A Hierarchy of Tractable Subsets for Computing Stable Models

R. Ben-Eliyahu

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A Hierarchy of Tractable Subsets for Computing Stable Models

R. Ben-Eliyahu

Artificial Intelligence · Submitted 1 Aug 1996

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper we present a hierarchy of classes of knowledge bases, Omega_1,Omega_2,..., with the following properties: first, Omega_1 is the class of all stratified knowledge bases; second, if a knowledge base Pi is in Omega_k, then Pi has at most k stable models, and all of them may be found in time O(lnk), where l is the length of the knowledge base and n the number of atoms in Pi; third, for an arbitrary knowledge base Pi, we can find the minimum k such that Pi belongs to Omega_k in time polynomial in the size of Pi; and, last, where K is the class of all knowledge bases, it is the case that union{i=1 to infty} Omega_i = K, that is, every knowledge base belongs to some class in the hierarchy.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.27 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.97 · compares with prior results 0.11 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Finding the stable models of a knowledge base is a significant computational problem in artificial intelligence. This task is at the computational heart of truth maintenance systems, autoepistemic logic, and default logic. Unfortunately, it is NP-hard. In this paper we present a hierarchy of classes of knowledge bases, Omega_1,Omega_2,..., with the following properties: first, Omega_1 is the class of all stratified knowledge bases; second, if a knowledge base Pi is in Omega_k, then Pi has at most k stable models, and all of them may be found in time O(lnk), where l is the length of the knowledge base and n the number of atoms in Pi; third, for an arbitrary knowledge base Pi, we can find the minimum k such that Pi belongs to Omega_k in time polynomial in the size of Pi; and, last, where K is the class of all knowledge bases, it is the case that union{i=1 to infty} Omega_i = K, that is, every knowledge base belongs to some class in the hierarchy.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Finding the stable models of a knowledge base is a significant computational problem in artificial intelligence.	background 1.00	0.06	0.02	asserted 1.00
s1	This task is at the computational heart of truth maintenance systems, autoepistemic logic, and default logic.	background 1.00	0.04	0.02	asserted 1.00
s2	Unfortunately, it is NP-hard.	background 0.93	0.08	0.02	asserted 1.00
s3	In this paper we present a hierarchy of classes of knowledge bases, Omega_1,Omega_2,..., with the following properties: first, Omega_1 is the class of all stratified knowledge bases; second, if a knowledge base Pi is in Omega_k, then Pi has at most k stable models, and all of them may be found in time O(lnk), where l is the length of the knowledge base and n the number of atoms in Pi; third, for an arbitrary knowledge base Pi, we can find the minimum k such that Pi belongs to Omega_k in time polynomial in the size of Pi; and, last, where K is the class of all knowledge bases, it is the case that union{i=1 to infty} Omega_i = K, that is, every knowledge base belongs to some class in the hierarchy.	contribution 0.94	0.93	0.93	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.88 · evidence from proof or derivation 0.99.

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Original abstract

Finding the stable models of a knowledge base is a significant computational problem in artificial intelligence. This task is at the computational heart of truth maintenance systems, autoepistemic logic, and default logic. Unfortunately, it is NP-hard. In this paper we present a hierarchy of classes of knowledge bases, Omega_1,Omega_2,..., with the following properties: first, Omega_1 is the class of all stratified knowledge bases; second, if a knowledge base Pi is in Omega_k, then Pi has at most k stable models, and all of them may be found in time O(lnk), where l is the length of the knowledge base and n the number of atoms in Pi; third, for an arbitrary knowledge base Pi, we can find the minimum k such that Pi belongs to Omega_k in time polynomial in the size of Pi; and, last, where K is the class of all knowledge bases, it is the case that union{i=1 to infty} Omega_i = K, that is, every knowledge base belongs to some class in the hierarchy.

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