# Trans-Boolean Cyber Physical Systems

Francisco Neves, Raul Rato, Manuel Ortigueira

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Francisco Neves, Raul Rato, Manuel Ortigueira. Trans-Boolean Cyber Physical Systems. 14th Doctoral Conference on Computing, Electrical and Industrial Systems (DoCEIS), Jul 2023, Monte Da Caparica, Portugal. pp.290-300, 10.1007/978-3-031-36007-7_22. hal-05473080

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# Trans-Boolean Cyber Physical Systems

Francisco Neves(✉), Raul Rato, and Manuel Ortigueira

School of Science and Technology, UNINOVA – CTS and LASI, Nova University of Lisbon,
2829-516 Caparica, Portugal
fs.neves@campus.fct.unl.pt, {r.rato,mdo}@fct.unl.pt

**Abstract.** This paper discusses the interplay between discrete time signal processing and symbolic computation under a general Turing Machine approach in the Cyber Physical systems framework. Discrete signals and symbols are analyzed for their Turing computing capabilities using either Kolmogorov axioms or quantum probability amplitudes. It analyzes the discrete time Trans Boolean gate $\sqrt{NOT}$ and its impact on conventional computational theories used for embedded systems and Internet of Things.

**Keywords:** Signal · Symbol · Qubit · Trans-Boolean · Turing · Computation · Cyber Physical Systems

## 1 Introduction

The integration of Cyber Physical (CP) systems into digital networks has played a crucial role in enabling embedded systems, where systems operate using a common digital computational framework that enables them to interact with peripheral devices. These embedded system applications rely both on software and hardware which is designed within the Turing Machine (TM) framework [1–3]. TMs are symbolic processors in the sense that operate and manipulate symbols belonging to a finite set Σ usually called Alphabet. It is a remarkable fact that the computational power of those TM devices does not depend on the Alphabet’s cardinality [4–6]. As a result of this fact, practical reasons led to the generalized use of computational machines based on two symbol alphabets. The so called and now ubiquitous digital binary Boolean (BB) systems do fully embody any theoretical TM.

However, Feynman’s quantum computational model proposed in [7] is capable of computing functions that cannot be effectively carried on by any BB or equivalently by any alphabet-based computing device. This paper uses the expression Trans-Boolean function (TB) for non-BB computable functions. Mind that Deutsch [8] showed that any classic Boolean computation can be performed by a quantum computer. This expresses the fact that any BB computation can be performed by a TB system.

In this paper we show that there is at least one trans-Boolean computation, the so called $\sqrt{NOT}$, that no BB system is able to compute, either effectively or with an exponential slowdown. A word of caution is in order. Anyone acquainted with the real
number system knows that $\pi$ or $\sqrt{2}$ are irrationals that no fraction is able to represent. That does not diminish the practical importance of such irrational numbers, albeit all our practical computations are performed with rational numbers. The discovery of numbers like $\pi$, $\sqrt{2}$ showed concrete and specific differences between the rational number system and the real number system.

The presented perspective reveals that there is at least one concrete and specific difference between any BB system and at least one TB system where the later can compute the $\sqrt{\sqrt{\text{NOT}}}$ function which the former is unable to do. Nevertheless, as in the rational vs real number case, a BB system can be used to compute $\sqrt{\text{NOT}}$ up to a desired precision. This is the basis for the practical proof of concept put forward by Filipe Velasques in [9]. That work was the driving force behind this theoretical endeavor.

This computational nonequivalence has major consequences for embedded systems. Not only it spawns deep theoretical reasons for researching computational devices beyond the current digital paradigm given quantum computing paradigm is a reversible one, it also enables the use of the Landauer computational limit [10, 11] as a design goal for the research of efficient and novel computational systems both for Cyber-Physical systems and general applications.

This paper explores the possibilities of Quantum Computation within the framework of Discrete Time Signal Processing (DSP) Theory. Since both theories make extensive use of Hilbert spaces [12, 13] our research questions whether it is possible to use the algebraic signal processing theory approach [14, 15] to implement reversible logic and quantum computations in order to achieve efficient and novel computational systems both for CP systems and general applications, unlocking the potential for faster computations and improved communication devices [16, 17].

The second section of this paper relates our research to Connected Cyber Physical Spaces and emphasizes the importance of a broader symbolic conceptualization for embedded systems. The third section reviews the interplay between of signals and symbols, while the fourth section showcases the trans-Boolean function $\sqrt{\text{NOT}}$. The fifth section discusses the non-equivalence between the conceptual computational framework of Turing machines and trans-Boolean systems.

