Leen Torenvliet | University of Amsterdam (original) (raw)
Papers by Leen Torenvliet
A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its o... more A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomial-time hierarchy from polynomial space by showing that all Turing-complete sets for certain levels of the exponential-time hierarchy are autoreducible but there exists some Turing-complete set for doubly exponential space that is not. Although we already knew how to separate these classes using diagonalization, our proofs separate classes solely by showing they have di erent structural properties, thus applying Post's Program to complexity theory. We feel such techniques may prove unknown separations in the future. In particular, if we could settle the question as to whether all Turing-complete sets for doubly exponential time are autoreducible, we would separate either polynomial time from polynomial space, and nondeterministic logarithmic space from nondeterministic polynomial time, or else the polynomial-time hierarchy from exponential time. We also look at the autoreducibility of complete sets under nonadaptive, bounded query, probabilistic and nonuniform reductions. We show how settling some of these autoreducibility questions will also lead to new complexity class separations.
arXiv (Cornell University), Jun 30, 1998
A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its o... more A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomial-time hierarchy from polynomial space by showing that all Turing-complete sets for certain levels of the exponential-time hierarchy are autoreducible but there exists some Turing-complete set for doubly exponential space that is not. Although we already knew how to separate these classes using diagonalization, our proofs separate classes solely by showing they have di erent structural properties, thus applying Post's Program to complexity theory. We feel such techniques may prove unknown separations in the future. In particular, if we could settle the question as to whether all Turing-complete sets for doubly exponential time are autoreducible, we would separate either polynomial time from polynomial space, and nondeterministic logarithmic space from nondeterministic polynomial time, or else the polynomial-time hierarchy from exponential time. We also look at the autoreducibility of complete sets under nonadaptive, bounded query, probabilistic and nonuniform reductions. We show how settling some of these autoreducibility questions will also lead to new complexity class separations.
Workshop on Graph-Theoretic Concepts in Computer Science, Jun 20, 1995
We prove that the local search optimization problem TSP 2Opt |though not known to be PLS-complete... more We prove that the local search optimization problem TSP 2Opt |though not known to be PLS-complete|shares an important infeasibility property with other PLS-complete sets.
Springer eBooks, 1984
ABSTRACT
Lecture Notes in Computer Science, 1986
In the present paper an overview is presented of diagonalisation methods which have been used for... more In the present paper an overview is presented of diagonalisation methods which have been used for the construction of oracle sets relative to which complexity classes in the P-Time Hierarchy are separated structurally. A comparison of the methods is made on the basis of inherent properties. A characterisation of these properties leads to a first attempt for a taxonomy for
Springer eBooks, 2008
and the author, except for brief excerpts in connection with reviews or scholarly analysis. Use i... more and the author, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Cover illustration: Created by Sven Geier of the California Institute of Technology. The image, an example of fractal art, is entitled "Deep Dive."
Cambridge University Press eBooks, Mar 21, 2017
In this expository paper, we investigate the structure of complexity classes and the structure of... more In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions.
Let f: {O, l}n x {O, l}n-+ {O, l}. Assume Alice has Xi, ... , Xk E {0, l}n, Bob has Y1, ... , Yk ... more Let f: {O, l}n x {O, l}n-+ {O, l}. Assume Alice has Xi, ... , Xk E {0, l}n, Bob has Y1, ... , Yk E {0, l}n, and they want to compute f(xi, Y1) • • • f(xk, Yk) communicating as few bits as possible. The Direct Sum Conjecture of Karchmer, Raz, and Wigderson, states that the obvious way to compute it (computing j(x 1, y1), then j(x2, Y2), etc.) is, roughly speaking, the best. This conjecture arose in the study of circuits since a variant of it implies NC 1 i= NC 2 • We consider three related problems.
Journal of Computer and System Sciences, Oct 1, 1996
We show that any p-selective and self-reducible set is in P. As the converse is also true, we obt... more We show that any p-selective and self-reducible set is in P. As the converse is also true, we obtain a new characterization of the class P. A generalization and several consequences of this theorem are discussed. Among other consequences, we show that under reasonable assumptions auto-reducibility and self-reducibility differ on NP, and that there are non-p-T-mitotic sets in NP.
