| { |
| "paper_id": "C88-1009", |
| "header": { |
| "generated_with": "S2ORC 1.0.0", |
| "date_generated": "2023-01-19T12:16:18.354775Z" |
| }, |
| "title": "Feature Graphs and Abstract Data Types: A Unifying Approach", |
| "authors": [ |
| { |
| "first": "Christoph", |
| "middle": [], |
| "last": "Beierle", |
| "suffix": "", |
| "affiliation": { |
| "laboratory": "", |
| "institution": "IBM Deutschland GmbH Science and Technology -LILOG", |
| "location": { |
| "postBox": "P.O. Box 80 08 80", |
| "postCode": "7000", |
| "settlement": "Stuttgart 80", |
| "country": "West Germany" |
| } |
| }, |
| "email": "" |
| }, |
| { |
| "first": "Udo", |
| "middle": [], |
| "last": "Pletat", |
| "suffix": "", |
| "affiliation": { |
| "laboratory": "", |
| "institution": "IBM Deutschland GmbH Science and Technology -LILOG", |
| "location": { |
| "postBox": "P.O. Box 80 08 80", |
| "postCode": "7000", |
| "settlement": "Stuttgart 80", |
| "country": "West Germany" |
| } |
| }, |
| "email": "" |
| } |
| ], |
| "year": "", |
| "venue": null, |
| "identifiers": {}, |
| "abstract": "", |
| "pdf_parse": { |
| "paper_id": "C88-1009", |
| "_pdf_hash": "", |
| "abstract": [], |
| "body_text": [ |
| { |
| "text": "Feature graphs appearing in unification--based grammar formalisms and algebraic specifications of abstract data types (ADTs) are both used for defining a collection of objects together with functions between these object sets. In this paper we investigate the relationship between feature graphs on the one hand and algebraic specifications of abstract data types (ADTs) on the other hand. There is a natural correlation between both these areas since a feature graph as well as an abstract data type defines a collection of objects together with functions relating the objects. We present a formal semantics for feature graphs by assigning to each feature graph G an equational ADT specification ~(G), called fg--specification.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "", |
| "sec_num": null |
| }, |
| { |
| "text": "This We say that E semantically implies an equation e if every <~,E>-algebra satisfies e. It is well-known that this model-theoretic notion of satisfaction coincides with the proof-theoretic notion of deduction (Birkhoff theorem) where e can be proved from E iff e can be deduced from E using the rules of the equational calculus (e.g. [EM 85]). We let E e denote the deductive closure of E.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "", |
| "sec_num": null |
| }, |
| { |
| "text": "The following theorem is one of the central results of ADT theory and forms the basis fo]:\" defining the semantics of a specification.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "", |
| "sec_num": null |
| }, |
| { |
| "text": "For each algebraic specification SP = <~, E> there is an initial algebra Tsp satisfying the equations in E. T~, is the so-called ~uotient term alqebr ~ consisting of congruence classes -obtained by factorization according to the equations in E -of constant terms over Z.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Theorem:", |
| "sec_num": null |
| }, |
| { |
| "text": "The initial algebra Tsp is 'the ADT specified by SP. It can be characterized informally by two conditions: all its elements are denoted by some ground term (\"no junk\"), and it satisfies a ground equation e iff every other <Z,E>-algebra also satisfies e (\"no confusion\"). 2. The problem is that x should be quantified only over all objects described by the original feature graph. But f(x) is not neccessarily in this set, so we must find a way of avoiding such a substitution. A simple way of achieving this is to switch to another signature with an additional sort, say \"soi', denoting the \"sort of interest\" and comprising all objects described by a feature graph. The fol ).owing theorem shows the power ef initial models in the sense of shrinking the search space: only the initial model has to be considered.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Theorem:", |
