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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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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
Characterizing nondeterministic circuit size
 In Proceedings of the 25th STOC
, 1993
"... Consider the following simple communication problem. Fix a universe U and a family Ω of subsets of U. Players I and II receive, respectively, an element a ∈ U and a subset A ∈ Ω. Their task is to find a subset B of U such that A∩B  is even and a ∈ B. With every Boolean function f we associate a co ..."
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Consider the following simple communication problem. Fix a universe U and a family Ω of subsets of U. Players I and II receive, respectively, an element a ∈ U and a subset A ∈ Ω. Their task is to find a subset B of U such that A∩B  is even and a ∈ B. With every Boolean function f we associate a collection Ωf of subsets of U = f −1 (0), and prove that its (one round) communication complexity completely determines the size of the smallest nondeterministic circuit for f. We propose a linear algebraic variant to the general approximation method of Razborov, which has exponentially smaller description. We use it to derive four different combinatorial problems (like the one above) that characterize NP. These are tight, in the sense that they can be used to prove superlinear circuit size lower bounds. Combined with Razborov’s method, they present a purely combinatorial framework in which to study the P vs. NP vs. co − NP question.
The Fusion Method for Lower Bounds in Circuit Complexity
 Keszthely (Hungary
, 1993
"... This paper coins the term "The Fusion Method" to a recent approach for proving circuit lower bounds. It describes the method, and surveys its achievements, potential and challenges. 1 Introduction In a recent paper, Karchmer [6] suggested an elegant way in which one can view at the same time both t ..."
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This paper coins the term "The Fusion Method" to a recent approach for proving circuit lower bounds. It describes the method, and surveys its achievements, potential and challenges. 1 Introduction In a recent paper, Karchmer [6] suggested an elegant way in which one can view at the same time both the "approximation method" of Razborov [13] and the "topological approach" of Sipser [15] for proving circuit lower bounds. In Karchmer's setting the lower bound prover shows that a given circuit C is too small for computing a given function f by contradiction, in the following way. She tries to combine (or 'fuse', as we propose calling it) correct accepting computations of inputs in f \Gamma1 (1) by C into an incorrect accepting computation of an input in f \Gamma1 (0). It turns out that this "Fusion Method" reduces the dynamic computation of f by C into a static combinatorial cover problem, which provides the lower bound. Moreover, different restrictions on how we can fuse computations ...
Combinatorics of Monotone Computations
 Combinatorica
, 1998
"... Our main result is a combinatorial lower bounds criterion for a general model of monotone circuits, where we allow as gates: (i) arbitrary monotone Boolean functions whose minterms or maxterms (or both) have length 6 d, and (ii) arbitrary realvalued nondecreasing functions on 6 d variables. This r ..."
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Our main result is a combinatorial lower bounds criterion for a general model of monotone circuits, where we allow as gates: (i) arbitrary monotone Boolean functions whose minterms or maxterms (or both) have length 6 d, and (ii) arbitrary realvalued nondecreasing functions on 6 d variables. This resolves a problem, raised by Razborov in 1986, and yields, in a uniform and easy way, nontrivial lower bounds for circuits computing explicit functions even when d !1. The proof is relatively simple and direct, and combines the bottlenecks counting method of Haken with the idea of finite limit due to Sipser. We demonstrate the criterion by superpolynomial lower bounds for explicit Boolean functions, associated with bipartite Paley graphs and partial tdesigns. We then derive exponential lower bounds for cliquelike graph functions of Tardos, thus establishing an exponential gap between the monotone real and nonmonotone Boolean circuit complexities. Since we allow real gates, the criterion...
Finite Limits and Monotone Computations: The Lower Bounds Criterion
 Proc. of the 12th IEEE Conference on Computational Complexity
, 1997
"... Our main result is a combinatorial lower bounds criterion for monotone circuits over the reals. We allow any unbounded fanin nondecreasing realvalued functions as gates. The only requirement is their "locality ". Unbounded fanin AND and OR gates, as well as any threshold gate T m s (x 1 ; : : : ..."
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Our main result is a combinatorial lower bounds criterion for monotone circuits over the reals. We allow any unbounded fanin nondecreasing realvalued functions as gates. The only requirement is their "locality ". Unbounded fanin AND and OR gates, as well as any threshold gate T m s (x 1 ; : : : ; xm ) with small enough threshold value minfs; m \Gamma s + 1g, are simplest examples of local gates. The proof is relatively simple and direct, and combines the bottlenecks counting approach of Haken with the idea of finite limit due to Sipser. Apparently this is the first combinatorial lower bounds criterion for monotone computations. It is symmetric and yields (in a uniform and easy way) exponential lower bounds. 1. Introduction The question of determining how much economy the universal nonmonotone basis f; ; :g provides over the monotone basis f; g has been a long standing open problem in Boolean circuit complexity. The The work was supported by a DFG grant Me 1077/101. Preliminary...
