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Field in logic and theoretical computer science
theoretical computer science, and specifically proof theory and computational complexity theory, proof complexity is the field aiming to understand and analyse
Proof_complexity
Complexity class used to classify decision problems
problems in computer science In computational complexity theory, NP (nondeterministic polynomial time) is a complexity class used to classify decision problems
NP_(complexity)
Proof checkable by a randomized algorithm
In computational complexity theory, a probabilistically checkable proof (PCP) is a type of proof that can be checked by a randomized algorithm using a
Probabilistically checkable proof
Probabilistically_checkable_proof
Measure of algorithmic complexity
= 1 to infinity: if nth_proof_proves_complexity_formula(i) and complexity_lower_bound_nth_proof(i) ≥ n return string_nth_proof(i) Given an n {\displaystyle
Kolmogorov_complexity
American computer scientist
contributions to complexity theory include: the construction of a pseudorandom number generator from any one-way function, his proof of Yao's XOR lemma
Russell_Impagliazzo
American-Canadian computer scientist, contributor to complexity theory
who has made significant contributions to the fields of complexity theory and proof complexity. He is a university professor emeritus at the University
Stephen_Cook
Set of problems in computational complexity theory
machines, interactive proof systems, Boolean circuits, and quantum computers). The study of the relationships between complexity classes is a major area
Complexity_class
Boolean satisfiability is NP-complete and therefore that NP-complete problems exist
In computational complexity theory, the Cook–Levin theorem, also known as Cook's theorem, states that the Boolean satisfiability problem is NP-complete
Cook–Levin_theorem
Provides lower bounds on the circuit complexity of boolean functions
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
Natural_proof
Branch of mathematical logic
Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects,
Proof_theory
Abstract machine that models computation
In computational complexity theory, an interactive proof system is an abstract machine that models computation as the exchange of messages between two
Interactive_proof_system
these systems. The characterization of standard complexity classes and correspondence to propositional proof systems allows to interpret theories of bounded
Bounded_arithmetic
Unsolved problem in computer science
of mathematical proofs could be automated. The relation between the complexity classes P and NP is studied in computational complexity theory, the part
P_versus_NP_problem
Complexity of sending information in a distributed algorithm
In theoretical computer science, communication complexity studies the amount of communication required to solve a problem when the input to the problem
Communication_complexity
American computer scientist and mathematician
major contributions to the fields of mathematical logic, complexity theory and proof complexity. He is currently a professor at the University of California
Samuel_Buss
Inherent difficulty of computational problems
or by encoding their adjacency lists in binary. Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input
Computational complexity theory
Computational_complexity_theory
Theorem in computational complexity theory
checkable proofs (proofs that can be checked by a randomized algorithm) of constant query complexity and logarithmic randomness complexity (uses a logarithmic
PCP_theorem
propositional calculus and proof complexity a propositional proof system (pps), also called a Cook–Reckhow propositional proof system, is a system for proving
Propositional_proof_system
Propositional proof system
In proof complexity, a Frege system is a propositional proof system whose proofs are sequences of formulas derived using a finite set of sound and implicationally
Frege_system
Canadian-American computer scientist
focused on proof complexity, a branch of computational complexity theory that seeks upper and lower bounds on the lengths of mathematical proofs of logical
Toniann_Pitassi
Data structure for storing non-overlapping sets
Bernard A. Galler and Michael J. Fischer in 1964. In 1973, their time complexity was bounded to O ( log ∗ ( n ) ) {\displaystyle O(\log ^{*}(n))} , the
Disjoint-set_data_structure
Unproven computational hardness assumption
