On the expressibility of stable logic programming

Abstract

Schlipf (1995) proved that Stable Logic Programming (SLP) solves all $\mathit{NP}$ decision problems. We extend Schlipf's result to prove that SLP solves all search problems in the class $\mathit{NP}$. Moreover, we do this in a uniform way as defined in Marek and Truszczyński (1991). Specifically, we show that there is a single $\mathrm{DATALOG}^{\neg}$ program $P_{\mathit{Trg}}$ such that given any Turing machine $M$, any polynomial $p$ with non-negative integer coefficients and any input $\sigma$ of size $n$ over a fixed alphabet $\Sigma$, there is an extensional database $\mathit{edb}_{M,p,\sigma}$ such that there is a one-to-one correspondence between the stable models of $\mathit{edb}_{M,p,\sigma} \cup P_{\mathit{Trg}}$ and the accepting computations of the machine $M$ that reach the final state in at most $p(n)$ steps. Moreover, $\mathit{edb}_{M,p,\sigma}$ can be computed in polynomial time from $p$, $\sigma$ and the description of $M$ and the decoding of such accepting computations from its corresponding stable model of $\mathit{edb}_{M,p,\sigma} \cup P_{\mathit{Trg}}$ can be computed in linear time. A similar statement holds for Default Logic with respect to $\Sigma_2^\mathrm{P}$-search problems.The proof of this result involves additional technical complications and will be a subject of another publication.