Natural proof

Natural proof

In 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 in some sense "natural", it can be shown (assuming a widely believed conjecture on the existence of one-way functions) that no such proof can possibly be used to solve the P vs. NP problem.

The notion of natural proofs was introduced by Alexander Razborov and Steven Rudich in their article Natural proofs, first presented in 1994, and later published in 1997, for which they received the 2007 Gödel Prize.[1]

Specifically, natural proofs prove lower bounds on the circuit complexity of boolean functions. A natural proof shows, either directly or indirectly, that a boolean function has a certain natural combinatorial property. Under the assumption that one-way functions exist with "exponential hardness" as specified in their main theorem, Razborov and Rudich show that these proofs cannot separate certain complexity classes. Notably, assuming one-way functions exist, these proofs cannot separate the complexity classes P and NP.[2]

For example, their article states:

[...] consider a commonly envisioned proof strategy for proving P ≠ NP:
  • Formulate some mathematical notion of "discrepancy" or "scatter" or "variation" of the values of a Boolean function, or of an associated polytope or other structure. [...]
  • Show by an inductive argument that polynomial-sized circuits can only compute functions of "low" discrepancy. [...]
  • Then show that SAT, or some other function in NP, has "high" discrepancy.
Our main theorem in Section 4 gives evidence that no proof strategy along these lines can ever succeed.

A property of boolean functions is defined to be natural if it contains a property meeting the constructivity and largeness conditions defined by Razborov and Rudich. Roughly speaking, the constructivity condition requires that a property be decidable in (quasi-)polynomial time when the 2n-sized truth table of an n-input boolean function is given as input, asymptotically as n increases. This is the same as time singly exponential in n. Properties that are easy to understand are likely to satisfy this condition. The largeness condition requires that the property hold for a sufficiently large fraction of the set of all boolean functions.

A property is useful against a complexity class C if every sequence of boolean functions having the property infinitely often defines a language outside of C. A natural proof is a proof that establishes that a certain language lies outside of C and refers to a natural property that is useful against C.

Razborov and Rudich give a number of examples of lower-bound proofs against classes C smaller than P/poly that can be "naturalized", i.e. converted into natural proofs. An important example treats proofs that the parity problem is not in the class AC0. They give strong evidence that the techniques used in these proofs cannot be extended to show stronger lower bounds. In particular, AC0-natural proofs cannot be useful against AC0[m].

Razborov and Rudich also reproduce Avi Wigderson's unconditional proof that natural proofs cannot prove exponential lower bounds for the discrete logarithm problem.

There is strong current belief that the mechanism of this paper actually blocks lower-bound proofs against the complexity class TC0 of constant-depth, polynomial-sized threshold circuits, which is believed but not proven smaller than P/poly.[3] However, some researchers believe that the Razborov-Rudich limitations are actually good guidance for what a "super-natural" lower-bound proof might involve, such as properties hard or complete for exponential space.[4]

Notes

  1. ^ "ACM-SIGACT 2007 Gödel Prize". http://sigact.acm.org/prizes/godel/2007.html. 
  2. ^ A. A. Razborov and S. Rudich (1997). "Natural proofs". Journal of Computer and System Sciences 55: 24–35. doi:10.1006/jcss.1997.1494.  (Draft)
  3. ^ http://qwiki.stanford.edu/index.php/Complexity_Zoo:T#tc0
  4. ^ K. Regan (October 2002). "Understanding the Mulmuley-Sohoni Approach to P vs. NP" (PDF). Bulletin of the European Association for Theoretical Computer Science 78: 86–97. http://www.cse.buffalo.edu/~regan/papers/pdf/Reg02MSFD.pdf. 

References


Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Proof theory — is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively defined data structures such as plain lists, boxed… …   Wikipedia

  • Proof-theoretic semantics — is an approach to the semantics of logic that attempts to locate the meaning of propositions and logical connectives not in terms of interpretations, as in Tarskian approaches to semantics, but in the role that the proposition or logical… …   Wikipedia

  • Proof — • The establishment of a disputed or controverted matter by lawful means or arguments. Catholic Encyclopedia. Kevin Knight. 2006. Proof     Proof      …   Catholic encyclopedia

  • Natural History (I Am Kloot album) — Natural History Studio album by I Am Kloot Released 26 March 200 …   Wikipedia

  • Natural number — Natural numbers can be used for counting (one apple, two apples, three apples, ...) from top to bottom. In mathematics, the natural numbers are the ordinary whole numbers used for counting ( there are 6 coins on the table ) and ordering ( this is …   Wikipedia

  • Natural Steps, Arkansas — Natural Steps   Census designated place   …   Wikipedia

  • Natural Law — • In English this term is frequently employed as equivalent to the laws of nature, meaning the order which governs the activities of the material universe. Among the Roman jurists natural law designated those instincts and emotions common to man… …   Catholic encyclopedia

  • Natural organic matter — (NOM) is broken down organic matter that comes from plants and animals in the environment. [ Natural Organic Matter, GreenFacts, 22 Apr, 2007 …   Wikipedia

  • Natural Bridge (Virginia) — Natural Bridge U.S. National Register of Historic Places U.S. National Historic Landmark …   Wikipedia

  • Natural Elements (rap) — Natural Elements Also known as N.E. Origin 1993 Genres Hip Hop, Rap Labels Kings Link Recordz, Fortress Entertainment Associated acts Charlemagne …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”