Polynomial creativity

Joseph and Young define the set ${\displaystyle \mathrm {NP} ^{(k)}}$ to be the set of nondeterministic Turing machine programs ${\displaystyle p}$ that, for inputs ${\displaystyle x}$ that they accept, have an accepting path with a number of steps that is at most ${\displaystyle |p|(|x|^{k}+1)}$. This notation should be distinguished with that for the complexity class NP. The complexity class NP is a set of formal languages, while ${\displaystyle \mathrm {NP} ^{(k)}}$ is instead a set of programs that accept some of these languages. Every language in NP is recognized by a program in one of the sets ${\displaystyle \mathrm {NP} ^{(k)}}$; the parameter ${\displaystyle k}$ is (up to the factor ${\displaystyle |p|}$ in the bound on the number of steps) the exponent in the polynomial running time of the program.[1]

According to Joseph and Young's theory, a language ${\displaystyle L}$ in NP is k-creative if it is possible to find a witness showing that the complement of ${\displaystyle L}$ is not recognized by any program in ${\displaystyle \mathrm {NP} ^{(k)}}$. More formally, there should exist a polynomially computable function ${\displaystyle f}$ that, when given a nondeterministic program ${\displaystyle p}$ in ${\displaystyle \mathrm {NP} ^{(k)}}$, finds an input ${\displaystyle x=f(p)}$ that either belongs to ${\displaystyle L}$ and causes the program to accept ${\displaystyle x}$, or does not belong to ${\displaystyle L}$ and causes the program to reject ${\displaystyle x}$. The function ${\displaystyle f}$ is called a productive function for ${\displaystyle L}$. If this productive function exists, the given program does not produce the behavior on input ${\displaystyle x}$ that would be expected of a program for recognizing the complement of ${\displaystyle L}$.[1]

Every ${\displaystyle k}$-creative set is NP-complete. For, applying a padding argument, there exists a polynomial-time translation ${\displaystyle g}$ of the instances of any other language in NP into nondeterministic polynomial-time programs in ${\displaystyle \mathrm {NP} ^{(k)}}$, such that yes-instances are translated into programs that accept all inputs and no-instances are translated into programs that reject all inputs. The composition of this translation with the function ${\displaystyle f}$ is a polynomial-time many-one reduction from the given language to the ${\displaystyle k}$-creative set. It is also possible to prove more strongly that there exists an invertible parsimonious reduction to the ${\displaystyle k}$-creative set.[1]

Joseph and Young define a polynomial-time function ${\displaystyle f}$ to be polynomially honest if its running time is at most a polynomial function of its output length. This disallows, for instance, functions that take polynomial time but produce outputs of less than polynomial length. As they show, every one-to-one polynomially-honest function ${\displaystyle f}$ is the productive function for a ${\displaystyle k}$-creative language ${\displaystyle K_{f}^{k}}$.[1]

Given ${\displaystyle f}$, Joseph and Young define ${\displaystyle K_{f}^{k}}$ to be the set of values ${\displaystyle f(p)}$ for nondeterministic programs ${\displaystyle p}$ that have an accepting path for ${\displaystyle f(p)}$ using at most ${\displaystyle |p|(|f(p)|^{k}+1)}$ steps. This number of steps (on that input) would be consistent with ${\displaystyle p}$ belonging to ${\displaystyle \mathrm {NP} ^{(k)}}$. Then ${\displaystyle K_{f}^{k}}$ belongs to NP, for given an input ${\displaystyle f(p)}$ one can nondeterministically guess both ${\displaystyle p}$ and its accepting path, and then verify that the path is valid for ${\displaystyle p}$.[1]

Language ${\displaystyle K_{f}^{k}}$ is ${\displaystyle k}$-creative, with ${\displaystyle f}$ as its productive function, because every program ${\displaystyle p}$ in ${\displaystyle \mathrm {NP} ^{(k)}}$ is mapped by ${\displaystyle f}$ to a value ${\displaystyle f(p)}$ that is either accepted by ${\displaystyle p}$ (and therefore also belongs to ${\displaystyle K_{f}^{k}}$) or rejected by ${\displaystyle p}$ (and therefore also does not belong to ${\displaystyle K_{f}^{k}}$).[1]

The Berman–Hartmanis conjecture states that there exists a polynomial-time isomorphism between any two NP-complete sets: a function that maps yes-instances of one such set one-to-one into yes-instances of the other, takes polynomial time, and whose inverse function can also be computed in polynomial time. It was formulated by Leonard C. Berman and Juris Hartmanis in 1977, based on the observation that all NP-complete sets known at that time were isomorphic. An equivalent formulation of the conjecture is that every NP-complete set is paddable. This means that there exists a polynomial-time and polynomial-time-invertible one-to-one transformation ${\displaystyle h(x,y)}$ from yes-instances ${\displaystyle x}$ to larger yes-instances that encode the "irrelevant" information ${\displaystyle y}$.[3]

However, it is unknown how to find such a padding transformation for a ${\displaystyle k}$-creative language whose productive function is not polynomial-time-invertible. Therefore, if one-way permutations exist, the ${\displaystyle k}$-creative languages having these permutations as their productive functions provide candidate counterexamples to the Berman–Hartmanis conjecture.[1]

The (unproven) Joseph–Young conjecture formalizes this reasoning. The conjecture states that there exists a one-way length-increasing function ${\displaystyle f}$ such that ${\displaystyle K_{f}^{k}}$ is not paddable.[1] Alan Selman observed that this would imply a simpler conjecture, the encrypted complete set conjecture: there exists a one-way function ${\displaystyle f}$ such that ${\displaystyle \mathrm {SAT} }$ (the set of yes-instances for the satisfiability problem) and ${\displaystyle f(\mathrm {SAT} )}$ are non-isomorphic.[4] There exists an oracle relative to which one-way functions exist, both of these conjectures are false, and the Berman–Hartmanis conjecture is true.[5]