# Quantum Computing for Dealing with Inaccurate Knowledge Related to the Certainty Factors Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Certainty Factors Model

- If $p(h/e)>p\left(h\right)$, then there is an increase in the probability of the hypothesis h after e. In this case: $MB(h,e)>0$, $MD(h,e)=0$, and $MB(h,e)$ is defined as follows:$$MB(h,e)=\frac{p(h/e)-p\left(h\right)}{1-p\left(h\right)}$$According to this expression, $MB(h,e)$ represents a relative increase in the likelihood of the hypothesis h after evidence e.
- If $p\left(h\right)>p(h/e)$, then there is a decrease in the probability of the hypothesis h after e. In this case: $MB(h,e)=0$, $MD(h,e)>0$, and $MD(h,e)$ is defined as follows:$$MD(h,e)=\frac{p\left(h\right)-p(h/e)}{p\left(h\right)}$$According to this expression, $MD(h,e)$ represents a relative increase in the negation of the likelihood of the hypothesis h after evidence e.
- If $p(h/e)=p\left(h\right)$, then either the new evidence e is independent of the hypothesis h or there is no knowledge about an eventual causal relationship between h and e. In this case:$$MB(h,e)=MD(h,e)=0$$

- If $CF(h,{e}_{1})>0$ and $CF(h,{e}_{2})>0$, then:$$CF(h,{e}_{1}\wedge {e}_{2})=CF(h,{e}_{1})+CF(h,{e}_{2})-CF(h,{e}_{1})\times CF(h,{e}_{2})$$
- If $CF(h,{e}_{1})<0$ and $CF(h,{e}_{2})<0$, then:$$CF(h,{e}_{1}\wedge {e}_{2})=CF(h,{e}_{1})+CF(h,{e}_{2})+CF(h,{e}_{1})\times CF(h,{e}_{2})$$
- If $CF(h,{e}_{1})\times CF(h,{e}_{2})<0$, then:$$CF(h,{e}_{1}\wedge {e}_{2})=\frac{CF(h,{e}_{1})+CF(h,{e}_{2})}{1-min\left\{\right|CF(h,{e}_{1})|,|CF(h,{e}_{2})\left|\right\}}$$

## 3. An Overview of Quantum Computing

- The rows of the matrix correspond to the basis vectors of the input;
- The columns of the matrix correspond to the basis vectors of the output;.
- The position $(j,i)$ of the matrix corresponds to the coefficient of the j basis vector output in relation to the i basis vector input.

- Quantum simulation of the classical NOT gate:Figure 2 illustrates the quantum circuit for simulating a classical NOT gate, in which only one quantum line ${q}_{0}$ is necessary to perform the calculations and only one conventional line ${c}_{1}$ is necessary to record the result of the measurements. Gate H places the system in a state of superposition, and gate ⨁ is a quantum NOT.
- Quantum simulation of the AND gate:Figure 4 illustrates the quantum circuit for simulating a classical AND gate, in which three quantum lines are necessary to perform the calculations, and only one conventional line is necessary to record the result of the operations. In this case, we measure the three qubits in order to illustrate the inputs and the outputs of the gate, where ${c}_{0}$ is the result, and ${c}_{1}$ and ${c}_{2}$ are the inputs. Gates H place the system in a state of superposition, and gate $CCN$ is a quantum AND.
- Quantum simulation of the OR gate:Figure 6 illustrates the quantum circuit for simulating a classical OR gate, in which three quantum lines are necessary to perform the calculations, and only one conventional line is necessary to record the result of the operations. In this case, we measure the three qubits in order to illustrate the inputs and the outputs of the gate, where ${c}_{0}$ is the result, and ${c}_{1}$ and ${c}_{2}$ are the inputs. Gates H place the system in a state of superposition, and the set of gates $CCN$, $CN$, and $CN$ is a quantum OR.

