3.1. Negation and identity

The simplest Boolean logic operation is the unary negation (NOT) operation $\neg A$ , which serves as a good basic reference for comparison between Boolean and Bayesian logic. In the Boolean sense, the output state of this operation is the opposite state of the input.

For the case of a two-valued random variable that can take the state true or false, the conditional probability distribution, represented as a two-dimensional matrix, takes the same form as the truth table for the NOT operation, as shown in Fig. 4. This is because the truth table for the probability $p$ determines the value for $\text{P}(Q=\text{t}|\ldots )$ , and its complement $q$ determines the values for $\text{P}(Q=\text{f}|\ldots )$ . The conditional probability distribution for a random variable with a single parent is just a matrix that is multiplied in the manner of matrix multiplication with the distribution of the parent random variable to produce the inferred distribution. The conditional probability distribution can also be described as in Eq. (6).

(6) \begin{equation} \begin{cases} \text{P}(Q=\text{f}|A=\text{t}) = 1.0 \\[3pt] \text{P}(Q=\text{t}|A=\text{f}) = 1.0 \\[3pt] \text{P}(Q=\text{t}) = 0.0\, \text{otherwise} \end{cases} \end{equation}

To provide a comparison example, an “identity” conditional random variable (with the operation abbreviated as ID) is created that has the effect of the inferred distribution being identical to the parent random variable, shown in Fig. 5. For a random variable with a single parent $A$ , this distribution takes the form of an identity matrix as the inference operation is again a matrix multiplication.

(7) \begin{equation} \begin{cases} \text{P}(Q=\text{t}|A=\text{t},B=\text{t},\ldots =\text{t}) = 1.0 \\[5pt] \text{P}(Q=\text{t}) = 0.0\, \text{otherwise} \end{cases} \end{equation}

These examples indicate a useful property of the conditional probability distribution for Bernoulli random variables. The negation operation has the effect of reversing the conditional probability distribution along the axis of the values for the random variable as shown in Eq. (8).

(8) \begin{equation} \text{P}(Q) = \neg \text{P}(A) = 1-\text{P}(A) = 1-[p_{0,0}, p_{1,0}] \end{equation}

Since $p_{1,0} = 1-p_{0,0}$ , the operation of negation on $\text{P}(A)$ is as illustrated in Eq. (9).

(9) \begin{equation} \neg \text{P}(A) = 1-[p_{0,0}, 1-p_{0,0}] = [1-p_{0,0}, p_{0,0}] = [p_{1,0}, p_{0,0}] \end{equation}

Thus, the distribution terms are reversed on negation. Using this rule, the Bernoulli conditional probability distributions can easily be produced that are equivalent to inverted logic operations such as NAND, NOR, and XNOR, from the non-inverted AND, OR, and XOR operations.

Figure 4. Bayesian logical negation (NOT) operation.

Figure 5. Bayesian logical identity (ID) operation.

3.2. Conjunction

The conjunction (AND) operation $A \wedge B \wedge \ldots$ is implemented as a conditional probability distribution by recognizing that the inferred distribution should show a state of true only for the case where all priors are considered to be true. Following the previous example, the indices of the distribution for $\text{P}(Q=\text{t}|\ldots )$ are set as in Eq. (10).

(10) \begin{equation} \begin{cases} \text{P}(Q=\text{t}|A=\text{t},B=\text{t},\ldots =\text{t}) = 1.0 \\[5pt] \text{P}(Q=\text{t}) = 0.0\, \text{otherwise} \end{cases} \end{equation}

In all cases, $\text{P}(Q=\text{f}|\ldots ) = 1-\text{P}(Q=\text{t}|\ldots )$ . Figure 6 shows the resulting distribution for the case of a two-input AND operation. This method also applies to multiple-input AND operations, as shown by the example of three parent random variables in Fig. 7.

Figure 6. Bayesian logical conjunction (AND) operation.

Figure 7. Bayesian logical conjunction (AND) operation with three inputs.

3.3. Disjunction

The disjunction (OR) operation $A \vee B \vee \ldots$ is implemented in a complementary fashion to conjunction, by recognizing that the inferred distribution should be true in all cases except for that where all priors are considered to be false, as stated in Eq. (11).

(11) \begin{equation} \begin{cases} \text{P}(Q=\text{t}|A=\text{f},B=\text{f},\ldots =\text{f}) = 0.0 \\[5pt] \text{P}(Q=\text{t}) = 1.0 \,\text{otherwise} \end{cases} \end{equation}

The resulting distribution for a two-input OR operation is shown in Fig. 8 and again applies equally to the case of three or more inputs as shown in Fig. 9.

Figure 8. Bayesian logical disjunction (OR) operation.

Figure 9. Bayesian logical disjunction (OR) operation with three inputs.

3.4. Exclusive disjunction

The exclusive disjunction (XOR) operation $A \oplus B \oplus \ldots$ differs from that of OR in that there are two cases in which the inferred distribution should be false, when all priors are considered to be true and when all priors are considered to be false. Equation (12) describes the conditional probability distribution for this operation.

(12) \begin{equation} \begin{cases} \text{P}(Q=\text{t}|A=\text{t},B=\text{t},\ldots =\text{t}) = 0.0 \\[3pt] \text{P}(Q=\text{t}|A=\text{f},B=\text{f},\ldots =\text{f}) = 0.0 \\[3pt] \text{P}(Q=\text{t}) = 1.0\, \text{otherwise} \end{cases} \end{equation}

The resulting distribution for a two-input XOR operation is shown in Fig. 10 and again applies equally to the case of three or more inputs as shown in Fig. 11.

