Samik Sengupta - Academia.edu (original) (raw)

Papers by Samik Sengupta

Research paper thumbnail of Vector evaluation of assertions

Research paper thumbnail of On Higher Arthur-Merlin Classes

Ijfcs, 2004

We study higher Arthur-Merlin classes defined via several natural probabilistic operators BP, R a... more We study higher Arthur-Merlin classes defined via several natural probabilistic operators BP, R and coR. We investigate the complexity classes they define, and a number of interactions between these operators and the standard polynomial time hierarchy. We prove a hierarchy theorem for these higher Arthur-Merlin classes involving interleaving operators, and a theorem giving non-trivial upper bounds to the intersection of the complementary classes in the hierarchy.

Research paper thumbnail of Architectures for Arithmetic over GF(2

Research paper thumbnail of Disjoint NP-pairs

18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings., 2003

We study the question of whether the class DisNP of disjoint pairs (A, B) of NP-sets contains aco... more We study the question of whether the class DisNP of disjoint pairs (A, B) of NP-sets contains acomplete pair. The question relates to the question of whether optimal proof systems exist, and werelate it to the previously studied question of whether there exists a disjoint pair of NP-sets that isNP-hard. We show under reasonable hypotheses that nonsymmetric disjoint NP-pairs exist,

Research paper thumbnail of Properties of NP‐Complete Sets

SIAM Journal on Computing, 2006

Research paper thumbnail of Arithmetic Circuits and Polynomial Replacement Systems

SIAM Journal on Computing, 2004

This paper addresses the problems of counting proof trees (as introduced by Venkateswaran and Tom... more This paper addresses the problems of counting proof trees (as introduced by Venkateswaran and Tompa) and counting proof circuits, a related but seemingly more natural question. These problems lead to a common generalization of straight-line programs which we call polynomial replacement systems. We contribute a classification of these systems and we investigate their complexity. Diverse problems falling in the scope of this study include, for example, counting proof circuits, and evaluating {∪, +}-circuits over the natural numbers. A number of complexity results are obtained, including a proof that counting proof circuits is #P-complete.

Research paper thumbnail of Disjoint NP-Pairs

SIAM Journal on Computing, 2004

ABSTRACT

Research paper thumbnail of Architectures for Arithmetic over GF (2m)

VLSI Design, 1997. Proceedings., …, 1997

... First we state the following theorem. Theorem 1 Let X fala + . .. + a,,,-lam-' be the re... more ... First we state the following theorem. Theorem 1 Let X fala + . .. + a,,,-lam-' be the represen-tation of an element of GF(2"'), where a andp(t) am as above. ... 1989. Figure 2: 'A-Bad Multiplier for Standard Basis 17) MAHasan, MZwanz and VKBhargava, “A modified 1pq-e ...

Research paper thumbnail of Proving SAT does not have small circuits with an application to the two queries problem

Journal of Computer and System Sciences, 2008

We show that if SAT does not have small circuits, then there must exist a small number of satisfi... more We show that if SAT does not have small circuits, then there must exist a small number of satisfiable formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if P NP[1] = P NP[2] , then the polynomial-time hierarchy collapses to S p 2 ⊆ Σ p 2 ∩ Π p 2. Even showing that the hierarchy collapsed to Σ p 2 remained open prior to this paper.

Research paper thumbnail of On Higher Arthur-Merlin Classes

International Journal of Foundations of Computer Science, 2004

We study higher Arthur-Merlin classes defined via several natural probabilistic operators BP, R a... more We study higher Arthur-Merlin classes defined via several natural probabilistic operators BP, R and coR. We investigate the complexity classes they define, and a number of interactions between these operators and the standard polynomial time hierarchy. We prove a hierarchy theorem for these higher Arthur-Merlin classes involving interleaving operators, and a theorem giving non-trivial upper bounds to the intersection of the complementary classes in the hierarchy.

Research paper thumbnail of Reductions between disjoint NP-Pairs

Information and Computation, 2005

We prove that all of the following assertions are equivalent: There is a many-one complete disjoi... more We prove that all of the following assertions are equivalent: There is a many-one complete disjoint NP-pair; there is a strongly many-one complete disjoint NP-pair; there is a Turing complete disjoint NP-pair such that all reductions are smart reductions; there is a complete disjoint NP-pair for one-to-one, invertible reductions; the class of all disjoint NP-pairs is uniformly enumerable. Let A, B, C, and D be nonempty sets belonging to NP. A smart reduction between the disjoint NP-pairs (A, B) and (C, D) is a Turing reduction with the additional property that if the input belongs to A ∪ B, then all queries belong to C ∪ D. We prove under the reasonable assumption UP ∩ co-UP has a P-bi-immune set that there exist disjoint NP-pairs (A, B) and (C, D) such that (A, B) is truth-table reducible to (C, D), but there is no smart reduction between them. This paper contains several additional separations of reductions between disjoint NP-pairs. We exhibit an oracle relative to which DisjNP has a truth-table-complete disjoint NP-pair, but has no many-one-complete disjoint NP-pair.