## 4. Correlating Certainty Factors with a Quantum Environment

#### 4.1. A Classical Example

- R1: $A\wedge B\stackrel{C{F}_{R1}}{\to}X$
- R2: $X\vee C\stackrel{C{F}_{R2}}{\to}Y$
- R3: $Y\wedge (D\vee E)\stackrel{C{F}_{R3}}{\to}H$

#### 4.2. Introducing Quantum Inaccuracy

- $\delta =0\to $$\theta =0\to $ state is 0 → the associated statement is false;
- $\delta =\pi /2\to $$\theta =\pi \to $ state is 1 → the associated declaration is true;
- $0<\delta <\pi /2\to $$0<\theta <\pi \to $ state is in superposition → we are in a situation of coherent superposition, and the associated statement is neither true nor false, or—in an equivalent way—it is true and false simultaneously.

## 5. Experimentation and Results

- The circuit is designed according to the case we are testing;
- The data that are employed in the test are defined;
- The results for the classical (Shortliffe and Buchanan, S.B.) and quantum (Quantum Computing, Q.C.) approaches are obtained;
- The results are compared in order to find a possible correlation. Both the regression functions (dotted line) and the coefficients of determination (${R}^{2}$) are shown in the figures of each experiment.

#### 5.1. Experiment 1: Inaccuracy in Declarative Knowledge (Imprecision)

#### 5.2. Experiment 2: Inaccuracy in Procedural Knowledge (Uncertainty)

#### 5.3. Experiment 3: Inaccuracy in Declarative and Procedural Knowledge (Imprecision and Uncertainty)

## 6. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Smolensky, P. Connectionist AI, symbolic AI, and the brain. Artif. Intell. Rev.
**1987**, 1, 95–109. [Google Scholar] [CrossRef] - Jordan, M.I.; Mitchell, T.M. Machine learning: Trends, perspectives, and prospects. Science
**2015**, 349, 255–260. [Google Scholar] [CrossRef] [PubMed] - Shor, P.W. Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM Rev.
**1999**, 41, 303–332. [Google Scholar] [CrossRef] - Grover, L.K. A fast quantum mechanical algorithm for database search. In Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, Philadelphia, PA, USA, 22–24 May 1996; pp. 212–219. [Google Scholar]
- Havlíček, V.; Córcoles, A.D.; Temme, K.; Harrow, A.W.; Kandala, A.; Chow, J.M.; Gambetta, J.M. Supervised learning with quantum-enhanced feature spaces. Nature
**2019**, 567, 209–212. [Google Scholar] [CrossRef] [Green Version] - Biamonte, J.; Wittek, P.; Pancotti, N.; Rebentrost, P.; Wiebe, N.; Lloyd, S. Quantum machine learning. Nature
**2017**, 549, 195–202. [Google Scholar] [CrossRef] [PubMed] - Ciliberto, C.; Herbster, M.; Ialongo, A.D.; Pontil, M.; Rocchetto, A.; Severini, S.; Wossnig, L. Quantum machine learning: A classical perspective. Proc. R. Soc. A Math. Phys. Eng. Sci.
**2018**, 474, 20170551. [Google Scholar] [CrossRef] - Dunjko, V.; Briegel, H.J. Machine learning & artificial intelligence in the quantum domain: A review of recent progress. Rep. Prog. Phys.
**2018**, 81, 074001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gabor, T.; Sünkel, L.; Ritz, F.; Phan, T.; Belzner, L.; Roch, C.; Feld, S.; Linnhoff-Popien, C. The Holy Grail of Quantum Artificial Intelligence: Major Challenges in Accelerating the Machine Learning Pipeline. In Proceedings of the IEEE/ACM 42nd International Conference on Software Engineering Workshops, Seoul, Korea, 27 June–19 July 2020; Association for Computing Machinery: New York, NY, USA, 2020; pp. 456–461. [Google Scholar] [CrossRef]
- Soni, K.K.; Rasool, A. Pattern matching: A quantum oriented approach. Procedia Comput. Sci.
**2020**, 167, 1991–2002. [Google Scholar] [CrossRef] - Montanaro, A. Quantum pattern matching fast on average. Algorithmica
**2017**, 77, 16–39. [Google Scholar] [CrossRef] [Green Version] - Moret-Bonillo, V. Emerging technologies in artificial intelligence: Quantum rule-based systems. Prog. Artif. Intell.
**2018**, 7, 155–166. [Google Scholar] [CrossRef] - Moret-Bonillo, V.; Fernández-Varela, I.; Álvarez-Estévez, D. Uncertainty in Quantum Rule-Based Systems. Arch. Clin. Biomed. Res.
**2018**, 5, 42–60. [Google Scholar] [CrossRef] - Lindley, D.V. Understanding Uncertainty; John Wiley & Sons: Hoboken, NJ, USA, 2013. [Google Scholar] [CrossRef]
- Shortliffe, E.H.; Buchanan, B.G. A model of inexact reasoning in medicine. Math. Biosci.
