reduction even if the two terms are only partially known.", "cite_spans": [ { "start": 74, "end": 97, "text": "(Bodirsky et al., 2001)", "ref_id": "BIBREF0" } ], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "Standard -reduction has the form", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "\u00a3 T T \u00a1 S y \u00aa \u00a1 V V \u00ac \u00ab \u00a3 T \u00a1 \u00b0S \u00b2 \u00b1 V \u00b3 S free for y The reducing \u00a1 -term consists of context \u00a3 which contains a redex T \u00a1 S y \u00aa \u00a1 V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": ". The redex itself is an occurrence of an application of a \u00a1 -abstraction \u00a1 S y \u00aa \u00a1 with body \u00a1 to argument . -reduction then replaces all occurrences of the bound variable S in the body by the argument while preserving the context.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "We can partition both redex and reduct into argument, body, and context segments. Consider Fig. 5 . The \u00a1 -structure contains the reducing ", "cite_spans": [], "ref_spans": [ { "start": 91, "end": 97, "text": "Fig. 5", "ref_id": null } ], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "\u00a1 - term i T T \u00a1 S y \u00aa \u00a9 T k S 1 V V T q V V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "T\u1e83 D w l V \u00ab T\u00fdw y w l y # w y y y w l y o V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "for a body with holes (for the variables bound in the redex). The -reduction relation holds iff two conditions are met:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "T\u1e83 D w l V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "must form a reducing term, and the structural equalities that we have noted above must hold between the tree segments. The latter can be stated by the following group parallelism relation, which also represents the correct binding behaviour:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "T\u1e83 D w l w y y y n w l V T\u00b4 7 y w y w l 2 y # w y y y \u00ae w l 2 y o V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "Note that any \u00a1 -structure satisfying this relation must contain both the reducing and the reduced term as substructures. Incidentally, this allows us to accommodate for global variables in \u00a1 -terms; Fig. 5 shows this for the global variable \u00a9 . We now extend CLLS with -reduction constraints", "cite_spans": [], "ref_spans": [ { "start": 200, "end": 206, "text": "Fig. 5", "ref_id": null } ], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "T \u00a3 @ w \u00a1 \u00b6 w V \u00ab T \u00a3 y w \u00a1 y w y # w y y y \u00ae w y o V w", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "which are interpreted by the -reduction relation. The reduction steps in Section 2 can all be represented correctly by -reduction constraints. Consider e.g. the first step in Fig. 1 . This is represented by the constraint", "cite_spans": [], "ref_spans": [ { "start": 175, "end": 181, "text": "Fig. 1", "ref_id": null } ], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "T \" d # % ' & w % 7 3 % ' ( w % ' 4 V \u00ab T \" P 3 A C & w A C & A D ( w A C ( V .", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "The entire middle constraint in Fig. 1 is entailed by the -reduction literal. If we learn in addition that e.g.", "cite_spans": [], "ref_spans": [ { "start": 32, "end": 38, "text": "Fig. 1", "ref_id": null } ], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "% 1 0 v x w % ' & , the -reduction literal will entail A 0 v", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "x w A & because the segments must correspond. This correlation between parallel segments is the exact same effect (quantifier parallelism) that is exploited in the CLLS analysis of \"Hirschb\u00fchler