On Ultrafilters and NP
"... We further develop the Fusion Method by exploring its similarities with the Ultraproduct Construction in Model Theory. We use this analogy to reprove a result of Sipser regarding countable circuits, in a simpler way. In the finite case this analogy allows us to give a new characterization of coNP ..."
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We further develop the Fusion Method by exploring its similarities with the Ultraproduct Construction in Model Theory. We use this analogy to reprove a result of Sipser regarding countable circuits, in a simpler way. In the finite case this analogy allows us to give a new characterization of coNP in terms of the CLIQUE function. This gives a natural interpretation to the NPcompleteness of the CLIQUE function. Introduction In this paper we further develop the analogy between Razborov's Generalized Approximation Method and the Ultraproduct Construction in Model Theory [3]. This analogy, in all its variations, has been named the Fusion Method [10]. The Fusion Method has many different incarnations (see [10] for a survey). Razborov originally used it to characterize the classes P and NL [6, 7]. He also used it to give a superlinear lower bound for the complexity of the Majority function on SwitchingandRectifying Networks [7]. Karchmer [3] shows that the monotone size lower bounds for...
Finite Limits and Lower Bounds for Circuits Size
, 1994
"... The lower bounds problem in circuit complexity theory may be looked as the problem about the possibility to diagonalize over finite sets of computations. Our goal here is to show that Sipser's notion of "finite limit" is the right diagonal in different situations. ..."
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The lower bounds problem in circuit complexity theory may be looked as the problem about the possibility to diagonalize over finite sets of computations. Our goal here is to show that Sipser's notion of "finite limit" is the right diagonal in different situations.
COMBINATORICA Bolyai Society – SpringerVerlag COMBINATORICA 19 (1) (1999) 65–85 COMBINATORICS OF MONOTONE COMPUTATIONS STASYS JUKNA*
, 1996
"... Our main result is a combinatorial lower bounds criterion for a general model of monotone circuits, where we allow as gates: (i) arbitrary monotone Boolean functions whose minterms or maxterms (or both) have length ≤ d, and (ii) arbitrary realvalued nondecreasing functions on ≤ d variables. This r ..."
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Our main result is a combinatorial lower bounds criterion for a general model of monotone circuits, where we allow as gates: (i) arbitrary monotone Boolean functions whose minterms or maxterms (or both) have length ≤ d, and (ii) arbitrary realvalued nondecreasing functions on ≤ d variables. This resolves a problem, raised by Razborov in 1986, and yields, in a uniform and easy way, nontrivial lower bounds for circuits computing explicit functions even when d →∞. The proof is relatively simple and direct, and combines the bottlenecks counting method of Haken with the idea of finite limit due to Sipser. We demonstrate the criterion by superpolynomial lower bounds for explicit Boolean functions, associated with bipartite Paley graphs and partial tdesigns. We then derive exponential lower bounds for cliquelike graph functions of Tardos, thus establishing an exponential gap between the monotone real and nonmonotone Boolean circuit complexities. Since we allow real gates, the criterion also implies corresponding lower bounds for the length of cutting planes proof in the propositional calculus. 1.
The Expressive Power of Analog Recurrent Neural Networks on Infinite Input Streams
, 2012
"... We consider analog recurrent neural networks working on infinite input streams, provide a complete topological characterization of their expressive power, and compare it to the expressive power of classical infinite word reading abstract machines. More precisely, we consider analog recurrent neural ..."
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We consider analog recurrent neural networks working on infinite input streams, provide a complete topological characterization of their expressive power, and compare it to the expressive power of classical infinite word reading abstract machines. More precisely, we consider analog recurrent neural networks as language recognizers over the Cantor space, and prove that the classes of ωlanguages recognized by deterministic and nondeterministic analog networks correspond precisely to the respective classes of Π 0 2sets and Σ 1 1sets of the Cantor space. Furthermore, we show that the result can be generalized to more expressive analog networks equipped with any kind of Borel accepting condition. Therefore, in the deterministic case, the expressive power of analog neural nets turns out to be comparable to the expressive power of any kind of Büchi abstract machine, whereas in the nondeterministic case, analog recurrent networks turn out to machine, including the main cases of classical automata, 1counter automata, kcounter automata, pushdown automata, and Turing machines.