**1975**, 23, 351–379. [Google Scholar] [CrossRef] - Heckerman, D. Probabilistic Interpretations for Mycin’s Certainty Factors. In Uncertainty in Artificial Intelligence; Machine Intelligence and Pattern Recognition; Kanal, L.N., Lemmer, J.F., Eds.; North-Holland: Amsterdam, The Netherlands, 1986; Volume 4, pp. 167–196. [Google Scholar] [CrossRef]
- Zadeh, L. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Shafer, G. A Mathematical Theory of Evidence; Princeton University Press: Princeton, NJ, USA, 1976. [Google Scholar]
- Pearl, J. Fusion, propagation, and structuring in belief networks. Artif. Intell.
**1986**, 29, 241–288. [Google Scholar] [CrossRef] [Green Version] - Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum
**2018**, 2, 79. [Google Scholar] [CrossRef] - Moret-Bonillo, V. Can artificial intelligence benefit from quantum computing? Prog. Artif. Intell.
**2015**, 3, 89–105. [Google Scholar] [CrossRef] - Zhao, X.; Chen, W. GIS-Based Evaluation of Landslide Susceptibility Models Using Certainty Factors and Functional Trees-Based Ensemble Techniques. Appl. Sci.
**2020**, 10, 16. [Google Scholar] [CrossRef] [Green Version] - Hou, E.; Wang, J.; Chen, W. A comparative study on groundwater spring potential analysis based on statistical index, index of entropy and certainty factors models. Geocarto Int.
**2018**, 33, 754–769. [Google Scholar] [CrossRef] - Cui, K.; Lu, D.; Li, W. Comparison of landslide susceptibility mapping based on statistical index, certainty factors, weights of evidence and evidential belief function models. Geocarto Int.
**2017**, 32, 935–955. [Google Scholar] [CrossRef] - Pourghasemi, H.R.; Pradhan, B.; Gokceoglu, C.; Mohammadi, M.; Moradi, H.R. Application of weights-of-evidence and certainty factor models and their comparison in landslide susceptibility mapping at Haraz watershed, Iran. Arab. J. Geosci.
**2013**, 6, 2351–2365. [Google Scholar] [CrossRef] - Lucas, P.J. Certainty-factor-like structures in Bayesian belief networks. Knowl. Based Syst.
**2001**, 14, 327–335. [Google Scholar] [CrossRef] [Green Version] - Yanofsky, N.S.; Mannucci, M.A. Quantum Computing for Computer Scientists; Cambridge University Press: Cambridge, UK, 2008. [Google Scholar]
- Mermin, N.D. Extreme quantum entanglement in a superposition of macroscopically distinct states. Phys. Rev. Lett.
**1990**, 65, 1838. [Google Scholar] [CrossRef] [PubMed] - Mishra, N. Understanding the Basics of Measurements in Quantum Computation. Available online: https://towardsdatascience.com/understanding-basics-of-measurements-in-quantum-computation-4c885879eba0 (accessed on 15 November 2021).
- Feynman, R.P.; Hey, T.; Allen, R.W. Feynman Lectures on Computation; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Sutor, R.S. Dancing with Qubits: How Quantum Computing Works and How It Can Change the World; Packt Publishing Ltd.: Birmingham, UK, 2019. [Google Scholar]
- IBM. IBM Quantum. Available online: https://quantum-computing.ibm.com/ (accessed on 7 December 2021).
- Bloch, F. Nuclear induction. Phys. Rev.
**1946**, 70, 460. [Google Scholar] [CrossRef] - Atos. Quantum Application Toolset—myQLM Documentation Documentation. Available online: https://myqlm.github.io/ (accessed on 9 December 2021).
- NEASQC. NExt ApplicationS of Quantum Computing. Available online: https://www.neasqc.eu/ (accessed on 9 December 2021).
- Vapnik, V.; Izmailov, R. Knowledge transfer in SVM and neural networks. Ann. Math. Artif. Intell.
**2017**, 81, 3–19. [Google Scholar] [CrossRef] - Bremner, M.J.; Montanaro, A.; Shepherd, D.J. Average-Case Complexity Versus Approximate Simulation of Commuting Quantum Computations. Phys. Rev. Lett.
**2016**, 117, 080501. [Google Scholar] [CrossRef] - Nadaban, S. From Classical Logic to Fuzzy Logic and Quantum Logic: A General View. Int. J. Comput. Commun. Control
**2021**, 16. [Google Scholar] [CrossRef] - Vourdas, A. Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces. J. Math. Phys.
**2014**, 55, 082107. [Google Scholar] [CrossRef] [Green Version] - Borujeni, S.E.; Nannapaneni, S.; Nguyen, N.H.; Behrman, E.C.; Steck, J.E. Quantum circuit representation of Bayesian networks. Expert Syst. Appl.
**2021**, 176, 114768. [Google Scholar] [CrossRef]

