R. J. Lipton, Model theoretic aspects of computational complexity, in Proceedings of the 19th Annual Symposium on Foundations of Computer Science, IEEE Computer Society, 1978, pp. 193--200.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....language they refer to the complexity of a specific algorithm or to the complexity of languages in relativised models of computation. 2. Another line of attack , is to show independence of natural problems relative to a weak (and consequently of limited interest) proof systems. The work of Lipton [Li78] Joseph and Young [JY81, JY85] and Leivant [Le82] can be viewed as taking this approach. 3. More recently, several researchers have succeeded in applying the basic idea behind the Paris Harrington result to computer theoretic questions. As shall be elaborated below, once a (recursive) function ....

Lipton R.J., "Model theoretic aspects of computational complexity", FOCUS 19 (1978), 193-200.

....special cases of one elementary theorem, proved by well known techniques. The simpler approach provides nicer insights into the fundamental logical mechanisms involved, and actually proves stronger results. For other material on independence results of potential interest to Computer Science see [9][2] 1] 4] 6] 7] 3] 11] In particular, Lipton and deMillo [9] 2] 1] show independence of complexity theorems from Basic Number Theory, a very weak subtheory of Peano s arithmetic, and how lower bounds are equivalent to independence results. Joseph and Young [6] and Leivant [7] discuss problems in ....

....techniques. The simpler approach provides nicer insights into the fundamental logical mechanisms involved, and actually proves stronger results. For other material on independence results of potential interest to Computer Science see [9] 2] 1] 4] 6] 7] 3] 11] In particular, Lipton and deMillo [9][2] 1] show independence of complexity theorems from Basic Number Theory, a very weak subtheory of Peano s arithmetic, and how lower bounds are equivalent to independence results. Joseph and Young [6] and Leivant [7] discuss problems in relating Basic Number Theory to Computer Science. Fortune, ....

R. J. Lipton, Model-Theoretic Aspects of Computational Complexity. 19th Annual IEEE Symposium on Foundations of Computer Science, Ann Arbor Michigan (1978) 193-200.

.... [19] In later work, subsequent to Parikh s feasibility paper, the rudimentary predicates were studied extensively by Wrathall [44] Harrow [17] Nepomnjascii [27] and Wilkie [42] both Wrathall and Wilkie essentially proved that this class was equal to the linear time hierarchy, but Lipton [23] was the first to explicitly prove the fact that the set of Delta 0 predicates is equal to the linear time hierarchy. Parikh was definitely interested in connections between computational complexity and the Delta 0 formulas, but instead of discussing the linear time hierarchy (since, in any ....

R. J. Lipton, Model theoretic aspects of computational complexity, in Proceedings of the 19th Annual Symposium on Foundations of Computer Science, IEEE Computer Society, 1978, pp. 193--200.

....need the fact that the graphs of addition and multiplication are computable in lineartime and a constant number of alternations when numbers are represented in binary. For addition, the algorithm is straightforward and in fact, deterministic. For multiplication, the reader is referred to Lipton [Lip78, page 198] which makes use of the Chineese Remainder Theorem (see also Wilkie [Wil79, Lemma 3.1] Lemma 1 also shows that we can decide in linear time if the m th bit of n is one when m and n are represented in binary. B.2 Finite variants We need a technical lemma. We need to see that the ....

R. J. Lipton. Model theoretic aspects of computational complexity. In 19th Annual IEEE Symposium on Foundations of Computer Science, pages 193-- 200, 1978.

....diagonalization methods formulated by Ambos Spies, Fleischhack, and Huwig [AFH87, AFH88] Both the mechanism and its relation to non r.p. classes appear to have applications beyond the results in this paper. Section 6 considers provability in certain formal systems, studied by Lipton and DeMillo [Lip78, DL80] and Leivant [Lei82] that are not recursively axiomatizable. It concludes with the speculative possibility that certain proofs of I C = 2 P 0 3 may help to solve major open problems in complexity theory. Earlier versions of the results in sections 2 4 and 6 appeared in the conference paper ....

....class C provably belong to C. In this section we deal with a particular formal system, called T Pi 2 , which is sound but not recursively axiomatizable. T Pi 2 is a superset of several theories studied in the literature, including Basic Number Theory as described by Lipton and DeMillo [Lip78, DL80]. This is interesting because of the contention by the authors of these papers, rebutted in [Lei82] and [JY85] that all constructive methods that computer scientists would ever use are formalizable in this theory. Before defining T Pi 2 , we note some technical aspects of formalizing predicates ....

R. Lipton. Model-theoretic aspects of computational complexity. In Proc. 19th Annual IEEE Symposium on Foundations of Computer Science, pages 191--200, 1978.

....is the class of languages accepted by alternating Turing machines making O(1) alternations, and running for linear time. We will usually denote this class by S k Sigma k Time(n) The rudimentary sets were originally defined by Smullyan [Smu61] and they have been studied extensively (see, e.g. [Wra78, Lip78, Boo78, Vol83, PD80]) One result of [Wra78] shows that the rudimentary sets can also be characterized in terms of lineartime nondeterministic oracle Turing machines, in analogy to the usual definition of the polynomial time hierarchy. We will have reason to be interested in the relationships among the rudimentary ....

R. Lipton. Model theoretic aspects of computational complexity. In Proc. 19th IEEE Symposium on Foundations of Computer Science, pages 193--200, 1978.

....deterministic and nondeterministic time classes. Let E = Dtime(2 O(n) NE = Ntime(2 O(n) and let RUD denote the class of Rudimentary sets. RUD is the linear time analog of the polynomial hierarchy. It was defined by Smullyan [Smu61] and has been studied extensively (see, e.g. Wra78, Lip78, Boo78, PD80, Vol83] One result of [Wra78] shows that the rudimentary sets can also be characterized in terms of linear time nondeterministic oracle Turing machines, in analogy to the usual definition of the polynomial hierarchy. It is not known if there is a set in E that is immune to RUD. The ....

R. Lipton. Model theoretic aspects of computational complexity. In Proc. 19th IEEE Symposium on Foudations of Computer Science, pages 193--200, 1978.

....appropriate coding tricks, and verify that each bit of it is correct. To prove the special case of this Lemma (numbers of length O(log n) used in Proposition 11.3 and in multiplication of O(log n) bit numbers, it suffices to use a simpler technique suggested by Sam Buss. He notes that Lipton [Li78] has shown how to multiply binary integers with an alternating Turing machine using a constant number of alternations and time linear in the length of the product. If the product is O(log n) bits, then, this computation is in LH and thus in FO by Corollary 8.3. Lipton s technique of carrying out ....

R. J. Lipton, "Model theoretic aspects of computational complexity", in 19th IEEE FOCS Symp. (1978), 193-200.

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R. J. Lipton, Model theoretic aspects of computational complexity, in Proceedings of the 19th Annual Symposium on Foundations of Computer Science, IEEE Computer Society, 1978, pp. 193--200.

No context found.

R. J. Lipton, Model theoretic aspects of computational complexity, in Proceedings of the 19th Annual Symposium on Foundations of Computer Science, IEEE Computer Society, 1978, pp. 193--200.

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