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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
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
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
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)
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
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
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
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
Subfield of automated reasoning and mathematical logic
Ramanujan machine Computer-aided proof Formal verification Logic programming Proof checking Model checking Proof complexity Computer algebra system Program
Automated_theorem_proving
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
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
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
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
these systems. The characterization of standard complexity classes and correspondence to propositional proof systems allows to interpret theories of bounded
Bounded_arithmetic
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
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
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
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
Indian computer scientist
includes publications in proof complexity, algebraic circuit complexity, small-space complexity classes, parameterized complexity, and algorithms for planar
Meena_Mahajan
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
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
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
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
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
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
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
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
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
Sufficient evidence/argument for truth
proposition Proof procedure Proof complexity Standard of proof Proving a negative Proof of impossibility – Category of mathematical proof Proof and other
Proof_(truth)
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
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
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
Israeli computer scientist and mathematician
areas including randomized computation, cryptography, circuit complexity, proof complexity, parallel computation, and our understanding of fundamental graph
Avi_Wigderson
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
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
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
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)
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
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
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)
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
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
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
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 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
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)
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)
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
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
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
Subunit of a computational problem
In computational complexity theory, a gadget is a subunit of a problem instance that simulates the behavior of one of the fundamental units of a different
Gadget_(computer_science)
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
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
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
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
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
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
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
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
The polynomial hierarchy is contained in probabilistic Turing machine in polynomial time
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
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
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)
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
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
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
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)
Existential second order logic captures NP
oldest result of descriptive complexity theory, a branch of computational complexity theory that characterizes complexity classes in terms of logic-based
Fagin's_theorem
Technique for proving sets have equal size
admit bijective proofs are not limited to binomial coefficient identities. As the complexity of the problem increases, a bijective proof can become very
Bijective_proof
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)
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
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)
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
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
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
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
Algorithm that employs a degree of randomness as part of its logic or procedure
Carlo algorithms are considered, and several complexity classes are studied. The most basic randomized complexity class is RP, which is the class of decision
Randomized_algorithm
Canadian logician, philosopher of mathematics
philosophical relevance of proof theory. In mathematical logic, he has made contributions to proof theory (epsilon calculus, proof complexity) and to modal and
Richard_Zach
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
Limitative results in mathematical logic
method of producing independent sentences, based on Kolmogorov complexity. Like the proof presented by Kleene that was mentioned above, Chaitin's theorem
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
problems, and whose security thus follows from rigorous mathematical proofs, complexity theory and formal reduction. These functions are called provably secure
Security of cryptographic hash functions
Security_of_cryptographic_hash_functions
Proof in set theory
existence of arbitrarily hard complexity classes and played a key role in early attempts to prove P does not equal NP. The above proof fails for W. V. Quine's
Cantor's_diagonal_argument
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
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
Task of computing complete subgraphs
basis for proofs of W[1]-hardness of many other problems, and thus serves as an analogue of the Cook–Levin theorem for parameterized complexity. Chen et
Clique_problem
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
arithmetic and first order bounded arithmetic", Arithmetic, Proof Theory and Computational Complexity, vol. 23, pp. 247–277, doi:10.1093/oso/9780198536901.003
Switching_lemma
Mathematical proof at least partially generated by computer
program correct does not appeal to computer proof skeptics, who see it as adding another layer of complexity without addressing the perceived need for human
Computer-assisted_proof
ACC, is a class of computational models and problems defined in circuit complexity, a field of theoretical computer science. The class is defined by augmenting
ACC0
Complexity class
algorithms. Christos Papadimitriou (1994). "On the complexity of the parity argument and other inefficient proofs of existence" (PDF). Journal of Computer and
PPAD_(complexity)
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
Establishment of a theorem using inference from the axioms
In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (known as well-formed formulas when relating to formal language)
Formal_proof
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
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)
of complexity classes in computational complexity theory. For other computational and complexity subjects, see list of computability and complexity topics
List_of_complexity_classes
Israeli computer scientist
in Jerusalem. Safra's research areas include complexity theory and automata theory. His work in complexity theory includes the classification of approximation
Shmuel_Safra
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)
Classification of computer problems
Geometric complexity theory (GCT), is a research program in computational complexity theory proposed by Ketan Mulmuley and Milind Sohoni. The goal of the
Geometric_complexity_theory
PROOF COMPLEXITY
PROOF COMPLEXITY
Boy/Male
Arabic, Muslim
Evidence; Proof
Boy/Male
Muslim
Evidence. Proof.
