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  "body": "\n## Definition\n\nLet $\\Sigma$ be a finite alphabet, and let $\\mathrm{TAUT} \\subseteq \\Sigma^*$ be the set of [[tautologies|0D1AHYPM8N]] of [[propositional logic|09K9MWD2QQ]], coded as strings.\nA *propositional proof system*, in the sense of Cook and Reckhow, is a [[function|0XHDTE5YMZ]] $P \\colon \\Sigma^* \\to \\Sigma^*$ such that:\n\n1. $P$ is computable in time polynomial in the length of its input;\n2. the image of $P$ is exactly $\\mathrm{TAUT}$.\n\nA string $w$ with $P(w) = \\varphi$ is a *$P$-proof* of $\\varphi$.\nThe system $P$ is *polynomially bounded* if there is a polynomial $p$ such that every $\\varphi \\in \\mathrm{TAUT}$ has a $P$-proof of length at most $p(\\abs{\\varphi})$.\n*Proof complexity* is the study of the lengths of proofs in such systems, and of the relations between systems.\n\n## Remarks\n\nExample: the truth table system is a proof system.\nLet $P$ send a string made of a formula $\\varphi$ and its full truth table, with all rows true, to $\\varphi$.\nLet $P$ send every other string to the fixed tautology $p \\vee \\neg p$.\nThe check takes time polynomial in the length of the input, and the image of $P$ is $\\mathrm{TAUT}$.\nThis system is not polynomially bounded, because the table of a formula with $n$ variables has $2^n$ rows.\nNon-example: the map that sends each string to itself is not a proof system, because its image is all of $\\Sigma^*$, not $\\mathrm{TAUT}$.\nCook and Reckhow proved that a polynomially bounded proof system exists if and only if $\\mathrm{NP} = \\mathrm{coNP}$.\nHaken proved in 1985 that resolution is not polynomially bounded, using the pigeonhole principle.\nFor stronger systems such as Frege systems, no superpolynomial lower bound is known.\n",
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