P. Chew and M. Machtey, \A note on structure and looking back applied to the relative complexity of computable sets," J. Comput. Sys. Sci., Vol. 22 (1981), 53-59. 10  Home/Search   Document Not in Database   Summary   Related Articles   Check

....0 1 . For each k 0, we define PF = DTimeF(x jxj ) the class of functions computable in O(n ) time on a deterministic multi tape TM, and PF = k 0 PF = the class of poly time computable functions. Let LSlow = f f 2 PF f is nondecreasing and unboundedg. By standard results [CM81,LLR81], for each recursive, increasing, unbounded f , there is an s 2 LSlow that grows slower than the inverse of f in the sense that, for all x, s(f(x) x. QF s = DTimeF(x jxj (log jxj) Delta s(jxj) for k 0 and s 2 LSlow. By standard results from [HS65,HU79] for all k 0 and s 2 LSlow, ....

P. Chew and M. Machtey, A note on structure and looking back applied to the relative complexity of computable functions, Journal of Computer and System Sciences 22 (1981), 53--59.

....set in P such as ; These alternations may take a very long time and thus cause B to look like an easy set for very long sequences of input lengths. This technique is ubiquitous in the area. For instance, see Balc azar et al. 7] Balc azar and Diaz [6] Landweber et al. 18] Chew and Machtey [9], Homer [12] Regan [21] Schoning [22] and Ambos Spies [1, 3] It is even the case that most applications of the so called speedup technique have at their heart a Ladner type strategy. We refer to, for instance, Downey [10] Shinoda Slaman [23] Ambos Spies [2] or Shore Slaman [24] One of ....

P. Chew and M. Machtey, "A note on structure and looking back applied to the relative complexity of computable sets," J. Comput. Sys. Sci., Vol. 22 (1981), 53-59.

....Lipton, and Robertson [15] further improved on this: actually any recursive set strictly above ; bounds a minimal pair. In the sequel these results were extended and the presentation of their proofs was substantially improved by several authors including Mehlhorn [17, 18] Chew and Machtey [11], Balc azar and D az [3] and Sch oning [30, 31] Then Ambos Spies [1] showed a general embedding theorem that comprises several preceding results as special cases: if r is equal to P T , to P m , or to one of several variants of polynomially time bounded truth table reducibility P ....

Paul Chew and Michael Machtey. A note on structure and looking back applied to the relative complexity of computable functions. Journal of Computer and System Sciences 22:53-59, 1981. 44

....Lipton, and Robertson [14] further improved on this: actually any recursive set strictly above ; bounds a minimal pair. In the sequel these results were extended and the presentation of their proofs was substantially improved by several authors including Mehlhorn [16, 17] Chew and Machtey [11], Balc azar and D iaz [3] and Schoning [28, 29] Then Ambos Spies [1] showed a general embedding theorem that comprises several preceding results as special cases: if r is equal to P T , to P m , or to one of several variants of polynomially time bounded truth table reducibility P tt , ....

Paul Chew and Michael Machtey. A note on structure and looking back applied to the relative complexity of computable functions. Journal of Computer and System Sciences, 22:53--59, 1981.

.... sets and other complexity classes (e.g. NP and P) under Part of this work was done when the author was a PhD student at the University of Heidelberg under the direction of Professor Ambos Spies 1 polynomial time reductions were studied by Ambos Spies [1, 2, 3, 4, 5] Schoning [18, 19] Chew [9], Schmidt [17] and others in a series of papers. Berman and Hartmanis [7] made a careful analysis of the p m reducibility in the complexity class NP and then conjectured that all p m complete sets for NP are polynomial time isomorphic. This conjecture has been one of the most important questions ....

P. Chew and M. Machtey. A note on structure and looking-back applied to the relative complexity of computer science. J. Comput. System Sci., 22:53--59, 1981.

....set in P such as ; These alternations may take a very long time and thus cause B to look like an easy set for very long sequences of input lengths. This technique is ubiquitous in the area. For instance, see Balcazar et al. 7] Balcazar and Diaz [6] Landweber et al. 20] Chew and Machtey [10], Homer [14] Regan [23] Schoning [24] and Ambos Spies [1, 3] It is even the case that most applications of the so called speedup technique have at their heart a Ladner type strategy. We refer to, for instance, Downey [11] Shinoda Slaman [25] Ambos Spies [2] or Shore Slaman [26] One of ....

P. Chew and M. Machtey, "A note on structure and looking back applied to the relative complexity of computable sets," J. Comput. Sys. Sci., Vol. 22 (1981), 53-59.