| "sec_num": null |
| }, |
| { |
| "text": "The fg-speei ficaT-ion SP is constant consistent, constant/complex consistent, or acyclic iff the initial algebra Ts~, has the respective property.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Theorem :", |
| "sec_num": null |
| }, |
| { |
| "text": "The above pz:opert:ies can be proven for the initial model by using the deductive closure E* of a set of equations E. The equivalences established by these two theorems show us that the consistency of a fg-specification <Z, E> can be tested by inspecting tile deductive closure E* for the absence of certain equations, depending on which consistency aspects one is interested ill. Since E* may be too large for performing these tests efficiently it would be desirable to be able to perform the consistency tests on E only.", |
| "cite_spans": [], |
| "ref_spans": [ |
| { |
| "start": 118, |
| "end": 379, |
| "text": "The equivalences established by these two theorems show us that the consistency of a fg-specification <Z, E> can be tested by inspecting tile deductive closure E* for the absence of certain equations, depending on which consistency aspects", |
| "ref_id": null |
| } |
| ], |
| "eq_spans": [], |
| "section": "Theorem :", |
| "sec_num": null |
| }, |
| { |
| "text": "ill the next section we develop a completion procedure for the set of equations E which transforms E into a normalized set E\" such that it Js sufficient to check E'. The complet.io~), procedure thus provides a simple and fast decision procedure for our consistency constraints. Let SP : <Z,E> be a fg-specification. We assume that E does not contain any trivia] equations of the form t :: t (otherwise we can just eliminate such equations from E).", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Theorem :", |
| "sec_num": null |
| }, |
| { |
| "text": "For all 1 = r ~ E we have either 1 <~ r or r <.~ ].. The next theorem shows that stepping from SP to SP\" simplifies this task:", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Le~na:", |
| "sec_num": null |
| }, |
| { |
| "text": "instead of inspecting the deductive closure E* it suffices to inspect the set of equations E'.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Le~na:", |
| "sec_num": null |
| }, |
| { |
| "text": "Let SP = <Z, E> be a fg-specification and SP\" = <Z, E'> be the result of running the completion procedure CP on SP. The proof of this theorem is based on the fact that E\" is a confluent and terminating set of rewrite rules. Since the atoms are smaller than any non-atomic term with respect to the term order <~, for any equation a --> t in E\" with an atom a, t must also be an atom~ Therefore, any equation holding between atoms must be contained directly in E', implying the constant consistency property of the theorem.", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Theorem:", |
| "sec_num": null |
| }, |
| { |
| "text": "The other parts of 'the theorem follow from similar observations. Thus, let X-consistent be either \"constant consistent\", \"constant/complex consistent\", \"acyclic\", or any combination thereof. Let GI and G2 be feature graphs. graph-unify(G~,G~) = let (Zi,Ei) = ~(Gt) in let (Z,E) = CP(Z~ u Z,, El u E2) in ~-I (Z,E) if (Z,E) is X-consistent fail if (Z,E) is not X-consistent", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Theorem:", |
| "sec_num": null |
| }, |
| { |
| "text": "We have presented a mathematical semantics of feature graphs and feature graph unification in terms of ADT specifications. ", |
| "cite_spans": [], |
| "ref_spans": [], |
| "eq_spans": [], |
| "section": "Conclusions", |
