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systematically in combinatorics via the probabilistic method. They are particularly used for non-constructive proofs. Normal numbers exist. Moreover, computable
List of probabilistic proofs of non-probabilistic theorems
List_of_probabilistic_proofs_of_non-probabilistic_theorems
Applications of logic under uncertainty
Probabilistic logic (also probability logic and probabilistic reasoning) involves the use of probability and logic to deal with uncertain situations. Probabilistic
Probabilistic_logic
Nonconstructive method for mathematical proofs
Interactive proof system Las Vegas algorithm Incompressibility method Method of conditional probabilities Probabilistic proofs of non-probabilistic theorems Random
Probabilistic_method
Proof checkable by a randomized algorithm
whole proof, always accepts correct proofs and rejects incorrect proofs. However, what makes them interesting is the existence of probabilistically checkable
Probabilistically checkable proof
Probabilistically_checkable_proof
Reasoning for mathematical statements
BCE) gave some of the first known proofs of theorems in geometry. Eudoxus (408–355 BCE) and Theaetetus (417–369 BCE) formulated theorems but did not prove
Mathematical_proof
functions List of mathematical identities List of mathematical proofs List of misnamed theorems List of scientific laws List of theories Most of the results
List_of_theorems
Theorem in mathematics
1112/plms/s2-7.1.14. MR 1575669. Di Crescenzo, A. (1999). "A Probabilistic Analogue of the Mean Value Theorem and Its Applications to Reliability Theory". J. Appl
Mean_value_theorem
Extremal graph theory bound on clique-free graph edges
{n^{2}}{2}}} . Aigner & Ziegler (2018) list five different proofs of Turán's theorem. Many of the proofs involve reducing to the case where the graph is a complete
Turán's_theorem
Method of deriving conclusions
lines of a proof from preceding lines. Proofs involve a series of inferential steps and often use various rules of inference to establish the theorem they
Rule_of_inference
Intelligence of machines
of proving a new statement (conclusion) from other statements that are given and assumed to be true (the premises). Proofs can be structured as proof
Artificial_intelligence
Proving validity without revealing other data
except for trivial proofs of BPP problems. In the common random string and random oracle models, non-interactive zero-knowledge proofs exist. The Fiat–Shamir
Zero-knowledge_proof
Concept in computer science
complexity theory, ZPP (zero-error probabilistic polynomial time) is the complexity class of problems for which a probabilistic Turing machine exists with these
ZPP_(complexity)
Branch of mathematics concerning probability
Form of modelling that uses statistics to predict outcomes Probabilistic logic – Applications of logic under uncertainty Probabilistic proofs of non-probabilistic
Probability_theory
1998 mathematics book by Aigner and Ziegler
Proofs from THE BOOK is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler, first published in 1998. The book is inspired by and named
Proofs_from_THE_BOOK
Method of statistical inference
Crisis of Modern Science. Columbia University Press. ISBN 978-0-231-55335-3. The following books are listed in ascending order of probabilistic sophistication:
Bayesian_inference
Theorem in political science
for societies. The theorem was first derived by Duncan Black in 1948, and independently by Kenneth Arrow. Similar median voter theorems exist for rules like
Median_voter_theorem
problem Independent set problem Probabilistic algorithm, randomized algorithm Las Vegas algorithm Non-determinism Non-deterministic Turing machine Interactive
List of computability and complexity topics
List_of_computability_and_complexity_topics
Mathematical formula for the number of Young tableaux
many alternate proofs. Hillman and Grassl gave the first proof that illuminates the role of hooks in 1976 by proving a special case of the Stanley hook-content
Hook_length_formula
Field of knowledge
established results. These results, called theorems, include previously proved theorems, axioms, and—in case of abstraction from nature—some basic properties
Mathematics
Statement in mathematical combinatorics
the proof of the theorem, and other arguments give lower bounds. (The first exponential lower bound was obtained by Paul Erdős using the probabilistic method
