Contains reports on one research project.U. S. Air Force (Electronics Systems Division) under Contract AF 19(628)-2487Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 36-039-AMC-03200(E)National Science Foundation (Grant GK-835)National Institutes of Health (Grant 2 PO1 MH-04737-06)National Aeronautics and Space Administration (Grant NsG-496

Kuroda, S. -Y.

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XIX. LINGUISTICSAcademic and Research StaffR. JakobsonA. N. ChomskyJ. A. FodorM. HalleJ. J. KatzR. P. V. KiparskyG. R. Bedell IVT. G. BeverE. W. BrowneR. J. CarterP. G. ChapinJanet P. DeanR. P. G. DeRijkR. C. DoughertyJ. E. EmondsJ. L. FidelholtzProf. E. S. KlimaProf. J. KurlyowiczProf. G. H. MatthewsProf. Krystyna PomorskaDr. S. BrombergerDr. M. F. GarrettDr. J. S. GruberGraduate StudentsM. L. GeisR. GoldfieldJ. W. HarrisT. R. HofmannR. S. JackendoffL. JenkinsR. S. KayneJ. P. KimballR. L. MendelsohnDr. S.-Y. KurodaDr. A. SchwartzDr. D. E. WalkerG. B. GraggP. L. PetersonJ. J. ViertelAmy E. MyersA. J. NaroD. M. PerlmutterP. S. Peters, Jr.C. B. QuallsJ. R. RossM. S. SnowCarol A. SpielmanR. J. StanleyNancy H. WooA. A CHARACTERIZATION OF ESSENTIALLY CONTEXT-SENSITIVE LANGUAGESThe object of this report is to give a characterization of essentially context-sensitivelanguages (i. e. , context-sensitive languages that are not context-free) and, for thatmatter, a characterization of context-free languages among context-sensitive languages.We begin by introducing a restricted type of linear-bounded automata, linear-boundedautomata with erasure:1Definition. A lba M is said to be a lba with erasure if M contains in its alphabeta special symbol c that does not affect computation in any way, i. e. , if M scans asquare with the symbol j it will not replace it by another symbol, will not change itsstates, and will continue to move in the same direction as it has moved into the scannedsquare (namely, continue to move to the right (left) if it has moved to the right (left) orstay on the scanned square if it has stayed there). More exactly, the set of states ofM is divided into three disjoint subsets, S+ , SL , and S in such a way that M is in one-1' -- 1' -oof the states in S+1 (S 1 , or S 0 ) just after it has moved to the right (left or not moved),and further for every instruction of the form(, S) - (a, T, D)This work was supported principally by the U. S. Air Force (Electronics SystemsDivision) under Contract AF 19(628)-2487; and in part by the Joint Services Elec-tronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under ContractDA 36-039-AMC-03200(E), the National Science Foundation (Grant GK-835), the NationalInstitutes of Health (Grant 2 PO1 MH-04737-06), and the National Aeronautics and SpaceAdministration (Grant NsG-496).QPR No. 82Prof.Prof.Prof.Prof.Prof.Prof.215(XIX. LINGUISTICS)a is 4 , T is S, and D is +1, -1, or 0 according as S is inS , S_ , or SWe define the notion of acceptance by a iba with erasure M as follows: a string xis accepted by M if there is a computation of M which accepts x in the usual sense ofacceptance by a iba and if, moreover, at the end of the computation the tape containsonly 4 's. Acceptance by a iba with erasure in this sense may be referred to by accept-ance with erasure. In spite of the apparent specialization of the notion of iba, it is easyto see that lba with erasure are as powerful as iba in the general sense; we state withoutproof the following proposition.Proposition. If a language L is accepted by a lba, there is a iba with erasure thataccepts L with erasure.Now, let M be a iba with erasure and for each pair of states (Si, S ) of M let [Si, Sj]be the set of strings accepted by the lba with erasure Mi j which is the same as Mexcept that its initial and final states are redefined as Si and Sj, respectively.Let x = yz be in [Si, Sj]. We shall say that x is partitioned into y and z in [Si, Sj]if (i), given x as an input, there is a computation C of M with S. and S as the initial1 jand final states, respectively, which accepts x with erasure, and (ii) C is divided intotwo successive subcomputations C1 and C 2 in such a way that, during C 1 and C 2 , Mscans, respectively, only the subtapes on which y and z are printed originally. Inother words, at the beginning of the computation C, M is scanning the leftmost elementof x (it is also the leftmost element of y) in the state Si; at some moment M will scanthe leftmost element of z for the first time, say, in the state Sk; this must be the endof the subcomputation C 1 and at the same time the beginning of the subcomputation C2;at this moment y has all been replaced by 4 's; after this M will never go to the leftof the square that it is then scanning.If x is partitioned into y and z in [Si, Si], then y and z are elements of [Si, Sk] and[Sk, Sj], respectively, for some Sk . Conversely, if x is in [Si, Sj] , x = yz, and y andz are in [Si, Sk] and [Sk, Sj] with some Sk, respectively, then x is partitioned into yand z in [Si, Sj].Let x be again in [Si, Sj] and let x = vyw, where v, y, and w are non-null. We saythat x is decomposed into an interior y and two boundaries v and w in [Si, Sj] if thereis a computation C with the initial and final states S. and S. which accepts x witherasure and will be divided into three consecutive subcomputations C 1 , C 2, and C 3 inthe following way. At the beginning of the first subcomputation C 1 (i. e. , actually at thebeginning of the entire computation) M is scanning the leftmost element of v (i. e. , theleftmost element of x). During the subcomputation C 1, M stays inside the subtape onwhich v is originally written. At the end of C1 , M will drop off the right edge of thatsubtape and scan the leftmost element of y in the state, say, Sk . This is at the sametime the beginning of the subcomputation C 2 . During C2, M stays entirely inside thesubtape on which y is originally written. At the end of C 2 , this subtape contains only 4'sQPR No. 82 216(XIX. LINGUISTICS)and M drops off the right edge of this subtape, scanning the leftmost element of w inthe state, say, S . Then the final subcomputation C 3 begins. During C3, M may scanany square of the entire tape, but the subtape on which y is originally written does notaffect the computation in any way, since it now contains only c 's. At the end of C 3 (i. e. ,at the end of the entire computation) the entire tape contains only 's and M drops offthe right edge of the entire tape in the state S..JIn the sequel we are interested only in such cases for which the two boundaries v andw of a decomposition are elements (i. e. , strings of length 1).We have the following theorem.Theorem. Let M be a lba with erasure. If for any pair of states (Si , S ) and for anystring x of length greater than 1 in [St, Sj], x can be either partitioned into two sub-strings or decomposed into an interior and two boundaries of length 1, then the languageL(M) accepted by M is context-free.Proof. For each pair of states (Si., S.), let us now consider [S., S.] as a nonterminalsymbol for a grammar G, and let[Si, Sj] - [Si, Sk][Sk, Sj]be a rule of G for any i, j, and k. Further, let[Si, S j ] - a[Sk, S2 ]bbe a rule if there exists x in the set [Si. Sj] which can be decomposed into an interior yand boundaries a and b in [Si, Sj] , where y is in [Sk, S ] . Further, if an element a isin [S, Sj], let[Si, Sj] -abe a rule of G. Finally, let S be the initial symbol of G and for each final state Sf ofM letS - [S0, Sf]be a rule of G, where S is the initial state of M. Then, L(M) is just the language gen-erated by G.From the theorem we obtain a well-known result that was first due to Chomsky andSchiitzenberger:2Corollary 1. If a language is accepted by a pushdown storage automaton, it iscontext-free.Proof. A pushdown storage automaton can be simulated by a real-time pushdownstorage automaton. 3 It is easy to see that a real-time pushdown storage automaton canbe regarded as a lba with erasure with the property stated in the theorem.Actually, Haines derived this result from the equivalence of pushdown storageQPR No. 82 217(XIX. LINGUISTICS)automata to real-time pushdown storage automata. 3 In his construction of a context-freegrammar equivalent to a real-time pushdown storage automaton he refers to the instruc-tions of the automaton rather than to its states as we did above in the proof of ourtheorem, and it would seem that the significance of the correspondence of the grammarand the automaton is not sufficiently clarified. Furthermore, he did not relate the notionof real-time pushdown storage automata to that of linear-bounded automata in an explicitway. Hence, although our theorem is a fairly straightforward generalization of hisresult, it may be of some interest, in particular, when we relate it to the fact thatlinear-bounded automata are the automata-theoretic counterpart of context-sensitivegrammars, that is, that a language is context-sensitive if and only if there is a linear-bounded automaton that accepts it. 4 In fact, our theorem will give a certain character-ization of essentially context-sensitive languages. Let us first put it in the followingway.Corollary 2. A context-sensitive language L is not context-free if and only if forany lba M with erasure that accepts it there exists a string in it which cannot beaccepted by a completely localized computation of M.The precise meaning of completely localized computation is as follows. Given astring x, consider a computation C starting at the left end of x with a state S. and1ending at the right end of x with a state S , and with the tape filled entirely with 's.This computation is said to be localized if (i) x is partitioned into y and z in [Si, Sj],where y is in [Si, Sk] and z is in [Sk, Sj] for some Sk, and the computation C consistsof two successive subcomputations C 1 and C 2 which correspond to this partitioning ofx into y and z, or (ii) x is decomposed into an interior y and two boundaries a andb and C consists of three