| "sec_num": "7." |
| } |
| ], |
| "back_matter": [], |
| "bib_entries": {}, |
| "ref_entries": { |
| "FIGREF0": { |
| "type_str": "figure", |
| "text": "== -<}:~ ]!',> be cl .fg-;~pe<:ificatien. A <}]~ E>-al.gebra A is ~ cpn[~tt~.g'h ~ g0nsiste~c Jff for all a, b ~. Atoms(Z) with a ~: b we have : u cons tagt/<'.omp.].ex cons.isten tiff for all a~ b ~ Atoms(Z) and a]] pl, p2, q ~ Feature~;(Z): tlhere exists ral element o ~ A,,,~ ,,\u00a2:iti~: 7p]~ (i~ (o)) :: a,, :: > p2,,(p]., (i, (o))) + q^(i~(o)) & )/,2a(pl~ (i.(o))) ~-b^ ac yc l'i,/ iff for all p~ q ~ Features(Z) \u00f7 there exists an element o ~ A,o\u00b1 with: p~(q~(i^(o))) + q^(i^(o", |
| "num": null, |
| "uris": null |
| }, |
| "FIGREF1": { |
| "type_str": "figure", |
| "text": "t :)2 , q ~ Features(Z)\" %'\u00a5,~ have : pl('i(const)) = ;~. ~ E';,\" it~p].ies p2(pl(i(const))) ::: q(i(const)) / E* and p2(pl(i(const))) = b / E* I acyclic iff for all p, q e Features(Z) ~ we have : p(q(i(const))) --: q(i(const)) / E*", |
| "num": null, |
| "uris": null |
| }, |
| "FIGREF2": { |
| "type_str": "figure", |
| "text": "Let a, b ~ ATOMS, f~, g~ e FEATURES, and t ...(f~ (t))...) <.r g, (...(g~ (t))...) -<-T is the reflexive and transitive closure of <.~.", |
| "num": null, |
| "uris": null |
| }, |
| "FIGREF3": { |
| "type_str": "figure", |
| "text": "whose lefthand side is an atom a ~ Atoms(Z) constant/complex consistent iff E\" does not contain an equation in which a term f(a) occurs where f ~ Features(Z) and a ~ Atoms(Z) acyclic iff E\" does not contain an equation p(t) -> t where p ~ Features(Z)* 44", |
| "num": null, |
| "uris": null |
| }, |
| "TABREF1": { |
| "text": "", |
| "num": null, |
| "content": "<table><tr><td colspan=\"12\">where each value is either an atomic graph or again a complex one. Two paths starting at the root and ending at the same node are said to co-.refer. Feature graph unification is a binary eperation taking two graphs and returning[ a graph containing exactly the informat J.on of both graphs if they are unifiabl~. ~, and fails otherwise. An atomic graph is unifiable only with itself and the empty graph. A complex graph G~ is unifiable with the, empty graph, and G~ is unifiable with a complex graph G2 if for all features in both G: and G2 the respective vahles are unifiab](~\u00b0</td><td>3We data information can be gathered introduce the basic notions of type specifications. More in [GTW 78] abstract detailed or [EM 85].</td></tr><tr><td colspan=\"12\">Several been suggested\u00b0 notations The graphical representation for feature graphs have of</td><td>A s.~gt_ure is a pair Z = <S, O> where S is a set of sorts and O = <O > a family of sets of ~erators w.r.t. S. We write op: s~ ...s~ -> s for an operator whose</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>i-th</td><td>argument</td><td>is</td><td>of</td><td>sort</td><td>s~</td><td>and</td><td>which</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>delivers a result of sort s.</td><td>The well-fermed</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>terms w.r.t. Z and S-sorted variables V</td><td>form</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>an</td><td>S-indexed family Tz (V),</td><td>and</td><td>an equation</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>over ~ and V is of the form i = r where 1 and</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>r are terms</td><td>of the</td><td>same sort.