this paper are strengthenings of the results in [4] and [9] for Turing machines. The results in [4] and [9] hold for SAT but our results hold for 2-SAT also, since the formulae we reduce the language L to belong to 2-SAT. Therefore our techniques are less promising if the ultimate goal is to prove that SAT does not belong to P, since it is known that 2-SAT belongs to P. Moreover we obtain the same lower bounds for NTMs as for DTMs, which indicates that our techniques may not be useful in separating nondeterministic time and deterministic time

We give the first extension of the result due to Paul, Pippenger, Szemeredi and Trotter [24] that deterministic linear time is distinct from nondeterministic linear time. We show that N T IM E(n

. We investigate the complexity of the fixed-points of bounded formulas in the context of finite set theory; that is, in the context of arbitrary classes of finite structures that are equipped with a built-in BIT predicate, or equivalently, with a built-in membership relation between hereditarily finite sets (input relations are allowed). We show that the iteration of a positive bounded formula converges in polylogarithmically many steps in the cardinality of the structure. This extends a previously known much weaker result. We obtain a number of connections with the rudimentary languages and deterministic polynomial-time. Moreover, our results provide a natural characterization of the complexity class consisting of all languages computable by bounded-depth, polynomialsize circuits, and polylogarithmic-time uniformity. As a byproduct, we see that this class coincides with LH(P), the logarithmic-time hierarchy with an oracle to deterministic polynomial-time. Finally, we dis...

Propositional Proof Complexity is an active area of research whose main focus is the study of the length of proofs in propositional logic. There are several motivations for such a study, the main of which is probably its connection to the P vs NP problem in Computational Complexity. The experience

We survey the recent lower bounds on the running time of general-purpose random-access machines that solve satisfiability in a small amount of work space, and related lower bounds for satisfiability in nonuniform models.

A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD6-SAT, Majority-of-Majority-SAT, and Tautologies, to name a few. These lower bound proofs all follow a certain diagonalization-based proof-by-contradiction strategy. A pressing open problem has been to determine how powerful such proofs can possibly be. We propose an automated theorem-proving methodology for studying these lower bound problems. In particular, we prove that the search for better lower bounds can often be turned into a problem of solving a large series of linear programming instances. We describe an implementation of a small-scale theorem prover and discover surprising experimental results. In some settings, our program provides strong evidence that the best known lower bound proofs are already optimal for the current framework, contradicting the consensus intuition; in others, the program guides us to improved lower bounds where none had been known for years.

We give a modern historical and philosophical discussion of diagonalization as a tool to prove lower bounds in computational complexity. We will give several examples and discuss four possible approaches to use diagonalization for separating logarithmic-space from nondeterministic polynomial-time. 1 Introduction The greatest embarrassment in computational complexity theory comes from our inability to achieve signicant complexity class separations. In recent years we have seen many interesting results come from an old technique|diagonalization. Deceptively simple, diagonalization, combined with techniques for collapsing classes, can yield quite interesting lower bounds on computation. In 1874, Cantor [Can74] rst used diagonalization for showing the set of reals is not countable. The proof worked by assuming an enumeration of the reals and designing a set that one-by-one is dierent from every set in the enumeration. Drawn as a table this process considers the diagonal set and re...

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Nepomnjaˇsčiǐ’s Theorem states that for all 0 ≤ ǫ < 1 and k> 0 the class of languages recognized in nondeterministic time n k and space n ǫ, NTISP[n k, n ǫ], is contained in the linear time hierarchy. By considering restrictions on the size of the universal quantifiers in the linear time hierarchy, this paper refines Nepomnjaˇsčiǐ’s result to give a subhierarchy, Eu-LinH, of the linear time hierarchy that is contained in NP and which contains NTISP[n k, n ǫ]. Hence, Eu-LinH contains NL and SC. This paper investigates basic structural properties of Eu-LinH. Then the relationships between Eu-LinH and the classes NL, SC, and NP are considered to see if they can shed light on the NL = NP or SC = NP questions. Finally, a new hierarchy, ξ-LinH, is defined to reduce the space requirements needed for the upper bound on Eu-LinH. Mathematics Subject Classification: 03F30, 68Q15 Keywords: structural complexity, linear time hierarchy, Nepomnjaˇsčiǐ’s Theorem