Another interesting application of similar techniques, due to [a4], is in proving -completeness of approximation problems (cf. Their nodes are enumerated by binary strings (addresses) $x$. Nodes are interpreted as elements of a field (a finite field or a segment of the field of rationals) and the colouring is a polynomial on it. List of previous changes on EOM, EoM Project Talk See Proof theory. This form has a remarkable property: the presence of any errors (essential deviations from the form or other requirements) is instantly apparent after checking just a negligible fraction of bits of the proof.

(In finite fields, constructing the equation to check is trickier.) Articles giving mathematical proofs within a physical model. The neighbours of $x$ are obtained by simple manipulations of its bits: adding $1$ to the first (or last) bit or shifting bits left. also Error-correcting code) based on low-degree polynomials and their Fourier transforms were a major area of research since the 1950s. Academic Publishers in 2002. will find it useful and will be motivated to update those topics that fall Surprisingly, these techniques were not used in general computer theory until the middle of the 1980s; one of the first such uses was for generating pseudo-random strings in [a6]. These results were significantly extended in a number of subsequent papers. Any proof is relative, since it is based on certain unprovable assumptions. to existing articles within encyclopediaofmath.org will come under the Creative Commons Attribution Share-Alike License. Some known mathematical proofs are so huge that no single human has been able to verify them. [a7], [a8], and, especially, [a2] were the most relevant papers. Even more problematic seems the verification of large computations. Holographic proofs came as a result of a number of previous major advances. It makes certain random choices, and any single round of verification has a chance of overlooking essential errors due to an unlucky choice. This requires some background from topology, algebra, geometry, etc. Jump to: navigation, search. Verifying holographic proofs takes a poly-logarithmic, i.e., a constant power of the logarithm of $n$, number of bit operations. A closer set of preceding results was a number of remarkable papers on the relations between high-level complexity classes associated with interactive proof systems. At first blush, mathematics appears to study abstractentities. If these problems are regarded as intractable, then onemight try to see if mathematical objects can somehow belong to theconcrete world after all. This is easy to do since low-degree polynomials cannot differ only in a small fraction of points: e.g., two different straight lines must differ in all points but one. Fourth, the claim which the proof is to support (or the input/output, the matching of which the computation is to confirm) also must be given in error-correcting form. Transforming arbitrary proofs into a holographic form starts with reducing an arbitrary proof system to a standard (not yet holographic) one: the domino pattern. The nodes are renamed $1$ and $2$, so that the domino is the same wherever it appears in the graph.

The extension is done using the same expression, i.e., without increasing the degree of the colouring polynomial. The original With more than 8,000 entries,

The graphs are taken from a standard family: only sizes and colourings can differ.
Despite these caveats the result is surprising. The European Mathematical Society, Move of The Encyclopedia of Mathematics from Springer Verlag to EMS Press. A proof system is an algorithm that verifies a proof (given as input) and outputs the proven statement. Second, only essential errors (i.e., not correctable from the context) have this probabilistic detection guarantee. For instructions on how this can be achieved, please see the Help page. From Encyclopedia of Mathematics Jump to: navigation , search A reasoning conducted according to certain rules in order to demonstrate some proposition (statement, theorem); it is based on initial statements (axioms). within their own expertise or add new topics enabling the wiki to become yet

The above description is just a general sketch. This exciting development was initiated by N. Nisan in a 1989 electronically distributed article. The error-correcting codes (cf. This is a tiny fraction: the binary logarithm of the number of atoms in the known Universe is under $300$. The condition that all dominos are restricted to given types is also expressed as equality to $0$ of a low-degree polynomial $P(x)$ of a node $x=x_1,\dots,x_k$, its neighbours, and their colours.