</td><td>An al~@hraic</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>~gcifiqation is a pair SP = <Z, E> where Z</td></tr><tr><td/><td/><td/><td colspan=\"2\">NP</td><td/><td/><td/><td/><td/><td/><td>is a over Z and some family of variables V. signature and E is a set of equations</td></tr><tr><td>could</td><td colspan=\"6\">al~o be represented</td><td/><td colspan=\"3\">in matrix</td><td>form:</td><td>Besides</td><td>these</td><td>syntactical</td><td>concepts of</td><td>ADT</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>specifications</td><td>we</td><td>provide</td><td>the</td><td>basic</td></tr><tr><td/><td/><td/><td/><td>:</td><td/><td>VP</td><td/><td/><td/><td/><td>semantical concepts of heterogenous algebras:</td></tr><tr><td/><td/><td colspan=\"2\">~Cat</td><td/><td/><td/><td/><td/><td>I</td><td/></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>Given a signature ~ =</td><td><S, O>,</td><td>a ~-algebra A</td></tr><tr><td/><td/><td/><td/><td/><td/><td colspan=\"2\">Agr</td><td/><td/><td/><td>consists of a family and for each operator op e O .... there is a of sets A = < A, > ....</td></tr><tr><td/><td/><td colspan=\"2\">I Agr</td><td/><td>< 1 ></td><td/><td/><td/><td/><td/><td>function opA : A, equation 1 = r if</td><td>---> for</td><td>A~ . A each</td><td>satisfies an assignment of</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>values from A to the variables of 1 and r the</td></tr><tr><td colspan=\"12\">in indicated by numbers the matrix notation enclosed coreference in brackets. is Another notation, which for instance is</td><td>model-theoretic evaluation of 1 and r element of A. A is a characterizations in A yields the <Z,E>-algebra satisfies every e ~ E.</td><td>have same if A</td></tr><tr><td colspan=\"3\">employed</td><td>in</td><td colspan=\"2\">PATR,</td><td colspan=\"2\">uses</td><td colspan=\"3\">special</td><td>equations</td><td>proof-theoretic</td><td>counterparts</td><td>in</td><td>terms</td><td>of</td></tr><tr><td colspan=\"3\">indicati[Lg</td><td colspan=\"4\">coreference</td><td/><td>of</td><td colspan=\"3\">paths and atomic</td><td>syntactic conditions on the deductive closure</td></tr><tr><td colspan=\"12\">values aL the end of of paths, respectively:</td><td>of the set of equations of r(G)o</td></tr><tr><td/><td/><td colspan=\"5\">< Cat > == VP</td><td/><td/><td/><td/><td>Although</td><td>the</td><td>proof-theoretic</td><td>consistency</td></tr><tr><td/><td/><td colspan=\"6\">< Subj Cat > == NP</td><td/><td/><td/><td>characterizations</td><td>are of syntactic nature, a</td></tr><tr><td/><td/><td/><td colspan=\"7\">Subj Agr > : < Agr ></td><td/><td>test</td><td>of</td><td>their validity requires to</td><td>examine</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>the deductive closure of the set of equations</td></tr><tr><td>In</td><td>the</td><td/><td colspan=\"3\">following</td><td colspan=\"2\">we</td><td colspan=\"2\">will</td><td colspan=\"2\">discuss</td><td>an</td><td>of a fg-specification.</td><td>Our</td><td>objective is to</td></tr><tr><td colspan=\"8\">equatJon~l representation</td><td>of</td><td colspan=\"3\">feature graphs</td><td>restrict</td><td>consistency</td><td>checks</td><td>to</td><td>equations</td></tr><tr><td>in</td><td colspan=\"3\">greater</td><td colspan=\"8\">detail. This representation will</td><td>explicitly</td><td>mentioned</td><td>in a fg-specificationo</td></tr><tr><td>be</td><td colspan=\"3\">oriented</td><td/><td colspan=\"3\">towards</td><td colspan=\"2\">the</td><td colspan=\"2\">equational</td><td>In the ADT-world there is</td><td>a well-known tool</td></tr><tr><td colspan=\"4\">specification</td><td/><td>of</td><td colspan=\"6\">abstract data types,</td><td>thus</td><td>for such</td><td>tasks:</td><td>the Knuth-Bendix</td><td>algorithm</td></tr><tr><td colspan=\"2\">making</td><td colspan=\"4\">~.vai lable</td><td>the</td><td/><td colspan=\"2\">powerful</td><td/><td>machinery</td><td>([KB</td><td>70]).</td><td>We present</td><td>a Knuth-Bendix like</td></tr><tr><td colspan=\"10\">developed for such specifications\u00b0</td><td/><td>completion</td><td>procedure</td><td>transforming</td><td>any</td></tr><tr><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td/><td>fg-specification</td><td>into a reduced normal form.