advertisement

From: AAAI-88 Proceedings. Copyright ©1988, AAAI (www.aaai.org). All rights reserved. Nonmonotonic Inheritance and Generic Reflexives Richmond H. Thomason Intelligent Systems Program University of Pittsburgh Pittsburgh, PA 15260 David S. Touretzky Computer Science Department Carnegie Mellon University Pittsburgh, PA 15213 Generic reflexive statements such as Elephants have traditionally been formalized using dove themselves purposes some variant coreferentiality. of predicate logic, with variables to mark We present a radically different semantics is presented here. The need for alternative for reflexives, based limited Abstract. extension system on nonmonotonic to Touretzky’s can derive as statements predicate about logic-based use variables; human inferential new generic this ordering. And unlike our formalism Our as well of a non-classical for analyzing commonsense are common in the world’s lanlinguistic theories subsume reflex- but no independent meaning, other expressions, thereby are meaningful. The leading current tics of reflexives’ no human linguistic language in some way to to larger units that theories to interpret uses variables marking “logical at surface forms” This be Though are introduced, they that contain them. however, genforms like the morphology, to say whether is a deep linguistic our only having one semantic a logical theory that mathematical Sentences be formalized (Many z)(Politiciun as in En- notations discovery theory was originally rather between languages the encode or an artifact of of reflexivizationdesigned than natural to explain language. For difficulties have Bowever, Many relatively quantifier call natural power. like Most politicians using forms such as z --f honest CC) led to a theory of generalized conservative are totally extensions inadequate pbzlrul, and so can’t generic to logical such for handling deal as a what with sen- can’t be used as they formalisms in interpreting language because they are so limited in expressive This paper doesn’t offer a solution to the general problem, but we do show that we can account for systematic interactions between generics and reflexives, using techniques from nonmonotonic inheritance theory. This suggests an alternative representation according to which reflexives-though bound still to a relation anaphoric because by appearing they in a path must be containing a single relational symbol R-seem to resemble individual nodes in many ways, and to have a greater measure of semantic independence than the variables of logical representations. 2 Structure Figures 1 and appear in the nodes, of the Paper 2 contain paper. [Fahlman representing all the network Our graphical 19791. There individuals primitives that notation is a variare several kinds of (Clyde), classes (elephant), instances of the term “self” (denoted by o), and instances of “other” (denoted by 0). There are also several kinds of links. IS-A and information, and Clyde is not a herbivore. links (3 links IS-NOT-A taxonomic tional ‘See [Thomason 19761 and [Th omason 19831 for the Montague Grammar approach, and [May 19851 for the Government-Binding approach. by the quantifiers2 ant of NETL the discrepancy form and the way in which human reflexives level- watch is self-winding. It is hard logical or in the verb so- must a sentence. Human languages that mark reflexive, erally do so either with special pronominal themselves, of the semanreflexives, “co-referentiality” in the course of parsing differ on how these variables theories agree in producing (along have are bound contributing use variables or indices these constructions as expressions that introduced glish is intensified logic to cope with the phenom- language. stand as an alternative ivization under anaphoru, treating with expressions like each other) English such to have alter- One such alternative theories of first-order can’t be useful of reflexive. tences like Elephants are gruy.3 Nonmonotonic semantic networks Reflexive constructions guages. Contemporary variables and linguists Motivation since ability are honest it would theories ena of natural knowlreasoning phenomena. 1 of comparison, semantic the leading does not to ac tual in structure of this work for AI is it closer significance of the benefits its demonstration and an statements approaches, brings edge representation distance reflexive individuals. The languages. inheritance native and $) such (-t as that Positive represent binary and #+) express elephants are gray and negative relations rela- between 2For example, the work of Altham [1971], and van Benthem & ter Meulen [1985]. 