Contains reports on two research projects.National Science Foundation (Grant GN-244

Darlington, J. L.

Klima, E. S.

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XVI. MECHANICAL TRANSLATIONV. H. Yngve Ursula Funke Elizabeth LandersCarol M. Bosche Elfriede Goody W. A. LeaElinor K. Charney T. R. Hofmann Ellie Reed LewisJ. L. Darlington L. Jenkins F. LiuJ. M. Dolan Muriel S. Kannel B. K T'souR. S. Fabry E. S. Klima R. J. ZatorskitA. REVISED PROOF PROCEDURE PROGRAMIn Quarterly Progress Report No. 76 (pages 260-270), we described a COMIT pro-gram that proves logical arguments expressed in a restricted form of ordinary English.Two examples were given of arguments that the logical evaluation part of the programwas unable to prove, even though it was verified by hand computation that the translationsinto logical symbolism were correct and that the arguments were valid. A more efficient1logical proof procedure program has since been devised. It not only proves argumentsthat the previous program was unable to prove, but it also substantially reduces thecomputation time (in some cases by a factor of 10) for some arguments and theoremsthat the previous program was able to prove. While the previous program was based2to a large extent on the Davis-Putnam proof procedure algorithm, the new program isan amalgam of the "one-literal clause rule" of Davis-Putnam and the "matching algo-rithm" of J. R. Guard. 3 Taking its point of departure from the Davis-Putnam algorithm,the program negates an input formula and puts the negated formula into prenex normalform and the matrix into conjunctive normal form. At this point, instead of manufac-turing generations of quantifier-free substitution instances (QFSI) and testing for con-sistency at certain specified points after the manner of Davis-Putnam, the programattempts to produce a proof on the basis of the "one-literal clauses" (i. e., conjunctscontaining no disjunction operators) in the matrix. By using Guard's matching algorithm,which enables one to test whether two clauses could generate any common QFSI, theprogram tests the one-literal clauses in the matrix for consistency. If two such clausesmatch negatively ("N-match"), i. e., if the two clauses could generate contradictoryQFSI, then the matrix is inconsistent and the original formula is valid. If the one-literalclauses are consistent, then the program attempts to generate additional one-literalclauses by N-matching the existing one-literal clauses against the polyliteral clauses.For example, the one-literal clauseF(y, P(x, y))- or FYPXY, in COMIT notation - N-matches the polyliteral clausenot-F(x, y) V G(y, x)This work was supported in part by the National Science Foundation (Grant GN-244).tVisiting from the University of Melbourne, Australia.QPR No. 77 315(XVI. MECHANICAL TRANSLATION)- or NFXY + GYX, in COMIT notation - to produce the one-literal clauseG(P(x, y), y))- or GPXYY, in COMIT notation. Whenever the N-matching procedure produces a newone-literal clause, it is tested for consistency against the existing one-literal clauses.If it is inconsistent with the others, it means that the original formula is valid, and therun is terminated. If one or more polyliteral clauses are produced by this N-matchingprocedure, they will have one less term than the original clause, and may thereforeproduce one-literal clauses during a further N-matching. If all possible N-matchingsfail to produce a contradictory set of one-literal clauses, or if there were no one-literalclauses to start with, then the program reverts to the older Davis-Putnam method ofgenerating QFSI. The program tests the QFSI for inconsistency after each generation,using an algorithm based on the "one-literal clause rule," which is the most efficient ofthe three Davis-Putnam rules for propositional logic, and which may be stated as follows:Given a formula C in conjunctive normal form, and a one-literal clause P of C, allconjuncts containing P are deleted from C, and all single occurrences of not-P aredeleted from C, thereby producing a reduced formula C' that is consistent if and onlyif C is consistent.Since the deletion of single occurrences of not-P may produce new one-literalclauses, the one-literal clauses must be tested for consistency after each applicationof the rule. The condition of inconsistency is the occurrence of two contradictoryone-literal clauses; and the condition of consistency is the deletion of the entireformula C. If the one-literal clause rule fails to prove C either consistent orinconsistent, then C is split into two branches by assuming a term P first true andthen false, and making appropriate cancellations according to a method describedby Quine. 