Figure 10. Bayesian logical exclusive disjunction (XOR) operation.

Figure 11. Bayesian logical exclusive disjunction (XOR) operation with three inputs.

3.5. Material implication

Material implication (which is abbreviated here as IMP) differs from the logical operations above that are used for computational logic in that the special cases for a multiple-input operation are not simply those of all-true or all-false. This means that the operation is not commutative – the ordering and identity of the priors matters, and one input is distinct as the hypothesis. For the case of implication ( $A \rightarrow B$ ), Eq. (13) provides the conditional probability distribution.

(13) \begin{equation} \begin{cases} \text{P}(Q=\text{t}|A=\text{t},B=\text{f}) = 0.0 \\[4pt] \text{P}(Q=\text{t}) = 1.0 \,\text{otherwise} \end{cases} \end{equation}

Figure 12 shows the distribution that results. The reverse case of converse implication (CIMP) $A \leftarrow B$ is provided by simply setting $\text{P}(Q=\text{t}|A=\text{t},B=\text{f}) = 0.0$ in Eq. (13).

Figure 12. Bayesian logical implication (IMP) operation.

Although the Boolean implication operation is not defined for multiple inputs, the extension of Eq. (13) to multiple variables is proposed in this methodology as an appropriate definition of multiple-input implication. This naturally produces the operation $A \rightarrow (B \vee C \vee \ldots )$ in which all input random variables other than the one serving as hypothesis are treated as a single disjunction. By defining the operation as false only if $A$ is true and all other variables are false, only one tensor entry for $\text{P}(Q=\text{t}|\ldots )$ needs to be defined as false, and the conditional probability distribution takes the form of Eq. (14).

(14) \begin{equation} \begin{cases} \text{P}(Q=\text{t}|A=\text{t},B=\text{f},\ldots =\text{f}) = 0.0 \\[4pt] \text{P}(Q=\text{t}) = 1.0 \,\text{otherwise} \end{cases} \end{equation}

Using this new proposed formulation, the distribution shown in Fig. 13 is produced. Any parent random variable (the first parent input $A$ is suggested here for convenience) can be selected as the implication input for this operation, which when true will require at least one of the other inputs to also be true for the operation output to be true. The converse implication operation for multiple variables ceases to have distinct meaning in this interpretation as it is simply the use of a different input variable. The negation of implication however (abbreviated here as NIMP) still a distinct operation that can again be accomplished by reversing the conditional probability distribution as described above.

Figure 13. Bayesian logical implication (IMP) operation extended to three inputs as proposed by $A \rightarrow (B \vee C \vee \ldots )$ .

While implementing the multiple-input IMP operation as $A \rightarrow (B \wedge C \wedge \ldots )$ is possible, in which all input random variables other than the hypothesis are treated as a single conjunction, it is both less efficient to generate and less useful in the sense that the output is identical to the hypothesis in all cases other than when all variables are true and thus requires setting conditional probabilities separately for $L$ parents rather than setting a single case.

Having defined Boolean operations in terms of Bayesian inference, more detail in their properties can now be explored with respect to Boolean logic systems that can propagate probabilistic information.

4.1. Product forms

The single-input identity and negation (NOT) operations are simple in their application, so instead multiple-input operations with more than one parent random variable are considered here. To begin with, consider the example of a two-input logical conjunction (AND) operation. Applying the expanded tensor inner product of Eq. (4) to the AND operation tensor constructed in Eq. (10) results in the expansion as follows, here shown with $\text{P}(Q=\text{t}|A,B)$ on the “top” of column vectors and $\text{P}(Q=\text{f}|A,B)$ on the “bottom” of Eq. (15).

(15) \begin{align} \text{P}(Q_{\text{AND}}|A,B){} & = \left[\begin{array}{c} (1.0) A_{\text{t}} B_{\text{t}} + (0.0) A_{\text{f}} B_{\text{t}} + (0.0) A_{\text{t}} B_{\text{f}} + (0.0) A_{\text{f}} B_{\text{f}} \\[5pt] (0.0) A_{\text{t}} B_{\text{t}} + (1.0) A_{\text{f}} B_{\text{t}} + (1.0) A_{\text{t}} B_{\text{f}} + (1.0) A_{\text{f}} B_{\text{f}} \\[5pt] \end{array}\right] \nonumber \\[5pt] & = \left[\begin{array}{c} A_{\text{t}} B_{\text{t}} \\[5pt] A_{\text{f}} B_{\text{t}} + A_{\text{t}} B_{\text{f}} + A_{\text{f}} B_{\text{f}} \\[5pt] \end{array}\right] \end{align}

The same pattern can be seen in the expansion of the logical disjunction (OR) operation constructed in Eq. (11) that is given in Eq. (16), exclusive disjunction (XOR) operation constructed in Eq. (12) that is given in Eq. (17), and logical implication (IMP) operation constructed in Eq. (13) that is given in Eq. (18).