3See [Carlson 19821 for detailed arguments. Touretzky and Thomason 433 classes The or individuals, asertion themselves 2 indicates celebrities royal by a $; like gray things. elephants link from to a 0 (read “self”) node. as a dashed line in network an “other” term 1 that is expressed phant node link, drawn nects such as Herbivores in Figure node to its referent. that opera stars other than themselves.) celebrities” is written “other do not like the royal For example, admire other ele- A fifth type of notation, conFigure celebrities In path notation (i.e., the @ : c. 3 Notation Let + and links. Let denote nonmonotonic and F denote nodes to their Taxonomic paths are such as ~1 + composed purely y- over individual then z + P 9 taxonomic Elephant paths, or, and bind IS-A and/or of abutting links, of “other” x4. Positive while negative link at the end. Taxonomic and class nodes; they contain no instances of 0 or 0. Lowercase Greek letters Gray Thing Herbivore that ~2 ---f x3 f, of IS-A paths include an IS-NOT-A paths contain only individual IS-NOT-A positive I’ is a collection plus the links referents. paths are sequences links, and IS-A nonmonotonic A network negative relational links. these four types of links, IS-NOT-A Likes ft 2 such as a and r will range in the degenerate case, single or class nodes, or the null path. If Q is null u -+ y should be read as x ---f y, and x ---f c j+ y should be read as z f, y. The notation ~1 3 . . . -+ xn refers to a path of length whose ith element is zi. Other occurrences variables do not imply a continuous chain Likes P Royal Elephant ample, -0 case \ 9 P Figure don’t 1: Herbivores like themselves. like gray things, that elephants They -------> ... + Celebrity and paths taxonomic paths, Figure paths written the any nodes of form yl, where 5 ... t yi respectively, d -% 5 or 0 form th e 3 $ as ~1 -+ link may be u and T must be positive ym (or x, the 8 relational ym) must path E + be T + e + T t C, which says that Ernie is royal elephants are elephants, elephants herbivores gray, royal elephants like gray things, are elephants (again), royal elephant, so Ernie likes Clyde. The definition of inheritability governs Movie Star in common notations, in expanded ... t and x, e 3 h 3 g + a royal elephant, Rock Star a path . . . ----f ym and ? = ym + 1 generates are herbivores, first backward Tl le components by $;. whose it is not necessarily u has are often R Xn ----f ym + in I’. denotes ~~-1. r = yr + are replaced x, subsequence Relational ‘i. CT+ are x1 and z,; to be the forward for the same path. ... + 0 the We define but royal x1 3 elements with 22 + @ Ernie Clyde the path and last n of subscripted of xi’s; for ex- elephants are and Clyde is a the way paths may be extended to form new paths. A set of paths @ is perfect iff every one of its elements is inheritable in @ and no path Beverly sions ofr. Frank not in @ is inheritable of a network Finally, Figure don’t 2: Opera admire The next stars admire other celebrities, but they rock stars. few sections introduce a notation for inheri- tance paths and extensions (nonmonotonic theories), followed by axioms for nonmonotonic multiple inheritance with relations. reflexive with and irreflexive an evaluation reflexives, 434 We will then extend statements. of the The inheritance-based and some linguistic Knowledge this system Representation observations. paper to handle concludes approach to if K: is a path, first link removed, removed. 4 in a. then We now present a nonmonotonic and to that [Touretzky of [Touretzky, with is K with is K with multiple individuals. et ad. 19871, ward reasoner, supersets the the last link Inheritance tem lar the ezten- perfect ButFirst and ButLast Taxonomic for classes We define I’ to be the minimal 19861. the system coupled The inheritance definition In the terminology is a credulous, extensions. sys- is simi- But of down- it uses OE which Sandewall on Touretzky’s path preemption, an improvement state only the axioms the negative path appropriate Basis for inheritability axioms substitution step: x + Induction: The [1986] original can of positive be derived from and “ft” links. of “+” y is inheritable path has proposed as definition. We paths; these by in @ iff x -+ y E @. K = x1 --+ . . . + ButFirst E ip. T2. ButLast E a. T3. There is no path x1 + T4. There is no w such notions of the tradiction both that E (a. (Contradiction) xl classic of contradiction nonmonotonic keeps being paths present lows subclasses + r1 ---f w --f r2 + and preemption Con- conflicting