Input | Output |
---|---|

|0〉 | |0〉 |

|1〉 | |1〉 |

|0〉 | |1〉 | |
---|---|---|

|0〉 | 1 | 0 |

|1〉 | 0 | 1 |

Input A | Input B | Output A | Output B |
---|---|---|---|

|0〉 | |0〉 | |0〉 | |0〉 |

|0〉 | |1〉 | |0〉 | |1〉 |

|1〉 | |0〉 | |1〉 | |1〉 |

|1〉 | |1〉 | |1〉 | |0〉 |

|00〉 | |01〉 | |10〉 | |11〉 | |
---|---|---|---|---|

|00〉 | 1 | 0 | 0 | 0 |

|01〉 | 0 | 1 | 0 | 0 |

|10〉 | 0 | 0 | 0 | 1 |

|11〉 | 0 | 0 | 1 | 0 |

|000〉 | |001〉 | |010〉 | |011〉 | |100〉 | |101〉 | |110〉 | |111〉 | |
---|---|---|---|---|---|---|---|---|

|000〉 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

|001〉 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

|010〉 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

|011〉 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

|100〉 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

|101〉 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |

|110〉 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |

|111〉 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |

**Table 6.**Comparison between the results of the a priori probability of the hypothesis H with the classical method and the quantum approach.