4 The first branch is formed by deleting all conjuncts containing P andall single occurrences of not-P, and the second branch is formed by deletingall conjuncts containing not-P and all single occurrences of P. The second branchis stored at the front of a shelf, and the first branch remains in the workspacewhere an attempt is made to apply the one-literal clause rule to it. If this attemptfails, the formula in the workspace is split again in the same way, the firstbranch remaining in the workspace and the second branch being stored at thefront of the shelf. This procedure continues until a branch is obtained to whichthe one-literal clause rule can be applied. The consistency test terminates whenone branch is proved consistent (i. e., is entirely deleted), in which case theentire formula is consistent, or when all branches are proved inconsistent (i. e.,contain contradictory one-literal clauses), in which case the entire formula isinconsistent.J. L. DarlingtonQPR No. 77 316(XVI. MECHANICAL TRANSLATION)References1. The revised program is described in greater detail in a memorandum, "A proofprocedure program based on the 'one-literal clause rule' and the 'matching algorithm',"Mechanical Translation Group, Research Laboratory of Electronics, MassachusettsInstitute of Technology, March 1965.2. M. Davis and H. Putnam, A computing procedure for quantification theory,J. ACM 7, 201-215 (1960).3. J. R. Guard, Automated Logic for Semi-automated Mathematics, ScientificReport No. 1, Contract No. AF19(628)-3250, Project No. 5632, Task No. 563205,Applied Logic Corporation, Princeton, New Jersey, March 30, 1964.4. W. V. Quine, Methods of Logic (Henry Holt Company, New York, 1959),pp. 26-27.B. CONSTITUENT STRUCTURE AND WORD-ORDER RULES FOR GERMANThe following rules represent a segment of the grammar of contemporary standardGerman. This miniature grammar is directed especially toward the solution of threeclassical problems in German syntax: (a) the order of nominals (subject and objects)and adverbials with respect to one another and to the finite verb; (b) the way in whichindependent declarative sentences differ, in particular, from subordinate clauses withrespect to this order; and (c) the occurrence of positional es, as in "es kommen vieleLeute."The aspect of the constituent structure which is original is the treatment of the sub-ject nominal (Nom 0 ) as the first among equals in a series of verbal specifiers whichincludes the object nominals (including the direct object Noml and the indirect objectNom 2 ), as well as the adverbial adjuncts (among others, adverbials of location Loc andof manner Mod). All of these specifiers of the verb are immediately dominated by asingle common constituent labeled Sp.The constituent structure presented below differs from those assumed in previousstudies in the transformational syntax of German I by having an empty constituent (Topic)into which some immediate constituent of Sp may, or under certain conditions must, beincorporated by a transformational rule of Topicalization.The constituent structure resembles that assumed by Matthews and Rogovin, byBach, and by Bierwisch, in that the verbal complex originates at the end of the sentence,with finite verb last. A transformational rule of Finite verb placement correctly locatesthe finite verb in second position (i. e. , after the Topic or after the interrogative markerin independent sentences).The symbols used here have the following interpretations: S sentential clause, Winterrogative marker, Kn subordinating conjunction, R relative conjunction, Nom nom-inal (with subscripts 0,1' 2 referring to nominative, accusative, and dative, respectively),QPR No. 77 317(XVI. MECHANICAL TRANSLATION)Loc adverbial of location, Mod adverbial of manner, Adj predicate adjective, Sp speci-fier of the