# 2 Signals and Symbols in Cyber Physical Systems

With the emergence of advances in CP systems, such as IOT and Artificial Intelligence (AI), human connection to digital spaces has become ubiquitous across all daily activities [18]. The increasingly complex and intelligent systems available today enabled the advent of elaborated and efficient collaborative networks empowered by a diversity of computing and communications technologies distributed through the physical world [19, 20].

## 2.1 Signals in Cyber Physical Systems

At the deepest level all CP systems interact by way of physical phenomena. These are formalized as band limited signals and regarded as inputs and outputs of the system
[21–23]. It is well known that any band limited signal can be described by a discrete time signal.

A discrete time signal is an ordered value sequence associated with a phenomenon deemed of interest [24]. The numerical value assigned to each element stands for a quantified assessment of the phenomenon. Order is set by at least one independent variable, typically time which assigns a common coordinate system to the phenomena under study [25].

Kindred physical facts generate sets of related signals. These can be detected as events or classified as states. In CP systems the detection or classification procedure starts with a discrete signal whose values can sometimes be called observations or measurements. This signal is assumed to consist of relevant components embedded in more or less noise. The CP system processes the aforementioned signal to decide whether or not either a particular event has occurred or a state is present. Many current DSP techniques and concepts, such as the matching filter, are intrinsic to the methods used to detect or classify signals in the presence of noise.

## 2.2 Symbols in Cyber Physical Systems

The concept of a symbol is a useful abstraction for developing CP systems and conceptualizing real-world events or states. It is extensively used in the design of CP systems. Each and every symbol is a physical fact, as expressed by Patee: “Even the most abstract symbols must have a physical embodiment” [26].

In the abstract mathematical communication theory realm it is assumed that there exists an information source that supplies symbols with meaning not related to its representation or physical embodiment but whose collection is limited by the number of available signals [27–29]. In CP systems information sources are physical process, where symbols emerge from signal detection procedures [30, 31].

## 2.3 Trans-Boolean Functions in Cyber Physical Systems

Among the main concerns in the design of embedded systems are the security and interference within the network, with artificial intelligence being the main source of leaked private information [32]. These systems need extensive raw data storage and their security is limited by encryption technology. Attacks over these networks can directly interfere with the communication path between sensors and actuators, potentially falsifying data, and altering the behavior of the network.

In the past few decades, quantum computing has been considered a possible solution or threat to secure communications due to the intrinsic properties of quantum measurements [33]. There are algorithms that can jeopardize our current encryption methods due to the computational speedup provided by quantum computing, enabling fast factorization without the exponential slowdown [34]. On the other hand, these algorithms can be used to speed up computations, thus significantly improving the performance of artificial intelligence and machine learning algorithms.

Quantum computing also offers new encryption methods such as the Quantum Key Distribution algorithm that relies on the physical properties of entanglement to ensure
secure communications in digital networks with a focus on either computational speed or security levels [35]. However, the inherent quantum and classical information theory that encompasses both models are a symbolic abstraction from the physical phenomena [33].

The arithmetic properties of qubits follow axioms established in the Hilbert Space and require operators to be defined by reversible logic [36]. This conceptualization, which is shared by signal processing theory, allows for the implementation of functions with decreased power consumption enabled by reversible logic [37]. The design of signal processing units that allow for low power consumption can reduce the environmental impact and the cost of embedded systems. Low power dissipation is a major constraint in reducing the size of devices, such as the size of transistors and batteries. Thus, power management is significantly important in increasing the battery life of each device in digital networks, and the consequent dependency on traditional power sources [38]. Our research focuses on a comprehensive computations based on trans-Boolean functions for Cyber Physical Systems. Incorporating trans-Boolean function and reversible logic into computational theory is a promising field of research that may bring valuable contributions to embedded systems.

## 3 The Interplay Between Signals and Symbols

It is well known that CP systems interact with humans, the environment and in between themselves. These interactions must be properly addressed, not only to develop good applications both at the software and hardware level, but also to optimize resources and reduce waste.

It is possible to assume the perspective that CP systems interactions are symbol based. This is the TM perspective, where either digital hardware design or software development is BB based. It ignores all signal processing subtleties since the final abstraction detail is at the bit level. There is no need to care how bits are implemented. For CP systems this is good, practical and straightforward, specially when dealing with digital information.