sentences\", where ellipses and scope interact (Egg et al., 2001 ).", "cite_spans": [ { "start": 241, "end": 258, "text": "(Egg et al., 2001", "ref_id": "BIBREF3" } ], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "-reduction constraints also represent the problematic example in Fig. 3 ", "cite_spans": [], "ref_spans": [ { "start": 65, "end": 71, "text": "Fig. 3", "ref_id": "FIGREF0" } ], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "T | w Q V else pick a formula redex\u2022 T \u00a3 @ w \u00a1 \u00b6 w V in | that is unreduced, with Q c T \u00a3 V in | add T \u00a3 @ w \u00a1 \u00b6 w V \u00ab T \u00a3 y w \u00a1 y w y # w y y y \u00ae w y o V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "T | V do usb T | y w T \u00a3 y V V end", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Beta reduction constraints", "sec_num": "4" }, { "text": "Having introduced -reduction constraints, we now show how to process them. In this section, we present the procedure usb, which performs a sequence of underspecified -reduction steps on CLLS descriptions. This procedure is parameterized by another procedure solve for solvingreduction constraints, which we discuss in the following section.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Underspecified Beta Reduction", "sec_num": "5" }, { "text": "A syntactic redex in a constraint | is a subformula of the following form: Figure 6 . It starts with a constraint | and a variable Q , which denotes the root of the current \u00a1 -term to be reduced. (For example, for the redex in Fig. 2 , this root would be \" & .) The procedure then selects an unreduced syntactic redex and adds a description of its reduct at a disjoint position. Then the solve procedure is applied to resolve thereduction constraint, at least partially. If it has to disambiguate, it returns one constraint for each reading it finds. Finally, usb is called recursively with the new constraint and the root variable of the new \u00a1 -term.", "cite_spans": [], "ref_spans": [ { "start": 75, "end": 83, "text": "Figure 6", "ref_id": "FIGREF1" }, { "start": 227, "end": 233, "text": "Fig. 2", "ref_id": null } ], "eq_spans": [], "section": "Underspecified Beta Reduction", "sec_num": "5" }, { "text": "redex\u2022 T \u00a3 @ w \u00a1 \u00b6 w V c df 2 T \u00a3 V m \u00a2 T % w T V V % m \u00a3\u00a4 \u00a6 \u00a5 d T T \u00a1 V V \u00a1 # T % V C c \u00b9 5 T \u00a1 V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Underspecified Beta Reduction", "sec_num": "5" }, { "text": "Intuitively, the solve procedure adds entailed literals to | , making the new -reduction literal more explicit. When presented with the left-hand constraint in Fig. 1 and the root variable \" d # , usb will add a -reduction constraint for the redex at % \u00ae #", "cite_spans": [], "ref_spans": [ { "start": 160, "end": 166, "text": "Fig. 1", "ref_id": null } ], "eq_spans": [], "section": "Underspecified Beta Reduction", "sec_num": "5" }, { "text": "; then solve will derive the middle constraint. Finally, usb will call itself recursively with the new root variable \" P 3 and try to resolve the redex at A D ( , etc. The partial solving steps do essentially the same as the na\u00efve graph rewriting approach in this case; but the new algorithm will behave differently on problematic constraints as in Fig. 3 .", "cite_spans": [], "ref_spans": [ { "start": 349, "end": 355, "text": "Fig. 3", "ref_id": "FIGREF0" } ], "eq_spans": [], "section": "Underspecified Beta Reduction", "sec_num": "5" }, { "text": "In this section we present a procedure solve for solving -reduction constraints. We go through several examples to illustrate how it works. We have to omit some details for lack of space; they can be found in (Bodirsky et al., 2001) .", "cite_spans": [ { "start": 209, "end": 232, "text": "(Bodirsky et al., 2001)", "ref_id": "BIBREF0" } ], "ref_spans": [], "eq_spans": [], "section": "A single reduction step", "sec_num": "6" }, { "text": "The aim of the procedure is to make explicit information that is implicit in -reduction constraints: it introduces new corresponding variables and copies constraints from the reducing term to the reduced term.