| 释义 |
recursive, a. (and n.)|rɪˈkɜːsɪv|
[f. L. recurs- (see recursant a.) + -ive.]
1. Periodically or continually recurring. Now rare or Obs.
1790Loiterer 13 Mar. 7 Till your ear be so attuned to one particular measure, that your ideas may be spontaneously absorbed into the same revolving eddy of recursive harmony.
2. a. Math. and Logic. [after similar uses of G. rekurrent (D. Hilbert 1904, in Verhandl. des dritten Internat. Math. Köngr.), rekursiv (K. Gödel 1931, in Monatshefte f. Math. u. Physik XXXVIII. 179).] Involving or being a repeated procedure such that the required result at each step except the last is given in terms of the result(s) of the next step, until after a finite number of steps a terminus is reached with an outright evaluation of the result; recursive definition, a definition (of a function) which is either primitive recursive or (now usu.) general recursive; recursive function, a function which has or which may be given a recursive definition; recursive relation, a property of, or relation between, natural numbers whose truth value for all arguments is a recursive function; recursive set, a set of natural numbers whose defining property is recursive; general recursive adj. phr., applied to a function or relation which is recursive and is defined for all natural number values of its arguments; partial recursive adj. phr., applied to a function defined by a recursive process which for some or all values of the arguments does not terminate, leaving the value of the function undefined; primitive recursive adj. phr., applied to a function or relation which can be generated by primitive recursion and substitution from the zero, successor, and identity functions.
1934Kleene & Rosser Gödel's Undecidable Propositions Formal Math. Syst. (typescript) 3 We define the class of recursive functions to be the totality of functions which can be generated by substitution..and recursion..from the successor function x + 1, constant functions.., and identity functions. Ibid., A relation R shall be recursive if the representing function is recursive. Ibid., Recursive functions have the important property that, for each given set of values of the arguments, the value of the function can be computed by a finite procedure. Similarly, recursive relations (classes) are decidable in the sense that, for each given set of natural numbers, it can be determined by a finite procedure whether the relation holds or does not hold... The functions x + y, xy, xy and x! are clearly recursive. 1936Math. Ann. CXII. 727 In this paper we offer several observations on general recursive functions, using essentially Gödel's form of the definition. Ibid., Ordinary or ‘primitive’ recursive functions. Ibid. 729 A recursive function (relation) in the sense of Gödel..will now be called a primitive recursive function (relation). 1938Jrnl. Symbolic Logic III. 151 If we omit the requirement that the computation process always terminate, we obtain a more general class of functions, each function of which is defined over a subset (possibly null or total) of the n-tuples of natural numbers... These functions we call partial recursive. 1940Mind XLIX. 240 Preliminary considerations, such as..the exact specification of the rules for the use of recursive definitions. 1943Mind LII. 268 Quite elementary theorems, requiring for their proofs recursive arguments to take care of the indefinite number of variables involved. 1943Trans. Amer. Math. Soc. LIII. 44 A system E of equations defines recursively a general recursive function of n variables if, for each set x1,{ddd}, xn of natural numbers, an equation of the form f(x1,{ddd}, xn) = x, where f is the principal function symbol of E, and where x1,{ddd}, xn are the numerals representing the natural numbers x1,{ddd}, xn, is derivable from E..for exactly one numeral x. 1944Bull. Amer. Math. Soc. L. 285 In the present paper, ‘recursive function’ means ‘general recursive function’. Ibid. 288 Closely related to the technical concept [of a] recursively enumerable set of positive integers is that of a recursive set of positive integers. This is a set for which there is a recursive function f(x) such that f(x) is say 2 when x is a positive integer in the set, 1 when x is a positive integer not in the set. We may also make this the definition of the set being recursively soluble. For 2 and 1 may be regarded as the two possible truth-values, true, false, of the proposition ‘positive integer x is in the set’. 1962R. B. Braithwaite in B. Meltzer tr. Gödel's Formally Undecidable Propositions 12 An arithmetical function is recursive if it is the last term in a finite sequence of functions in which each function is recursively defined by a rule involving two functions preceding it in the sequence (or is the successor function or a constant or obtained by substitution from a preceding function). 1964E. Mendelson Introd. Math. Logic 125 Relations obtained from primitive recursive (or recursive) relations by means of the propositional connectives and the bounded quantifiers are also primitive recursive (or recursive). 1965Herman & Plassman tr. H. Hermes' Enumerability, Decidability, Computability i. 29 Today it is generally believed that every system of algorithms can be defined by recursive functions. This gives a deeper meaning to Gödel's result. Ibid. iii. 82 The essence of Ackermann's proof of the existence of a computable function which is not primitive recursive consists in defining a computable function which increases in a certain sense faster than any primitive recursive function. 1967Encycl. Philos. VII. 92/1 This Herbrand–Gödel–Kleene notion of general recursive function can be put in the context of instructions and computations discussed above. 1970Nature 19 Dec. 1234/1 Turing formulated his concept of an abstract computing machine; the functions computable by these machines are exactly the recursive functions. 1974A. Kenny tr. Wittgenstein's Philos. Gram. 34 Is there a further step from writing the recursive proof to the generalization? Doesn't the recursion schema already say all that is to be said?