conclusions from extension, example. as in Reiter’s Preemption the properties from superclasses, even in the presence links. See [Touretzky et al. 19871 and [Horty If Fred crook, there dislike Dick. time then return that cept for the use of off-path Ethering- ton, who with Reiter preemption. David was the first logic to translate [Etherington recently produced a version of this system as well [Etherington 1987131. Basis step: x -% y is inheritable ButFirst E @. R2. ButLast E @. R3. There is no path x1 ---f u $ in default ... + 5; t tax1987a], in Cp iff x 5 Induction step: The path K: = x1 + . . . + yr is inheritable in @ iff: Rl. logic y E a. Z, -% ym t There are no w, w’ such that X~ E ip, yr W F + ri --f x:1 ---f rl + 20’ + ~1 or xn, respectively, w + that and similarly 71 or 74 (Preemption) Fred will like of contradictory relational but Fred dislikes gray things. elephants because directly, hand, double such are derived back like gray and elephants, the section, in greater as John and detail. is a philoso- ordinary we can Elephants relational For example, are herbivores things hence from on themselves. they’re yr = xi for some 6.1 plicit The following which in and are gray, from derive Elephants like The like themselves. lat- i, 1 < i 5 n. eflexive rule creates can be inherited Paths a new implicit reflexive path by lower nodes: Let K be a relational path of form xl --, . . . + x, 3 ym + . . . + yl where x1 = yi . Then the implicit reflexive path K t SRl. ButFirst E a. SR2. ButLast E a. SR3. There is no path diction) zr -+ u fi 5 t yl E @. SR4. There is no path diction) x1 ----f Q $ 7 0 0 Notice present an is inheritable paths is: Fred that “other” green the in Qi. The in Q iff: things, infer can still rule Amazon parrots Amazon infer does reason statement. and Amazon can’t An example Readdefini- section to the discussion the definitions 1, since elephants 72 + may be null and w’ may equal yi or ym, respectively. likes animals statements. with the preceding These paths confuse themselves. R . . . ---f x, ---f 0. Implicit reflexive paths, xi + Herbivores 74 --) ym E a’, and w’ E @‘, where 71 or r2 may be null and w may equal reflexive (Contra- y1 E <p. (Contra- diction) R4. proceed “self” other means into default an elected ter conclusion would be written e ---+ h 3 g t e t 0. Implicit reflexive paths take the general form zr + . . . + R xn --) ym + . . . t y1 + 0, where the doubling back relations. 19861, ex- Ray and Dick and in it Fred does not to skim this and the following here to study are of form on the Figure Inheritance inheritance citizen and philosophers in- We next present inheritance axioms for binary Again, this definition is similar to [Touretzky, onomic of a relational path is Citicitizens don’t dislike elected Statements through, mention pher, would et al. 19871 two extension There will be two kinds of reflexive paths in our system. Explicit reflexive paths are derived from statements paths of redundant These each other. is a gullible tions are advised first gray. in the same is only one extension, Reflexive al- and examples. Relational they contradict they’re appear An example of preemption zens dislike crooks, but gullible is what they herit 5 are the definition. with to override for more details because never ers who are not yet comfortable inheritance in the same Nixon/pacifist can We are now ready to introduce r f+ x, (Preemption) The like them paths because 6 x+1 E Q, and w f, x, E a, where 71 or 72 may be null, and zu may equal x1 or ~~-1, respectively. heart he won’t reasoning crooks. xn is inheritable in @ ifX Tl. other Amazon not is that For parrots parrots E de. (Contra- require K itself to be K can be preempted example, given are parrots don’t like other parrots t and Amazon like Amazon by Parrots like are green, parrots, parrots, but we we like themselves, In one extension animals; in the Tour&&y and Tbomason 435 6.2 Inheritance Let K be a reflexive R - xn ym explicit + of Reflexive path, ... + reflexive i.e. y1 + path; Paths a path otherwise of reflexive To allow a reflexive to be preempted relational link, preempting d -+ x, we require relational E a. Induction is reflected x 3 yr t direct link x, path. below. by an ordinary and tail nodes in clause x1 + S5 below. in <P iff x 3 K = x1 + m possibly -% 0; path See Irreflexive Let 2 JJi @ : y mean y must struct otherwise, . . . -+ x, 3 ym t is a with x1 # yr) is inheritable stars E ip. ... t s2. ButLast E a. gray things, ate implicit s3. There is no x1 ---) CT$ s4. There is no w such that @, ---) 71 - and w $; statement.) s5. There 72 - 0 are no ~1,202 w2 + 73 - ’+ 73 6.3 ordinary Statements 72 + by explicit such that E a, x1 + 7; - In order to make the individual 2, by more about R - If a * Q -+ r +- @ E @, then inheritable in @. In Figure 1 > this axiom derives Clyde from the inherited path Clyde 6.4 Modification R - a ---f (T 4 r t Clyde does not like does not like himself. Relations of a conlike gray things but Royal elephants we should relations by adding do not like themselves, like royal elephants. To achieve the rule for inheriting ordinary an additional are still free to infer the slightly Royal elephants Knowledge like other restriction, more restricted royal elephants. Representation R5. path back ... - x, p > 0. is of 5 @ : c), star Beverly admires is generated on itself. In Figure Bev- (hence a Frank when an ordi- Implicit ... t zp t and elephants paths for both love other (o 5 is a movie that paths take %1 ‘@:ym 1, since + herbivores love are gray herbivores, we generElephants love themselves and The elephants. K be a path Irreflexive of form xi + ~1 + y, where y = x1 and individual. Then the implicit xn 3 zp + ... t ~1 t ORl. ButFirst E Q,. OR2. ButLast E a. OR3. There latter path is written Paths ... + x, -% zp t ... t y is a class rather than an irreflexive path ~1 --f . . . - @ : y is inheritable in Q, ifE But we is no path x1 + c $; ?- t y E @. (Contradic- tion) OR4. There is no path x1 -+ (T $ 71 t @ : i-2 t y E Q. (Contradiction) As was the case with implicit require to Ordinary conclude Implicit Let a is A reflexive statement should block inheritance tradictory ordinary relation. Thus, if Herbivores not infer Royal elephants this behavior we modify 7.1 “self” agree with its statements about a, we add the following axiom. Note that it is an implication, not an equivalence: SI. con- path wr $ Individuals a’s statements irreflexive E 0), 7-i --j and either y’s” for this [email protected]:e. w1 -+ Wl - wr E <p. (Preemption relation.) About Elephants reflexive x1 ---f 71 + ym E Q, (or m = 0), w2 E @ or w2 + specific x, w + ym E @ (or m = (P reemption E a. 202 - x1 ---f 71 + w ---f 74 + R to other celebrities, and Frank doubles yr , with xn E Forexample, star, 11 + 72 - statement.) an individual, explicit other irreflexive relation ButFirst Xl not An admire ([email protected]:ctmtF). An implicit nary (Contradiction) reflexive “x’s are in relation sense. we may Sl. @ E a. w + formxl+...+x,[email protected]:y,t...tyr. the form 7 t 71 - ym E @, and w $; @ E @. Statements be a class, to make celebrity), in @ ifE 436 7 Node x1 - 7; + w - by explicit erly is an opera @ E a. 0, in which case there 7; - (Preemption if opera The path 0 (with of the for an explanation forthcoming] @ is inheritable step: ... t implicit is no w such that @, Xl - K is an appears link to be on the same This step: paths the head [Touretzky & Thomason, of why this is necessary. Basis 0 then it is an The rule for inheritability path There of form x1 ---r . . . ---f If m = 0. R5. 7.2 K to be present in a; Inheritance reflexive it could paths, we do not be preempted. of Irreflexive Paths The 0 node never stands nected to a node indicating alone; it always appears conthe referent of the word “other.” To simplify below, ture the definition 0 : y as a single [email protected]:y) x, we will treat In particular, the struc- ButLast(a 2 isaZF. Let K be a path 21 + node. O:ym -% @:ym.) + of form x1 ---f . . . + .. . + The path statement, 01. ButFirst E <p. 02. ButLast E a. y1. x, (If p = 0 there K: is inheritable -% zp t is a direct in @ iE ... t link 03. There is no path diction) ~1 + There x1 + u 8 5 t y1 E a. constructibility (Contra- fident 04. is no path 0 $ 71 t @ : 72 t yr E @. (Contradiction) There 05. are no w, w’ such that x1 + ~1 + w + 72 - 7-i zp E Q (or, if E a, y1 --) 71 --f w’ = 0, then y1 -+ 7; + w’ ---f ri + ym E a), and P $ 8 : w’ E G’, where 7-r or 72 may be null and w may equal x1 or x,, respectively, and similarly r[ or 7; may be null and w’ may equal yr or zp (or W ym ifp = O>, respectively. irreflexive statement.) There (Preemption are no w, w’ such that x1 + by explicit 71 + W & w’ E a, where 71 or r2 may be null and w and similarly ri or may equal x1 or xn, respectively, r2/ may be null and w’ may equal yr or zp (or yna if respectively. by more specific then x1 # yr. does opera due to 06. cause do not Mick; this admire We do no derive Modification Irreflexive statements tradictory ordinary like gray things, (Non- Beverly admires relations. For example, Wild elephants do not like other the inference that There