J. Hartmanis. Independence results about context-free languages and lower bounds, Information Proc. Lett. 20(5):241--248, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[FLO83] and Loebl and Nesetril [LN88] have found problems in the areas of Semantics of Programming Languages and of Data Structures (respectively) that give rise to such functions and, consequently, to non provable (as far as finitistic methods go) true statements in these fields. Hartmanis [Ha85] and Kurtz O donnell and Royer [KOR87] similarly prove that, for every recursive proof system, the existence of non trivial languages that are not provably infinite. 4. Kurtz O donnell and Royer chose to attack the problem of provability by strengthening a natural proof system (rather than taking ....

Hartmanis J., "Independence results about Context Free Languages and lower bounds", Information processing Letters 20 (1985) 241-248.

....not in P, etc. Under slightly stronger conditions, such L 0 s may be found within every L 2 R Gamma S. 1 Introduction In a recent paper, Hartmanis shows how to use diagonalization techniques to obtain logical independence results for automata theoretic and complexitytheoretic facts [5]. These results are representation independent in the sense that they do not depend on the choice of particular opaque programs to represent sets involved in the statements of these facts. For example, he shows that, for each effectively axiomatized theory, there exists a contextfree language that ....

....than each provably total function. Then the assertion A is infinite, which may be expressed in Pi 2 form as (8x9y) y x y 2 A] is not provable, since its witness function grows essentially as fast as f . Hartmanis independence results (Corollaries 2.4 and 3.1 and Theorems 4.2 and 4. 3 of [5]) are weakened special cases of the following theorem. Definition 4.1 Let f : N Gamma N . A set A N is f (co)emaciated if, for infinitely many i, the ith (non)member of A is bigger than f(i) A class S of sets of integers is superrecursively (co)emaciable if for every total computable ....

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J. Hartmanis, Independence Results About Context-Free Languages and Lower Bounds. Information Processing Letters 20:3 (1985) 241-248.

....to gain insight into theorems of Ladner, Schoning and Ambos Spies [Lad, Sch, Amb] in structural complexity theory and to obtain independence results about complexity. These latter results are in the style of independence results due to Regan, Kowalczyk, Hartmanis, and Kurtz, O Donnell, and Royer [Rega, Kow, Har, Regb, Regc, KOR]. Theorem 8 Suppose C and D are disjoint subsets of S. Suppose there are programs c and d such that, for every finite set A, d j A [ c j A ) 2 C and ( c j A [ d j A ) 2 D. Then: SC is effectively Delta 0 2 inseparable from SD . The quantifier 8 1 from [Blu] means for all ....

J. Hartmanis. Independence results about context-free languages and lower bounds. Information Processing Letters, 20:241--248, 1985.

....it) For instance, for each fixed recursively axiomatizable proof system there is a low complexity language that is infinite, but for no Turing machine accepting the language can the proof system prove that that Turing machine accepts an infinite language. See [Reg96,Reg87,Reg88] and also [Har85,KOR87] 2. BS96] Let A be a counting property of circuits. The counting problem for A, Counting(A) is the set of all circuits c such that #(c) 2 A. 3. see [GJ79] For each complexity class C and each set B Sigma , we say B is C hard if (8L 2 C) L p T B] where as is standard p ....

J. Hartmanis. Independence results about context-free languages and lower bounds. Information Processing Letters, 20:241--248, 1985.

....such that for all Turing machines M accepting E, F cannot prove the first order arithmetical sentence L(M) 2 LOGSPACE. Schoning [Sch82] observed an analogous result for unprovable non membership of NP languages in P, on hypothesis NP 6= P. For further results of this type, see Hartmanis [Har85] and also [KOR87, Reg88] Author s current address: Computer Science Department, 226 Bell Hall, UB North Campus, Buffalo, NY 14260 2000. Email: regan cs.buffalo.edu y Author s current address: Theoretische Informatik, Universitat Wurzburg, Am Exerzierplatz 3, D97072, Germany. Email: ....

J. Hartmanis. Independence results about context-free languages and lower bounds. Inf. Proc. Lett., 20:241--248, 1985.

....such that for all Turing machines M accepting E, F cannot prove the first order arithmetical sentence L(M) 2 LOGSPACE. Schoning [Sch82] observed an analogous result for unprovable non membership of NP languages in P, on hypothesis NP 6= P. For further results of this type, see Hartmanis [Har85] and also [KOR87, Reg88] The languages E constructed above are commonly known as gap languages. To determine which language classes C admit construction of such gap languages, Schmidt Supported in part by the National Science Foundation under Grant CCR 9409104. Author s current address: ....

J. Hartmanis. Independence results about context-free languages and lower bounds. Inf. Proc. Lett., 20:241--248, 1985.

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J. Hartmanis. Independence results about context-free languages and lower bounds, Information Proc. Lett. 20(5):241--248, 1985.

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