(16) \begin{eqnarray} \text{P}(Q_{\text{OR}}|A,B){}&=& \begin{bmatrix} (1.0) A_{\text{t}} B_{\text{t}} + (1.0) A_{\text{f}} B_{\text{t}} + (1.0) A_{\text{t}} B_{\text{f}} + (0.0) A_{\text{f}} B_{\text{f}} \\[5pt] (0.0) A_{\text{t}} B_{\text{t}} + (0.0) A_{\text{f}} B_{\text{t}} + (0.0) A_{\text{t}} B_{\text{f}} + (1.0) A_{\text{f}} B_{\text{f}} \\[5pt] \end{bmatrix} \nonumber \\[5pt] &=&\begin{bmatrix} A_{\text{t}} B_{\text{t}} + A_{\text{f}} B_{\text{t}} + A_{\text{t}} B_{\text{f}} \\[5pt] A_{\text{f}} B_{\text{f}} \\[5pt] \end{bmatrix} \end{eqnarray}

(17) \begin{eqnarray} \text{P}(Q_{\text{XOR}}|A,B){}&= & \begin{bmatrix} (0.0) A_{\text{t}} B_{\text{t}} + (1.0) A_{\text{f}} B_{\text{t}} + (1.0) A_{\text{t}} B_{\text{f}} + (0.0) A_{\text{f}} B_{\text{f}} \\[5pt] (1.0) A_{\text{t}} B_{\text{t}} + (0.0) A_{\text{f}} B_{\text{t}} + (0.0) A_{\text{t}} B_{\text{f}} + (1.0) A_{\text{f}} B_{\text{f}} \\[5pt] \end{bmatrix} \nonumber \\[5pt] &=&\begin{bmatrix} A_{\text{f}} B_{\text{t}} + A_{\text{t}} B_{\text{f}} \\[5pt] A_{\text{t}} B_{\text{t}} + A_{\text{f}} B_{\text{f}} \\[5pt] \end{bmatrix} \end{eqnarray}

(18) \begin{eqnarray} \text{P}(Q_{\text{IMP}}|A,B){}&= & \begin{bmatrix} (1.0) A_{\text{t}} B_{\text{t}} + (0.0) A_{\text{f}} B_{\text{t}} + (1.0) A_{\text{t}} B_{\text{f}} + (1.0) A_{\text{f}} B_{\text{f}} \\[5pt] (0.0) A_{\text{t}} B_{\text{t}} + (1.0) A_{\text{f}} B_{\text{t}} + (0.0) A_{\text{t}} B_{\text{f}} + (0.0) A_{\text{f}} B_{\text{f}} \\[5pt] \end{bmatrix} \nonumber \\[5pt] &=&\begin{bmatrix} A_{\text{t}} B_{\text{t}} + A_{\text{f}} B_{\text{t}} + A_{\text{f}} B_{\text{f}} \\[5pt] A_{\text{t}} B_{\text{f}} \\[5pt] \end{bmatrix} \end{eqnarray}

This expansion demonstrates how probabilistic logic with Bernoulli random variables can differ from Boolean logic. It is clear that due to the imposed Boolean logic structure of all conditional probability distribution terms being either $1.0$ or $0.0$ , and the probabilities along the random variable distribution summing to $1.0$ in all cases, each combination of input product $AB$ terms appears only in one place in the inferred distribution. Note that a negation (NAND, NOR, XNOR) has the effect already explained of reversing the probabilities of the inferred distribution.

This leads to a sum-of-products expression for each of the Boolean-derived probabilistic logic operations that can be used as an efficient way to produce these logic operations numerically and makes the result of a hybrid logic calculation clear to see. This sum-of-products expression is equivalent to a Boolean sum-of-products expression, indicating that if a set of purely Boolean values $\{1.0, 0.0\}$ are used as inputs, only one of the product terms will result in a probability of $\mathrm{P}(Q=\text{t}|A,B) = 1.0$ . Due to the conceptual parallel with the Boolean sum-of-products form, these products within the sum are also referred to as “minterms," and validate that the proposed methodology produces the same results as an equivalent Boolean logic system.

Of much greater interest is the result if probabilities that do not indicate certainty and lie in the interval $[0, 1]$ are used, for input random variables of the form $\mathrm{P}(A=t) = A_{\text{t}} = [0.0, 1.0]$ , $\text{P}(A=f) = A_{\text{f}} = 1.0 - A_{\text{t}}$ . If either $A_{\text{t}} = 0$ or $B_{\text{t}} = 0$ in an AND operation, it is clear from the minterms that $A_{\text{t}} B_{\text{t}}$ will be $0$ and the output will be definite with $\text{P}(Q_{\text{AND}}=\text{t}|A,B) = 0$ , which reflects the removal of uncertainty by the ultimate dependence on both inputs.

Similarly, in the OR operation, if either $A_{\text{t}} = 1$ or $B_{\text{t}} = 1$ , it is clear from the minterms that $A_{\text{f}} B_{\text{f}}$ will be $0$ and the output will be definite with $\text{P}(Q_{\text{OR}}=\text{t}|A,B) = 1$ .

Logical implication as presented in this work is similar to AND with one minterm determining the probability that the result is false, but indicates the specific values of inputs that will produce $\text{P}(Q_{\text{IMP}}=\text{t}|A,B) = 0$ , and definite truth is implied if any of these input values are $0$ , and uncertainty if not.

An XOR operation is different in that there are two minterms for each state, and given the definite $A_{\text{t}} = 0$ , then $\text{P}(Q_{\text{XOR}}=\text{t}|A,B) = B_{\text{t}}$ , while $A_{\text{t}} = 1$ , results in $\text{P}(Q_{\text{XOR}}=\text{t}|A,B) = B_{\text{f}}$ . Both $A$ and $B$ must be definite to produce a definite output in XOR, and like the pure Boolean operation, it effectively serves as a selectable negation operation when one input is definite while uncertainty is passed through the other. These minterms clearly indicate how both certainty and uncertainty are propagated together through hybrid logic operations.

In summary, the different logic operations as proposed in this methodology provide different “tolerances” to uncertainty in one variable with other variables being definite, as indicated by analysis of Eqs. (15)–(18).

