NUMBER OF SOLUTIONS OF SATISFIABILITY INSTANCES—APPLICATIONS TO KNOWLEDGE BASES

1989 ◽  
Vol 03 (01) ◽  
pp. 53-65 ◽  
Author(s):  
J.C. SIMON ◽  
O. DUBOIS
Keyword(s):  
Artificial Intelligence ◽  
Propositional Logic ◽  
Statistical Approach ◽  
Knowledge Bases ◽  
Zero Order ◽  
Number Of Solutions ◽  
Exponential Process ◽  
Complete Problems ◽  
Np Complete ◽  
Logical Rules

In propositional logic (zero order) a system of logical rules may be put under the form of a conjunction of disjunction, i.e. a “satisfiability” or SAT-problem. SAT is central to NP-complete problems. Any result obtained on SAT would have consequences for a lot of problems important in artificial intelligence. We deal with the question of the number N of solutions of SAT. Firstly, any system of SAT clauses may be transformed in a system of independent clauses by an exponential process; N may be computed exactly. Secondly, by a statistical approach, results are obtained showing that for a given SAT-instance, it should be possible to find an estimate of N with a margin of confidence in polynomial time. Thirdly, we demonstrate the usefulness of these ideas on large knowledge bases.

scholarly journals Complexity of n-Queens Completion (Extended Abstract)

2018 ◽  
Author(s):  
Ian P. Gent ◽  
Christopher Jefferson ◽  
Peter Nightingale
Keyword(s):  
Artificial Intelligence ◽  
Phase Transition ◽  
Decision Problem ◽  
Completion Problem ◽  
Complete Problems ◽  
Random Instances ◽  
Np Complete

The n-Queens problem is to place n chess queens on an n by n chessboard so that no two queens are on the same row, column or diagonal. The n-Queens Completion problem is a variant, dating to 1850, in which some queens are already placed and the solver is asked to place the rest, if possible. We show that n-Queens Completion is both NP-Complete and #P-Complete. A corollary is that any non-attacking arrangement of queens can be included as a part of a solution to a larger n-Queens problem. We introduce generators of random instances for n-Queens Completion and the closely related Blocked n-Queens and Excluded Diagonals Problem. We describe three solvers for these problems, and empirically analyse the hardness of randomly generated instances. For Blocked n-Queens and the Excluded Diagonals Problem, we show the existence of a phase transition associated with hard instances as has been seen in other NP-Complete problems, but a natural generator for n-Queens Completion did not generate consistently hard instances. The significance of this work is that the n-Queens problem has been very widely used as a benchmark in Artificial Intelligence, but conclusions on it are often disputable because of the simple complexity of the decision problem. Our results give alternative benchmarks which are hard theoretically and empirically, but for which solving techniques designed for n-Queens need minimal or no change.

Complexity, computational

2018 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Propositional Logic ◽  
Common Type ◽  
Combinatorial Games ◽  
Np Completeness ◽  
Complete Problems ◽  
Np Complete ◽  
Feasible Solutions ◽  
Np Problems

The theory of computational complexity is concerned with estimating the resources a computer needs to solve a given problem. The basic resources are time (number of steps executed) and space (amount of memory used). There are problems in logic, algebra and combinatorial games that are solvable in principle by a computer, but computationally intractable because the resources required by relatively small instances are practically infeasible. The theory of NP-completeness concerns a common type of problem in which a solution is easy to check but may be hard to find. Such problems belong to the class NP; the hardest ones of this type are the NP-complete problems. The problem of determining whether a formula of propositional logic is satisfiable or not is NP-complete. The class of problems with feasible solutions is commonly identified with the class P of problems solvable in polynomial time. Assuming this identification, the conjecture that some NP problems require infeasibly long times for their solution is equivalent to the conjecture that P≠NP. Although the conjecture remains open, it is widely believed that NP-complete problems are computationally intractable.

scholarly journals Complexity of n-Queens Completion

2017 ◽  
Vol 59 ◽  
pp. 815-848 ◽  
Author(s):  
Ian P. Gent ◽  
Christopher Jefferson ◽  
Peter Nightingale
Keyword(s):  
Artificial Intelligence ◽  
Phase Transition ◽  
Decision Problem ◽  
Completion Problem ◽  
Complete Problems ◽  
Random Instances ◽  
Np Complete