It is also possible to assume the perspective that CP systems interactions are signal based. This is the DSP perspective, where signals must be processed and detectors must be used in order to symbols to emerge. There is the need to care how bits are implemented. This is also good, practical and straightforward, specially when translating information to and from the digital CP systems realm.

As signals and symbols are considered disparate and it is also assumed that the operations appropriate for one realm have the same computational power those ones in the other, either perspective is used according to the problem at hand practicalities, since all problems are deemed computationally equivalent in the two realms. Both conceptualizations are interdependent and commonly utilized in CP systems to establish a connection between the physical environment and the relevant digital framework.

Existing a difference in computational power between the realms then some operation can be performed in one that will have no counterpart in the other. This will have a practical impact for all CP project designs, since some problems can be solved only in one realm. Moreover, there can be instances where there will be solutions for problems that seem unsolvable with current technology and know-how.
Another word of caution is in order. We have already addressed the rational/real number issue. Now it is necessary to address the ink/write issue. We need ink for writing characters, digits, or other symbols, which we can use to write about ink. But we all know that the writing is not the ink, even if it is about ink. We cannot manipulate ink just by writing. But we can write just by manipulating ink. In the same way we need signals to produce symbols, which we can use to compute and process a signal. Those symbols are not the signals. We cannot change the signal just with symbol manipulation and processing. But we do change symbols with just and only signal processing and manipulation.

<figure><img src="image_4.png" /><figcaption>Fig. 1. Event to decision model.</figcaption></figure>

DSP systems are formalized with theories and axioms emanant from vector spaces theory that differ from those in the symbolic computation formalization, emanant from the TM theory.

**Communication: From Symbol to Signal to Symbol.** An adequate approach to this problematic makes extensive use of sets. It is the event to decision model within a signals and symbols framework, as depicted in Fig. 1.

**Computation and Signals.** For most purposes, signals can be deemed as vectors [39]. Thus, vector space theory is a suitable tool for signal analysis. As an abstract entity, a vector is a descriptive tool that assigns arithmetic properties to the physical processes under study.

**Computation and Symbols.** A symbolic processor executes functions without the need to specify the physical operations underlying them. The communication channel established between the symbolic processor and the signal detector is considered to be an identity function with no additional noise in the system. In the TM conceptualization, the head scans for a symbol and either finds a corresponding symbol or discards the obtained value and imprints a new one on the tape. Boolean functions describe the possible transformations that occur at a symbolic level in the channel. Nevertheless, it's worth noting that trans-boolean transformations can also occur as signals are the physical embodiment of symbols.
## 4 Trans-Boolean Computations

The methods use for measurement and classification by a DSP system converts a discrete variable into a symbol. If the observable is deterministic, the estimator is simply a classifier which transforms a numerical input signal into a symbol. If the observable is probabilistic as in quantum computing theory, then the mapping realized in the transition from qubit to symbol converts a continuous variable randomly chosen into an estimation of the symbolic values. Since signal processing and quantum computing theory both give extensive use of Hilbert Spaces then both concepts can abstractly be used interchangeably though differing in meaning [40].

Although the measurements are probabilistic in nature, it is possible to define a composition of functions that is deterministic instead. Let us first consider the set of all possible Boolean transformations of one argument, as given in Table 1. The variables $x$ and $y$ represent the input and output of the table, respectively. This truth table corresponds to the Cartesian product between the symbol 1 and 0 with the possible symbolic transformations given by the relations $f_0(y = \vec{F})$, $f_1(y = x)$, $f_2(y = \bar{x})$ and $f_3(y = \vec{V})$.

<table><caption>**Table 1.** Boolean Functions of One Argument.</caption><thead><tr><th>x</th><th>y0</th><th>y1</th><th>y2</th><th>y3</th></tr></thead><tbody><tr><td>0</td><td>0</td><td>0</td><td>1</td><td>1</td></tr><tr><td>1</td><td>0</td><td>1</td><td>0</td><td>1</td></tr></tbody></table>

Let us map the truth values $\vec{V}$ and $\vec{F}$ to a Euclidean vector space. These symbols are arbitrarily represented by the orthogonal vectors $[0 \ 1]^T$ and $[1 \ 0]^T$, respectively. The function has a one-to-one mapping if $p = 1 - q$, $\forall p, q \in \{0, 1\}$ for the generic vector $[p \ q]^T$. The matrix representation of each function $f_i$ is given by the outer product between the input and output vector. Since the input and output vectors are orthogonal, each function $f_i$ can be represented by $M_i$ with $\langle x | y_i \rangle = x^T M_i y_i = 0$:

$$f_0 = \begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix}, \quad f_1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \quad f_2 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \quad f_3 = \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}. \qquad (1)$$

Let us consider that each function represents an independent event with a probability $p_o = P(f_i)$ of occurring. The possible transformations that can occur with given probability are represented in Fig. 1. Suppose $i = 1,2,3,4$ are chosen in order for $f_i$ to generate the NOT function as a composition of two functions belonging to the set $Y$. Thus, the elements of the set $F$ will be chosen from the set $\mathbb{R}$.

