On the Expressibility of Stable Logic Programming

Abstract

(We apologize for pidgin LaTeX) Schlipf \cite{sch91} 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 \cite{mt99}. 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 $σ$ of size $n$ over a fixed alphabet $Σ$, there is an extensional database $\mathit{edb}_{M,p,σ}$ such that there is a one-to-one correspondence between the stable models of $\mathit{edb}_{M,p,σ} \cup P_{\mathit{Trg}}$ and the accepting computations of the machine $M$ that reach the final state in at most $p(n)$ steps. Moreover, $\mathit{edb}_{M,p,σ}$ can be computed in polynomial time from $p$, $σ$ and the description of $M$ and the decoding of such accepting computations from its corresponding stable model of $\mathit{edb}_{M,p,σ} \cup P_{\mathit{Trg}}$ can be computed in linear time. A similar statement holds for Default Logic with respect to $Σ_2^\mathrm{P}$-search problems\footnote{The proof of this result involves additional technical complications and will be a subject of another publication.}.