1. AND definite false if one input is definite false, otherwise propagates uncertainty in other inputs

2. OR definite true if one input is definite true, otherwise propagates uncertainty in other inputs

3. XOR not definite unless all inputs definite; one definite input being true will invert the other with two inputs

4. IMP definite false if hypothesis input is definite true and all others false, otherwise propagates other inputs

4.2. Number of inputs

The number of inputs $L$ as parent random variables to the operation is highly relevant to the resulting inferred distribution. In the case of AND, OR, and implication operations, only one minterm will appear in one of the inferred distribution state probabilities (either for true or for false) and for an XOR operation two minterms, regardless of the number of inputs. The remaining terms appear in the opposite state probability. As Bernoulli random variables are restricted to two states, this can be seen to heavily bias the “output” inferred distribution of the operation in favour of the state in which more terms appear.

This behaviour is a consequence of the uncertainty that is built into the concept of a random variable, as seen in the context of a logical operation. A Boolean operation with $L$ independent and uncertain inputs that all have a probability $p$ of state true and probability $q=1-p$ of state false can be modelled by a binomial distribution. In the case of a Boolean AND operation, the probability of the operation resulting in true is $p^L$ , while in the case of a Boolean OR operation, the probability of the operation resulting in true is $1-q^L$ , which typically is a larger value for large $L$ . As the XOR operation has two terms in which the result is false, the probability is $p^L + q^L$ in this case. Even in the case of entirely uncertain input state probabilities $p=0.5$ and $q=0.5$ , the output will be biased depending on the type of logical operation to a degree proportional to the number of inputs.

Since the probability distributions of parent random variables (inputs) to a logical operation will vary, a more general description of the truth probability of the inferred distribution (output) $\text{P}(Q=\text{true})$ is preferred and takes the form of a product of the truth probability of the parent random variables $X_l$ in $\text{Pa}(Q)$ to the logical operation random variable $Q$ . For clarity of notation, the $L$ probabilities of truth for each parent random variable in $\text{Pa}(Q)$ are denoted as $p_l$ and the corresponding probabilities of falsity are denoted as $q_l = 1 - p_l$ as stated in Eqs. (19) and 20.

(19) \begin{eqnarray} p_l = \text{P}(X_l=\text{true}), X_l \in \text{Pa}(Q) \end{eqnarray}

(20) \begin{eqnarray} q_l = \text{P}(X_l=\text{false}) = 1 - p_l, X_l \in \text{Pa}(Q) \end{eqnarray}

Table I provides a summary of the probability of these truth probabilities for the operations discussed here as a product of input truth probabilities from $\text{Pa}(Q)$ . If all these probabilities are identical and represented by $p_l$ , the probability of truth can be calculated with respect to $L$ , as on the right side of this table. As it is not commutative, the logical implication operation (IMP) that is here defined as $A \rightarrow (B \vee C \vee \ldots )$ for multiple inputs and its negation (NIMP) must specify one variable $A$ to be the hypothesis, and this variable is considered here for simplicity to be the first indexed parent at $l = 1$ .

Table I. The dependence of inferred probability of truth $\text{P}(Q=\text{true})$ on the number of parent inputs $\text{Pa}(Q)$ for probabilistic logic operations. Assuming that all $L$ probabilities of truth $p_l = 1-q_l$ in $\text{Pa}(Q)$ are the same, the scaling of $p$ by the number of inputs $L$ is calculated.

Note: For IMP and CIMP, the input at l = 1 is the hypothesis.

Although this scaling is a significant factor in the design of logical programmes that involve uncertainty, it is important to note that if the inputs (probability distributions of parent random variables) are traditional Boolean quantities and therefore certain in state, then $p$ must be either $0.0$ or $1.0$ . In this case, the minterms in Eqs. (15)–(18) will also reduce to either $0.0$ or $1.0$ , and the certainty of state implied in Boolean logic will continue to propagate throughout the system until a probability value $1.0 \gt p \gt 0.0$ is introduced into a variable, at which point uncertainty will cause the number of parent variables to affect the probability distribution. This proves that the certainty of Boolean logic and consequential invariance with number of inputs is retained in the proposed methodology, while allowing uncertainty to act on the system in a means consistent with Bayesian inference.

5.1. Programming structure

The framework for programming Bayesian inference as described here makes use of arithmetic operations in fixed-point arithmetic for high efficiency and C structures with the following members that realize Bayesian network nodes following the methodology described above [Reference Post32]:

• name: A unique name string for identification of the node

• numParents: The integer number $L$ of node parents

• parent: An ordered array of integer node numbers indicating the parents of the node

• numVals: The integer number of states or discrete node distribution size ( $N$ )

• distSize: The total size in words of the order $L+1$ distribution tensor

• conditionals: A fixed-point array for the size $N$ inferred (conditional) distribution

• distributions: A fixed-point array for the order $L+1$ distribution tensor

• distFunction: A pointer to a single-input single-output fixed-point function used to populate the probability distribution if a special operation is needed (e.g., a continuous distribution function or a connection to a sensor or actuator)

Linear array indexing with an array declared with fixed size is used to store the list of node structures and the order $L+1$ tensor that represents the conditional probability distribution, and operations are bounds-checked to prevent pointer arithmetic errors. The tensor is indexed in row-major order, with the first dimension of the tensor sized to represent the internal probability distribution of the Bayesian node, and subsequent dimensions sized to represent the distributions of sizes $M_l$ of the parents $l = 1 \ldots L$ . The index $i$ into the linear array is calculated for each access of the tensor.