Research paper thumbnail of Polylogarithmic-round interactive proofs for coNP collapse the exponential hierarchy

Theoretical Computer …, 2007

If every language in coNP has constant round interactive proof system, then the polynomialtime hi... more If every language in coNP has constant round interactive proof system, then the polynomialtime hierarchy collapses [BHZ87]. On the other hand, the well-known LFKN protocol gives O(n)-round interactive proof systems for all languages in coNP [LFKN92]. We consider the question whether it is possible for coNP to have interactive proof systems with polylogarithmic round complexity. We show that this is unlikely by proving that if a coNP-complete set has a polylogarithmic-round interactive proof system then the exponential-time hierarchy collapses. We also consider exponential versions of the Karp-Lipton theorem and Yap's theorem. * Some of the results of this paper were presented at the 19th IEEE Conference on Computational Complexity theory by second and third author.

Research paper thumbnail of Polylogarithmic-round interactive proofs for coNP collapse the exponential hierarchy

Theoretical Computer …, 2007

If every language in coNP has constant round interactive proof system, then the polynomialtime hi... more If every language in coNP has constant round interactive proof system, then the polynomialtime hierarchy collapses [BHZ87]. On the other hand, the well-known LFKN protocol gives O(n)-round interactive proof systems for all languages in coNP [LFKN92]. We consider the question whether it is possible for coNP to have interactive proof systems with polylogarithmic round complexity. We show that this is unlikely by proving that if a coNP-complete set has a polylogarithmic-round interactive proof system then the exponential-time hierarchy collapses. We also consider exponential versions of the Karp-Lipton theorem and Yap's theorem. * Some of the results of this paper were presented at the 19th IEEE Conference on Computational Complexity theory by second and third author.

Research paper thumbnail of Vector evaluation of assertions

Research paper thumbnail of On Higher Arthur-Merlin Classes

Ijfcs, 2004

We study higher Arthur-Merlin classes defined via several natural probabilistic operators BP, R a... more We study higher Arthur-Merlin classes defined via several natural probabilistic operators BP, R and coR. We investigate the complexity classes they define, and a number of interactions between these operators and the standard polynomial time hierarchy. We prove a hierarchy theorem for these higher Arthur-Merlin classes involving interleaving operators, and a theorem giving non-trivial upper bounds to the intersection of the complementary classes in the hierarchy.

Research paper thumbnail of Architectures for Arithmetic over GF(2

Research paper thumbnail of Disjoint NP-pairs

18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings., 2003

We study the question of whether the class DisNP of disjoint pairs (A, B) of NP-sets contains aco... more We study the question of whether the class DisNP of disjoint pairs (A, B) of NP-sets contains acomplete pair. The question relates to the question of whether optimal proof systems exist, and werelate it to the previously studied question of whether there exists a disjoint pair of NP-sets that isNP-hard. We show under reasonable hypotheses that nonsymmetric disjoint NP-pairs exist,

Research paper thumbnail of Properties of NP‐Complete Sets

SIAM Journal on Computing, 2006

Research paper thumbnail of Arithmetic Circuits and Polynomial Replacement Systems

SIAM Journal on Computing, 2004

This paper addresses the problems of counting proof trees (as introduced by Venkateswaran and Tom... more This paper addresses the problems of counting proof trees (as introduced by Venkateswaran and Tompa) and counting proof circuits, a related but seemingly more natural question. These problems lead to a common generalization of straight-line programs which we call polynomial replacement systems. We contribute a classification of these systems and we investigate their complexity. Diverse problems falling in the scope of this study include, for example, counting proof circuits, and evaluating {∪, +}-circuits over the natural numbers. A number of complexity results are obtained, including a proof that counting proof circuits is #P-complete.