peer-reviewed. OpenAI stated that future review of the proof would be done in collaboration with complexity researchers Russell Impagliazzo, Ramamohan Paturi
Exponential_time_hypothesis
Index of articles associated with the same name
proof, a proof that can be verified by making a small number of queries to the bits of the proof Quantum complexity theory#Quantum query complexity,
Query_complexity
Indian computer scientist
includes publications in proof complexity, algebraic circuit complexity, small-space complexity classes, parameterized complexity, and algorithms for planar
Meena_Mahajan
Subfield of automated reasoning and mathematical logic
Ramanujan machine Computer-assisted proof Formal verification Logic programming Proof checking Model checking Proof complexity Computer algebra system Program
Automated_theorem_proving
Given more time, a Turing machine can solve more problems
In computational complexity theory, the time hierarchy theorems are important statements about time-bounded computation on Turing machines. Informally
Time_hierarchy_theorem
Complexity class from interactive proofs
computational complexity theory, the class IP (which stands for interactive proof) is the class of problems solvable by an interactive proof system. It is
IP_(complexity)
Concept in the philosophy of mathematics
Troelstra Predicative Arithmetic by Edward Nelson Logical Foundations of Proof Complexity by Stephen A. Cook and Phuong The Nguyen Bounded Reverse Mathematics
Ultrafinitism
Topics referred to by the same term
true Proof complexity, computational resources required to prove statements Proof procedure, method for producing proofs in proof theory Proof theory
Proof
Sufficient evidence/argument for truth
Proof procedure Proof complexity Standard of proof Proving a negative (disambiguation) Proof of impossibility – Category of mathematical proof Proof and
Proof_(truth)
Estimate of time taken for running an algorithm
the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly
Time_complexity
Branch of mathematical logic
between complexity and the logic of finite structures allows results to be transferred easily from one area to the other, facilitating new proof methods
Descriptive_complexity_theory
Abstract machine used to study decision problems
In complexity theory and computability theory, an oracle machine is an abstract machine that can query a black box called an oracle, which is able to give
Oracle_machine
Israeli computer scientist and businessman
and computational complexity theory. In the early 2000s, Ben-Sasson published a series of articles on short, efficiently testable proofs, including quasi-linear
Eli_Ben-Sasson
Russian mathematician and computer scientist
transformation used in SAT solvers, Tseitin tautologies used in the proof complexity theory, and for his work on Algol 68. Tseitin studied mathematics at
Grigori_Tseitin
Amount of resources to perform an algorithm
In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus
Computational_complexity
Class of problems solvable in polynomial time
In computational complexity theory, P, also known as PTIME or DTIME(nO(1)), is a fundamental complexity class. It contains all decision problems that can
P_(complexity)
Interactive proof system in computational complexity theory
In computational complexity theory, an Arthur–Merlin protocol, introduced by Babai (1985), is an interactive proof system in which the verifier's coin
Arthur–Merlin_protocol
Spanish computer scientist
computer scientist interested in logic in computer science, including proof complexity and algorithms for the maximum satisfiability problem. She is a professor
María_Luisa_Bonet
Proving validity without revealing other data
In cryptography, a zero-knowledge proof (also known as a ZK proof or ZKP) is a protocol in which one party (the prover) can convince another party (the
Zero-knowledge_proof
Systematic method for producing proofs
procedure will diverge (not terminate). Automated theorem proving Proof complexity Deductive system Willard Quine 1982 (1950). Methods of Logic. Harvard
Proof_procedure
Complexity class in computer science
Reingold et al. in 2005. A proof of this is the holy grail of the efforts in the field of unconditional derandomization of complexity classes. A major step
RL_(complexity)
1995 publication in mathematics
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves
Wiles's proof of Fermat's Last Theorem
Wiles's_proof_of_Fermat's_Last_Theorem