xn We may are no w, w’ such that x1 + rr + w --, 72 + r; w’ --, 7-2’---+ ym E 4p, and 0:w’ E a. generics in classical admit first exceptions, order logic. they cannot We therefore a statements such as Elephants are gray. We the system by adding axioms for reflexive tleties in the phrasing system Although there of the new axioms us to go into, the general should allowed with resent generic then extended not permit that started inheritance statements. system be expressed nonmonotonic and irreflexive us to rep- are some which space does nature of the extended tance the problem with default definition 1987b]. logic for- which replaced rules, to filter This did indeed still relied on the set of exten- was necessary override dis- to reflexive default inheritance, of our new system to ensure superclasses. into default A similogic would to be straightforward. Another advantage it does not of our require path-based logic-based ive pronouns map of reflexives. and the phrase paths by extracting Elephants Reflex- “other may be translated the first The like themselves. is to constrain mapping between inwhich does not exist treatments to @ nodes, Inheritance sentences formulation the use of variables There is a natural and surface structure for predicate node, interior y’s” to En- the relation, g t e t of the path @ as serves as an argument or justification for the statement. In conclusion, there is no a priori reason why the rich structure of human language should predicate logic-based representation. developed to describe mathematics. power of classical modal operators, map conveniently to a Logic was originally One can increase the logic by adding nonstandard and extra truth values, but other formalisms may in some We find path-based formalisms cases prove more natural. convenient for inheritance reasoning, of reflexives than 9 and their logic- based Some treatment more natural formalisms. Linguistic There are two possible other word “other.” of extensions. Etherington’s notation [Etherington the inferential of preemption) relational distance subclasses glish of extending statements. of ordinary formalisms. for them as well, since we (the determiner and irreflexive that semantic will be valuable ordering stars detest be clear. the existence) of Observations sub- pretation One thing we have not yet done is prove the constructibility (or at least have solved expressive quantifiers, Discussion Since than current and the last node, as when we read e ---f h 2 E a’, y1 - W p+ 8 ele- Wild elephants are no instances network power maps to @:y. if Herbivores This is accomplished by R6 below. like elephants. still infer Wild elephants like themselves. R6. of con- even if there pressive co-referentiality. heritance paths can also block the inheritance but be- Beverly about classes systems based This difference becomes more are added to the language, be- like themselves, Our formulation appear 07. relations reflexive statements. From Herbivores like gray for example, we can derive the generic conclusion lar translation Beverly of preemption constraint, irreflexive to operate within a logic has greater ex- sions to Ordinary we want to block phants, stars, is an instance of the non-coreferentiality 7.3 rock and ordi- We are con- elephants in the network. Some researchers may still prefer default logic framework, since default that stars not admire explicit things, our inferential coreferentiality.) Since of reflexive only 19861. cause relational paths that double back on themselves can generate reflexive paths even when a network contains no our path-based yr are individuals containing to constructibility. on default logic cannot. apparent when reflexives mulation relation.) If x1 and 07. (Preemption addition no obstacle Elephants w + 72 + xn E a, y1 71 w’ r; zp E @ (or, if p = 0, then yr --f ri + w’ + r.$ --) yrra E a), and P = 0), ordinary that presents for networks was given in [Touretzky, Our system can derive new statements as well as about individuals. Inheritance xn 06. proof nary relations However, a of the sentence depending other which than is that himself each rock star or herself. Touretzky Rock on the scope of the We have so far been using the narrow of “other,” all celebrities interpretations celebrities, This and Thomason interdetests is imple- 437 mented by clause 07 in the definition of inheritability for irreflexive paths. The alternate, broad interpretation Rock stars detest other celebrities is that rock stars detest celebrities other than rock stars. We will not formalize this second interpretation here, but