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Natural Proof
In computational complexity theory, a natural proof is a certain kind of proof establishing that one complexity class differs from another one. While these proofs are in some sense "natural", it can be shown (assuming a widely believed conjecture on the existence of pseudorandom functions) that no such proof can possibly be used to solve the P vs. NP problem. Overview The notion of natural proofs was introduced by Alexander Razborov and Steven Rudich in their article "Natural Proofs", first presented in 1994, and later published in 1997, for which they received the 2007 Gödel Prize. Specifically, natural proofs prove lower bounds on the circuit complexity of boolean functions. A natural proof shows, either directly or indirectly, that a boolean function has a certain natural combinatorial property. Under the assumption that pseudorandom functions exist with "exponential hardness" as specified in their main theorem, Razborov and Rudich show that these proofs cannot separate cert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Complexity Class
In computational complexity theory, a complexity class is a set (mathematics), set of computational problems "of related resource-based computational complexity, complexity". The two most commonly analyzed resources are time complexity, time and space complexity, memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time complexity, time or space complexity, memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements. For instance, the class P (complexity), P is the set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g. Counting problem (complexity), counting problems and function problems) and using other models of computation (e.g. probabil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Pseudorandom Function Family
In cryptography, a pseudorandom function family, abbreviated PRF, is a collection of efficiently-computable functions which emulate a random oracle in the following way: no efficient algorithm can distinguish (with significant advantage) between a function chosen randomly from the PRF family and a random oracle (a function whose outputs are fixed completely at random). Pseudorandom functions are vital tools in the construction of cryptographic primitives, especially secure encryption schemes. Pseudorandom functions are not to be confused with pseudorandom generators (PRGs). The guarantee of a PRG is that a ''single'' output appears random if the input was chosen at random. On the other hand, the guarantee of a PRF is that ''all its outputs'' appear random, regardless of how the corresponding inputs were chosen, as long as the ''function'' was drawn at random from the PRF family. A pseudorandom function family can be constructed from any pseudorandom generator, using, for exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
P Versus NP Problem
The P versus NP problem is a major unsolved problem in theoretical computer science. Informally, it asks whether every problem whose solution can be quickly verified can also be quickly solved. Here, "quickly" means an algorithm exists that solves the task and runs in polynomial time (as opposed to, say, exponential time), meaning the task completion time is bounded above by a polynomial function on the size of the input to the algorithm. The general class of questions that some algorithm can answer in polynomial time is " P" or "class P". For some questions, there is no known way to find an answer quickly, but if provided with an answer, it can be verified quickly. The class of questions where an answer can be ''verified'' in polynomial time is "NP", standing for "nondeterministic polynomial time".A nondeterministic Turing machine can move to a state that is not determined by the previous state. Such a machine could solve an NP problem in polynomial time by falling into t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Alexander Razborov
Aleksandr Aleksandrovich Razborov (; born February 16, 1963), sometimes known as Sasha Razborov, is a Soviet and Russian mathematician and computational theorist. He is Andrew McLeish Distinguished Service Professor at the University of Chicago. Research In his best known work, joint with Steven Rudich, he introduced the notion of '' natural proofs'', a class of strategies used to prove fundamental lower bounds in computational complexity. In particular, Razborov and Rudich showed that, under the assumption that certain kinds of one-way functions exist, such proofs cannot give a resolution of the P = NP problem, so new techniques will be required in order to solve this question. Awards * Nevanlinna Prize (1990) for introducing the "approximation method" in proving Boolean circuit lower bounds of some essential algorithmic problems, * Erdős Lecturer, Hebrew University of Jerusalem, 1998. * Corresponding member of the Russian Academy of Sciences (2000) * Gödel Prize (2007, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Steven Rudich
Steven Rudich (; October 4, 1961 – October 29, 2024) was an American computational theorist. He was a professor in the Carnegie Mellon School of Computer Science. In 1994, he and Alexander Razborov proved that a large class of combinatorial arguments, dubbed natural proofs, was unlikely to answer many of the important problems in computational complexity theory. For this work, they were awarded the Gödel Prize in 2007. He also co-authored a paper demonstrating that all currently known NP-complete problems remain NP-complete even under AC0 or NC0 reductions. Amongst Carnegie Mellon students, he is best known as the teacher of the class "Great Theoretical Ideas in Computer Science" (formerly named "How to Think Like a Computer Scientist"), often considered one of the most difficult classes in the undergraduate computer science curriculum. He was a long-time editor of the ''Journal of Cryptology'', as well as an accomplished magician. His Erdős number is 2. Leap@CMU Rudich ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gödel Prize