This is a list of computability and complexity topics, by Wikipedia page. Computability theory is the part of the theory of computation that deals with
List of computability and complexity topics
List_of_computability_and_complexity_topics
Israeli computer scientist and mathematician
areas including randomized computation, cryptography, circuit complexity, proof complexity, parallel computation, and our understanding of fundamental graph
Avi_Wigderson
Category of mathematical proof
computational complexity theory, techniques like relativization (the addition of an oracle) allow for "weak" proofs of impossibility, in that proofs techniques
Proof_of_impossibility
Class of problems in computer science
In complexity theory, PP, or PPT is the class of decision problems solvable by a probabilistic Turing machine in polynomial time with an error probability
PP_(complexity)
Quantum Merlin Arthur
abbreviation for Quantum Merlin Arthur, refers to a complexity class in computational complexity theory. It is the set of all formal languages that satisfy
QMA
Model of computational complexity
In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according
Circuit_complexity
Branch of computational complexity theory
In computer science, parameterized complexity is a branch of computational complexity theory that focuses on classifying computational problems according
Parameterized_complexity
Concept in computer science
In computational complexity theory, a branch of computer science, bounded-error probabilistic polynomial time (BPP) is the class of decision problems solvable
BPP_(complexity)
Complexity class
computational complexity theory, the class QIP (which stands for Quantum Interactive Proof) is the quantum computing analogue of the classical complexity class
QIP_(complexity)
Approach to the study of finite semigroups and automata
Krohn-Rhodes complexity long motivated much work in semigroup theory. In June 2024, Stuart Margolis, John Rhodes, and Anne Schilling announced a proof that the
Krohn–Rhodes_theory
Implicit computational complexity (ICC) is a subfield of computational complexity theory that characterizes programs by constraints on the way in which
Implicit computational complexity
Implicit_computational_complexity
Japanese mathematician (1926–2017)
ISBN 978-981-238-279-5, MR 1984952 Sam Buss (2017-05-10). "[Proof Complexity] Gaisi Takeuti". Proof-Complexity mailing list. Retrieved 2019-01-13. Takeuti 2013.
Gaisi_Takeuti
Type of search algorithm
unsatisfiable instances correspond to tree resolution refutation proofs. Proof complexity Herbrandization General Davis, Martin; Putnam, Hilary (1960). "A
DPLL_algorithm
Formal proof
prove it independently. A famous network of conditional proofs is the NP-complete class of complexity theory. There is a large number of interesting tasks
Conditional_proof
Overview of and topical guide to algorithms
The study of algorithms includes their design, proof of correctness, efficiency, computational complexity, and implementation in computer programs. Algorithm
Outline_of_algorithms
Reasoning for mathematical statements
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The
Mathematical_proof
Problem in formal logic
2307/2268661. Stephen Cook; Phuong Nguyen (2010). Logical foundations of proof complexity. Cambridge University Press. p. 224. ISBN 978-0-521-51729-4. (Author's
Horn-satisfiability
Bounded-error probabilistic polynomial time is contained in the polynomial time hierarchy
In computational complexity theory, the Sipser–Lautemann theorem or Sipser–Gács–Lautemann theorem states that bounded-error probabilistic polynomial (BPP)
Sipser–Lautemann_theorem
Mathematical model of computation
that includes several important complexity classes is allowing for an error probability of 1/3. For instance, the complexity class BPP is defined as the class
Probabilistic_Turing_machine
Algorithmic runtime requirements for matrix multiplication
computational complexity as matrix multiplication. The proof does not make any assumptions on matrix multiplication that is used, except that its complexity is O
Computational complexity of matrix multiplication
Computational_complexity_of_matrix_multiplication
Parser algorithm for languages
such algorithms, and provides uniform results regarding correctness proofs, complexity with respect to grammar classes, and optimization techniques. The
GLR_parser
Russian mathematician
