# How Hard Is It to Detect Surveillance? A Formal Study of Panopticons and Their Detectability Problem

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## Abstract

**:**

## 1. Introduction

## 2. Definitions and Notation

**Definition**

**1.**

**The Halting Problem**

**Input:**A string $x=<M,w>$ which is actually the encoding (description) of a Turing machine <M> and its input w.

**Output:**If the input Turing M machine halts on w, output True. Otherwise, output False.

**Definition**

**2**

## 3. The Panopticon Detection Problem and Our Approach

**Definition**

**3**

**Definition**

**4.**

**The Deductive Panopticon Detection Problem 1**

**Input:**A description of a Turing machine (program).

**Output:**If the input Turing machine operates like a Panopticon according to Definition 3 output True. Otherwise, output False.

**The Deductive Panopticon Detection Problem 2**

**Input:**A description of a Turing machine (program).

**Output:**If the input Turing machine operates like a Panopticon according to Definition 4 output True. Otherwise, output False.

**Theorem**

**1.**

**Theorem**

**2.**

## 4. Impossibility of Detecting Behavioural Panopticons

**Theorem**

**3.**

**Proof.**

## 5. Impossibility of Detecting Deductive Panopticons

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

- (Simulation A) over all the t${q}_{?}$-${q}_{y}$ queries in the examined computation of ${M}^{{S}_{1}^{\prime}}$, trying to locate a string accepted by a queried Turing Machine ${T}_{i}$ and one of the Turing Machines ${M}_{1}^{1},{M}_{2}^{1},\cdots ,{M}_{k}^{1}$ in ${S}_{1}^{\prime}$, and
- (Simulation B) over ${M}^{{S}_{1}^{\prime}}$ and the Turing Machines ${M}_{1}^{2},{M}_{2}^{2},\cdots ,{M}_{k}^{2}$ in ${S}_{2}^{\prime}$, with the same input w, trying to discover whether w, which is accepted by the examined (by ${M}^{{}^{\prime}}$) computation of ${M}^{{S}_{1}^{\prime}}$, if valid, is, also, accepted by one of the Turing Machines ${M}_{1}^{2},{M}_{2}^{2},\cdots ,{M}_{k}^{2}$ in ${S}_{2}^{\prime}$.

## 6. Impossibility of Proving Panopticon Status within Formal Systems

**Theorem**

**6.**

**Theorem**

**7.**

**Proof.**

## 7. Philosophical Remarks on Panopticon Detectability

#### 7.1. Detectability of Panopticons by Their Thermal Emissions

#### 7.2. Infinite Time Turing Machine Hypercomputation Model

#### 7.3. Detecting Deductive Panopticons with Ittm Computation Model

**Definition**

**5.**

- 1.
- The representation theorem: For any relation R, R is in the arithmetical hierarchy iff R is definable in elementary arithmetic.
- 2.
- The strong hierarchy theorem:
- $R\in {\Delta}_{n+1}^{0}$ iff R is recursive in ${\varnothing}^{(n)}$ (i.e., the nth jump of the empty set).
- $R\in {\mathrm{\Sigma}}_{n+1}^{0}$ iff R is recursively enumerable in ${\varnothing}^{(n)}$.
- $R\in {\mathrm{\Pi}}_{n+1}^{0}$ iff R is recursively enumerable in ${\varnothing}^{(n)}$.

