is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind...

computational problems are constructive proofs, i.e., a computational problem is proved to be solvable by showing an algorithm that solves it; a computational...

Such a proof is non-constructive, since the whole approach may not lend itself to construction. In terms of algorithms, purely theoretical existence theorems...

constructions given in § Existence (constructive proof) or § Existence (direct proof). The Chinese remainder theorem can be generalized to non-coprime moduli....

assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves...

A non-constructive proof might show a solution exists without specifying either an algorithm to obtain it or a specific bound. Even if the proof is constructive...

mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction...

ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without...

Brouwer reduced the debate to the use of proofs designed from "negative" or "non-existence" versus "constructive" proof: According to Brouwer, a statement that...

problem is phrased as follows: Yang–Mills Existence and Mass Gap. Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists...

about intuitionistic proofs to be transferred back to classical proofs. Recent developments in proof theory include the study of proof mining by Ulrich Kohlenbach...

believe that lengthy computer-assisted proofs should be regarded as calculations, rather than proofs: the proof algorithm itself should be proved valid, so...

(x\in X).{\big (}Q(x)\lor \neg Q(x){\big )}} is provable. Non-constructive axioms may enable proofs that formally claim decidability of such P {\displaystyle...

determining existence. He provided six such NP-complete search problems, or universal problems. Additionally he found for each of these problems an algorithm that...

rendering more precise the concept of algorithm emerges, however, in connection with the problem of a constructive foundation for mathematics....[p. 3,...

{A}}}(1-x(A)).} The Lovász Local Lemma is non-constructive because it only allows us to conclude the existence of structural properties or complex objects...

the existence of a propositional proof system that admits polynomial size proofs for all tautologies is equivalent to NP=coNP. Contemporary proof complexity...

systematic method for converting non-constructive probabilistic existence proofs into efficient deterministic algorithms that explicitly construct the desired...

integers and therefore a rational number. Dov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that ab is...

most commonly used in the probabilistic method, in particular to give existence proofs. There are several different versions of the lemma. The simplest and...

nonconstructive proofs just as sound as constructive ones, they are often considered less elegant, intuitive, or practically useful. Euclid's constructive proofs often...

mathematical proof that was essentially impossible to verify by humans due to the enormous size of the program's calculation (such proofs are called non-surveyable...

construction closely resembles Peano's axioms. In type theory, proofs are types whereas in set theory, proofs are part of the underlying first-order logic. Proponents...

induction is an inference rule used in formal proofs, and is the foundation of most correctness proofs for computer programs. Despite its name, mathematical...

The Misra & Gries edge-coloring algorithm is a polynomial-time algorithm in graph theory that finds an edge coloring of any simple graph. The coloring...

enumerates the proofs within S and we specify a procedure P which takes as an input an integer L and prints the strings x which are within proofs within S of...

self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical study...

"effective computability" as follows: "Clearly the existence of CC and RC (Church's and Rosser's proofs) presupposes a precise definition of 'effective'...

theory of ZFC, the axiom of choice enables nonconstructive proofs in which the existence of a type of object is proved without an explicit instance being...

completely verified by proof assistant software. Gödel's original proofs of the incompleteness theorems, like most mathematical proofs, were written in natural...