Algorithmica, Jun 19, 2018
It is well-known that the class of sets that can be computed by polynomial size circuits is equal... more It is well-known that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in EXP NP , or even in EXP that are not computable by polynomial size circuits and hence are not reducible to a sparse set. In this paper we study this question in a more restricted setting: what is the computational complexity of sparse sets that are selfreducible? It follows from earlier work of Lozano and Torán (in: Mathematical systems theory, 1991) that EXP NP does not have sparse selfreducible hard sets. We define a natural version of selfreduction, tree-selfreducibility, and show that NEXP does not have sparse tree-selfreducible hard sets. We also construct an oracle relative to which all of EXP is reducible to a sparse tree-selfreducible set. These lower bounds are corollaries of more general results about the computational complexity of sparse sets that are selfreducible, and can be interpreted as super-polynomial circuit lower bounds for NEXP.
Electronic Colloquium on Computational Complexity, 2004
A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitel... more A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitely many elements including h(x). f is a k(n)-enumerator if for every input x of length n, h(x) is among the first k(n) elements enumerated by f. If there is a k(n)-enumerator for h then h is called k(n)-enumerable. We also consider enumerators which are only A-recursive for some oracle A. We determine exactly how hard it is to enumerate the Kolmogorov function, which assigns to each string x its Kolmogorov complexity: • For every underlying universal machine U , there is a constant a such that C is k(n)-enumerable only if k(n) ≥ n/a for almost all n. • For any given constant k, the Kolmogorov function is k-enumerable relative to an oracle A if and only if A is at least as hard as the halting problem. • There exists an r.e., Turing-incomplete set A such for every non-decreasing and unbounded recursive function k, the Kolmogorov function is k(n)-enumerable relative to A. The last result is obtained by using a relativizable construction for a nonrecursive set A relative to which the prefix-free Kolmogorov complexity differs only by a constant from the unrelativized prefix-free Kolmogorov complexity. Although every 2-enumerator for C is Turing hard for K, we show that reductions must depend on the specific choice of the 2-enumerator and there is no bound on the quantity of their queries. We show our negative results even for strong 2-enumerators as an oracle where the querying machine for any x gets directly an explicit list of all hypotheses of the enumerator for this input. The limitations are very general and we show them for any recursively bounded function g: • For every Turing reduction M and every non-recursive set B, there is a strong 2-enumerator f for g such that M does not Turing reduce B to f. • For every non-recursive set B, there is a strong 2-enumerator f for g such that B is not wtt-reducible to f. Furthermore, we deal with the resource-bounded case and give characterizations for the class S p 2 introduced by Canetti and independently Russell and Sundaram and the classes PSPACE, EXP.
Springer eBooks, 2006
It is well-known that the class of sets that can be computed by polynomial size circuits is equal... more It is well-known that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in EXP NP , or even in EXP that are not computable by polynomial size circuits and hence are not reducible to a sparse set. In this paper we study this question in a more restricted setting: what is the computational complexity of sparse sets that are selfreducible? It follows from earlier work of Lozano and Toran [1] that EXP NP does not have hard sparse selfreducible sets. We define a natural version of selfreduction, tree-selfreducibility, and show that NEXP does not have hard sparse tree-selfreducible sets. We also show that this result is optimal with respect to relativizing proofs, by exhibiting an oracle relative to which all of EXP is reducible to a sparse tree-selfreducible set. These lower bounds are corollaries of more general results about the computational complexity of sparse sets that are selfreducible, and can be interpreted as super-polynomial circuit lower bounds for NEXP.
Kolmogorov complexity has proven to be a very useful tool in simplifying and improving proofs tha... more Kolmogorov complexity has proven to be a very useful tool in simplifying and improving proofs that use complicated combinatorial arguments. In this paper we use Kolmogorov complexity for oracle construction. We obtain separation results that are much stronger than separations obtained previously even with the use of very complicated combinatorial arguments. Moreover the use of Kolmogorov arguments almost trivializes the construction itself. In particular we construct relativized worlds where: 1. NP \ CoNP = 2 P=poly. 2. NP has a set that is both simple and NP \ CoNP-immune. 3. CoNP has a set that is both simple and NP \ CoNP-immune. 4. p 2 has a set that is both simple and p 2 \ p 2-immune.