introduced the notion of natural proofs, a class of strategies used to prove fundamental lower bounds in computational complexity. In particular, Razborov and
Alexander_Razborov
Concept in computer science
In complexity theory, ZPP (zero-error probabilistic polynomial time) is the complexity class of problems for which a probabilistic Turing machine exists
ZPP_(complexity)
Computational complexity class of problems
APPROX-QCIRCUIT-PROB's completeness makes it useful for proofs showing the relationships between other complexity classes and BQP. Given a description of a quantum
BQP
have made increasing progress in generating mathematical proofs at research level. Most such proofs have been constructed using models from OpenAI and Anthropic
List of mathematical discoveries made with artificial intelligence
List_of_mathematical_discoveries_made_with_artificial_intelligence
Complexity class (logarithmic space)
In computational complexity theory, L (also known as LSPACE, LOGSPACE or DLOGSPACE) is the complexity class containing decision problems that can be solved
L_(complexity)
Concept in cryptography theory
correctness is a lower bound in terms of Arthur-Merlin proof complexity, considered in a communication complexity setting. This suggests that proving linear lower
Conditional disclosure of secrets
Conditional_disclosure_of_secrets
Mathematical technique used in proof theory
property Feferman–Schütte ordinal Bachmann–Howard ordinal Complexity class Gentzen's consistency proof 1.^For 1 < n ≤ ω {\displaystyle 1<n\leq {\mathsf {\omega
Ordinal_analysis
Class in computational complexity theory
}{=}}{\mathsf {P}}} More unsolved problems in computer science In computational complexity theory, the class NC (for Nick's class) is the set of decision problems
NC_(complexity)
Subfield of computational complexity theory
Fine-grained complexity is a subfield of computational complexity theory that studies the time required to solve computational problems, especially those
Fine_grained_complexity
Complexity class
In computational complexity theory, SNP (from Strict NP) is a complexity class containing a limited subset of NP based on its logical characterization
SNP_(complexity)
Computational input that relies on the length but not content of the input
In computational complexity theory, an advice string is an extra input to a Turing machine that is allowed to depend on the length n of the input, but
Advice_(complexity)
Formal language that can be expressed using a regular expression
S2CID 14677270. Cook, Stephen; Nguyen, Phuong (2010). Logical foundations of proof complexity (1. publ. ed.). Ithaca, NY: Association for Symbolic Logic. p. 75.
Regular_language
Type of error-correcting codes
In coding theory, expander codes form a class of error-correcting codes that are constructed from bipartite expander graphs. Along with Justesen codes
Expander_code
Both deterministic and nondeterministic machines can solve more problems given more space
In computational complexity theory, the space hierarchy theorems are separation results that show that both deterministic and nondeterministic machines
Space_hierarchy_theorem
Quality of an algorithm being correct with respect to a specification
its partial correctness and its termination. The latter kind of proof (termination proof) can never be fully automated, since the halting problem is undecidable
Correctness (computer science)
Correctness_(computer_science)
Concept in computational complexity theory
In computational complexity theory, BPL (Bounded-error Probabilistic Logarithmic-space), sometimes called BPLP (Bounded-error Probabilistic Logarithmic-space
BPL_(complexity)
Algorithmic complexity class
In computational complexity theory, the complexity class EXPTIME (sometimes called EXP or DEXPTIME) is the set of all decision problems that are solvable
EXPTIME
Closure of nondeterministic space under complementation
In computational complexity theory, the Immerman–Szelepcsényi theorem states that nondeterministic space complexity classes are closed under complementation
Immerman–Szelepcsényi_theorem
Proof technique in computational complexity theory
sometimes used for space complexity classes, alternating classes, and bounded alternating classes. Theorem. If P = NP then EXP = NEXP. Proof. E X P ⊆ N E X P
Padding_argument
Form of mathematical proof
up to the next one (the step). — Concrete Mathematics, page 3 margins. A proof by induction consists of two cases. The first, the base case, proves the
Mathematical_induction