## 8. Discussion and Directions for Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Bentham, J. Panopticon or the Inspection House. Written as a Series of Letters in 1787. Available online: https://www.ics.uci.edu/~djp3/classes/2012_09_INF241/papers/PANOPTICON.pdf+ (accessed on 15 January 2022).
- Foucault, M. Discipline and Punish: The Birth of the Prison; Random House: New York, NY, USA, 1977. [Google Scholar]
- Cohen, F. Computer Viruses. Ph.D. Thesis, University of Southern California, Los Angeles, CA, USA, 1985. [Google Scholar]
- Cohen, F. Computer Viruses: Theory and Experiments. Comput. Secur.
**1987**, 6, 22–35. [Google Scholar] [CrossRef] - Turing, A.M. On Computable Numbers, with an Application to the Entscheidungs problem. Proc. Lond. Math. Soc.
**1936**, 2, 230–265. [Google Scholar] - Turing, A.M. Systems of logic based on ordinals. Proc. Lond. Math. Soc.
**1939**, 45, 161–228. [Google Scholar] [CrossRef] [Green Version] - Hopcroft, J.; Ullman, J.D. Introduction to Automata Theory, Languages, and Computation; Addison-Wesley Series in Computer Science; Addison-Wesley: Boston, MA, USA, 1979. [Google Scholar]
- Hartmanis, J.; Hopcroft, J.E. Structure of undecidable problems in automata theory. In Proceedings of the 9th Annual IEEE Symposium on Switching and Automata Theory (SWAT 1968), Schenedtady, NY, USA, 15–18 October 1968; pp. 327–333. [Google Scholar]
- Hartmanis, J.; Hopcroft, J.E. Independence results in computer science. ACM Sigact News
**1976**, 8, 13–24. [Google Scholar] [CrossRef] - Bennett, C.H. The Thermodynamics of Computation—A Review. Int. J. Theor. Phys.
**1982**, 21, 905–940. [Google Scholar] [CrossRef] - Hamkins, J.D. Infinite Time Turing Machines. Minds Mach.
**2002**, 12, 521–539. [Google Scholar] [CrossRef] - Evans, D. Introduction to Computing: Explorations in Language, Logic, and Machines; Eleven Learning: Delhi, India, 2011. [Google Scholar]
- Davis, M. The Universal Computer: The Road from Leibniz to Turing, 3rd ed.; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Post, E.L. Formal reductions of the general combinatorial decision problem. Am. J. Math.
**1943**, 65, 197–215. [Google Scholar] [CrossRef] - Post, E.L. Recursively enumerable sets of positive integers and their decision problems. Bull. Am. Math. Soc.
**1944**, 50, 284–316. [Google Scholar] [CrossRef] [Green Version] - Post, E.L. Degrees of recursive unsolvability: Preliminary report (abstract). Bull. Am. Math. Soc.
**1948**, 54, 641–642. [Google Scholar] - Kleene, S.C.; Post, E.L. The upper semi-lattice of degrees of recursive unsolvability. Ann. Math.
**1954**, 59, 379–407. [Google Scholar] [CrossRef] - Rice, H.G. Classes of Recursively Enumerable Sets and Their Decision Problems. Trans. Am. Math. Soc.
**1953**, 74, 358–366. [Google Scholar] [CrossRef] - Bennett, C.H. Logical Reversibility of Computation. IBM J. Res. Dev.
**1973**, 17, 525–532. [Google Scholar] [CrossRef] - Leff, H.; Rex, A.F.; Hilger, A. (Eds.) Maxwell’s Demon: Entropy, Information, Computing; Princeton University Press: Princeton, NJ, USA, 1990. [Google Scholar]
- Abbott, A.A.; Calude, C.S.; Svozil, K. On Demons and Oracles. In Asia Pacific Mathematics Newsletter; World Scientific: Singapore, 2012; Volume 2. [Google Scholar]
- Aoun, M.A. Advances in Three Hypercomputation Models. Electron. J. Theor. Phys. (EJTP)
**2016**, 13, 169–182. [Google Scholar] - Davis, M. The Myth of Hypercomputation. In Alan Turing: Life and Legacy of a Great Thinker; Teuscher, C., Ed.; Springer: Berlin/Heidelberg, Germany, 2004; pp. 195–211. [Google Scholar]
- Davis, M. Why there is no such discipline as hypercomputation. Appl. Math. Comput.
**2006**, 178, 4–7. [Google Scholar] [CrossRef] - Kleene, S.C. Recursive predicates and quantifiers. Trans. Am. Math. Socl.
**1943**, 53, 41–73. [Google Scholar] [CrossRef]

**Figure 1.**The Arithmetical and the Analytical Hierarchies. (

**a**) The Arithmetical Hierarchy; (

**b**) The Analytical Hierarchy.

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**MDPI and ACS Style**

Liagkou, V.; Nastou, P.E.; Spirakis, P.; Stamatiou, Y.C.
How Hard Is It to Detect Surveillance? A Formal Study of Panopticons and Their Detectability Problem. *Cryptography* **2022**, *6*, 42.
https://doi.org/10.3390/cryptography6030042

**AMA Style**

Liagkou V, Nastou PE, Spirakis P, Stamatiou YC.
How Hard Is It to Detect Surveillance? A Formal Study of Panopticons and Their Detectability Problem. *Cryptography*. 2022; 6(3):42.
https://doi.org/10.3390/cryptography6030042

**Chicago/Turabian Style**

Liagkou, Vasiliki, Panayotis E. Nastou, Paul Spirakis, and Yannis C. Stamatiou.
2022. "How Hard Is It to Detect Surveillance? A Formal Study of Panopticons and Their Detectability Problem" *Cryptography* 6, no. 3: 42.
https://doi.org/10.3390/cryptography6030042