(21) \begin{align}{c} i & = m_1 + m_2 M_1 + m_3 M_2 M_1 + \ldots + m_{L+1} \prod _{l=1}^L M_l \nonumber\\[4pt]&= \sum _{n=1}^{L+1} \left ( m_n \prod _{l=1}^{n-1} M_l \right ). \end{align}

This requires an array of total size $N \prod _{l=1}^L M_l$ . For subsuming Boolean logic using Bernoulli random variables as in the proposed methodology, $M_l = 2$ in all cases except when a parent node is not a Bernoulli random variable, but instead is a general random variable with more than two states (shown in Eq. 21).

The advantages of programming behaviours in the manner of a flexible Bayesian network include that the programme is robust against unexpected system states and logical faults (such as division by zero or undesired negative results) as all calculation is done entirely by tensor-based multiplication operations on a field with known size of defined random variable states. The behavioural programme can also be changed quickly and easily by changing the tensor values that represent the conditional probability distribution of the nodes, as well as the structure of the network itself, which makes it practical to change behavioural programmes and implement machine learning methods that operate on probabilistic data.

5.2. Robot behaviour logic programme

To illustrate how the use of hybrid Boolean and Bayesian logic can process sensor information and produce system behaviours, I created a simulation scenario within the CoppeliaSim 4.2.0 robotics simulator environment. The scene included a simple differential drive mobile robot with multiple sensor inputs, an obstacle, and a goal location with a beacon that the robot can sense. I implemented the logic programme for navigation using the C programming language. The scenario and starting position of the robot are illustrated in Fig. 14. A mobile robot with a sensor that can identify the approximate direction of a target beacon must travel to the target and also avoid an obstacle that lies between the robot and the target. The robot must identify from sensor signals how to turn so as to avoid the obstacle while travelling in the direction of the target. As the focus of this analysis is to exemplify a hybrid logic programme and compare Boolean and hybrid logic results based on the proposed methodology rather than to design a useable hybrid logic control system, no closed-loop control or robotic movement model is used.

Figure 14. The scenario for the analysis of robotic navigational sensor processing. A mobile robot with sensor arcs $\{L, FL, F, FR, R\}$ must reach a target that is sensed by means of a beacon signal with some uncertainty, while avoiding an obstacle in the way that is sensed by rangefinders and a physical collision detector. The outline arrows show the movement of the robot in the time period considered.

Figure 15. The navigation programme for mobile robot control, as a network of Boolean logic gates. The decision for forward (“Fwd”) is an OR operation, speed (“Spd”) is an AND operation, and direction control (“Dir”) is a set of AND gates representing the desired direction of movement, turning.

Three kinds of external sensors are used. First, an obstacle collision sensor produces a Boolean output of true to indicate a collision has occurred. Obstacle avoidance is implemented using obstacle range sensors with an infrared range measurement model to detect obstacles and produce a probability distribution that indicates the relative probability of obstacle presence across direction states. Navigation is driven by a direction sensor that is modelled on a directional infrared beacon, that provides a Gaussian probability distribution over states of the directional movement. Boolean logic is used to determine the most appropriate action to take based on the sensor input in the same manner that logical rules are used to govern many types of robotic movement. However, when they are realized as Bayesian nodes, they propagate the uncertainty estimates in the sensors through the entire system so that it can be used as part of further decision-making processes. The actuators for the robot are modelled after a typical differential drive robot, and consist of two motors opposite each other that produce forward speed when rotating in the same direction, and turning based on the difference between their rotation speeds. The actuators of the robot are only the left and right motors, which can only be switched forward or reverse using Boolean logic, but using inference can be scaled in speed proportional to the probability value of the relevant random variables for forward and reverse movement, assuming higher certainty should result in higher movement speed.

The Boolean logic programme is written in C the language using the software framework described in [Reference Post32] with the logic layout shown in Fig. 15. This system is based on sensor inputs to drive behaviour with Boolean two-state information in traditional fashion. While it is very simplistic compared to most practical robotic control programmes, it is used here only for the purpose of comparing the differences between Boolean and hybrid Bayesian logic. The sensory responses provided by Boolean logic are compared to those of a system that uses the proposed methodology to replace Boolean logic with Bayesian inference for the same logic structure and operations, as shown in Fig. 16. The implemented programme is shown exported directly as a Bayesian network in Fig. 17. All are created using the common framework detailed above so that a single programming methodology is followed throughout. All decisions are implemented as inference operations into linked random variables, each with posterior probability values for the states {false, true}. The conditional probability distributions are set for each type of logic operation as described in Eq. (6) and Eqs. (10)–(14), and the process of inference is done as in Eq. (1) using the tensor indexing from Eq. (21). To compare the use of purely Boolean logic to Bayesian in an appropriate context, it is assumed that a probability $p \gt = 0.5, q \lt = 0.5$ corresponds to a logic true; otherwise, a logic false is propagated. Therefore, if a state of the target sensor, visual obstacle detection, or direction estimate are above $0.5$ , it is considered in the Boolean interpretation to be true. Only the probability of truth $p$ is plotted against time in time steps $t$ in Figs. 19–26 as the complementary state $q = 1 - p$ is implied at all times. The programme is run in separate test cases using Boolean logic and then using hybrid Bayesian logic. In both cases, the programme runs its main loop at a frequency of $10$ Hz, which means that the states of all logic gates and random variables on the robot are updated every “time step” of $0.1$ s. Comparing operation under Boolean logic to that under Bayesian, the robot follows a similar trajectory, but actions are performed differently due to the use of uncertainty information (for example, in setting motor speeds to intermediate values) leading to a timing difference between runs (the robot reaches the obstacle faster in the Boolean case but takes more time to avoid it). Example images indicating the path that the robot takes by illustration of positions at the associated time step numbers are shown in Fig. 18.

