On the Expressibility of Stable Logic Programming
<|reference_start|>On the Expressibility of Stable Logic Programming: (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 $\sigma$ 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\footnote{The proof of this result involves additional technical complications and will be a subject of another publication.}.<|reference_end|>
@article{marek2003on,
    title={On the Expressibility of Stable Logic Programming},
    author={Victor W. Marek and Jeffrey B. Remmel},
    journal={TCLP 3(2003), pp. 551-567q},
    year={2003},
    archivePrefix={arXiv},
    eprint={cs/0312053},
    primaryClass={cs.AI}
}