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25
The History and Status of the P versus NP Question
, 1992
"... this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the re ..."
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Cited by 50 (0 self)
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this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the realization that certain problems are algorithmically unsolvable. At around this time, forerunners of the programmable computing machine were beginning to appear. As mathematicians contemplated the practical capabilities and limitations of such devices, computational complexity theory emerged from the theory of algorithmic unsolvability. Early on, a particular type of computational task became evident, where one is seeking an object which lies
Making Nondeterminism Unambiguous
, 1997
"... We show that in the context of nonuniform complexity, nondeterministic logarithmic space bounded computation can be made unambiguous. An analogous result holds for the class of problems reducible to contextfree languages. In terms of complexity classes, this can be stated as: NL/poly = UL/poly Lo ..."
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Cited by 37 (10 self)
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We show that in the context of nonuniform complexity, nondeterministic logarithmic space bounded computation can be made unambiguous. An analogous result holds for the class of problems reducible to contextfree languages. In terms of complexity classes, this can be stated as: NL/poly = UL/poly LogCFL/poly = UAuxPDA(log n; n O(1) )/poly
Relationships Among PL, L, and the Determinant
, 1996
"... Recent results byToda, Vinay, Damm, and Valianthave shown that the complexity of the determinantischaracterized by the complexity of counting the number of accepting computations of a nondeterministic logspacebounded machine. #This class of functions is known as #L.# By using that characterizati ..."
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Cited by 33 (8 self)
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Recent results byToda, Vinay, Damm, and Valianthave shown that the complexity of the determinantischaracterized by the complexity of counting the number of accepting computations of a nondeterministic logspacebounded machine. #This class of functions is known as #L.# By using that characterization and by establishing a few elementary closure properties, we giveavery simple proof of a theorem of Jung, showing that probabilistic logspacebounded #PL# machines lose none of their computational power if they are restricted to run in polynomial time.
NonCommutative Arithmetic Circuits: Depth Reduction and Size Lower Bounds
 Theoretical Computer Science
"... We investigate the phenomenon of depthreduction in commutativeand noncommutative arithmetic circuits. We prove that in the commutative setting, uniform semiunbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree (and unrestricted dept ..."
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Cited by 28 (10 self)
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We investigate the phenomenon of depthreduction in commutativeand noncommutative arithmetic circuits. We prove that in the commutative setting, uniform semiunbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree (and unrestricted depth); earlier proofs did not work in the uniform setting. This also provides a unified proof of the circuit characterizations of the class LOGCFL and its counting variant #LOGCFL. We show that AC 1 has no more power than arithmetic circuits of polynomial size and degree n O(log log n) (improving the trivial bound of n O(logn) ). Connections are drawn between TC 1 and arithmetic circuits of polynomial size and degree. Then we consider noncommutative computation. We show that over the algebra (\Sigma ; max, concat), arithmetic circuits of polynomial size and polynomial degree can be reduced to O(log 2 n) depth (and even to O(log n) depth if unboundedfanin gates are allowed) . This...
Determinant: Combinatorics, Algorithms, and Complexity
, 1997
"... We prove a new combinatorial characterization of the determinant. The characterization yields a simple combinatorial algorithm for computing the determinant. Hitherto, all (known) algorithms for the determinant have been based on linear algebra. Our combinatorial algorithm requires no division, a ..."
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Cited by 27 (6 self)
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We prove a new combinatorial characterization of the determinant. The characterization yields a simple combinatorial algorithm for computing the determinant. Hitherto, all (known) algorithms for the determinant have been based on linear algebra. Our combinatorial algorithm requires no division, and works over arbitrary commutative rings. It also lends itself to e#cient parallel implementations. It has been known for some time now that the complexity class GapL characterizes the complexity of computing the determinant of matrices over the integers. We present a direct proof of this characterization.
Boolean Circuits, Tensor Ranks, And Communication Complexity
 SIAM J. ON COMPUTING
, 1997
"... We investigate two methods for proving lower bounds on the size of small depth circuits, namely the approaches based on multiparty communication games and algebraic characterizations extending the concepts of the tensor rank and rigidity of matrices. Our methods are combinatorial, but we think that ..."
