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**11 - 17**of**17**### Instance Complexity of NP-hard sets

, 1999

"... Instance complexity was introduced by Orponen, Ko, Schöning, and Watanabe [7, 14, 15] as a measure of the complexity of individual instances of a decision problem. Comparing instance complexity to Kolmogorov complexity (i.e. the inherent complexity of a string), they introduced the notion of p-h ..."

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Instance complexity was introduced by Orponen, Ko, Schöning, and Watanabe [7, 14, 15] as a measure of the complexity of individual instances of a decision problem. Comparing instance complexity to Kolmogorov complexity (i.e. the inherent complexity of a string), they introduced the notion of p-hard instances, and conjectured that every set not in P has p-hard instances. Whereas this conjecture is still unsettled, Fortnow and Kummer [4] proved that NP-hard sets have p-hard instances, unless P=NP. The unbounded version of the conjecture was proven wrong by Kummer [9], and therefore it cannot be used to characterize e.g. the class recursive sets. In [13] a slightly weaker notion of hard instances was introduced. In the unbounded version, it is used to characterize the classes of recursively enumerable resp. recursive sets. In time-bounded versions, the class P could be characterized- Hard instances are shown to be stronger than complexity cores (introduced by Lynch [12]). N...

### On reductions to sets that avoid EXPSPACE

"... M. Mundhenk x{ AsetB is called EXPSPACE-avoiding, ifevery subset of B in EXPSPACE is sparse. Sparse sets and sets of high information density (called HIGH sets in [5]) are shown to be EXPSPACE-avoiding. Investigating the complexity of sets A in EXPSPACE that honestly reduce to EXPSPACE-avoiding sets ..."

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M. Mundhenk x{ AsetB is called EXPSPACE-avoiding, ifevery subset of B in EXPSPACE is sparse. Sparse sets and sets of high information density (called HIGH sets in [5]) are shown to be EXPSPACE-avoiding. Investigating the complexity of sets A in EXPSPACE that honestly reduce to EXPSPACE-avoiding sets, we show that if the reducibility used has a property, calledrange-constructibility, then A must also reduce to a sparse set under the same reducibility. Keywords: Computational Complexity, Reducibilities, Sparse Sets. 1

### Monotonic Oracle Machines and Binary Search Reductions

, 1996

"... Polynomial-time oracle machines being restricted to perform a certain search technique in the oracle are considered. These search techniques (e.g. binary search, prefix search) are expressed as monotonicity properties of the queries computed by the oracle machine. The power of different kinds of res ..."

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Polynomial-time oracle machines being restricted to perform a certain search technique in the oracle are considered. These search techniques (e.g. binary search, prefix search) are expressed as monotonicity properties of the queries computed by the oracle machine. The power of different kinds of restricted machines is systematically investigated. It turns out that restrictions are comparable in different ways if the class of oracles is restricted to NP resp. to sparse sets, or if restricted machines being additionally positive (as defined by [Sel82]) are considered. 1 Introduction The notion of a polynomial-time oracle machine is a basic tool in complexity theory. Different types of machines having restricted access to the oracle are well-studied (see e.g. [LLS75, Wag90]). For example, non-adaptive oracle machines generate their queries independent from the oracle; bounded oracle machines may ask not more than a certain number of queries; conjunctive oracle machines accept exactly whe...

### Monotonous Oracle Machines

, 1995

"... Polynomial-time oracle machines being restricted to perform a certain search technique in the oracle are considered. These search techniques (e.g. binary search, prefix search) are expressed as monotony properties of the queries computed by the oracle machine. The power of different kinds of restric ..."

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Polynomial-time oracle machines being restricted to perform a certain search technique in the oracle are considered. These search techniques (e.g. binary search, prefix search) are expressed as monotony properties of the queries computed by the oracle machine. The power of different kinds of restricted machines is systematically investigated. It turns out that restrictions are comparable in different ways if the class of oracles is restricted to NP resp. to sparse sets, or if restricted machines being additionally positive (cf. [Sel82]) are considered. 1 Introduction The notion of a polynomial-time oracle machine is a basic tool in complexity theory. Different types of machines having restricted access to the oracle are well-studied (see e.g. [LLS75, Wag90]). For example, non-adaptive oracle machines generate their queries independent from the oracle; bounded oracle machines may ask not more than a certain number of queries; conjunctive oracle machines accept exactly when all queries ...

### On reductions to sets that avoid EXPSPACE (Extended Abstract)

, 1993

"... A set B is called EXPSPACE-avoiding, if every subset of B in EXPSPACE is sparse. For example, sets of high information density (called HIGH sets in [5]) are shown to be EXPSPACE-avoiding. Investigating the complexity of sets A in EXPSPACE that honestly reduce to EXPSPACE-avoiding sets, we show that ..."

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A set B is called EXPSPACE-avoiding, if every subset of B in EXPSPACE is sparse. For example, sets of high information density (called HIGH sets in [5]) are shown to be EXPSPACE-avoiding. Investigating the complexity of sets A in EXPSPACE that honestly reduce to EXPSPACE-avoiding sets, we show that if the reducibility used has a property, called range-constructibility, then A must also reduce to a sparse set under the same reducibility. 1 Introduction The study of reductions to sets having low information content has received much attention in recent research in structural complexity theory. Informally, a class of sets is of low information content if it cannot contain hard sets for intractable complexity classes, unless there is an unlikely collapse of complexity classes. The class of sparse sets is a well-studied example of low information content sets [8, 10, 11]. A research trend is to identify and study different classes of low information content sets. Book and Lutz [5] introduc...

### Learning Weak Reductions to Sparse Sets

"... We study the consequences of NP having non-uniform polynomial size circuits of various types. We continue the work of Agrawal and Arvind [1] who study the consequences of Sat being many-one re-ducible to functions computable by non-uniform circuits consisting of a single weighted threshold gate. (Sa ..."

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We study the consequences of NP having non-uniform polynomial size circuits of various types. We continue the work of Agrawal and Arvind [1] who study the consequences of Sat being many-one re-ducible to functions computable by non-uniform circuits consisting of a single weighted threshold gate. (Sat ≤pm LT1). They claim that as a consequence P = NP follows, but unfortunately their proof was incorrect. We take up this question and use results from computational learn-ing theory to show that if Sat ≤pm LT1 then PH = PNP. We furthermore show that if Sat disjunctive truth-table (or major-ity truth-table) reduces to a sparse set then Sat ≤pm LT1 and hence a collapse of PH to PNP also follows. Lastly we show several interesting consequences of Sat ≤pdtt SPARSE. 1

### Learning Reductions to Sparse Sets

"... Abstract. We study the consequences of NP having non-uniform polynomial size circuits of various types. We continue the work of Agrawal and Arvind [1] who study the consequences of Sat being many-one reducible to functions computable by non-uniform circuits consisting of a single weighted threshold ..."

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Abstract. We study the consequences of NP having non-uniform polynomial size circuits of various types. We continue the work of Agrawal and Arvind [1] who study the consequences of Sat being many-one reducible to functions computable by non-uniform circuits consisting of a single weighted threshold gate. (Sat ≤ p m LT1). They claim that P = NP follows as a consequence, but unfortunately their proof was incorrect. We take up this question and use results from computational learning theory to show that if Sat ≤ p m LT1 then PH = P NP. We furthermore show that if Sat disjunctive truth-table (or majority truth-table) reduces to a sparse set then Sat ≤ p m LT1 and hence a collapse of PH to P NP also follows. Lastly we show several interesting consequences of Sat ≤ p dtt SPARSE.