Ramsey's_theorem
science, the method of conditional probabilities is a systematic method for converting non-constructive probabilistic existence proofs into efficient deterministic
Method of conditional probabilities
Method_of_conditional_probabilities
Square of a triangular number
mathematicians have studied and provided proofs of Nicomachus's theorem. Stroeker (1995) claims that "every student of number theory surely must have marveled
Squared_triangular_number
Syllogism with conditional premise(s)
(P\to R))} An example of the proofs of these theorems in such systems is given below. We use two of the three axioms used in one of the popular systems
Hypothetical_syllogism
Inherent difficulty of computational problems
lists in binary. Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input encoding, one tries to keep the
Computational complexity theory
Computational_complexity_theory
Unsolved problem in computer science
prove theorems, and some proofs have taken decades or even centuries to find after problems have been stated—for instance, Fermat's Last Theorem took over
P_versus_NP_problem
(LSH): a method of performing probabilistic dimension reduction of high-dimensional data Naive Bayes classifier: a family of probabilistic classifiers based
List_of_algorithms
Set of problems in computational complexity theory
defined using interactive proof systems. Interactive proofs generalize the proofs definition of the complexity class NP and yield insights into cryptography
Complexity_class
18 mathematical problems stated in 1998
3} ! Formanek, Edward (2011). "Theorems of W. W. Stothers and the Jacobian Conjecture in two variables". Proceedings of the American Mathematical Society
Smale's_problems
Algorithm for determining whether a number is prime
counterexample or proof that there is none, with the prize now due from the Number Theory Foundation.[citation needed] Probabilistic tests are more rigorous
Primality_test
Logical problem studied in computer science
dependently typed language that uses Z3 to find proofs; the compiler carries these proofs through to produce proof-carrying bytecode. The Viper verification
Satisfiability modulo theories
Satisfiability_modulo_theories
Mathematics award
chair of the Fields Medal Committee, Yuri I. Manin, with the first-ever IMU silver plaque in recognition of his proof of Fermat's Last Theorem. Don Zagier
Fields_Medal
Number divisible only by 1 and itself
of Groups. Dover Books on Mathematics. Courier Dover Publications. ISBN 978-0-486-81690-6. For the Sylow theorems see p. 43; for Lagrange's theorem,
Prime_number
Kth smallest value in a statistical sample
fundamental tools in non-parametric statistics and inference. Important special cases of the order statistics are the minimum and maximum value of a sample, and
Order_statistic
Open problem on 3x+1 and x/2 functions
strongest possible proof, particularly when compared to proofs that have only been verified by humans. In July 2026, a disproof of the Collatz conjecture
Collatz_conjecture
Complexity class used to classify decision problems
A major result of complexity theory is that NP can be characterized as the problems solvable by probabilistically checkable proofs where the verifier
NP_(complexity)
List of mathematical probabilists Nuisance variable Probabilistic encryption Probabilistic logic Probabilistic proofs of non-probabilistic theorems Pseudocount
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Area of discrete mathematics
The introduction of probabilistic methods in graph theory, especially in the study of Erdős and Rényi of the asymptotic probability of graph connectivity
Graph_theory
Set of problems solved by small circuits
Mahaney's Theorem. CS 810: Introduction to Complexity Theory. The University of Wisconsin–Madison. September 18, 2003. Adleman, L. M. (1978), "Two theorems on
P/poly
Statistical model used in machine learning
sometimes used in machine learning for post-processing of the (class posterior) outputs of a probabilistic n {\displaystyle n} -class classifier, uses the softmax
Flow-based_generative_model
Turing machine that halts for any input
The proof of their totality either rests on some assumptions or require another proof system. As one can enumerate all the proofs in the proof system
Decider_(Turing_machine)
Logic error due to ignoring the base rate
