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Algebraic properties for selector functions
 SIAM Journal on Computing
"... The nondeterministic advice complexity of the Pselective sets is known to be exactly linear. Regarding the deterministic advice complexity of the Pselective sets—i.e., the amount of Karp– Lipton advice needed for polynomialtime machines to recognize them in general—the best current upper bound is ..."
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Cited by 6 (4 self)
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The nondeterministic advice complexity of the Pselective sets is known to be exactly linear. Regarding the deterministic advice complexity of the Pselective sets—i.e., the amount of Karp– Lipton advice needed for polynomialtime machines to recognize them in general—the best current upper bound is quadratic [Ko83] and the best current lower bound is linear [HT96]. We prove that every associatively Pselective set is commutatively, associatively Pselective. Using this, we establish an algebraic sufficient condition for the Pselective sets to have a linear upper bound (which thus would match the existing lower bound) on their deterministic advice complexity: If all Pselective sets are associatively Pselective then the deterministic advice complexity of the Pselective sets is linear. The weakest previously known sufficient condition was P = NP. We also establish related results for algebraic properties of, and advice complexity of, the nondeterministically Selman [Sel79,Sel81,Sel82a,Sel82b] defined the Pselective sets about twenty years ago. In addition to being of interest in their own right, they have recently had some surprising applications. For example, selectivity is a powerful tool in the study of search versus decision problems [HNOS96a], and
The Complexity of Kings
, 2006
"... A king in a directed graph is a node from which each node in the graph can be reached via paths of length at most two. There is a broad literature on tournaments (completely oriented digraphs), and it has been known for more than half a century that all tournaments have at least one king [Lan53]. Re ..."
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Cited by 5 (3 self)
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A king in a directed graph is a node from which each node in the graph can be reached via paths of length at most two. There is a broad literature on tournaments (completely oriented digraphs), and it has been known for more than half a century that all tournaments have at least one king [Lan53]. Recently, kings have proven useful in theoretical computer science, in particular in the study of the complexity of the semifeasible sets [HNP98,HT05] and in the study of the complexity of reachability problems [Tan01,NT02]. In this paper, we study the complexity of recognizing kings. For each succinctly specified family of tournaments, the king problem is known to belong to Π p 2 [HOZZ]. We prove that this bound is optimal: We construct a succinctly specified tournament family whose king problem is Π p 2complete. It follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs (which need not
Pselectivity, immunity, and the power of one bit
 In Proceedings of the 32nd International Conference on Current Trends in Theory and Practice of Computer Science. SpringerVerlag Lecture Notes in Computer Science
, 2006
"... We prove that Psel, the class of all Pselective sets, is EXPimmune, but is not EXP/1immune. That is, we prove that some infinite Pselective set has no infinite EXPtime subset, but we also prove that every infinite Pselective set has some infinite subset in EXP/1. Informally put, the immunity ..."
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Cited by 3 (2 self)
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We prove that Psel, the class of all Pselective sets, is EXPimmune, but is not EXP/1immune. That is, we prove that some infinite Pselective set has no infinite EXPtime subset, but we also prove that every infinite Pselective set has some infinite subset in EXP/1. Informally put, the immunity of Psel is so fragile that it is pierced by a single bit of information. The above claims follow from broader results that we obtain about the immunity of the Pselective sets. In particular, we prove that for every recursive function f, Psel is DTIME(f)immune. Yet we also prove that Psel is not Π p 2 /1immune. 1
Generalizations of the Hartmanis–Immerman–Sewelson Theorem and Applications to Infinite Subsets of PSelective Sets
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY
, 2008
"... The Hartmanis–Immerman–Sewelson theorem is the classical link between the exponential and the polynomial time realm. It states that NE = E if, and only if, every sparse set in NP lies in P. We establish similar links for classes other than sparse sets: 1. E = UE ⇐ ⇒ all functions f: {1} ∗ → Σ ∗ in ..."
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The Hartmanis–Immerman–Sewelson theorem is the classical link between the exponential and the polynomial time realm. It states that NE = E if, and only if, every sparse set in NP lies in P. We establish similar links for classes other than sparse sets: 1. E = UE ⇐ ⇒ all functions f: {1} ∗ → Σ ∗ in NPSVg lie in FP. 2. E = NE ⇐ ⇒ all functions f: {1} ∗ → Σ ∗ in NPFewV lie in FP. 3. E = E NP ⇐ ⇒ all functions f: {1} ∗ → Σ ∗ in OptP lie in FP. 4. E = E NP ⇐ ⇒ all standard left cuts in NP lie in P. 5. E = EH ⇐ ⇒ PH ∩ P/poly = P. We apply these results to the immunity of Pselective sets. It is known that they can be biimmune, but not Π p 2 /1immune. Their immunity is closely related to topToda languages, whose complexity we link to the exponential realm, and also to king languages. We introduce the new notion of superkings, which are characterized in terms of ∃∀predicates rather than ∀∃predicates, and show that king languages cannot be Σ p 2immune. As a consequence, /1immune and, if EΣp2 = E, not even P/1immune. Pselective sets cannot be Σ p 2 1