The n-Queens problem is to place n chess queens on an n by n chessboard so that no two queens are on the same row, column or diagonal. The n-Queens Completion problem is a variant, dating to 1850, in which some queens are already placed and the solver is asked to place the rest, if possible. We show that n-Queens Completion is both NP-Complete and #P-Complete. A corollary is that any non-attacking arrangement of queens can be included as a part of a solution to a larger n-Queens problem. We introduce generators of random instances for n-Queens Completion and the closely related Blocked n-Queens and Excluded Diagonals Problem. We describe three solvers for these problems, and empirically analyse the hardness of randomly generated instances. For Blocked n-Queens and the Excluded Diagonals Problem, we show the existence of a phase transition associated with hard instances as has been seen in other NP-Complete problems, but a natural generator for n-Queens Completion did not generate consistently hard instances. The significance of this work is that the n-Queens problem has been very widely used as a benchmark in Artificial Intelligence, but conclusions on it are often disputable because of the simple complexity of the decision problem. Our results give alternative benchmarks which are hard theoretically and empirically, but for which solving techniques designed for n-Queens need minimal or no change.

Application of formal languages in polynomial transformations of instances between NP-complete problems

2013 ◽  
Vol 14 (8) ◽  
pp. 623-633
Author(s):  
Jorge A. Ruiz-Vanoye ◽  
Joaquín Pérez-Ortega ◽  
Rodolfo A. Pazos Rangel ◽  
Ocotlán Díaz-Parra ◽  
Héctor J. Fraire-Huacuja ◽  
...  
Keyword(s):  
Formal Languages ◽  
Complete Problems ◽  
Polynomial Transformations ◽  
Np Complete

NP vs QMA_\log(2)

2010 ◽  
Vol 10 (1&2) ◽  
pp. 141-151
Author(s):  
S. Beigi
Keyword(s):  
Quantum Algorithms ◽  
Np Hard ◽  
Complete Problems ◽  
Hard Problems ◽  
Np Complete ◽  
Np Hard Problems

Although it is believed unlikely that $\NP$-hard problems admit efficient quantum algorithms, it has been shown that a quantum verifier can solve NP-complete problems given a "short" quantum proof; more precisely, NP\subseteq QMA_{\log}(2) where QMA_{\log}(2) denotes the class of quantum Merlin-Arthur games in which there are two unentangled provers who send two logarithmic size quantum witnesses to the verifier. The inclusion NP\subseteq QMA_{\log}(2) has been proved by Blier and Tapp by stating a quantum Merlin-Arthur protocol for 3-coloring with perfect completeness and gap 1/24n^6. Moreover, Aaronson et al. have shown the above inclusion with a constant gap by considering $\widetilde{O}(\sqrt{n})$ witnesses of logarithmic size. However, we still do not know if QMA_{\log}(2) with a constant gap contains NP. In this paper, we show that 3-SAT admits a QMA_{\log}(2) protocol with the gap 1/n^{3+\epsilon}} for every constant \epsilon>0.

A Decision Support System for Artificial Recharge Plant

1991 ◽  
Vol 24 (9) ◽  
pp. 331-342 ◽  
Author(s):  
C. Masciopinto ◽  
V. Palmisano ◽  
F. Tangorra ◽  
M. Vurro
Keyword(s):  
Artificial Intelligence ◽  
Decision Support ◽  
Artificial Recharge ◽  
Groundwater Resources ◽  
Treatment Plant ◽  
Knowledge Bases ◽  
Wastewater Discharge ◽  
Training Tool ◽  
Qualitative And Quantitative ◽  
Expert System Shell

The need for artificial recharge plants is the result of the qualitative and quantitative worsening of groundwater resources due to increased pumping and wastewater discharge. This paper described a system that uses artificial intelligence techniques for designing an artificial recharge plant. The system can be used as a training tool for new engineers, as well as an aid in the choices for expert engineers. The system is an application of an expert system shell running on a common p.c. machine. The model is made up of two knowledge bases, respectively denoted as Quantity artificial recharge and Quality artificial recharge. The former is related to the quantitative aspects, such as geology, climate and land availability, the latter to qualitative aspects, such as water use and treatment plant. Two case studies have been implemented in order to confirm the validity of this kind of systemic approach.

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