certain norms of probabilistic reasoning, such as Bayes' theorem. The conclusion drawn from this line of research was that human probabilistic thinking is
Base_rate_fallacy
Thesis on the nature of computability
Rosser, J. B. (1939). "An Informal Exposition of Proofs of Godel's Theorem and Church's Theorem". The Journal of Symbolic Logic. 4 (2): 53–60. doi:10.2307/2269059
Church–Turing_thesis
Counterintuitive result in probability
used the theorem to illustrate the timescales implicit in the foundations of statistical mechanics. There is straightforward proof of this theorem. As an
Infinite_monkey_theorem
precisely in the sense that the probabilistic behavior of the channel is well known and the probability of occurrence of too many or too few errors is low
List_decoding
Israeli computer scientist (born 1957)
Complexity: A Conceptual Perspective (2008), and Modern Cryptography, Probabilistic Proofs and Pseudorandomness (1998). Goldreich received the Knuth Prize in
Oded_Goldreich
Conjecture on zeros of the zeta function
non-trivial zeros must lie in the interior of the critical strip 0 < Re(s) < 1. This was a key step in their first proofs of the prime number theorem
Riemann_hypothesis
Branch of discrete mathematics
the pigeonhole principle. In probabilistic combinatorics, the questions are of the following type: what is the probability of a certain property for a random
Combinatorics
Foundations of probability theory
Probability Theory. New York: Macmillan. pp. 13–28. Formal definition of probability in the Mizar system, and the list of theorems formally proved about it.
Probability_axioms
Type of software system
process. Theorem provers use automated reasoning techniques to determine proofs of mathematical theorems. They may also be used to verify existing proofs. In
Reasoning_system
Concept in computer science
using proof of work and a difficulty adjustment function, in which participants compete to solve cryptographic hash puzzles, and probabilistically earn
Consensus_(computer_science)
Complexity class used in circuit complexity
(1984). "A theorem on probabilistic constant depth Computations". Proceedings of the sixteenth annual ACM symposium on Theory of computing - STOC '84.
TC0
Field of economics and game theory
"Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions". Journal of Economic
Mechanism_design
Algorithm to be run on quantum computers
speedup, since a classical probabilistic algorithm can solve the problem with a constant number of queries with small probability of error. The algorithm determines
Quantum_algorithm
Study of abstract machines and automata
count of the number of states in a minimal machine for the language. The pumping lemma for regular languages, also useful in regularity proofs, was proven
Automata_theory
variant of Chesterton's fence. Double counting – counting events or occurrences more than once in probabilistic reasoning, which leads to the sum of the probabilities
List_of_fallacies
Complexity class from interactive proofs
the presented proof is correct. The prover is assumed to be infinite in computation and storage, while the verifier is a probabilistic polynomial-time
IP_(complexity)
Number
Mathematics: Proof Techniques and Mathematical Structures. World Scientific. p. 34. ISBN 978-981-02-4088-2. Cummings, Jay (2021). Proofs: a long-form
0
Subfield of information theory and computer science
measures of algorithmic information as particular cases of axiomatically defined measures of algorithmic information. Instead of proving similar theorems, such
Algorithmic information theory
Algorithmic_information_theory
Electoral system with lottery among ballots
Ex-post efficiency. A probabilistic version of independence of irrelevant alternatives. Subsequent research have provided alternative proofs, as well as various
Random_ballot
Branch of pure mathematics
Selberg. The term is somewhat ambiguous. For example, proofs based on complex Tauberian theorems, such as Wiener–Ikehara, are often seen as quite enlightening
Number_theory
Exploring properties of the integers with complex analysis
(1896). Both proofs used methods from complex analysis, establishing as a main step of the proof that the Riemann zeta function ζ(s) is non-zero for all
Analytic_number_theory
probability Probabilistic causation Probabilistic design Probabilistic forecasting Probabilistic latent semantic analysis Probabilistic metric space