Boy/Male
Arabic
Proof; Evidence
Girl/Female
Muslim
Guide, Proof
Boy/Male
Afghan, Arabic, Hindu, Indian, Muslim
Proof
Girl/Female
Indian
Witness; Proof
Girl/Female
Arabic, Muslim
Guide; Proof
Boy/Male
Muslim
Argument, Reasoning, Proof
Boy/Male
Muslim
Proof
Boy/Male
Arabic, Muslim
The Proof
Boy/Male
Arabic, French, German, Gujarati, Hindu, Indian, Malaysian, Muslim, Turkish
Proof; Evidence
Boy/Male
Muslim
Proof
Girl/Female
Muslim
Proof
Boy/Male
Indian
Argument, Reasoning, Proof
Boy/Male
Indian
Proof
Girl/Female
Muslim/Islamic
Guide Proof
Boy/Male
Muslim/Islamic
Proof
Boy/Male
Indian
Proof
Boy/Male
Muslim/Islamic
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.
PROOF COMPLEXITY
PROOF COMPLEXITY
Boy/Male
Arabic
Precious; Talented
Boy/Male
Indian
Presence of the foremost one
Boy/Male
British, English, German
Leader; The People's Ruler
Girl/Female
Afghan, Arabic, German, Gujarati, Hindu, Indian, Kannada, Kurdish, Malayalam, Marathi, Muslim, Parsi, Punjabi, Sikh, Sindhi
A Diamond; Adamant; Brightness
Surname or Lastname
English
English : probably a habitational name from a place in Devon named Bowditch, from the Old English phrase būfan dīce ‘above the ditch’.The surname Bowditch is well known in New England. Nathaniel Bowditch (1773–1838), author of The Practical Navigator (1772), a standard work that went through more than sixty editions, was born in Salem, MA, the son of a shipmaster. The family can be traced back, via a clothier who settled in New England in 1671, to Thorncombe in Devon in the early 16th century.
Boy/Male
French American
A title name ranking below duke and above earl.
Female
English
Feminine form of English Dean, DEANA means "dean, head, leader."
Boy/Male
Indian
Face
Girl/Female
Indian
Goddess Durga
Female
Gypsy/Romani
 Perhaps a Romani form of the biblical Hebrew name Yael (English Jael), JAELLE means "chamois," "ibex," or "mountain goat."Â
PROOF COMPLEXITY
PROOF COMPLEXITY
PROOF COMPLEXITY
PROOF COMPLEXITY
PROOF COMPLEXITY
n.
Proof; trial.
n.
Demonstration; proof.
n.
Proof; evidence.
n.
Proof.
n.
A trial impression, as from type, taken for correction or examination; -- called also proof sheet.
v. t.
Armor of excellent or tried quality, and deemed impenetrable; properly, armor of proof.
n.
Proof.
n.
Trial; proof.
a.
Firm or successful in resisting; as, proof against harm; waterproof; bombproof.
a.
Used in proving or testing; as, a proof load, or proof charge.
a.
Proof against proofs; obstinate in the wrong.
v. t.
To cover with a roof.
n.
The cover of any building, including the roofing (see Roofing) and all the materials and construction necessary to carry and maintain the same upon the walls or other uprights. In the case of a building with vaulted ceilings protected by an outer roof, some writers call the vault the roof, and the outer protection the roof mask. It is better, however, to consider the vault as the ceiling only, in cases where it has farther covering.
a.
Highly rectified; very strongly alcoholic; as, high-proof spirits.
n.
Proof.
n.
That which resembles, or corresponds to, the covering or the ceiling of a house; as, the roof of a cavern; the roof of the mouth.
n.
Proof.
v. t.
To arm with proof armor; to arm securely; as, to proof-arm herself.