Circuit Satisfiability Problem for circuits of small complexity

Abstract

The following problem is considered. A Turing machine $M$, that accepts a string of fixed length $t$ as input, runs for a time not exceeding a fixed value $n$ and is guaranteed to produce a binary output, is given. It's required to find a string $X$ such that $M(X) = 1$ effectively in terms of $t$, $n$, the size of the alphabet of $M$ and the number of states of $M$. The problem is close to the well-known Circuit Satisfiability Problem. The difference from Circuit Satisfiability Problem is that when reduced to Circuit Satisfiability Problem, we get circuits with a rich internal structure (in particular, these are circuits of small Kolmogorov complexity). The proof system, operating with potential proofs of the fact that, for a given machine $M$, the string $X$ does not exist, is provided, its completeness is proved and the algorithm guaranteed to find a proof of the absence of the string $X$ in the case of its actual absence is presented (in the worst case, the algorithm is exponential, but in a wide class of interesting cases it works in polynomial time). We present an algorithm searching for the string $X$, for which its efficiency was neither tested, nor proven, and it may require serious improvement in the future, so it can be regarded as an idea. We also discuss first steps towards solving a more complex problem similar to this one: a Turing machine $M$, that accepts two strings $X$ and $Y$ of fixed length and running for a time that does not exceed a fixed value, is given; it is required to build an algorithm $N$ that builds a string $Y = N(X)$ for any string $X$, such that $M(X, Y) = 1$ (details in the introduction).