Figure 16. The navigation programme for mobile robot control using the proposed methodology for Boolean logic within a Bayesian network. The same number of elements is used for behaviour determination, but inputs and outputs are provided as separate random variables for each state, and probability values are propagated.

Figure 17. The navigation programme for mobile robot control. Fully implemented using the proposed methodology, in the form of a Bayesian network.

Figure 18. The positions of the robot during its path through the simulation. Positions illustrated are at time step numbers $0$ (a), $160$ (b), $280$ (c), $380$ (d), $470$ (e), and $580$ (f).

5.3. Sensory inputs

Information from direction-dependent sensors is split into five directional components to approximate angular position as Boolean variables. The facing direction $\theta$ that the sensed quantity refers to in the body frame of the robot is denoted by $F$ (front), $FL$ (front-left), $FR$ (front-right), $L$ (left), and $R$ (right), such that $\theta \in \{L, FL, F, FR, R\}$ .

In a purely Bayesian or probabilistic inference approach, this information can be stored as a single random variable with five states, using a tensor of order 6 to store the conditional probability distribution. However, as the aim of this work is to show how probabilistic inference can operate as Boolean two-state logic, these states are actually implemented as separate Bernoulli random variables as indicated above. It should be noted that in this system, the use of multiple Bernoulli probability values in a set of states such as “direction” is not conceptually the same as a single random variable with multiple states, because the total of all probabilities for the directions $F$ , $FL$ , $FR$ , $L$ , and $R$ does not necessarily sum to $1$ . Instead, $p + q = 1$ for two states. In comparison to using a single random variable as above, the probability $p$ of each sensor direction can be independently in the range $[0\ldots 1]$ without being normalized with respect to the probabilities of other sensor directions. Therefore, the numerical outputs of the sensor array will be different and it is important to consider this distinction with respect to how the sensor programming of the robot is interpreted logically.

The navigation target is named “Target” and is assumed to contain a directional beacon that the robot can sense to serve as a reference point for navigation. The target direction sensing model assumes that the probability distribution of the sensor follows a discretized Gaussian “normal” distribution $\mathcal{N}(\mu,\sigma ^2)$ , and it is assumed that the mean of the distribution is located at the angle to target $\mu = \theta _T$ and that the direction sense grows stronger and more directional the closer the robot is to the sensor, such that the standard deviation $\sigma = r_T/2$ decreases with range to target $r_T$ . The sensor directional components are located at angles $\theta = [{-}\pi/3, -\pi/6, 0, \pi/6, \pi/3]$ and sample the probability distribution of the beacon at these values. A Boolean logic direction estimate to the target is obtained by the probability of the target being greater than $0.3$ $(p_{\theta } \gt 0.3)$ in these directions. More than one sensing direction can be true at a time.

The values for the “Target” nodes as the robot navigates through the simulation are shown in Fig. 19. In the Boolean logic case, logic values for the forward directions overlap due to the spread of the target distribution and can be seen to cycle rapidly as the robot turns left and right in small increments to track the location of the beacon in the interval between $t=[80,180]$ time steps while the robot is turning towards the target. This rapid cycling behaviour is seen often in Boolean logic cases where discrete feedback thresholds are used. In the hybrid Bayesian logic case, a gradual change in probabilities occurs as the robot tracks the target. The probability values are seen to diverge as the robot gets closer to the target and the spread of the Gaussian sensor distribution decreases, making it clearer over time which direction dominates. This allows the robot to quantify directional probabilities with gradual changes, avoiding sudden logic changes as seen in the Boolean case and propagating this information through the entire system to control actuators in a similarly gradual fashion.

Figure 19. Sensor inputs to the navigation programme. Outputs of target direction sensor states in the five forward directions, for Boolean (top) and hybrid Bayesian (bottom) control cases, indicating the direction of the target point from the robot. As a probabilistic sensor, uncertainty information is affected by both the direction of the robot and the range from the target.

Figure 20. Sensor inputs to the navigation programme. Output of obstacle sensor, interpreted as the probability of no obstacles being detected (a clear path) in the five forward directions. The Boolean case (top) only indicates the point at which an obstacle is unlikely to be in the sensor arc ( $p\lt 0.5$ ) to be in a given direction while the Bayesian case (bottom) includes clear information about how likely or close the obstacle may be.

Five obstacle presence sensors designed to simulate infrared rangefinders also produce estimates of obstacle presence probability for the corresponding five forward directions. A probabilistic model is used for each sensor, in which the probability of obstacle presence is $0.2/r_O$ , where $r_O$ is the range to the target. For simplicity of combination with the navigation target information, the logical complement value is used and is named “not-Obstacle." The Boolean value for a given obstacle direction $\theta$ is set to the complement of whether the obstacle probability is greater than $0.1$ ( $\neg (p_{\theta } \gt 0.1)$ ) to increase sensitivity to obstacles, while the Bayesian value is the probability $p_{\theta }$ itself.

The obstacle detection values for “not-Obstacle” for the simulation are shown in Fig. 20. As the robot gets closer to the obstacle, the obstacle probability increases as the sensor becomes more accurate and certain, with eventual changes to false in Boolean values, and a clear decrease in $\neg Obstacle$ for hybrid Bayesian values until certainty of an obstacle for sensors in the $F$ , $FR$ , and $R$ directions is achieved at $t=300$ time steps.