List_of_statistics_articles
Counting technique in combinatorics
then The combinatorial and the probabilistic version of the inclusion–exclusion principle are instances of (2). Proof Take m _ = { 1 , 2 , … , m } {\displaystyle
Inclusion–exclusion_principle
Mathematics of varieties with integer coordinates
geometry. Four theorems of fundamental importance in Diophantine geometry are: Mordell–Weil theorem Roth's theorem Siegel's theorem Faltings' theorem Another
Diophantine_geometry
Processing of natural language by a computer
tree using a probabilistic context-free grammar (PCFG) (see also stochastic grammar). Lexical semantics What is the computational meaning of individual
Natural_language_processing
Relationship where one statement follows from another
concept in terms of proofs and via models. The study of the syntactic consequence (of a logic) is called (its) proof theory whereas the study of (its) semantic
Logical_consequence
Branch of machine learning
universal approximation theorem or probabilistic inference. The classic universal approximation theorem concerns the capacity of feedforward neural networks
Deep_learning
Common communications channel model
stated in the theorem. The decoding error probability is exponentially small. The theorem can be proved directly with a probabilistic method. Consider
Binary_symmetric_channel
Algorithm for supervised learning of binary classifiers
essentially a special case of the theorems by George Cybenko and Kurt Hornik. Perceptrons (Minsky and Papert, 1969) studied the kind of perceptron networks necessary
Perceptron
Even integers as sums of two primes
crude version of the heuristic probabilistic argument (for the strong form of the Goldbach conjecture) is as follows. The prime number theorem asserts that
Goldbach's_conjecture
Puzzle in logic and mathematics
Non-Probabilistic Two Envelope Paradox" (PDF). Analysis. 62 (2): 157–160. doi:10.1093/analys/62.2.157. Katz, Bernard; Olin, Doris (2007). "A tale of two
Two_envelopes_problem
Probabilistic problem-solving algorithm
open-source Monte Carlo software List of software for Monte Carlo molecular modeling Mean-field particle methods – Probabilistic problem-solving algorithms
Monte_Carlo_method
Computation model defining an abstract machine
functions" is on Turing machine proofs of computability of recursive functions, etc. Knuth, Donald E. (1973). The Art of Computer Programming, Vol. 1: Fundamental
Turing_machine
Empirical law on the variance of species in a habitat
hypothesis has been proposed based on the Tweedie distributions, a family of probabilistic models that express an inherent power function relationship between
Taylor's_law
Study of the semantics, or interpretations, of formal and natural languages
logics of (finite) partially ordered quantification, which were originally investigated by Leon Henkin, who studied Henkin quantifiers. Probabilistic semantics
Semantics_(logic)
Methods in artificial intelligence research
possibility and necessity; and probabilistic logics to handle logic and probability together. Examples of automated theorem provers for first-order logic
Symbolic artificial intelligence
Symbolic_artificial_intelligence
Model of algorithmic learning
successful for PAC learning in the presence of errors, probabilistic concepts, function learning and Markovian non-independent examples. Computational learning
Occam_learning
Overview of and topical guide to algorithms
and probabilistic reasoning Algorithm engineering Outline of artificial intelligence Outline of computer science Procedure (computer science) List of data
Outline_of_algorithms
Branch of applied probability theory
probabilities of various events, whereas non-probabilistic rules, such as minimax, are robust in that they do not make such assumptions. A general criticism of decision
Decision_theory
Topic in computer science
subset of at most εn2 edges." Property testing algorithms are central to the definition of probabilistically checkable proofs, as a probabilistically checkable
Property_testing
Computer science field
systems Prism: a probabilistic symbolic model checker Roméo: an integrated tool environment for modelling, simulation, and verification of real-time systems
Model_checking
Distance between two statistical objects