verb, V principal verb, sep separable prefix, Aux verbal auxiliary, M modalverb, T finite tense and person marker, VP verb phrase, # sentence boundary, +pp pastparticipial form, Pro personal pronoun, Reflex reflexive pronoun, inf infinitival form.Constituent structure:S --- # Vorfeld - VP #rW(Kn)Vorfeld -s. (R)KnTopicVP - (Sp) V - AuxSp - (Nom 0 )(Nom) (Noml)r(Pas sive)(M)(M)Aux - (Passive) Perfect(M)(Perfect)Topic --- esTransformational structure({od}) (Loc)(M) TI. Topicalization - conditional:1 2 3p31 2 3 41 4 3 055where [... sp... ]Sp signifies any constituent immediately dominated by SP.IIa. Intraspecifier word order for personal and reflexive pronouns - conditional:# X 1 X2 Pro X3 X 4 #XP ReflexJ Sp1 2 3 41 3 2 4Obligatory if: # X X2 Nom1 ProNom 2 Reflexi3 4p 4#4Optional if: #XI[X 2 - Nom 0 - Reflex- X 3]s X 4 #QPR No. 77 318(XVI. MECHANICAL TRANSLATION)IIb. Intraspecifier word order for newly introduced versus already mentioned speci-fiers - conditional:# X1[X - sp - sp2 X3 S X41 2 3 43 2 4The transformation has least conditions on its applicability when spl is a non-unique, not previously mentioned specifier (e. g., with indefinite article), and sp2is an already mentioned specifier (e. g., with definite article). The transformationis nonapplicable if spl is a pronoun and sp2 is a noun.III. Placement of the perfect auxiliary of modal verbs - obligatory (A similaroptional rule operates on sehen, hoeren in place of M.):# X (sep) V - M - Perfect - T #10 2 3 4==>1 3 2 0 4IV. Creation of past participles and finite verb:X V PerfectX jerfect Passive X1 2 3--- 1 2+pp 3X M T-XPerfect T -Passive1 2 3 4-- 1 2+3 0 4V. Conversion of uninflected verbs into infinitives - obligatory:VX M -XPerfectPassive1 2 3S 2 + inf 3where simple concatenation and affixation (indicated by +)are differentiated.QPR No. 77 319(XVI. MECHANICAL TRANSLATION)VI. Finite verb placement:VX +T-X#Topic Vorfeld PassierfecPassive1 0 2 3 41 3 2 0 4CommentsAmong the conditions on Topicalization are the following: (a) if no other constituentimmediately dominated by Sp has been topicalized, then a pronoun subject must betopicalized ("es kommen viele Leute" without topicalization and "viele Leute kommen"with topicalization, but no "es kommt er," only "er kommt"); (b) under certain con-ditions, a noun already mentioned in a preceding sentence and thus determined bya definite article or its equivalent must be topicalized ("der Mann kam an" versus"es kam der Mann an"); (c) certain pronoun objects may not be topicalized ( "sichereignete ein Unglueck" versus "es ereignete sich ein Unglueck" with positionales and without topicalization, on the one hand, and "ein Unglueck ereignete sich,"on the other hand).The rules for Intraspecifier word order account for the fact that the permissible ini-tial constituents following the interrogative word, if there is one, in questions and fol-lowing the conjunction in subordinate clauses do not coincide with the constituents thatmay occur as initial constituent (i. e., as Topic) in independent declarative sentences.Thus, im Zimmer may occur initially as Topic in an independent declarative sentenceeven if the sentence has a pronoun subject ("im Zimmer spielte er Karten"), but not ina subordinate clause ( "weil im Zimmer er Karten spielte,.. . "), whereas if the subjectis a noun (with an indefinite article, particularly), the permutation is possible even insubordinate clauses ("weil im Zimmer ein Gast Karten spielte,... "). Moreover, in asubordinate clause, a reflexive pronoun may occur initially ("weil sich ein Unglueckereignete, ... "). As we have mentioned, topicalization of a reflexive pronoun is notpossible ( "sich ereignete ein Unglueck").Transformational rules III, IV, and V account for the following facts: (a) in sub-ordinate clauses the finite perfect auxiliary (habe, hat, etc.) forming the periphrasticperfect of a modal verb occurs at the head, rather than at the end, of the verbal complex,i.e., in front of the principal verb and its separable prefix ("weil er den brief hatschreiben wollen,..." versus "weil er den Brief schreiben wollen wird, ... ," "weil erden Brief geschrieben hat,.. ." or "weil der Brief geschrieben worden ist,.. ."); thisorder holds only if the modal verb is preceded by an