In contrast to the probabilistic sensors, a collision sensor is created as a purely Boolean device named “Collision," which produces a single transition from false ( $p = 0$ ) to true ( $p = 1$ ) at the time of the collision $t=280$ time steps as shown in Fig. 21. The effect of the collision sensor is to immediately inform the robot that forward speed is not possible and reverse the motors fully to free the robot while allowing turning toward the target. The effect of a certain value of $p = 0$ or $p = 1$ on hybrid Bayesian logic is to immediately cause corresponding dependent states to become certain also, a type of “saturation” effect which can be used to quickly change the system’s response in an emergency.

Figure 21. Sensor inputs to the navigation programme: output of collision sensor. As a sensor that only registers “collision” or “no collision," only a Boolean output is considered for this sensor, though it is propagated through Bayesian logic as probabilities $p=0.0$ and $p=1.0$ . No collision occurred in the Bayesian control case for this simulation.

5.4. Decision variables and actuators

Two intermediate variables are represented by logic operations in the navigation programme to represent whether the robot can move forward, and which directions are permissible given the target direction and estimate of obstacle presence. These consist of a scalar quantity referred to as “Speed” which controls forward movement, and a quantity represented again as separate logical quantities $F$ (no turn), $FL$ (slow turn left), $FR$ (slow turn right), $L$ (fast turn left), and $R$ (fast turn right) and collectively referred to as “Direction." “Speed” implies forward movement if it is true in the Boolean case, and higher overall speed is assumed with higher speed probability in the hybrid Bayesian case. As such, the logic for speed is a disjunction of the forward, forward-left, and forward-right target directions, but is false if an obstacle collision occurs. A turning direction is assumed to be valid if both the target is in that direction, and if there is no obstacle in that direction by conjunction. The logic for these variables can be described as in Eq. (22).

(22) \begin{eqnarray} \text{Speed} &=& (\text{Target}_{FL} \vee \text{Target}_{F} \vee \text{Target}_{FR}) \wedge \neg \text{Collision} \nonumber \\ \forall \theta, \text{Direction}_\theta &=& (\text{Collision}_\theta \wedge \neg \text{Obstacle}_\theta ). \end{eqnarray}

Figure 22 shows the values of the speed variable in the simulation. In the Boolean case, immediate decisions to start and stop forward motion are made based on target direction and obstacle proximity, which causes frequent cycling between states, particularly after $t=300$ time steps as the robot approaches the goal. In the hybrid Bayesian case, a gradual increase and decrease in probability and therefore speed is seen with no cycling and only a rapid decrease in speed after turning around the obstacle at $t=480$ time steps, which is possible due to the additional information propagated through the logic by the Bayesian approach.

Figure 22. Navigation programme control results using the proposed methodology. Logically produced forward speed decision. In the Boolean case (top), only stop and go movement is possible based on two logic states with frequent cycling. The Bayesian case (bottom) produces gradual change in forward speed.

Figure 23. Navigation programme control results using the proposed methodology. Logically produced best-turning direction of travel. Using Boolean logic for the five directions (top) produces sudden judgements in the optimal direction. Using Bayesian inference on Bernoulli random variables (bottom) indicates more clearly how relatively safe each direction is at a given point in time and can be used to vary turning and movement speed with respect to the certainty provided.

Figure 23 shows the performance of the set of “Direction” states. In the Boolean logic case, more than one direction may be true at a time, and the most appropriate turning direction given a set of states is less clear, leading to less precise directional control. The robot stops for a short time between $t=178$ time steps and $t=263$ time steps due to the obstacle momentarily obscuring the forward direction, until small changes in sensor data result in the robot continuing movement forward at $t=290$ time steps and turning around the obstacle at $t=325$ time steps. In the Bayesian case, a divergence of probabilities over time results from higher certainty in both target direction sensing and obstacle presence sensing. The robot starts its turn around the object gradually at $t=150$ time steps, makes a sharp turn away from the obstacle at $t=290$ time steps and then shows smooth movement around the obstacle until aligning itself with the goal at $t=480$ time steps. It is clear in the Bayesian case how directions should be chosen, as a marginal MAP query can easily pick out the direction with highest probability.

For simplicity, the right and left motors of the robot are driven forward by a disjunction of the three left direction values ( $L$ , $FL$ , and $F$ ) and a disjunction of the three right direction values ( $R$ , $FR$ , and $F$ ) respectively so that the robot will turn toward the indicated state of “Direction." These two variables are named “Turn” and are shown for the simulation in Fig. 24. The right motor is used more heavily to turn to the left as the target is to the left and the obstacle predominantly to the right. In the Boolean case (top), coarse steering control is performed with frequent cycling during turns and near the obstacle and target. In the Bayesian case (bottom), steering is smooth for the most part to the left and right, with sudden changes at $t=290$ time steps as the robot turns away from the obstacle, and again at $t=480$ time steps as the robot aligns to travel to the target.

Figure 24. Navigation programme control results using the proposed methodology. Logic for control of left and right motors based on direction. Using only Boolean logic (top) only coarse steering is possible and a stop occurs, while using Bayesian inference can cause smooth steering without a stop.