Let $p$ and $q$ belong to the set of real positive values, each representing the probability of obtaining a logic vector as a truth or false vector, respectively. While preserving the condition $p + q = 1, \forall p, q \in \mathbb{R}^+$, each function $f_i$ sets the transition matrix as:

$$ \begin{aligned} p_{00} &= p_{01} = 0 \rightarrow f_0, & p_{01} &= p_{10} = 0 \rightarrow f_1. \\ p_{00} &= p_{11} = 0 \rightarrow f_2, & p_{10} &= p_{11} = 0 \rightarrow f_3. \end{aligned} \qquad (2) $$
To achieve a deterministic output, it is necessary to force the probability of a given pair of transitions to be zero, which does not represent a random event. In the classical theory of transition probabilities, the probability of correctly reading a symbol is determined by the successive composition of functions. This composition yields:

$$ 
\begin{bmatrix} p' \\ q' \end{bmatrix} = \begin{bmatrix} p_{00} & p_{01} \\ p_{01} & p_{11} \end{bmatrix} \cdot \begin{bmatrix} p_{00} & p_{01} \\ p_{01} & p_{11} \end{bmatrix} \cdot \begin{bmatrix} p \\ q \end{bmatrix} = \begin{bmatrix} p_{00}^2 + p_{01}p_{10} & p_{00}p_{10} + p_{10}p_{11} \\ p_{00}p_{10} + p_{10}p_{11} & p_{11}^2 + p_{01}p_{10} \end{bmatrix} \cdot \begin{bmatrix} p \\ q \end{bmatrix} \\
\begin{bmatrix} p' \\ q' \end{bmatrix} = \begin{bmatrix} P_{00} & P_{01} \\ P_{01} & P_{11} \end{bmatrix} \cdot \begin{bmatrix} p \\ q \end{bmatrix}. \qquad (3)
$$

with,

$$ 
\begin{align} 
P_{00} &= p_{00}^2 + p_{01}p_{10}, & P_{01} &= p_{01}p_{11} + p_{01}p_{11}. \tag{4} \\ 
P_{11} &= p_{11}^2 + p_{01}p_{10}, & P_{10} &= p_{00}p_{10} + p_{10}p_{11}. \nonumber 
\end{align} 
$$

Since the possible values for each $p_{ij}$ belong to the set of real numbers, the metric space $L_1$ is insufficient to define a transformation that when applied twice, it results in the NOT transformation. Let us assume the Hilbert vector space $L_2$ is sufficient to describe the $\sqrt{\text{NOT}}$ transformation. By setting $\alpha = |p|l^2$ and $\beta = |q|l^2$, with $\alpha, \beta \in \mathbb{C}$, there is a function $f' = \sqrt{f_3}$.

$$ 
\begin{align} 
C_{00} &= c_{00}^2 + c_{01}c_{10} = 0, & C_{01} &= c_{01}c_{11} + c_{01}c_{11} = 1. \tag{5} \\ 
C_{11} &= c_{11}^2 + c_{01}c_{10} = 0, & C_{10} &= c_{00}c_{10} + c_{10}c_{11} = 1. \nonumber 
\end{align} 
$$

By setting $c_{00} = c_{11}$, the possible solutions for the equation system are symmetric, with the square root of the function given by:

$$ C_0 = \sqrt{-c_1^2} = ic_1, \quad c_1 = \pm \frac{1 \mp i}{2}. \qquad (6) $$

The resulting matrix belongs to the SU(2) group, in which any combination of possible roots results in $f' = f_3$. The obtained amplitude of probabilities for both truth values is given by:

$$ 
\begin{bmatrix} p' \\ q' \end{bmatrix} = \pm \frac{1}{2} \begin{bmatrix} (1 \pm i) & (1 \mp i) \\ (1 \mp i) & (1 \pm i) \end{bmatrix} \cdot \begin{bmatrix} p \\ q \end{bmatrix} = \pm \frac{1}{2} \begin{bmatrix} p \cdot (1 \pm i) + q \cdot (1 \mp i) \\ p \cdot (1 \mp i) + q \cdot (1 \pm i) \end{bmatrix}. \qquad (7) $$