infinitive ("weil er den Brief gewollthat,..."); (b) the modal verb itself, when immediately preceded by an infinitive, assumesQPR No. 77 320(XVI. MECHANICAL TRANSLATION)its infinitival rather than its past participial form in this periphrastic perfect construc-tion; this is the case in independent sentences, as well as in subordinate clauses ("er hatden Brief schreiben wollen" versus "er hat den Brief gewollt"); (c) the ungoverned cita-tion form of verbs, as well as this special perfect, is the infinitive. The grammaraccounts for these characteristics by a particular interpretation of German syntax.Rule IV (Creation of past participles and finite verb) interprets verbal government asregressive; i. e., the form of the verb is determined by the constituent to its right. Thus,geben assumes the form gab if one particular form of T is to its right, namely third per-son singular past; geben assumes the form gegeben if either the Passive (werden) or thePerfect (haben or sein) is to the right. By rule V, any verbal stem that has not under-gone some process of suffixation (the incorporation of a marker of tense, a past parti-cipial marker, or, though not included in the present miniature grammar, a nominalizingor a present participial marker) automatically assumes its infinitival form. The factthat under certain conditions the modal verbs have infinitival form in the perfect is inter-preted as a result of the relative ordering of rules. The rule that places the perfectauxiliary at the head of the verbal complex (rule III) precedes the rules whereby the finiteor participial form of a verb, if it is to occur in one of those forms, is determined bythe constituent that follows it. The result of this ordering of rules is that the modal verb,since it is no longer followed by the auxiliary of the perfect, is therefore not subject toconversion into its past participial form and thus by rule V becomes an infinitive.The rule of Finite verb placement in independent sentences (VI) is ordered after therule for the Placement of the perfect auxiliary of modal verbs and has in its domain notonly any finite verb, including the auxiliary of the perfect, which occurs at the end of theverbal complex but also the perfect auxiliary relocated by rules III at the head of the ver-bal complex. In this way the occurrence of the infinitival perfect form of modal verbs inindependent sentences is also accounted for (not only "er hat den Brief gewollt" from anunderlying [er]Topic den-Brief-woll-Perfect-T, but also "er hat den Brief schreibenwollen" from an intermediate [er]Topic den-Brief-Perfect-T-schreib-woll).The following fact about German grammar lends support to the assumption (a) thatFinite verb placement in independent sentences (rule VI) occurs after the Placement ofthe perfect auxiliary of modal verbs (rule III) and after the rule for the Creation of pastparticiples (IV), and (b) that rule VI applies also to the relocated finite perfect auxiliarythat heads the verbal complex: namely, for the perfect of certain semimodal verbs likehoeren, the perfect auxiliary is found in subordinate clauses both at the end of theverbal complex, in which case the semimodal must be in its past participial form,gehoert, and at the head of the verbal complex, in which case the semimodal mustbe in its infinitival form ("weil er ihn abfahren gehoert hat, . . ." and "weil er ihnhat abfahren hoeren,... "). The double source of the finite perfect auxiliary correctlypredicts the occurrence of both infinitival and past participial forms in independentQPR No. 77 321(XVI. MECHANICAL TRANSLATION)sentences of otherwise identical form ("er hat ihn abfahren gehoert" from [er]Topic ihn-ab-fahr-hoer-Perfect-T and "er hat ihn abfahren hoeren" from an optional intermediate[er]Topic ihn-Perfect-T -ab-fahr-hoer).E. S. KlimaReferences1. G. H. Matthews and Syrell Rogovin, German sentence recognition, MechanicalTranslation 5, 114-20 (1958); R. B. Lees, Structural grammars, Mechanical Trans-lation 4, 5-10 (1957; E. Bach, The order of elements in a transformational grammar ofGerman, Language 38, 263-9 (1962); M. Bierwisch, Grammatik des deutschen Verbs,Studia Grammatica II (Berlin, 1963).QPR No. 77 322

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