symmetric measure of the overlap of two distributions. Probabilistic metric space Randomness extractor Similarity measure Zero-knowledge proof Dodge, Y. (2003)—entry
Statistical_distance
Function related to statistics and probability theory
Mascarenhas restates their proof using the mountain pass theorem. In the proofs of consistency and asymptotic normality of the maximum likelihood estimator
Likelihood_function
Deviations from local realism
reconstruction of quantum mechanics via Generalized Probabilistic Theories. The works above rely on the implicit assumption that any physical set of correlations
Quantum_nonlocality
Probability equation in mathematics
variable is small, in terms of its first two moments. The inequality was proved by Raymond Paley and Antoni Zygmund. Theorem: If Z ≥ 0 is a random variable
Paley–Zygmund_inequality
of a statement or theorem, based on axioms, definitions, and previously established theorems. proof by cases A proof technique that divides the proof
Glossary_of_logic
German mathematical physicist (born 1945)
quantum (statistical) mechanics obtained through functional-analytic and probabilistic techniques, jointly with his (former) students and other co-workers
Hajo_Leschke
Theory in number theory
combinatorial ideas of the arithmetic of braid groups and their Lie algebras of Makoto Matsumoto et al., then later in Mochizuki's proofs of the Grothendieck
Anabelian_geometry
Mathematical model of the time dependence of a point in space
Theory A Probabilistic Approach to Dynamical Systems, Ch3, ergodic theorems Alex Blumenthal Lai-Sang Young Ergodic Theory A Probabilistic Approach to
Dynamical_system
Branch of algebraic geometry
exist if non-zero rational solutions exist. In the 1850s, Leopold Kronecker formulated the Kronecker–Weber theorem, introduced the theory of divisors
Arithmetic_geometry
Australian and American mathematician (born 1975)
theorems in Fourier analysis, his work on wave maps, his global existence theorems for KdV-type equations, and for his solution with Allen Knutson of
Terence_Tao
Equalities for combinations of sets
combinatorics Set (mathematics) – Collection of mathematical objects Simple theorems in the algebra of sets – Basic set identities like commutative and
List of set identities and relations
List_of_set_identities_and_relations
Recursive integer sequence
combinatorial problems listed above. The first proof below uses a generating function. The other proofs are examples of bijective proofs; they involve literally
Catalan_number
Computer science award
Sanjeev; Safra, Shmuel (1998), "Probabilistic checking of proofs: a new characterization of NP" (PDF), Journal of the ACM, 45 (1): 70–122, doi:10.1145/273865
Gödel_Prize
Method in number theory
algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over the field F p {\displaystyle \mathbb {F}
Berlekamp–Rabin_algorithm
Abstract mathematics problem
warranted this problem to be named the Ross–Littlewood paradox. Ross's probabilistic version of the problem extended the removal method to the case where whenever
Ross–Littlewood_paradox
Concept in game theory
Distributional values are an extension of the Shapley value and related value operators designed to preserve the probabilistic output of predictive models in machine
Shapley_value
Theorem in statistics
Basu's theorem states that any boundedly complete and sufficient statistic is independent of any ancillary statistic. This is a 1955 result of Debabrata
Basu's_theorem
Study of correct reasoning
A proof system is a collection of rules to construct formal proofs. It is a tool to arrive at conclusions from a set of axioms. Rules in a proof system
Logic
Branch of statistical computational learning theory
first step of the proofs, and since it is used in many machine learning proofs on bounding empirical loss functions (including the proof of the VC inequality
Vapnik–Chervonenkis_theory
LIST OF-PROBABILISTIC-PROOFS-OF-NON-PROBABILISTIC-THEOREMS
LIST OF-PROBABILISTIC-PROOFS-OF-NON-PROBABILISTIC-THEOREMS
LIST OF-PROBABILISTIC-PROOFS-OF-NON-PROBABILISTIC-THEOREMS
LIST OF-PROBABILISTIC-PROOFS-OF-NON-PROBABILISTIC-THEOREMS
LIST OF-PROBABILISTIC-PROOFS-OF-NON-PROBABILISTIC-THEOREMS
LIST OF-PROBABILISTIC-PROOFS-OF-NON-PROBABILISTIC-THEOREMS
LIST OF-PROBABILISTIC-PROOFS-OF-NON-PROBABILISTIC-THEOREMS
LIST OF-PROBABILISTIC-PROOFS-OF-NON-PROBABILISTIC-THEOREMS
LIST OF-PROBABILISTIC-PROOFS-OF-NON-PROBABILISTIC-THEOREMS