b. Linguistics. Applied to a grammatical feature or element which may be involved in a procedure whereby that feature or element is repeatedly reintroduced; applied to a grammatical rule in which part of the output serves as input to the same rule.
1955N. Chomsky Logical Struct. Linguistic Theory (microfilm, Mass. Inst. Technol.) vi. 248 We will find many other reasons to question the validity of the extension of the notion of production to recursive production. 1957― in Janua Linguarum IV. 57 Bar-Hillel has suggested..that Pike's proposals can be formalized without the circularity that many sense in them by the use of recursive definitions. 1968J. Lyons Introd. Theoret. Linguistics vii. 326 The adverb is a recursive category..in the sense that one adverb may modify another. 1970― Chomsky viii. 90 It will be observed that rules (2), (3) and (4) are recursive, but in different ways. Rule (2) is left recursive; rule (3) is right recursive; and rule (4) is self-embedding. 1972R. A. Palmatier Gloss. Eng. Transformational Gram. 142 A recursive rule is a rule which reapplies indefinitely to its own output... The recursive power of a grammar, which resides entirely in the syntactic component, is its ability to generate an infinity of sentences... The recursive mechanism is the system of rules which account for the infinite properties of language... A recursive element is one from which strings can be derived that contain the same element... The recursive property of a grammar..is its provision for embedding sentences within other sentences. 1977Word 1972 XXVIII. 336 Logicians of the first half of the century had developed and used recursive grammars with such clarity that Chomsky's ‘application’ can hardly be regarded as a tour de force.
c. Computing. Applied to a statement, definition, subroutine, or the like, some part of which makes use of the whole of itself, so that its explicit interpretation requires in general many successive executions; applied also to languages, compilers, etc., which allow of such techniques.
Quot. 1958 uses the word in a context where ‘iterative’ would now be usual.
1958Commun. Assoc. Computing Machinery Aug. 10 The idea of recursive curve fitting has been in use for some time as a graphical technique for fitting curves ‘by eye’ to observational data. 1959Numerische Math. I. 45 The definition of expressions, and their constituents, is necessarily recursive. 1960Ibid. II. 312 It is then impossible to call in a subroutine while one or more previous activations of the same subroutine have not yet come to an end... We intend to describe..a means of removing the..restriction..; hence the name ‘recursive programming’. 1973C. W. Gear Introd. Computer Sci. v. 233 We can understand recursive procedures by imagining that many different copies of the procedure are available. 1979Page & Wilson Introd. Computational Combinatorics vi. 136 Since backtrack programming is closely related to tree searching we can consider using recursive techniques in our implementations.
3. Phonetics. A term sometimes used to refer to consonants accompanied by glottal closure or implosion. Also as n.
1924R. L. Turner in Bull. School Oriental Stud. III. 304 According to one of my informants, an m accompanied by glottal closure and distinguished from ordinary m, exists in Magarkurā, one of the Mongolian languages of Nepal. Prince Troubetzkoy refers to consonants in the Caucasian languages accompanied by complete closure of the glottis. These he calls ‘recoursives’, a convenient term I have anglicized as ‘recursives’; he indicates them by a dot above or below the letter. 1934Webster, Recursive, adj.,..formed with an inward movement of air caused by lowering the larynx with closed glottis;—said of certain consonants in Sindhi (g, j, d, b). 1974Encycl. Brit. Macropædia IX. 448/1 One major feature distinguishing Sindhi from the rest of the northwest group is the development of a series of imploded stops (also called suction stops and recursive stops), for b, d, j, and g. |