We study the robustness of complete sets for various complexity classes. A complete set A is robu... more We study the robustness of complete sets for various complexity classes. A complete set A is robust if for any f (n)-dense set S ∈ P, A − S is still complete, where f (n) ranges from log(n), polynomial, to subexponential. We show that robustness can be used to separate complexity classes: • for every ≤ p m-complete set A for EXP and any subexponential dense sets S ∈ P, A − S is still Turing complete and under a reasonable hardness assumption even ≤ p m-complete. • For EXP and the delta levels of the exponential hierarchy we show that for every Turing complete set A and any log-dense set S ∈ P, A − S is still Turing complete.
Springer eBooks, 1994
ABSTRACT
Sir, we are truly six special and interesting characters. Believe us. However we have gone lost.-... more Sir, we are truly six special and interesting characters. Believe us. However we have gone lost.-"Six Characters in Search of an Author,''
We study FP NP k , the class of functions that can be computed in polynomial time with nonadaptiv... more We study FP NP k , the class of functions that can be computed in polynomial time with nonadaptive queries to an NP oracle. This is motivated by the question of whether it is possible to compute witnesses for NP sets within FP NP k. The known algorithms for this task all require sequential access to the oracle. On the other hand, there is no evidence known yet that this should not be possible with parallel queries. We de ne a class of optimization problems based on NP sets, where the optimum is taken over a polynomially bounded range (NPbOpt). We s h o w t h a t i f s u c h an optimization problem is based on one of the known NP-complete sets, then it is hard for FP NP k. Moreover, we will characterize FP NP k as the class of functions that reduces to such optimization functions. We will call this property strong hardness. The main question is whether these function classes are complete for FP NP k. That is, whether it is possible to compute an optimal value CWI.
A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its o... more A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomial-time hierarchy from polynomial space by showing that all Turing-complete sets for certain levels of the exponential-time hierarchy are autoreducible but there exists some Turing-complete set for doubly exponential space that is not. Although we already knew how to separate these classes using diagonalization, our proofs separate classes solely by showing they have di erent structural properties, thus applying Post's Program to complexity theory. We feel such techniques may prove unknown separations in the future. In particular, if we could settle the question as to whether all Turing-complete sets for doubly exponential time are autoreducible, we would separate either polynomial time from polynomial space, and nondeterministic logarithmic space from nondeterministic polynomial time, or else the polynomial-time hierarchy from exponential time. We also look at the autoreducibility of complete sets under nonadaptive, bounded query, probabilistic and nonuniform reductions. We show how settling some of these autoreducibility questions will also lead to new complexity class separations.
arXiv (Cornell University), Jun 30, 1998
A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its o... more A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomial-time hierarchy from polynomial space by showing that all Turing-complete sets for certain levels of the exponential-time hierarchy are autoreducible but there exists some Turing-complete set for doubly exponential space that is not. Although we already knew how to separate these classes using diagonalization, our proofs separate classes solely by showing they have di erent structural properties, thus applying Post's Program to complexity theory. We feel such techniques may prove unknown separations in the future. In particular, if we could settle the question as to whether all Turing-complete sets for doubly exponential time are autoreducible, we would separate either polynomial time from polynomial space, and nondeterministic logarithmic space from nondeterministic polynomial time, or else the polynomial-time hierarchy from exponential time. We also look at the autoreducibility of complete sets under nonadaptive, bounded query, probabilistic and nonuniform reductions. We show how settling some of these autoreducibility questions will also lead to new complexity class separations.
Workshop on Graph-Theoretic Concepts in Computer Science, Jun 20, 1995
We prove that the local search optimization problem TSP 2Opt |though not known to be PLS-complete... more We prove that the local search optimization problem TSP 2Opt |though not known to be PLS-complete|shares an important infeasibility property with other PLS-complete sets.