Proof by Alan Turing
Turing's proof is a proof by Alan Turing submitted on 12 November 1936 and first published in 1937 with the title "On Computable Numbers, with an Application
Turing's_proof
Type of computational problem
In computational complexity theory and computability theory, a counting problem is a type of computational problem that is obtained by strengthening a
Counting_problem_(complexity)
Theorem in computational complexity theory
Simple Proof of Toda's Theorem". Theory of Computing. 5: 135–140. doi:10.4086/toc.2009.v005a007. Arora, Sanjeev; Barak, Boaz (2009). "17. Complexity of counting"
Toda's_theorem
Notion in combinatorial game theory
Combinatorial game theory measures game complexity in several ways: State-space complexity (the number of legal game positions from the initial position)
Game_complexity
Method for finding minimum spanning trees
to find the minimum spanning forest. In terms of their asymptotic time complexity, these three algorithms are equally fast for sparse graphs, but slower
Prim's_algorithm
Mathematical game
Foundations of Computer Science, Japan. Jakob Nordström. Pebble Games, Proof Complexity, and Time-Space Trade-offs. Logical Methods in Computer Science, volume
Pebble_game
Mathematical proof expressed visually
In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident
Proof_without_words
Unsolved problem in computational complexity theory
checkable proof for problems in NP. A unique game can be viewed as a special kind of nonadaptive probabilistically checkable proof with query complexity 2, where
Unique_games_conjecture
If there is a polynomial time algorithm for unambiguous-SAT, then NP equals RP
belongs to the promise version of the complexity class UP (the class UP as such is only defined for languages). The proof of the Valiant–Vazirani theorem consists
Valiant–Vazirani_theorem
Problem of determining if a Boolean formula could be made true
an NP-complete problem did not even exist. The proof shows how every decision problem in the complexity class NP can be reduced to the SAT problem for
Boolean satisfiability problem
Boolean_satisfiability_problem
Computational complexity
in computer science In computational complexity theory, NL (Nondeterministic Logarithmic-space) is the complexity class containing decision problems that
NL_(complexity)
Prize in foundations of computer science
2019), Knuth Prize 2019 Awarded For Contributions To Complexity Theory "Optimization, Complexity and Math ... using Gradient" – Knuth Prize Lecture, STOC
Knuth_Prize
Relation between deterministic and nondeterministic space complexity
In computational complexity theory, Savitch's theorem, proved by Walter Savitch in 1970, gives a relationship between deterministic and non-deterministic
Savitch's_theorem
1651. Provides information on the connection between various complexity classes. Proof of BPQ in AWPP. Fenner, Stephen A. (2003). "PP-lowness and a simple
AWPP
Ordinal-indexed family of rapidly increasing functions
In computability theory, computational complexity theory and proof theory, a fast-growing hierarchy (also called an extended Grzegorczyk hierarchy, or
Fast-growing_hierarchy
Problem in mathematics
other values, and the chance of guessing them correct is very low. The complexity of the protocol is O ( d 2 ) {\displaystyle O(d^{2})} . Alice constructs
Yao's_Millionaires'_problem
travel, tourism, insurance
PROOF COMPLEXITY
PROOF COMPLEXITY
Girl/Female
Muslim
Guide, Proof
Boy/Male
Muslim
Proof
Boy/Male
Arabic, Muslim
The Proof
Boy/Male
Muslim/Islamic
Proof
Boy/Male
Muslim
Argument, Reasoning, Proof
Surname or Lastname
English
English : variant of Rolfe.German : from Ruffo, a short form of a personal name formed with hrÅd ‘renown’, ‘victory’.Probably an Americanized spelling of German Ruf and Ruff.
Boy/Male
Indian
Proof
Boy/Male
Arabic, French, German, Gujarati, Hindu, Indian, Malaysian, Muslim, Turkish
Proof; Evidence
Boy/Male
Arabic, Muslim
Evidence; Proof
Girl/Female
Muslim
Proof
Girl/Female
Arabic, Muslim
Guide; Proof
Boy/Male
Indian
Proof
Boy/Male
Muslim
Evidence. Proof.
Girl/Female
Muslim/Islamic
Guide Proof
Boy/Male
Muslim
Proof
Boy/Male
Muslim/Islamic
Proof
Girl/Female
Indian
Witness; Proof
Boy/Male
Afghan, Arabic, Hindu, Indian, Muslim
Proof
Boy/Male
Indian
Argument, Reasoning, Proof
Boy/Male
Arabic
Proof; Evidence
PROOF COMPLEXITY
PROOF COMPLEXITY
PROOF COMPLEXITY
PROOF COMPLEXITY
PROOF COMPLEXITY
PROOF COMPLEXITY
PROOF COMPLEXITY
travel, tourism, insurance