# 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

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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 |

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**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