Research paper thumbnail of Disjoint NP-Pairs

SIAM Journal on Computing, 2004

ABSTRACT

Research paper thumbnail of Architectures for Arithmetic over GF (2m)

VLSI Design, 1997. Proceedings., …, 1997

... First we state the following theorem. Theorem 1 Let X fala + . .. + a,,,-lam-' be the re... more ... First we state the following theorem. Theorem 1 Let X fala + . .. + a,,,-lam-' be the represen-tation of an element of GF(2"'), where a andp(t) am as above. ... 1989. Figure 2: 'A-Bad Multiplier for Standard Basis 17) MAHasan, MZwanz and VKBhargava, “A modified 1pq-e ...

Research paper thumbnail of Proving SAT does not have small circuits with an application to the two queries problem

Journal of Computer and System Sciences, 2008

We show that if SAT does not have small circuits, then there must exist a small number of satisfi... more We show that if SAT does not have small circuits, then there must exist a small number of satisfiable formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if P NP[1] = P NP[2] , then the polynomial-time hierarchy collapses to S p 2 ⊆ Σ p 2 ∩ Π p 2. Even showing that the hierarchy collapsed to Σ p 2 remained open prior to this paper.

Research paper thumbnail of On Higher Arthur-Merlin Classes

International Journal of Foundations of Computer Science, 2004

We study higher Arthur-Merlin classes defined via several natural probabilistic operators BP, R a... more We study higher Arthur-Merlin classes defined via several natural probabilistic operators BP, R and coR. We investigate the complexity classes they define, and a number of interactions between these operators and the standard polynomial time hierarchy. We prove a hierarchy theorem for these higher Arthur-Merlin classes involving interleaving operators, and a theorem giving non-trivial upper bounds to the intersection of the complementary classes in the hierarchy.

Research paper thumbnail of Reductions between disjoint NP-Pairs

Information and Computation, 2005

We prove that all of the following assertions are equivalent: There is a many-one complete disjoi... more We prove that all of the following assertions are equivalent: There is a many-one complete disjoint NP-pair; there is a strongly many-one complete disjoint NP-pair; there is a Turing complete disjoint NP-pair such that all reductions are smart reductions; there is a complete disjoint NP-pair for one-to-one, invertible reductions; the class of all disjoint NP-pairs is uniformly enumerable. Let A, B, C, and D be nonempty sets belonging to NP. A smart reduction between the disjoint NP-pairs (A, B) and (C, D) is a Turing reduction with the additional property that if the input belongs to A ∪ B, then all queries belong to C ∪ D. We prove under the reasonable assumption UP ∩ co-UP has a P-bi-immune set that there exist disjoint NP-pairs (A, B) and (C, D) such that (A, B) is truth-table reducible to (C, D), but there is no smart reduction between them. This paper contains several additional separations of reductions between disjoint NP-pairs. We exhibit an oracle relative to which DisjNP has a truth-table-complete disjoint NP-pair, but has no many-one-complete disjoint NP-pair.

Research paper thumbnail of Polylogarithmic-round interactive proofs for coNP collapse the exponential hierarchy

Theoretical Computer …, 2007

If every language in coNP has constant round interactive proof system, then the polynomialtime hi... more If every language in coNP has constant round interactive proof system, then the polynomialtime hierarchy collapses [BHZ87]. On the other hand, the well-known LFKN protocol gives O(n)-round interactive proof systems for all languages in coNP [LFKN92]. We consider the question whether it is possible for coNP to have interactive proof systems with polylogarithmic round complexity. We show that this is unlikely by proving that if a coNP-complete set has a polylogarithmic-round interactive proof system then the exponential-time hierarchy collapses. We also consider exponential versions of the Karp-Lipton theorem and Yap's theorem. * Some of the results of this paper were presented at the 19th IEEE Conference on Computational Complexity theory by second and third author.

Research paper thumbnail of Polylogarithmic-round interactive proofs for coNP collapse the exponential hierarchy

Theoretical Computer …, 2007

If every language in coNP has constant round interactive proof system, then the polynomialtime hi... more If every language in coNP has constant round interactive proof system, then the polynomialtime hierarchy collapses [BHZ87]. On the other hand, the well-known LFKN protocol gives O(n)-round interactive proof systems for all languages in coNP [LFKN92]. We consider the question whether it is possible for coNP to have interactive proof systems with polylogarithmic round complexity. We show that this is unlikely by proving that if a coNP-complete set has a polylogarithmic-round interactive proof system then the exponential-time hierarchy collapses. We also consider exponential versions of the Karp-Lipton theorem and Yap's theorem. * Some of the results of this paper were presented at the 19th IEEE Conference on Computational Complexity theory by second and third author.