The Gödel Prize is an annual prize for outstanding papers in the area of theoretical computer science, given jointly by the European Association for Theoretical Computer Science (EATCS) and the Association for Computing Machinery Special Interest Group on Algorithms and Computational Theory ( ACM SIGACT). The award is named in honor of Kurt Gödel. Gödel's connection to theoretical computer science is that he was the first to mention the "P versus NP" question, in a 1956 letter to John von Neumann in which Gödel asked whether a certain NP-complete problem could be solved in quadratic or linear time. The Gödel Prize has been awarded since 1993. The prize is awarded alternately at ICALP (even years) and STOC (odd years). STOC is the ACM Symposium on Theory of Computing, one of the main North American conferences in theoretical computer science, whereas ICALP is the International Colloquium on Automata, Languages and Programming, one of the main Europe Europe is a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Circuit Complexity
In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according to the size or depth of the Boolean circuits that compute them. A related notion is the circuit complexity of a recursive language that is decided by a uniform family of circuits C_,C_,\ldots (see below). Proving lower bounds on size of Boolean circuits computing explicit Boolean functions is a popular approach to separating complexity classes. For example, a prominent circuit class P/poly consists of Boolean functions computable by circuits of polynomial size. Proving that \mathsf\not\subseteq \mathsf would separate P and NP (see below). Complexity classes defined in terms of Boolean circuits include AC0, AC, TC0, NC1, NC, and P/poly. Size and depth A Boolean circuit with n input bits is a directed acyclic graph in which every node (usually called ''gates'' in this context) is either an input node of in-degree 0 l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Boolean Function
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Boolean functions are the subject of Boolean algebra and switching theory. A Boolean function takes the form f:\^k \to \, where \ is known as the Boolean domain and k is a non-negative integer called the arity of the function. In the case where k=0, the function is a constant element of \. A Boolean function with multiple outputs, f:\^k \to \^m with m>1 is a vectorial or ''vector-valued'' Boolean function (an S-box in symmetric cryptography). There are 2^ different Boolean functions with k arguments; equal to the number of different truth tables with 2^k entries. Every k-ary Boolean function can be expressed as a propositional formula in k variables x_1,...,x_k, and two propositional formulas a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Boolean Satisfiability Problem
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) asks whether there exists an Interpretation (logic), interpretation that Satisfiability, satisfies a given Boolean logic, Boolean Formula (mathematical logic), formula. In other words, it asks whether the formula's variables can be consistently replaced by the values TRUE or FALSE to make the formula evaluate to TRUE. If this is the case, the formula is called ''satisfiable'', else ''unsatisfiable''. For example, the formula "''a'' AND NOT ''b''" is satisfiable because one can find the values ''a'' = TRUE and ''b'' = FALSE, which make (''a'' AND NOT ''b'') = TRUE. In contrast, "''a'' AND NOT ''a''" is unsatisfiable. SAT is the first problem that was proven to be NP-complete—this is the Cook–Levin theorem. This means that all problems in the complexity class NP (complexity), NP, which includes a wide range of natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Truth Table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, Boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables. In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid. A truth table has one column for each input variable (for example, A and B), and one final column showing all of the possible results of the logical operation that the table represents (for example, A XOR B). Each row of the truth table contains one possible configuration of the input variables (for instance, A=true, B=false), and the result of the operation for those values. A proposition's truth table is a graphical representation of its truth function. The truth function can be more useful for mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
P/poly
In computational complexity theory, P/poly is a complexity class that can be defined in both circuit complexity and non-uniform complexity. Since the two definitions are equivalent, this concept bridges the two areas. In the perspective of circuit complexity, P/poly is the class of problems that can be solved by small circuits. More precisely, it is the set of formal languages that have polynomial-size circuit families. In the perspective of non-uniform complexity, P/poly is defined in terms of Turing machines with advice, extra information supplied to the Turing machine along with its input, that may depend on the input length but not on the input itself. In this formulation, P/poly is the class of decision problems that can be solved by a polynomial-time Turing machine with advice strings of length polynomial in the input size. For example, the popular Miller–Rabin primality test can be formulated as a P/poly algorithm: the "advice" is a list of candidate values to test. It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