An example is the family of shuffle exchange graphs. Then comes the arithmetization stage. scientific authority over alterations and deletions. This function is then extended from its original domain to a larger one (a larger field or a larger segment of rationals). A form in which every proof or record of computation can be presented. This allows one to reduce the problem of finding a proof in any particular proof system to the above standard domino problem. Springer, in cooperation with the European Mathematical Society, (Of the names in the header, the phrase "probabilistically checkable" is somewhat misleading, since both holographic and traditional proofs can be checked either deterministically or probabilistically, though randomness does not speed up the checking of traditional proofs.). articles are from the online Encyclopaedia of Mathematics, published by Kluwer Such an algorithm can be efficiently simulated, first on a special form of a random access machine and then on a sorting network. Verifying holographic proofs takes a poly-logarithmic, i.e., a constant power of the logarithm of $n$, number of bit operations. In a holographic form, however, the verification time barely grows at all, even if the proof fills up the whole Universe. On the … One must add some set theory to formalize this background. In the actual construction, $x$ is broken into several variables, so it is convenient to shuffle bits just within the first variable and permute variables (cycle-shifts of all variables and of the first two suffice). This article was adapted from an original article by Leonid Levin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. https://encyclopediaofmath.org/index.php?title=Holographic_proof&oldid=33124, K. Appel, W. Haken, J Koch, "Every planar map is four colorable. also Graph, oriented) of two nodes coloured with a fixed (independent of the proof size) set of colours. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference.

Throw in some necessary logic, parsing, syntax procedures and one obtains a book instead of the above informal one-line formulation. This wiki is a MediaWiki that uses the MathJax extension, making it possible to insert mathematical equations in $\rm \TeX$ and $\rm \LaTeX$. The European Mathematical Society, transparent proof, instantly checkable proof, probabilistically checkable proof, PCP. The verification consists of statistical checking that all partial sums (with possibly only a small fraction of deviating points) are polynomials of low, for the extended domain, degree. The third caveat is trivial but often overlooked. has made the content of this Encyclopedia freely open to the public. This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098, https://encyclopediaofmath.org/index.php?title=Proof&oldid=18441. also Monte-Carlo method). This page was last edited on 7 February 2011, at 17:26. The claim must be formal and self-contained: one cannot just write a mathematical theorem in English. This chance never exceeds $50$%, regardless of the nature of errors, and vanishes as $1/2^k$ with $k$ independent rounds. illuminating nearly 50,000 notions in mathematics, the Encyclopaedia of This makes one wonder what the nature of mathematicalentities consists in and how we can have knowledge of mathematicalentities. www.springer.com It was quickly followed by improved theorems, which contained powerful techniques used later for the construction of holographic proofs. Springer, in cooperation with the European Mathematical Society, has made the content of this Encyclopedia freely open to the public. It is hoped that the mathematics community With more than 8,000 entries, illuminating nearly 50,000 notions in mathematics, the Encyclopaedia of Mathematics was the most up-to-date graduate-level reference work in the field of mathematics. In practice, however, it may also be based on previously demonstrated propositions. The other variables are taken with all values that exist in the extended domain. A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The translation to a formal system, e.g., Zermelo–Fraenkel set theory, may be quite involved. These operations (or their constant combinations) define the edges of the graph. Part II: Reducibility", L. Babai, L. Fortnow, C. Lund, "Non-deterministic exponential time has two-prover interactive protocols". How to categorize EoM pages (still under discussion), Help

This page was last edited on 22 May 2020, at 06:08. Traditional proofs are verifiable in time that is a constant power (say, quadratic) of the proof length $n$. Of course, the instant verification can only assure the success of this proofreading; it has no time to actually perform it for all bits. also Complexity theory). An enhanced version can be made very tolerant: it can guarantee that any errors affecting a small, say $5$%, fraction of bits will be inessential, correctable by the proofreader and tolerated by the verifier with high probability. This page was last edited on 24 August 2014, at 10:26. The European Mathematical Society. [a3] introduced holographic proofs (called transparent proofs there). The Encyclopedia of Mathematics wiki is an open access resource Of course, all parameters must be finely tuned and many other details addressed. These graphs may be replaced by any other family with edges expressed as linear transformations of variables, as long as it has sufficient connectivity to implement an efficient sorting network. Mathematical Society, monitors any changes to articles and has full again the most comprehensive and up-to-date online mathematics reference work.

Examples are the four-colour theorem (verified with an aid of a computer, see [a1] and Four-colour problem), the classification of simple finite groups (broken into many pieces, each supposedly verified by one member of a large group of researchers, see [a5] and Simple finite group), and others. One must give a number of concepts from advanced calculus just to explain what a curve is. There are four caveats: First, the verification is a so-called Monte-Carlo algorithm (cf. A domino is a directed graph (cf. These axioms assert the reflexivity of the equality relation and the possibility of substituting equals for equals.

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