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Cited by 24 (2 self)
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We investigate two methods for proving lower bounds on the size of small depth circuits, namely the approaches based on multiparty communication games and algebraic characterizations extending the concepts of the tensor rank and rigidity of matrices. Our methods are combinatorial, but we think that the main contribution concerns the algebraic concepts used in this area (tensor ranks and rigidity). Our main results are following. (i) An o(n) bit protocol for a communication game for computing shifts, which also gives an upper bound of o(n 2 ) on the contact rank of the tensor of multiplication of polynomials; this disproves some earlier conjectures. A related probabilistic construction gives o(n) upper bound for computing all permutations and O(n log log n) upper bound on the communication complexity of pointer jumping with permutations. (ii) A lower bound on certain restricted circuits of depth 2 which are related to the problem of proving a superlinear lower bound on the size of ...
On TC^0, AC^0, and Arithmetic Circuits
 Journal of Computer and System Sciences
, 2000
"... Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC [CMTV96], we study the class of functions . One way to define #AC is as the class of functions computed by constantdepth polynomialsize ..."
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Continuing a line of investigation that has studied the function classes #P [Val79b], #SAC [Val79a, Vin91, AJMV], #L [AJ93b, Vin91, AO94], and #NC [CMTV96], we study the class of functions . One way to define #AC is as the class of functions computed by constantdepth polynomialsize arithmetic circuits of unbounded fanin addition and multiplication gates. In contrast to the preceding # Part of this research was done while visiting the University of Ulm under an Alexander von Humboldt Fellowship.
A Combinatorial Algorithm for the Determinant
 In Proceedings of the 8th Annual ACMSIAM Symposium on Discrete Algorithms
, 1997
"... We show the first efficient combinatorial algorithm for the computation of the determinant. Hitherto, all (known) algorithms for determinant have been based on linear algebra. In contrast, our algorithm and its proof of correctness are totally combinatorial in nature. The algorithm requires no divis ..."
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Cited by 12 (1 self)
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We show the first efficient combinatorial algorithm for the computation of the determinant. Hitherto, all (known) algorithms for determinant have been based on linear algebra. In contrast, our algorithm and its proof of correctness are totally combinatorial in nature. The algorithm requires no division and works on arbitrary commutative rings. It also lends itself to efficient sequential and parallel implementations. 1 Introduction The Determinant has been the subject of study for over 200 years. Its history can be traced back to Leibnitz, Crammer, Vandermode, Binet, Cauchy, Jacobi, Gauss and others. Given its importance in linear algebra in particular and in geometry in general, it is not surprising that a galaxy of great mathematicians investigated the determinant from varied viewpoints. The algorithmic history of the determinant is as old as the mathematical concept itself. After all, the determinant was invented to solve systems of linear equations. Much of the initial effort was...
Hardness as randomness: A survey of universal derandomization
 in Proceedings of the International Congress of Mathematicians
, 2002
"... We survey recent developments in the study of probabilistic complexity classes. While the evidence seems to support the conjecture that probabilism can be deterministically simulated with relatively low overhead, i.e., that P = BPP, it also indicates that this may be a difficult question to resolve. ..."
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Cited by 11 (5 self)
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We survey recent developments in the study of probabilistic complexity classes. While the evidence seems to support the conjecture that probabilism can be deterministically simulated with relatively low overhead, i.e., that P = BPP, it also indicates that this may be a difficult question to resolve. In fact, proving that probalistic algorithms have nontrivial deterministic simulations is basically equivalent to proving circuit lower bounds, either in the algebraic or Boolean models.
Relationships Between Quantum and Classical SpaceBounded Complexity Classes
 In Proceedings of the 13th IEEE Conference on Computational Complexity
, 1998
"... This paper investigates the relative power of spacebounded quantum and classical (probabilistic) computational models. The following relationships are proved. 1. Any probabilistic Turing machine (PTM) which runs in space s and which halts absolutely (i.e. halts with certainty after a finite number o ..."
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Cited by 11 (1 self)
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This paper investigates the relative power of spacebounded quantum and classical (probabilistic) computational models. The following relationships are proved. 1. Any probabilistic Turing machine (PTM) which runs in space s and which halts absolutely (i.e. halts with certainty after a finite number of steps) can be simulated in space O(s) by a quantum Turing machine (QTM). If the PTM operates with bounded error, then the QTM may be taken to operate with bounded error as well, although the QTM may not halt absolutely in this case. In the unbounded error case, the QTM may be taken to halt absolutely. 2. Any QTM running in space s can be simulated by an unbounded error PTM running in space O(s). No assumptions on the probability of error or running time for the QTM are required, but it is assumed that all transition amplitudes of the quantum machine are rational. It follows that unbounded error, space O(s) bounded quantum Turing machines and probabilistic Turing machines are equivalent in...