Certainty Factors Model | Quantum Approach | |
---|---|---|

$P(H=FALSE)$ | $0.895$ | $0.892$ |

$P(H=TRUE)$ | $0.105$ | $0.108$ |

Input | Output | |||||
---|---|---|---|---|---|---|

CF(A) | CF(B) | CF(C) | CF(D) | CF(E) | S.B. | Q.C. |

1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

0.950 | 0.750 | 0.950 | 0.950 | 1.000 | 0.950 | 0.997 |

0.900 | 0.125 | 0.900 | 0.900 | 1.000 | 0.900 | 0.979 |

0.850 | 0.125 | 0.850 | 0.850 | 1.000 | 0.850 | 0.949 |

0.800 | 0.375 | 0.800 | 0.800 | 1.000 | 0.800 | 0.933 |

0.750 | 1.000 | 0.750 | 0.750 | 1.000 | 0.750 | 0.978 |

0.700 | 0.500 | 0.700 | 0.700 | 0.750 | 0.700 | 0.842 |

0.650 | 0.750 | 0.650 | 0.650 | 0.875 | 0.650 | 0.892 |

0.600 | 1.000 | 0.600 | 0.600 | 1.000 | 0.600 | 0.894 |

0.550 | 0.625 | 0.550 | 0.550 | 0.750 | 0.550 | 0.726 |

0.500 | 1.000 | 0.500 | 0.500 | 0.125 | 0.500 | 0.386 |

0.450 | 0.375 | 0.450 | 0.450 | 0.750 | 0.450 | 0.463 |

0.400 | 0.125 | 0.400 | 0.400 | 1.000 | 0.400 | 0.348 |

0.350 | 0.875 | 0.350 | 0.350 | 1.000 | 0.350 | 0.476 |

0.300 | 1.000 | 0.300 | 0.300 | 1.000 | 0.300 | 0.385 |

0.250 | 0.500 | 0.250 | 0.250 | 0.500 | 0.250 | 0.138 |

0.200 | 1.000 | 0.200 | 0.200 | 0.375 | 0.200 | 0.072 |

0.150 | 1.000 | 0.150 | 0.150 | 0.750 | 0.150 | 0.094 |

0.100 | 1.000 | 0.100 | 0.100 | 0.500 | 0.100 | 0.024 |

0.050 | 0.375 | 0.050 | 0.050 | 0.500 | 0.050 | 0.004 |

0.000 | 1.000 | 0.000 | 0.000 | 0.500 | 0.000 | 0.000 |

Input | Output | |||
---|---|---|---|---|

CF_{R1} | CF_{R2} | CF_{R3} | S.B. | Q.C. |

1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

0.950 | 0.950 | 0.950 | 0.902 | 0.988 |

0.900 | 0.900 | 0.900 | 0.810 | 0.953 |

0.850 | 0.850 | 0.850 | 0.722 | 0.893 |

0.800 | 0.800 | 0.800 | 0.640 | 0.817 |

0.750 | 0.750 | 0.750 | 0.562 | 0.734 |

0.700 | 0.700 | 0.700 | 0.490 | 0.619 |

0.650 | 0.650 | 0.650 | 0.423 | 0.516 |

0.600 | 0.600 | 0.600 | 0.360 | 0.435 |

0.550 | 0.550 | 0.550 | 0.303 | 0.322 |

0.500 | 0.500 | 0.500 | 0.250 | 0.240 |

0.450 | 0.450 | 0.450 | 0.203 | 0.157 |

0.400 | 0.400 | 0.400 | 0.160 | 0.102 |

0.350 | 0.350 | 0.350 | 0.122 | 0.075 |

0.300 | 0.300 | 0.300 | 0.090 | 0.057 |

0.250 | 0.250 | 0.250 | 0.062 | 0.030 |

0.200 | 0.200 | 0.200 | 0.040 | 0.016 |

0.150 | 0.150 | 0.150 | 0.022 | 0.000 |

0.100 | 0.100 | 0.100 | 0.010 | 0.001 |

0.050 | 0.050 | 0.050 | 0.003 | 0.000 |

0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

Input | Output | ||||||||
---|---|---|---|---|---|---|---|---|---|