successive subcomputations C1, C 2 , and C 3 which corre-spond to this decomposition of x into a, y, and b. A computation C of x is said tobe completely localized if (i) x is of length 1, or (ii) if C is localized with respectto a partitioning of x into y and z and the corresponding subcomputations C 1 andC2 are both completely localized, or (iii) C is localized with respect to a decom-position of x into a, y, and b, and the corresponding subcomputation of the interiory is completely localized. In brief, a computation is completely localized if x isprocessed by gradually partitioning and decomposing it until single elements arereached.Let us reformulate Corollary 2 in terms of (general) lba. We define the notion ofpartition and decomposition with respect to a lba similarly to those notions with respectto a lba with erasure, except that in this case we do not require that after each subcom-putation of a partition or after the subcomputation of the interior of a decomposition thecorresponding subtape is filled with 's, but we do require of a decomposition that what-ever is left on the interior subtape after the second subcomputation of the decompositiondoes not affect the computation during the last subcomputation of the decomposition. 5 WeQPR No. 82 218(XIX. LINGUISTICS)then define the notions of localized and completely localized computations with respectto a lba in the same way as with respect to a iba with erasure.Now, given a lba M, we can construct a lba with erasure M' which is equivalent toM in such a way that if x is accepted by a localized computation of M it is accepted bya localized computation of M'. Indeed, M' can be constructed, in brief, by equippingM with a device that erases symbols (i. e. , replaces them by c) on the tape. Thus, wehave the following corollary.Corollary 3. A context-sensitive language L is not context-free if and only if forany lba M that accepts L there is a string in L which cannot be accepted by a com-pletely localized computation of M.L and M being as in Corollary 3, the set of all strings of L which cannot be acceptedby a completely localized computation of M is obviously a subset of L which is respon-sible for the degree of complexity of L as a context-sensitive language and itself cannotbe, for example, context-free. 6S.-Y. KurodaFootnotes and References1. By linear-bounded automata (hereafter abbreviated as iba) we understand nondeter-ministic linear-bounded automata. For the definition of iba, see S.-Y. Kuroda,"Classes of Languages and Linear-bounded Automata," Information and Control 1,207-223 (1964). Note, however, that boundary symbol # is dispensable in the defi-nition of iba, contrary to the remark made in footnote 3 of this paper; seeS. Ginsburg and G. F. Rose, "Preservation of Languages by Transducers" (to appearin Information and Control).2. N. Chomsky, "Context-free Grammars and Pushdown storage," Quarterly ProgressReport No. 65, Research Laboratory of Electronics, M. I. T., April 15, 1962,pp. 187-194; M. P. Schiitzenberger, "Context-free Languages and Pushdown Auto-mata," Information and Control 6, 246-264 (1963).3. L. Haines, "Generation and Recognition of Formal Languages," Ph. D. Thesis,M. I. T. , 1965.4. S. -Y. Kuroda, op. cit.5. In this case the definitions of partition and decomposition are, in a sense, behavior-istic. Thus, for example, we know that x is decomposed as ayb with respect to acomputation only after having observed that the computation ended in the prescribedway, and not by means of a particular instantaneous description of computation thatappears during the computation. Assume that we define [Si, Sj], analogously to theprevious case of lba with erasure, as the set of strings that are accepted (this timein the general sense without erasure) with the initial and final states S. and S., respec-tively. Then, on the one hand, it remains true that the product set [Si, Sk][Sk, Sj] isa subset of the set [Si, Sj]. But, on the other hand, it does not hold that a[Sk, S]b isa subset of [St, Sj] if there exists a string x in [Si, Sj] which is decomposed as aybwith y in [Sk, S]. Thus, the proof of our theorem cannot be generalized directly tothe general case without erasure.QPR No. 82 219(XIX. LINGUISTICS)6. Actually, the notions of iba and context-sensitive grammars are not essential restric-tions in developing the ideas pursued in this report, and our theorem and corollariescan be generalized as statements in terms of Turing machines and unrestrictedrewriting systems. On the one hand, this generalization is rather straightforwardconceptually, but, on the other hand, its exact formulation would become greatlyinvolved, since we must formulate a type of Turing machine that is allowed (not onlyto add at the edges but also) to insert a piece of tape between two squares. Thus wehave restricted ourselves here to dealing with cases within the machinery that isreadily available at present.QPR No. 82 220

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