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "A single reduction step", "sec_num": "6" }, { "text": "We build upon the solver for -reduction constraints from (Bodirsky et al., 2001 ). This solver is complete, i.e. it can enumerate all solutions of a constraint; but it disambiguates a lot, which we want to avoid in underspecified -reduction. We obtain an alternative procedure solve by disabling all rules which disambiguate and adding some new non-disambiguating rules. This allows us to perform a complete underspecifiedreduction for many examples from underspecified semantics without disambiguating at all. In those cases where the new rules alone are not sufficient, we can still fall back on the complete solver.", "cite_spans": [ { "start": 57, "end": 79, "text": "(Bodirsky et al., 2001", "ref_id": "BIBREF0" } ], "ref_spans": [], "eq_spans": [], "section": "A single reduction step", "sec_num": "6" }, { "text": "Our constraint solver is based on saturation with a given set of saturation rules. Very briefly, this means that a constraint is seen as the set of its literals, to which more and more literals are added according to saturation rules. A saturation rule of the form | & \u00bd U o q b # | q says that we can add one of the | q to any constraint that contains at least the literals in | & . We only apply rules where each possible choice adds new literals to the set; a constraint is saturated under a set \u00be of saturation rules if no rule in \u00be can add anything else. solve returns the set of all possible saturations of its input. If the rule system contains nondeterministic distribution rules, with \u00c0 \u00bf \u00c1 , this set can be non-singleton; but the rules we are going to introduce are all deterministic propagation rules (with c ).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Saturation", "sec_num": "6.1" }, { "text": "The main problem in doing underspecifiedreduction is that we may not know to which part of a redex a certain node belongs (as in Fig. 1 ).", "cite_spans": [], "ref_spans": [ { "start": 129, "end": 135, "text": "Fig. 1", "ref_id": null } ], "eq_spans": [], "section": "Solving Beta Reduction Constraints", "sec_num": "6.2" }, { "text": "We address this problem by introducing underspecified correspondence literals of the form", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Solving Beta Reduction Constraints", "sec_num": "6.2" }, { "text": "co T 9 v \u00a6 T \u00a3 # w # V w y y y w T \u00a3 o w o V V TQ V c % y", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Solving Beta Reduction Constraints", "sec_num": "6.2" }, { "text": "Such a literal is satisfied if the tree segments denoted by the \u00a3 's and by the 's do not overlap properly, and there is an r for which co T \u00a3 q w q V TQ V e c % is satisfied. In Fig. 7 we present the rules UB for underspecified -reduction; the first five rules are the core of the algorithm. To keep the rules short, we use the following abbreviations (with $ s r s ):", "cite_spans": [], "ref_spans": [ { "start": 179, "end": 185, "text": "Fig. 7", "ref_id": null } ], "eq_spans": [], "section": "Solving Beta Reduction Constraints", "sec_num": "6.2" }, { "text": "beta c P \u00c2 \u00c3 \u00c5 \u00c4 AE T \u00a3 P w \u00a1 \u00a2 w V \u00ab T \u00a3 y w \u00a1 y w y # w y y y 2 w y o V coq c P \u00c2 \u00c3 \u00c5 \u00c4 co T 9 v \u00a6 T \u00a3 @ w \u00a4 \u00a3 y