</td></tr><tr><td colspan=\"2\">Above</td><td colspan=\"5\">we did not make</td><td>a</td><td colspan=\"2\">clear</td><td colspan=\"2\">distinction</td><td>We show that</td><td>the model-theoretic consistency</td></tr><tr><td colspan=\"4\">between the</td><td colspan=\"2\">syntax</td><td colspan=\"2\">for</td><td colspan=\"3\">describing</td><td>feature</td><td>characterizations for G are equivalent to the</td></tr><tr><td colspan=\"2\">graphs</td><td colspan=\"2\">and</td><td colspan=\"3\">feature</td><td colspan=\"2\">graphs</td><td>as</td><td colspan=\"2\">semantical</td><td>presence</td><td>resp. absence of</td><td>certain</td><td>types of</td></tr><tr><td colspan=\"12\">objects. In fact, such a distinction has been</td><td>equations in this reduced normal form.</td></tr><tr><td colspan=\"3\">omitted to</td><td colspan=\"9\">a large extent in the literature.</td></tr><tr><td>The</td><td colspan=\"8\">situation changed with</td><td colspan=\"3\">approaches</td><td>that</td><td>These</td><td>results</td><td>are</td><td>used</td><td>for</td><td>defining</td><td>the</td></tr><tr><td colspan=\"3\">formalize</td><td colspan=\"2\">the</td><td colspan=\"3\">concepts</td><td>of</td><td colspan=\"3\">feature</td><td>graphs</td><td>semantics of the unification</td><td>of</td><td>two feature</td></tr><tr><td colspan=\"3\">since such</td><td colspan=\"9\">a distinction is essential for a</td><td>graphs</td><td>GI</td><td>and</td><td>G2</td><td>as</td><td>the</td><td>(set-theoretic</td></tr><tr><td colspan=\"12\">formal treatment (see e.g. [KR 86], [Jo 87]).</td><td>componentwise)</td><td>union</td><td>of</td><td>~(GI)</td><td>and</td><td>x(G2 )</td></tr><tr><td colspan=\"6\">In the area of ADT</td><td colspan=\"6\">specifications the strict</td><td>followed by the normalization</td><td>process</td><td>using</td></tr><tr><td colspan=\"3\">separation</td><td colspan=\"9\">opens of the syntactic and the semantic the rich world</td><td>the completion</td><td>algorithm and the consistency</td></tr><tr><td colspan=\"2\">of level</td><td colspan=\"10\">ma'thematical has ulways beet* a central aspect. Our foundations of</td><td>ADT</td><td>check on the resulting set of equations\u00b0</td></tr><tr><td colspan=\"7\">specifications ADT-.based approach to (e.g</td><td colspan=\"5\">[GTW 78], feature graphs adopts [EM 85])</td><td>in</td></tr><tr><td colspan=\"12\">order to obtain a better -not only this two-level view in a natural way: Feature -formal</td></tr><tr><td colspan=\"12\">understanding graph specifications of means to describe feature graphs which the nature of are the syntactical feature are graphs. the models (or structures) of such</td><td>2. Featuregraphs</td></tr><tr><td colspan=\"6\">In particular, specifications.</td><td colspan=\"6\">we provide a model-theoretic</td></tr><tr><td colspan=\"6\">characterization</td><td colspan=\"2\">of</td><td colspan=\"3\">various</td><td>consistency</td><td>A</td><td>feature graph is a directed</td><td>graph</td><td>with a</td></tr><tr><td colspan=\"6\">conditions for reflecting the</td><td colspan=\"6\">feature consistency graph specifications concepts usually</td><td>distinguished graph are called features. An atomic graph is root node. The edges of the</td></tr><tr><td colspan=\"3\">imposed</td><td/><td>on</td><td colspan=\"3\">feature</td><td/><td colspan=\"2\">graphs</td><td>such</td><td>as</td><td>just a symbol;</td><td>it</td><td>contains no</td><td>features.</td><td>A</td></tr><tr><td colspan=\"6\">clash-freeness</td><td colspan=\"2\">and</td><td colspan=\"4\">acyclicity.</td><td>These</td><td>complex graph is a set of feature-value pairs</td></tr><tr><td colspan=\"2\">40</td><td/><td/><td/><td/><td/><td/><td/><td/><td/></tr></table>", |
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