This function is a possible wave function $\Psi$ and it is typically represented as:

$$ \Psi = \pm \frac{1 \pm i}{2} |\ddot{V}\rangle \pm \frac{1 \mp i}{2} |\ddot{F}\rangle \qquad (8) $$

where $|\ddot{V}\rangle$ and $|\ddot{F}\rangle$ represent the state vectors for the binary values of 1 and 0, respectively. The output vector of the $\sqrt{\text{NOT}}$ function becomes $\pm \frac{1}{2}[-i i]^T$, which has a many-to-one mapping from qubits to boolean bits.

The act of measurement becomes a decision problem in the vector space [41] with the probability of obtaining the truth symbol given by $|\alpha|l^2$.

The quantum model highlights the possibility to implement trans-Boolean functions through reversible logic. A symbol is the result of a conceptual decision process that occurs at the instant of measurement. It is not a requirement to set a one-to-one mapping between the possible outcomes and the possible binary boolean values.
# 5 Discussion

The hierarchy of abstraction levels enables computational and communication theories to be developed without the need to define further the notion of symbol [42]. In [1], the Turing machine scans a symbol from a tape and executes an abstract process, also known as computation. The physical execution of the computational process is deemed irrelevant to the symbolic operations performed by the machine. However, it must be implemented as a physical event culminating in a result mapped into a symbol by a categorical decider.

The computational model of a classic Turing machine can only accept or compute a finite alphabet of symbols to represent the natural numbers field without errors. The real numbers that can be computed through a Turing machine are calculated by boolean functions. In [43], the quantum model is able to compute the set of all boolean functions and additional trans-Boolean functions. However, in the measurement act, the qubit value at the measurement instant is mapped into the same conceptualization as a regular TM. Thus, the computable numbers by a Quantum Turing Machine are equivalent to a classic Turing machine [44].

A trans-Boolean function is a relation that can not be represented by a truth table. The decision problem of trans-boolean operations in a quantum machine is said to invalidate the computation when a measurement takes place [34]. Since a one-to-one mapping between the two machines cannot be established, the decision procedure is capable of producing stochastic sequences of symbols [8]. If a “Gedanken-experiment” is applied to test for the equivalence between the two machines, nothing could be concluded from the measurements as there are no random procedures between two equivalent machines [45].

The approach taken in [46] proves that any numeration of sets of output symbols can be done either by a deterministic machine and a probabilistic machine, if and only if the probability of the transition between states is a computable real number. In Quantum computing theory, the $\sqrt{NOT}$ has complex amplitude of transition probabilities which has a many-to-one mapping to the real valued probabilistic machines.

Furthermore, the function $\sqrt{NOT}$ is not computable by a Turing machine as the law of excluded middle does not accept partial symbol [47]. The probabilistic approach is also insufficient to describe this trans-Boolean function as the probability for each output symbol is perpetually mixed in further computations. Thus, quantum mechanics provided us a trans-Boolean function that is not realizable with the axioms of Boolean operations of a vector space, yet it describes a physical event that takes place independently of the abstraction theory.

When dealing with amplitude probabilities, the act of measurement maps a $L_2$ space with an unknown choice procedure to a binary value that can become a deterministic process depending on the composition of applied functions. There is no binary boolean function that is able to compute the $\sqrt{NOT}$ function. Thus, the quantum machine is equipped with an extended set of trans-boolean functions that includes the Turing conceptualization of computable numbers.
# 6 Conclusion

This paper demonstrated that there exists at least one trans-Boolean computation that no binary Boolean system can compute. This concrete difference has significant implications for the design of Cyber-Physical Systems. The √ NOT is an example of a computations that is not accessible to embedded systems.

A TB system makes extensive use of reversible logic to implement functions in the Hilbert vector space conceptualization, resulting in greater energy efficiency compared to current BB systems. These systems are also capable of implementing new cryptography algorithms that can enhance security in communication channels.

Future research will focus on investigating the potential applications of TB functions in embedded systems under the algebraic signal processing framework.

**Acknowledgments.** The authors would like to thank professor Dr. Luís Camarinha-Matos for the useful instructions. This work was supported in part by the Portuguese FCT program UIDB/00066/2020.

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