it appears straightforward to handle. mixed The two uses by introducing of “other” a new node can type even be inter- to denote broadly scoped “other .” Note that the distinction between narrow and broad scope disappears when the origin of the relational link is an individual, can only mean musicians while e.g., Trombone Hurry “musicians players is jealous other are jealous of other than of other himself,” is musicians ambiguous. Similarly, for relational ence the same sense, e.g., elephants class, e 2 other In English links whose head and tail refer- only the narrow @ :e could than one can substitute compare that Elephants the expression of a relational love other of the use of “other” part of the formal system requires membership. a subset path @ : c means For “politicians intimidate the that true any politicians English networks Finally, actually semantics to require accompanied is not Politicians that crooks by imposing any link of form by a link x ---f y, unless we acknowledge that We can get a restriction x 5 on 0 : y be x = y. our account of “other” is far from complete. For example, “other” has an existential interpretation as well as the universal one we have been using. A sentence means like Roger some women who is not his wife. sense is appropriate, For some speakers, other fools than Semantics around with other women his wife, not every woman mainly determines which but there may also be syntactic cues. Elephants love other elephants is pref- erentially understood as an existential because they expect the universal interpretation to be expressed Elephants love each other. Acknowledgements This research dation under to Jeff Horty was supported grant number for helpful by the National Science FounIRI-8700705. We are grateful discussions, whose NETL system provided investigation of reflexives. and to Scott the original Fahlman, impetus [Carlson 19821 sentences.” Carlson, Journal for our [Etherington Knowledge Representation terms of Philosophical 1987a] Etherington, nonmonotonic reasoning gence 31, pp. 41-85. [Etherington itance and D. and generic Logic 11, (1982), 1987b] Etherington, with inferential W. systems.” hierarchies D. W. “More exceptions: distance.” Proc. “Formalizing Artificial Intelli- on inher- default theories Seattle, AAAI-87, 352-357. 19791 Fahlman, E., S. NETL: a System for MIT Using Real- World Knowledge. Massachusetts, 1979. R. H., and [Horty et al. 19871 Horty, J. F., Thomason, Touretzky, D. S. “A skeptical theory of inheritance in nonmonotonic 87, pp. 358-363. May., semantic R., networks.” Logical MIT Derivation. Press, Form: Its Cambridge, Proc. AAAI- Structure and Massachusetts, 1985. [Sandewall 19861 Sandewall, ence rules for multiple Proc. [Thomason 19761 Montague E. “Non-monotonic inherinheritance with exceptions.” vol. 74, pp. 1345-1353. IEEE Thomason, grammar.” tee, ed., Academic R., “Some In Montague Press, extensions Grammar, New York, 1976, of B. Par- pp. 77-117. [Thomason 19831 Thomason, R., “On the semantic interpretation of the Thomason 1972 fragment.” Indiana University Linguistics Club, 1979. [Touretzky, 19861 Touretzky, D. S. The Mathematics Inheritance Systems. Morgan-Kauffman, 1986. of [Touretzky et al. 19871 Touretzky, D. S., Thomason, R. H., and Horty, J. F. “A clash of intuitions: the current state of nonmonotonic multiple inheritance systems.” Proc. IJCAI-87, Milan, pp. 476-482. [Touretzky & Thomason, forthcoming] Touretzky, and Thomason, R. II. “An inheritance-based of generic reflexives.” Forthcoming technical Carnegie Mellon D. S., theory report, University. [van Benthem & ter Meulen 19851 van Benthem, A. ter Meulen, eds., Generalized Quantifiers ral Language. 438 1985. “Generic G., MIT of Anaphora. pp. 145-182. [May 19851 in general, an individual politihimself.” It doesn’t are crooks. J., A Grammar [Aoun 19851 Aoun., Press, Cambridge, Massachusetts, Representing and Press, Cambridge, it usually example, [Altham 19711 Altham, J. E. J. The Logic of Plurality. Methuen and Company, Ltd., London, 1971. [Fahlman @:m) substituthat here is that but do not conclude from this that cian who is also a crook intimidates imply This in English presented are iden- (e 5 cannot be true unless Politicians our current set of axioms, the link intimidate other crooks are crooks is true. With p 3 love “each other” mammals with Elephants love each other (e -% 8 : e). tion is mandatory for some speakers. One aspect makes elephants themselves. when the first and last nodes tical: interpretation only mean References of Foris Publications, Dordrecht. J. and in Natu-