Springer eBooks, 1984
ABSTRACT
Lecture Notes in Computer Science, 1986
In the present paper an overview is presented of diagonalisation methods which have been used for... more In the present paper an overview is presented of diagonalisation methods which have been used for the construction of oracle sets relative to which complexity classes in the P-Time Hierarchy are separated structurally. A comparison of the methods is made on the basis of inherent properties. A characterisation of these properties leads to a first attempt for a taxonomy for
Springer eBooks, 2008
and the author, except for brief excerpts in connection with reviews or scholarly analysis. Use i... more and the author, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Cover illustration: Created by Sven Geier of the California Institute of Technology. The image, an example of fractal art, is entitled "Deep Dive."
Cambridge University Press eBooks, Mar 21, 2017
In this expository paper, we investigate the structure of complexity classes and the structure of... more In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions.
Let f: {O, l}n x {O, l}n-+ {O, l}. Assume Alice has Xi, ... , Xk E {0, l}n, Bob has Y1, ... , Yk ... more Let f: {O, l}n x {O, l}n-+ {O, l}. Assume Alice has Xi, ... , Xk E {0, l}n, Bob has Y1, ... , Yk E {0, l}n, and they want to compute f(xi, Y1) • • • f(xk, Yk) communicating as few bits as possible. The Direct Sum Conjecture of Karchmer, Raz, and Wigderson, states that the obvious way to compute it (computing j(x 1, y1), then j(x2, Y2), etc.) is, roughly speaking, the best. This conjecture arose in the study of circuits since a variant of it implies NC 1 i= NC 2 • We consider three related problems.
Journal of Computer and System Sciences, Oct 1, 1996
We show that any p-selective and self-reducible set is in P. As the converse is also true, we obt... more We show that any p-selective and self-reducible set is in P. As the converse is also true, we obtain a new characterization of the class P. A generalization and several consequences of this theorem are discussed. Among other consequences, we show that under reasonable assumptions auto-reducibility and self-reducibility differ on NP, and that there are non-p-T-mitotic sets in NP.
Algorithmica, Jun 19, 2018
It is well-known that the class of sets that can be computed by polynomial size circuits is equal... more It is well-known that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in EXP NP , or even in EXP that are not computable by polynomial size circuits and hence are not reducible to a sparse set. In this paper we study this question in a more restricted setting: what is the computational complexity of sparse sets that are selfreducible? It follows from earlier work of Lozano and Torán (in: Mathematical systems theory, 1991) that EXP NP does not have sparse selfreducible hard sets. We define a natural version of selfreduction, tree-selfreducibility, and show that NEXP does not have sparse tree-selfreducible hard sets. We also construct an oracle relative to which all of EXP is reducible to a sparse tree-selfreducible set. These lower bounds are corollaries of more general results about the computational complexity of sparse sets that are selfreducible, and can be interpreted as super-polynomial circuit lower bounds for NEXP.
Electronic Colloquium on Computational Complexity, 2004
A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitel... more A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitely many elements including h(x). f is a k(n)-enumerator if for every input x of length n, h(x) is among the first k(n) elements enumerated by f. If there is a k(n)-enumerator for h then h is called k(n)-enumerable. We also consider enumerators which are only A-recursive for some oracle A. We determine exactly how hard it is to enumerate the Kolmogorov function, which assigns to each string x its Kolmogorov complexity: • For every underlying universal machine U , there is a constant a such that C is k(n)-enumerable only if k(n) ≥ n/a for almost all n. • For any given constant k, the Kolmogorov function is k-enumerable relative to an oracle A if and only if A is at least as hard as the halting problem. • There exists an r.e., Turing-incomplete set A such for every non-decreasing and unbounded recursive function k, the Kolmogorov function is k(n)-enumerable relative to A. The last result is obtained by using a relativizable construction for a nonrecursive set A relative to which the prefix-free Kolmogorov complexity differs only by a constant from the unrelativized prefix-free Kolmogorov complexity. Although every 2-enumerator for C is Turing hard for K, we show that reductions must depend on the specific choice of the 2-enumerator and there is no bound on the quantity of their queries. We show our negative results even for strong 2-enumerators as an oracle where the querying machine for any x gets directly an explicit list of all hypotheses of the enumerator for this input. The limitations are very general and we show them for any recursively bounded function g: • For every Turing reduction M and every non-recursive set B, there is a strong 2-enumerator f for g such that M does not Turing reduce B to f. • For every non-recursive set B, there is a strong 2-enumerator f for g such that B is not wtt-reducible to f. Furthermore, we deal with the resource-bounded case and give characterizations for the class S p 2 introduced by Canetti and independently Russell and Sundaram and the classes PSPACE, EXP.