V w T \u00a1 \u00b6 w \u00a1 y V w T w y q V V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Solving Beta Reduction Constraints", "sec_num": "6.2" }, { "text": "The procedure solve consists of UB together with the propagation rules from (Bodirsky et al., 2001 ). The rest of this section shows how this procedure operates and what it can and cannot do. First, we discuss the five core rules. Rule (Beta) states that whenever the -reduction relation holds, group parallelism holds, too. (This allows us to fall back on a complete solver for group parallelism.) Rule (Var) introduces a new variable as a correspondent of a redex variable, and (Lab) and (Dom) copy labeling and dominance literals from the redex to the reduct. To understand the exceptions they make, consider e.g. Fig. 5 . For the variables that possess a correspondent, all dominance relations in the redex hold in the reduct too. The rule (\u00a1 .Inv) copies inverse \u00a1 binding literals, i.e. the information that all variables bound by a \u00a1 -binder are known. For now, Figure 7 : New saturation rules UB for constraint solving during underspecified -reduction.", "cite_spans": [ { "start": 76, "end": 98, "text": "(Bodirsky et al., 2001", "ref_id": "BIBREF0" } ], "ref_spans": [ { "start": 617, "end": 623, "text": "Fig. 5", "ref_id": null }, { "start": 869, "end": 877, "text": "Figure 7", "ref_id": null } ], "eq_spans": [], "section": "Solving Beta Reduction Constraints", "sec_num": "6.2" }, { "text": "EQUATION", "cite_spans": [], "ref_spans": [], "eq_spans": [ { "start": 0, "end": 8, "text": "EQUATION", "ref_id": "EQREF", "raw_str": "\u00c7 I \u00c8 7 \u00c9 \u00cb \u00ca U \u00c9 \u00cb \u00cc 7 \u00cd e \u00ce \u00cf \u00c7 \u00d0 \u00c8 7 \u00d1 \u00d2 \u00c9 k \u00ca 1 \u00d1 \u00d2 \u00c9 k \u00cc ' \u00d1 \u00d3 8 \u00c9 \u00c5 \u00d4 \u00b5 \u00d4 \u00c5 \u00d4 H \u00c9 k \u00cc ' \u00d1 \u00d5 \u00cd \u00cf \u00c7 \u00d0 \u00c8 7 \u00c9 t \u00ca U \u00c9 k \u00cc 2 \u00c9 \u00c5 \u00d4 \u00c5 \u00d4 9 \u00d4 \u00c9 \u00d2 \u00cc 1 \u00cd g \u00d6 \u00a2 \u00c7 I \u00c8 7 \u00d1 X \u00c9 \u00cb \u00ca 1 \u00d1 X \u00c9 k \u00cc 5 \u00d1 \u00d3 \u00c9 \u00c5 \u00d4 \u00b5 \u00d4 \u00b5 \u00d4 H \u00c9 \u00d2 \u00cc 5 \u00d1 \u00d5 \u00cd (Var) beta \u00d7 redex\u00d8 2 \u00c7 \u00d0 \u00c8 7 \u00c9 t \u00ca U \u00c9 k \u00cc 1 \u00cd \u00d7 \u00d9 \u00c7 \u00d0 \u00c8 W \u00cd I \u00da \u00a4 \u00db \u00dc \u00d7 \u00a6 \u00dc n \u00dd \u2022 \u00cf p \u00de \u00dc \u00d1\u00d4 c o\u00df t \u00c7 I \u00dc ' \u00cd \u00dc \u00d1 (Lab) beta \u00d7 redex\u00d8 \u00c7 \u00d0 \u00c8 7 \u00c9 t \u00ca U \u00c9 k \u00cc 1 \u00cd \u00d7 \u00a6 \u00dc g \u00e0 \u00e1\u00e2 H \u00c7 I \u00dc \u00d3 \u00c9 9 \u00d4 \u00c5 \u00d4 \u00c5 \u00d4 \u00c9 k \u00dc x \u00e3 9 \u00cd \u00d7 \u00a6 \u00e4 \u00e3 \u00e5 ae \u00e0 co\u00df \u00cb \u00c7 \u00d0 \u00dc \u00e5 \u00cd \u00dc 5 \u00d1 \u00e5 \u00d7 e \u00dc g \u00e0 \u00dd 1 \u00e7 \u00c7 \u00d0 \u00c8 W \u00cd H \u00d7 e \u00dc \u00a6 \u00e0 \u00e8 \u00e9 \u00e7 \u00ea \u00c7 \u00d0 \u00ca \u00ae \u00cd \u00cf \u00dc f \u00d1 \u00e0 \u00e1\u00e2 H \u00c7 I \u00dc f \u00d1 \u00d3 \u00c9 \u00b5 \u00d4 \u00c5 \u00d4 \u00b5 \u00d4 H \u00c9 k \u00dc 5 \u00d1 \u00e3 \u00cd (Dom) beta \u00d7 \u00e4 n \u00eb \u00e5 8 ae \u00d3 co\u00df \u00cb \u00c7 \u00d0 \u00dc \u00e5 \u00cd \u00dc 5 \u00d1 \u00e5 \u00d7 \u00a6 \u00dc \u00d3 \u00da \u00db \u00dc \u00eb \u00cf \u00dc 5 \u00d1 \u00d3 \u00da \u00db \u00dc f \u00d1 \u00eb (\u00ec .Inv) beta \u00d7 redex\u00d8 \u00c7 I \u00c8 7 \u00c9 \u00cb \u00ca n \u00c9 \u00d2 \u00cc 7 \u00cd 9 \u00d7 f \u00ec \u00ed \u00d3 \u00c7 I \u00dc g \u00e0 \u00cd \u00ae \u00ee \u00dc \u00d3 \u00c9 \u00c5 \u00d4 \u00b5 \u00d4 \u00b5 \u00d4 H \u00c9 k \u00dc g \u00ef D \u00f0 \u00d7 \u00e4 \u00ef \u00e5 