CF(A) | CF(B) | CF(C) | CF(D) | CF(E) | CF_{R1} | CF_{R2} | CF_{R3} | S.B. | Q.C. |

1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

0.950 | 0.750 | 0.950 | 0.950 | 1.000 | 0.950 | 0.950 | 0.950 | 0.857 | 0.992 |

0.900 | 0.125 | 0.900 | 0.900 | 1.000 | 0.900 | 0.900 | 0.900 | 0.729 | 0.933 |

0.850 | 0.125 | 0.850 | 0.850 | 1.000 | 0.850 | 0.850 | 0.850 | 0.614 | 0.858 |

0.800 | 0.375 | 0.800 | 0.800 | 1.000 | 0.800 | 0.800 | 0.800 | 0.512 | 0.749 |

0.750 | 1.000 | 0.750 | 0.750 | 1.000 | 0.750 | 0.750 | 0.750 | 0.422 | 0.707 |

0.700 | 0.500 | 0.700 | 0.700 | 0.750 | 0.700 | 0.700 | 0.700 | 0.343 | 0.540 |

0.650 | 0.750 | 0.650 | 0.650 | 0.875 | 0.650 | 0.650 | 0.650 | 0.275 | 0.461 |

0.600 | 1.000 | 0.600 | 0.600 | 1.000 | 0.600 | 0.600 | 0.600 | 0.216 | 0.326 |

0.550 | 0.625 | 0.550 | 0.550 | 0.750 | 0.550 | 0.550 | 0.550 | 0.166 | 0.204 |

0.500 | 1.000 | 0.500 | 0.500 | 0.125 | 0.500 | 0.500 | 0.500 | 0.125 | 0.091 |

0.450 | 0.375 | 0.450 | 0.450 | 0.750 | 0.450 | 0.450 | 0.450 | 0.091 | 0.074 |

0.400 | 0.125 | 0.400 | 0.400 | 1.000 | 0.400 | 0.400 | 0.400 | 0.064 | 0.040 |

0.350 | 0.875 | 0.350 | 0.350 | 1.000 | 0.350 | 0.350 | 0.350 | 0.043 | 0.028 |

0.300 | 1.000 | 0.300 | 0.300 | 1.000 | 0.300 | 0.300 | 0.300 | 0.027 | 0.006 |

0.250 | 0.500 | 0.250 | 0.250 | 0.500 | 0.250 | 0.250 | 0.250 | 0.016 | 0.002 |

0.200 | 1.000 | 0.200 | 0.200 | 0.375 | 0.200 | 0.200 | 0.200 | 0.008 | 0.000 |

0.150 | 1.000 | 0.150 | 0.150 | 0.750 | 0.150 | 0.150 | 0.150 | 0.003 | 0.000 |

0.100 | 1.000 | 0.100 | 0.100 | 0.500 | 0.100 | 0.100 | 0.100 | 0.001 | 0.000 |

0.050 | 0.375 | 0.050 | 0.050 | 0.500 | 0.050 | 0.050 | 0.050 | 0.000 | 0.000 |

0.000 | 1.000 | 0.000 | 0.000 | 0.500 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Moret-Bonillo, V.; Magaz-Romero, S.; Mosqueira-Rey, E.
Quantum Computing for Dealing with Inaccurate Knowledge Related to the Certainty Factors Model. *Mathematics* **2022**, *10*, 189.
https://doi.org/10.3390/math10020189

**AMA Style**

Moret-Bonillo V, Magaz-Romero S, Mosqueira-Rey E.
Quantum Computing for Dealing with Inaccurate Knowledge Related to the Certainty Factors Model. *Mathematics*. 2022; 10(2):189.
https://doi.org/10.3390/math10020189

**Chicago/Turabian Style**

Moret-Bonillo, Vicente, Samuel Magaz-Romero, and Eduardo Mosqueira-Rey.
2022. "Quantum Computing for Dealing with Inaccurate Knowledge Related to the Certainty Factors Model" *Mathematics* 10, no. 2: 189.
https://doi.org/10.3390/math10020189