Springer eBooks, 2006
It is well-known that the class of sets that can be computed by polynomial size circuits is equal... more It is well-known that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in EXP NP , or even in EXP that are not computable by polynomial size circuits and hence are not reducible to a sparse set. In this paper we study this question in a more restricted setting: what is the computational complexity of sparse sets that are selfreducible? It follows from earlier work of Lozano and Toran [1] that EXP NP does not have hard sparse selfreducible sets. We define a natural version of selfreduction, tree-selfreducibility, and show that NEXP does not have hard sparse tree-selfreducible sets. We also show that this result is optimal with respect to relativizing proofs, by exhibiting an oracle relative to which all of EXP is reducible to a sparse tree-selfreducible set. These lower bounds are corollaries of more general results about the computational complexity of sparse sets that are selfreducible, and can be interpreted as super-polynomial circuit lower bounds for NEXP.
Kolmogorov complexity has proven to be a very useful tool in simplifying and improving proofs tha... more Kolmogorov complexity has proven to be a very useful tool in simplifying and improving proofs that use complicated combinatorial arguments. In this paper we use Kolmogorov complexity for oracle construction. We obtain separation results that are much stronger than separations obtained previously even with the use of very complicated combinatorial arguments. Moreover the use of Kolmogorov arguments almost trivializes the construction itself. In particular we construct relativized worlds where: 1. NP \ CoNP = 2 P=poly. 2. NP has a set that is both simple and NP \ CoNP-immune. 3. CoNP has a set that is both simple and NP \ CoNP-immune. 4. p 2 has a set that is both simple and p 2 \ p 2-immune.
We study the robustness of complete sets for various complexity classes. A complete set A is robu... more We study the robustness of complete sets for various complexity classes. A complete set A is robust if for any f (n)-dense set S ∈ P, A − S is still complete, where f (n) ranges from log(n), polynomial, to subexponential. We show that robustness can be used to separate complexity classes: • for every ≤ p m-complete set A for EXP and any subexponential dense sets S ∈ P, A − S is still Turing complete and under a reasonable hardness assumption even ≤ p m-complete. • For EXP and the delta levels of the exponential hierarchy we show that for every Turing complete set A and any log-dense set S ∈ P, A − S is still Turing complete.
Springer eBooks, 1994
ABSTRACT
Sir, we are truly six special and interesting characters. Believe us. However we have gone lost.-... more Sir, we are truly six special and interesting characters. Believe us. However we have gone lost.-"Six Characters in Search of an Author,''
We study FP NP k , the class of functions that can be computed in polynomial time with nonadaptiv... more We study FP NP k , the class of functions that can be computed in polynomial time with nonadaptive queries to an NP oracle. This is motivated by the question of whether it is possible to compute witnesses for NP sets within FP NP k. The known algorithms for this task all require sequential access to the oracle. On the other hand, there is no evidence known yet that this should not be possible with parallel queries. We de ne a class of optimization problems based on NP sets, where the optimum is taken over a polynomially bounded range (NPbOpt). We s h o w t h a t i f s u c h an optimization problem is based on one of the known NP-complete sets, then it is hard for FP NP k. Moreover, we will characterize FP NP k as the class of functions that reduces to such optimization functions. We will call this property strong hardness. The main question is whether these function classes are complete for FP NP k. That is, whether it is possible to compute an optimal value CWI.