8 ae \u00e0 co \u00d3 \u00c7 I \u00dc \u00e5 \u00cd \u00dc \u00d1 \u00e5 \u00cf \u00ec \u00ed \u00d3 \u00c7 \u00d0 \u00dc \u00d1 \u00e0 \u00cd \u00ae \u00ee \u00dc \u00d1 \u00d3 \u00c9 \u00c5 \u00d4 9 \u00d4 \u00c5 \u00d4 \u00c9 \u00cb \u00dc \u00d1 \u00ef \u00f0 redex linear (Par.part) beta \u00d5 \u00d7 co\u00df \u00cb \u00c7 I \u00dc ' \u00cd \u00dc \u00d1 \u00d7 \u2022 \u00e9 \u00f1 \u00c7 \u00d0 \u00cc 7 \u00cd \u00d7 \u00a6 \u00dc f \u00da \u00a4 \u00db \u2022 \u00cf \u00dc \u00d1 \u00e8 \u00e9 \u00f2 \u00c7 \u00d0 \u00ca \u00d1\u00cd # W \u00f3 q \u00f3 o (Par.all) co\u00c7 \u00ee \u00c7 I \u00f4 e \u00c9 \u00cb \u00f4 \u00d1\u00cd \u00cb \u00c9 \u00c5 \u00d4 9 \u00d4 \u00c5 \u00d4 t \u00f0 8 \u00cd t \u00c7 \u00d0 \u00dc 7 \u00cd \u00dc \u00d1 \u00d7 \u00a6 \u00dc \u00e9 \u00f5 \u00f1 \u00c7 \u00d0 \u00f4 C \u00cd \u00cf \u00dc \u00d1 \u00e9 \u00f5 \u00f1 \u00c7 \u00d0 \u00f4 \u00d1\u00cd H \u00d7 co\u00c7 \u00d0 \u00f4 e \u00c9 \u00cb \u00f4 \u00d1\u00cd \u00cb \u00c7 I \u00dc ' \u00cd \u00dc \u00d1", "eq_num": "(Beta)" } ], "section": "Solving Beta Reduction Constraints", "sec_num": "6.2" }, { "text": "it is restricted to linear redexes; for the nonlinear case, we have to take recourse to disambiguation. It can be shown that the rules in UB are sound in the sense that they are valid implications when interpreted over \u00a1 -structures.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Solving Beta Reduction Constraints", "sec_num": "6.2" }, { "text": "To see what the rules do, we go through the first reduction step in Fig. 1 . The -reduction constraint that belongs to this reduction is", "cite_spans": [], "ref_spans": [ { "start": 68, "end": 74, "text": "Fig. 1", "ref_id": null } ], "eq_spans": [], "section": "Some Examples", "sec_num": "6.3" }, { "text": "EQUATION", "cite_spans": [], "ref_spans": [], "eq_spans": [ { "start": 0, "end": 8, "text": "EQUATION", "ref_id": "EQREF", "raw_str": "T \u00a3 @ w \u00a1 \u00b6 w V \u00ab T \u00a3 y w \u00a1 y w \u00b2 y # V with \u00a3 c \" d # \u00a4 % 7 & w \u00a1 c % \u00ae # \u00a4 % 7 ( w c % 5 4 w \u00a3 y c \" P 3 a A C & w \u00f6 \u00a1 y c A D & a A C ( w \u00b2 y # c A D", "eq_num": "(" } ], "section": "Some Examples", "sec_num": "6.3" }, { "text": "Now saturation can add more constraints, for example the following:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Some Examples", "sec_num": "6.3" }, { "text": "\u00c7 # \u00cd \u2022 a \u00f7 \u00dd \u2022 \u00d3 \u00c7 6 \u00cd \u2022 a \u00f7 \u00dd \u2022 a \u00f8 \u00c7 3 \u00cd \u2022 \u00f9 \u00dd \u2022 \u00d3 \u00c7 ) \u00cd \u00fa \u00dc \u00f7 \u00e1\u00fb \u00fc \u00c7 \u00d0 \u00dc \u00f9 \u00cd (Lab) \u00c7 ( \u00cd \u00de \u00dc \u00f7 \u00d4 c o \u00d3 \u00c7 \u2022 a \u00f7 \u00cd \u00dc \u00f7 (Var) \u00c7 0 \u00cd \u00fa \u00fd \u00eb \u00da \u00db \u00dc \u00f7 (Dom) \u00c7 4 \u00cd \u00de \u00dc \u00f9 \u00d4 c o \u00d3 \u00c7 \u2022 \u00f9 \u00cd \u00dc \u00f9 (Var)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Some Examples", "sec_num": "6.3" }, { "text": "We get (1), (2), (5) by propagation rules from (Bodirsky et al., 2001 ): variables bearing different labels must be different. Now we can apply (Var) to get (3) and (4), then (Lab) to get (6). Finally, (7) shows one of the dominances added by (Dom). Copies of all other variables and literals can be computed in a completely analogous fashion. In particular, copying gives us another redex starting at A D B , and we can continue with the algorithm usb in Figure 6 .", "cite_spans": [ { "start": 47, "end": 69, "text": "(Bodirsky et al., 2001", "ref_id": "BIBREF0" } ], "ref_spans": [ { "start": 456, "end": 464, "text": "Figure 6", "ref_id": "FIGREF1" } ], "eq_spans": [], "section": "Some Examples", "sec_num": "6.3" }, { "text": "Note what