....set in NP that is not NP complete. Proof: By [6, 14] if P6=NP then SAT is 2 p terse. Assuming P6=NP, Theorem 8.16 yields a set in NP that is 2 p terse but not NP complete. Ladner s techniques have been extended in many papers dealing with the structure of the p m and p T degrees [2, 3, 22, 29, 53, 55]. For example, it is known that the p degrees are dense. Ambos Spies codified the constructions in a particularly nice way, leading to easy proofs of virtually all the previous results, as well as some new results. We apply his techniques to yield many theorems about the structure of the p m ....

P. Chew and M. Machtey. A note on structure and looking back applied to the relative complexity of computable functions. J. Comput. Syst. Sci., 22:53--59, 1981.

....set in P such as ; These alternations may take a very long time and thus cause B to look like an easy set for very long sequences of input lengths. This technique is ubiquitous in the area. For instance, see Balc azar et al. 7] Balc azar and Diaz [6] Landweber et al. 18] Chew and Machtey [9], Homer [12] Regan [21] Schoning [22] and Ambos Spies [1, 3] It is even the case that most applications of the so called speedup technique have at their heart a Ladner type strategy. We refer to, for instance, Downey [10] Shinoda Slaman [23] Ambos Spies [2] or Shore Slaman [24] One of ....

P. Chew and M. Machtey, "A note on structure and looking back applied to the relative complexity of computable sets," J. Comput. Sys. Sci., Vol. 22 (1981), 53-59.

....by Turing machines that run in linear time and log space. Let C 1 ; C 2 be such that D C 1 [ C 2 and both D n C 1 and D n C 2 are scfv. Then at least one of C 1 ; C 2 is not recursively presentable. If C 1 is r.p. and FIN D n C 2 , then C 2 is not r.e. presentable either. Corollary 4. 6 ([LLR81, CM81]) If NP 6= P, then NP n P is Delta 0 2 presentable but not r.p. or r.e. presentable. In particular, J NPnP = 2 Q 0 2 . The same goes for REC n P and EXPTIME n P. We remark that every P 0 3 class is either r.e. presentable or the difference of an r.e. presentable class with the r.p. class ....

P. Chew and M. Machtey. A note on structure and looking-back applied to the relative complexity of computable functions. J. Comp. Sys. Sci., 22:53--59, 1981.

....refinements of Sipser s classes and notions of log time reductions, following on from recent work by Cai, Chen, and others. 1 Introduction Many theorems about the structure of familiar complexity classes such as P, NP, and PSPACE have been obtained by a technique called delayed diagonalization [Lad75, CM81, Sch82, MY85, Amb85a] (see also [BDG88] For instance, there are languages E in PSPACE such that E is not in LOGSPACE and E is not PSPACE complete under log space reductions ( log m ) Moreover, the structure of such languages E under log m embeds all countable partial orders. On hypothesis NP 6= P, there is a ....

P. Chew and M. Machtey. A note on structure and looking-back applied to the relative complexity of computable functions. J. Comp. Sys. Sci., 22:53--59, 1981.

....refinements of Sipser s classes and notions of log time reductions, following on from recent work by Cai, Chen, and others. 1 Introduction Many theorems about the structure of familiar complexity classes such as P, NP, and PSPACE have been obtained by a technique called delayed diagonalization [Lad75, CM81, Sch82, MY85, Amb85a] (see also [BDG88] For instance, there are languages E in PSPACE such that E is not in LOGSPACE and E is not PSPACE complete under log space reductions ( log m ) Moreover, the structure of such languages E under log m embeds all countable partial orders. On hypothesis NP 6= P, there is a ....

P. Chew and M. Machtey. A note on structure and looking-back applied to the relative complexity of computable functions. J. Comp. Sys. Sci., 22:53--59, 1981.

....in P such as ; These alternations may take a very long time and thus cause B to look like an easy set for very long sequences of input lengths. This technique is ubiquitous in the area. For instance, see Balc azar et al. 6] Balc azar and Diaz [5] Landweber et al. 16] Chew and Machtey [8], Homer [11] Regan [18] Schoning [19] and Ambos Spies [1, 3] It is even the case that most applications of the so called speedup technique have at their heart a Ladner type strategy. We refer to, for instance, Downey [9] Shinoda Slaman [20] Ambos Spies [2] or Shore Slaman [21] One of the ....

P. Chew and M. Machtey, "A note on structure and looking back applied to the relative complexity of computable sets," J. Comput. Sys. Sci., Vol. 22 (1981), 53-59.

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P. Chew and M. Machtey, \A note on structure and looking back applied to the relative complexity of computable sets," J. Comput. Sys. Sci., Vol. 22 (1981), 53-59. 10

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