Motor control is accomplished by creating four random variables “Left Motor Forward," “Right Motor Forward," “Left Motor Reverse," and “Right Motor Reverse” separately so as to simplify connection with other system components and stay consistent with the Boolean logic implementation. To produce a desired motor speed, the “Motor Reverse” variables are not simply defined as the negation or logical complement of “Motor Forward." The logic for these variables is defined in Eq. (23), with “Turn” implicit in $\text{Motor}F_L$ and $\text{Motor}F_R$ . The actual speed of the motors is set in the Boolean case by subtracting the logic value of “Motor Reverse” variables from “Motor Forward” variables for left and right motors, and motors are set to either $0$ if false or a fixed maximum speed of $0.2$ in CoppeliaSim if true. In the hybrid Bayesian case, motor speed is set by directly subtracting the probability values of “Motor Reverse” variables from “Motor Forward” variables. This allows information about both desired forward and reverse movement to be combined for control of each motor.

Movement of the right and left motors as shown in Fig. 25 is controlled by a conjunction of “Turn” and “Speed” variables. The “Left Motor Forward” and “Right Motor Forward” variables can be seen to closely follow the “Turn” variables. In the Boolean case (top) motor speed simply displays on-off behaviour, where frequent cycling provides coarse turning control of both motors to control the robot’s trajectory. In the Bayesian case (bottom), higher probability is mapped to higher speed as a percentage of maximum, allowing the useful effect of uncertainty to cause gradual speed increase and decrease, and also smooth turning control. Initially, a left turn is required using the right motor to avoid the obstacle due to the presence of the obstacle and the option to turn left as offered by the nonzero probability of “Target” in the $FL$ direction, but the left motor is run at lower speed due to nonzero forward movement probability. As the robot rounds the obstacle at $t=310$ time steps, the left motor is run at a higher speed and the turn angle increases as the obstacle is passed until the robot aligns with the target at $t=480$ time steps.

Figure 25. Navigation programme control results using the proposed methodology. Logically produced motor control commands for left and right motors. Using Boolean logic results in sudden changes in motor speed. Using Bayesian inference on Bernoulli random variables (bottom) speed can be controlled based on the likelihood of forward motor movement, resulting in a more complex set of behaviours.

Figure 26. Navigation programme control results using the proposed methodology. Logically produced motor control commands for left and right motors. Using Boolean logic results in sudden changes in motor speed. Using Bayesian inference on Bernoulli random variables (bottom) speed can be controlled based on the likelihood of forward motor movement, resulting in a more complex set of behaviours.

Reversing of the motors to cause backward robot movement is shown for the simulation in Fig. 26 is controlled very simply, as a disjunction of “Direction” on the same side as the motor to cause fast turns toward a target, and “Collision” to the left motor only so as to turn the robot away from an obstacle in front quickly if a collision occurs, as is seen in the Boolean case at $t=280$ time steps.

(23) \begin{align} \text{Motor}F_L & = \text{Speed} \wedge (\text{Direction}_R \vee \text{Direction}_{FR} \vee \text{Direction}_F) \nonumber \\ \text{Motor}F_R &= \text{Speed} \wedge (\text{Direction}_L \vee \text{Direction}_{FL} \vee \text{Direction}_F) \nonumber \\ \text{Motor}B_L &= \text{Collision} \vee \text{Direction}_L \nonumber \\ \text{Motor}B_R &= \text{Collision} \vee \text{Direction}_R \end{align}

6.1. Consideration of uncertainty in behaviours

Probabilistic information available at every stage of processing effectively allows decisions to be made based on a continuum of uncertainty in states that can be interpreted in a variety of ways, including the variation of actuator speed, maximum a priori estimation of states, and to facilitate smooth transitions between states. Motor speed control allows more responsive movement based on the information in Figs. 25 and 26.

6.2. More complete information about states

Thresholded logic can cause potentially useful state information to be overlooked. In Fig. 20, a fixed threshold determines how far an obstacle must be to be detected in the Boolean case, while the probabilistic sensor model allows distance information to affect decisions about motor speed in Figs. 25 and 26. Any logic that depends on the quantity thresholded (e.g., motor speed) will lose the capacity to incorporate uncertainty.

6.3. Combination of definite and probabilistic quantities

Figures 25 and 26 show that smooth state transitions of the target and obstacle sensors cause a similarly smooth response, but sudden changes in probability value such as that which occurs at time step $t=480$ or changes in a Boolean state variable such as the collision sensor can cause similarly immediate state transitions in Bayesian logic. This is useful in a robot as it allows the system to retain desirable properties of Boolean logic such as immediate response to definite quantities in time-critical situations, while allowing the flexibility of continuous interpretation characteristic of Bayesian inference.

6.4. Probabilistic data fusion from multiple inputs

Fusion of information into multiple-input logic such as the three direction states used in an OR gate configuration to control turning allows fusion of probabilistic information into each of these inputs while deemphasizing information from individual inputs using the concept of hybrid probabilistic logic. This facilitates mediation at the logic level between the direction probabilities for three directions at once in Fig. 23 to produce a single turning probability in Fig. 24.

6.5. Rejection of relatively small probability changes and state oscillation

Figure 23 and all related Figures show how relatively small changes in sensor data can cause fast cycling of Boolean states while the obstacle or target moves in and out of thresholded sensor arcs, and can also cause a full stop of the robot if the logic of “Target” is fully negated by low states of “not-Obstacle.” While the robot successfully can navigate using this information, it is undesirable as it causes oscillating stop-start movement that must be damped out with additional logic or programming so as to not cause mechanical strain on a physical robot. It also increases the chance that logic may be put in an unrecoverable state in a hard-to-find corner case of operation, immobilizing the robot. Using hybrid Bayesian inference, state changes are smooth as long as the input variable probabilities are continuous and not Boolean in nature, and the robot will in general continue moving as long as probability values do not saturate at $1$ or $0$ .