happens in case of a nonlinear redex, as in the left picture of Fig. 8 : as the redex is \u00fe ary, the rules produce two copies of the ! labeling constraint, one via co # and one via co 3 . The result is shown on the right-hand side of the figure. We will return to this example in a minute.", "cite_spans": [], "ref_spans": [ { "start": 74, "end": 80, "text": "Fig. 8", "ref_id": null } ], "eq_spans": [], "section": "Some Examples", "sec_num": "6.3" }, { "text": "The last two rules in Fig. 7 enforce consistency between scoping in the redex and scoping in the reduct. The rules use literals that were introduced in (Bodirsky et al., 2001) , of the forms", "cite_spans": [ { "start": 152, "end": 175, "text": "(Bodirsky et al., 2001)", "ref_id": "BIBREF0" } ], "ref_spans": [ { "start": 22, "end": 28, "text": "Fig. 7", "ref_id": null } ], "eq_spans": [], "section": "More Complex Examples", "sec_num": "6.4" }, { "text": "Q T V , Q \u00ff T \u00a1 V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "More Complex Examples", "sec_num": "6.4" }, { "text": ", etc., where , \u00a1 are segment terms. We take Q T V to mean that Q must be inside the tree segment denoted by , and we take Q \u00ff T \u00a1 V (i for 'interior') to mean that Q T \u00a1 V and Q denotes neither the root nor a hole of \u00a1 . As an example, reconsider ", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "More Complex Examples", "sec_num": "6.4" }, { "text": "i T ! T t p H T q V V V", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "More Complex Examples", "sec_num": "6.4" }, { "text": "because that would require the ! operator to be in \u00ff T \u00a1 y V . Similarly in Fig. 8 , where we have introduced two copies of the ! label. If the ! in the redex on the left ends up as part of the context, there should be only one copy in the reduct. This is brought about by the rule (Par.all) and the fact that correspondence is a function (which is enforced by rules from ) which are part of the solver in (Bodirsky et al., 2001) ). Together, they can be used to infer that A D & can have only one correspondent in the reduct context.", "cite_spans": [ { "start": 406, "end": 429, "text": "(Bodirsky et al., 2001)", "ref_id": "BIBREF0" } ], "ref_spans": [ { "start": 76, "end": 82, "text": "Fig. 8", "ref_id": null } ], "eq_spans": [], "section": "More Complex Examples", "sec_num": "6.4" }, { "text": "In this paper, we have shown how to perform an underspecified -reduction operation in the CLLS framework. This operation transforms underspecified descriptions of higher-order formulas into descriptions of their -reducts. It can be used to essentially -reduce all readings of an ambiguous sentence at once.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Conclusion", "sec_num": "7" }, { "text": "It is interesting to observe how our underspecified -reduction interacts with parallelism constraints that were introduced to model ellipses. Consider the elliptical three-reading example \"Peter sees a loophole. Every lawyer does too.\" Under the standard analysis of ellipsis in CLLS (Egg et al., 2001) , \"Peter\" must be represented as a generalized quantifier to obtain all three readings. This leads to a spurious ambigu- Figure 8 : \"Peter and Mary do not laugh.\" ity in the source sentence, which one would like to get rid of by -reducing the source sentence. Our approach can achieve this goal: Adding -reduction constraints for the source sentence leaves the original copy intact, and the target sentence still contains the ambiguity.", "cite_spans": [ { "start": 284, "end": 302, "text": "(Egg et al., 2001)", "ref_id": "BIBREF3" } ], "ref_spans": [ { "start": 424, "end": 432, "text": "Figure 8", "ref_id": null } ], "eq_spans": [], "section": "Conclusion", "sec_num": "7" }, { "text": "Under the simplifying assumption that all redexes are linear, we can show that it takes tim\u00eb T \u00a9 ( V to perform \u00a9 steps of underspecifiedreduction on a constraint with variables. This is feasible for large \u00a9 as long as , which should be sufficient for most reasonable sentences. If there are non-linear redexes, the present algorithm can take exponential time because subterms are duplicated. The same problem is known in ordinary \u00a1 -calculus; an interesting question to pursue is whether the sharing techniques developed there (Lamping, 1990 ) carry over to the underspecification setting.", "cite_spans": [ { "start": 528, "end": 542, "text": "(Lamping, 1990", "ref_id": "BIBREF8" } ], "ref_spans": [], "eq_spans": [], "section": "Conclusion", "sec_num": "7" }, { "text": "In Sec. 6, we only employ propagation rules; that is, we never disambiguate. This is conceptually very nice, but on more complex examples (e.g. in many cases with nonlinear redexes) disambiguation is still needed.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Conclusion", "sec_num": "7" }, { "text": "This raises both theoretical and practical issues. On the theoretical level, the questions of completeness (elimination of all redexes) and confluence still have to be resolved. To that end, we first have to find suitable notions of completeness and confluence in our setting. Also we would like to handle larger classes of examples without disambiguation. On the practical side, we intend to implement the procedure and disambiguate in a controlled fashion so we can reduce completely and still disambiguate as little as possible.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Conclusion", "sec_num": "7" } ], "back_matter": [], "bib_entries": { "BIBREF0": { "ref_id": "b0", "title": "Beta reduction constraints", "authors": [ { "first": "M", "middle": [], "last": "Bodirsky", "suffix": "" }, { "first": "K", "middle": [], "last": "Erk", "suffix": "" }, { "first": "A", "middle": [], "last": "Koller", "suffix": "" }, { "first": "J", "middle": [], "last": "Niehren", "suffix": "" } ], "year": 2001, "venue": "Proc. 12th Rewriting Techniques and Applications", "volume": "", "issue": "", "pages": "", "other_ids": {}, "num": null, "urls": [], "raw_text": "M. Bodirsky, K. Erk, A. Koller, and J. Niehren. 2001. Beta reduction constraints. 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Semantic Am- biguity and Underspecification. CSLI Press, Stan- ford.", "links": null } }, "ref_entries": { "FIGREF0": { "uris": null, "text": "Problems with rewriting of descriptions", "num": null, "type_str": "figure" }, "FIGREF1": { "uris": null, "text": "Underspecified -reduction satisfy the -reduction constraint, as the bodies would not correspond.", "num": null, "type_str": "figure" }, "FIGREF2": { "uris": null, "text": "Fig. 3: by rule (Par.part), the reduct (right-hand picture ofFig. 3) cannot represent the term", "num": null, "type_str": "figure" }, "TABREF5": { "html": null, "content": "
by requiring that , and\u00b4to\u00b4y , again modulo binding. This is l corresponds to l , 2 y y indeed true in the given to -structure, as we have \u00a1 seen above.
More generally, we define the -reduction re-
lation
starting at& . The reduced
term can be found at context, D w y for the body and & . Writing\u00b4w\u00b4y for the y l w l for the ar-2 y gument tree segments of the reducing and the re-
duced term, respectively, we find
c & \u00b5 # c 3 \u00b5 4 \u00b5 \u00b4 c ( l ' y c 1 y & 1 y # y c 1 y # 1 y ( l 2 y c 1 y (
", "text": "Because we have both the reducing term and the reduced term as parts of the same \u00a1 -structure, we can express the fact that the structure below y & can be obtained by -reducing the structure below &", "num": null, "type_str": "table" } } } }