Genomic Intelligence as Über BioCybersecurity: The Gödel Sentence in ImmunoCognitive Systems
Abstract
:1. Introduction
2. GTP Conditions and Major Evolutionary Developments in ImmunoCognitive Systems
GTP Conditions  Adaptive Immune System (AIS)  Brain/Neuronal System  

1  GTP Encoded Genomic Basal Information in Fixed Finite Language, Recursive Function Operations on Codes for Online Machine Executions: Self–Ref (SelfReference) and SelfAssembly  
(1a) Section 2.1 and Section 2.2  Unique identifiers aka Gödel numbers (gns) from smallest unit of programs/algorithms based on encoded information, Equations (2) and (3) Notation: Set G for gene codes in Equation (2); Set A for selfactions in Equation (3) 
 Unique identifiers for single neurons and neuronneuron interaction [82,83,84] 
(1b) Section 2.2  Self–Ref (Online): Diag(x) = ${\varphi}_{x}\left(x\right)$, Online halting (↓) selfassembly program x instructs machine ϕ to run code x as its input Equations (4) and (7) [17]  Online Basal Ribosomal and RNA Machine Execution of gene codes as 3D Self Assembly of digitized materials of morphology and regulatory networks Diag(g) = ${\varphi}_{g}\left(g\right)$↓, g ∊ G. [85] (p. 30) and [4,5,86]  Online Basal SelfActions with Canonical Neurons Firing in Sensorimotor Cortex Diag(a) = ${\varphi}_{a}\left(a\right)$↓, a ∊ A. [52] 
2  G.T.P Offline Mirror Systems with OneOne Mapping of Online Machine Execution in AIS of Gene Codes and Self Action in MNS: Self–Rep (SelfRepresentation): Rogers [16], (pp. 202–204)  
(2a) Section 2.3 



(2b) 

 
3  GTP Formal System of the Other and Novel Hostile Other, Fixed Point of Gödel Liar/Negation as Gödel Sentence (See Section 3)  
(3a) 

 
(3b) 



2.1. GTP Condition (1): Encoding and Recursive Function Operations on Codes
Unique Biotic Identifiers and Gödel Numbers
2.2. G.TP Gödel Numbering of Basal Information in Gene Codes and Sensorimotor Cortex
Online SelfAssembly/SelfRef Machinery
2.3. GTP Mirror/Meta Condition 2 and Evidence from Genomic Evolution of SelfRep
3. GTP BioInformatics for V(D)J Recombination and TCell Training for NonSelf Antigen Detection
3.1. Horizon Scanning and Astronomic Numbers in AIS
 In having mirrored/expressed ~85% of gene codes in mTECs, the V(D)J recombinations generate putative clones of nonself antigens in relation to these gene codes.
 This provides “an anticipatory system of defense” [87] of prodigious capacity. Ref [88] state that the capacity of the AIS for “somatic generation of immune recognition motifs of a system of practically unlimited (openended) information capacity” with orders of magnitude of “αβT cell receptors to be around 10^{15} to 10^{20} with such levels of diversity in a single individual that exceeds the size of the entire germline genome by several orders of magnitude.” [62] gives an even higher number for the V(D)J generated “individual antigen receptors computed to be approximately 10^{30}”.
 Ref [102] (Chapter 8) ask the following question in the context of BioInspired Computing and Cyber Security:
“Any paradigm for computer security that is based on the differentiation of self from nonself must imply some operational definition of self that represents normal and benign operation. It is clear that a good definition is matched to the signature of the threat being defended against, and hence the designer must be able to answer the question, “How would I know my system were under attack?”(Ibid, p. 263)
3.2. Halting SelfAssembly Gene Codes and Forbidden Codes of Antigens in GTP Formal System
3.3. Information Processing in GTP Meta Systems for V(D)J: Positive Selection of T Cell Receptors
 (i)
 A change in the program;
 (ii)
 A change in its input;
 (iii)
 A change in both program and input.
 σ(g, f◦g): Implying change in input of the basal Diag(g) = ${\varphi}_{g}\left(g\right)$.
 σ(f◦g, g): Implying change in program of the basal ${\varphi}_{g}\left(g\right)$ which is no longer a Diag(.) operation.
 σ(f◦g, f◦g): Implying a change in both program and input for Diag(g) and transforming it to Diag(f◦g).
3.3.1. Positive Selection of T Cell Receptor (TCR) Motifs {σ(g, f◦g), σ(f◦g, g), σ(f◦g, f◦g)}
3.3.2. Synchrony in Anticipatory TCell Receptors Clone of Tissue Specific Attacker with Peripheral MHC Record of Same in Online Attack
3.3.3. Negative Selection of TCells
3.3.4. Dangerous V(D)J Codes and Successful NonSelf Antigen Attacks
3.3.5. Negative Selection of Dangerous TCells Receptors with Motifs: σ(g_{n}^{¬}, g_{n}) in (10)
3.4. Precision Engineered Novel Antibodies Made Possible Only by Gödel Sentence σ(g_{n}^{¬}, g_{n}^{¬})
3.5. GTP Bioinformatics for COVID19 Pathology and Recovery
4. Conclusions
Funding
Acknowledgments
Conflicts of Interest
References and Notes
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 Ref [10] states “The resemblance of the Turing machine to naturally occurring molecular machineries that are capable of processing information, such as the ribosome and processive DNA (and RNA) polymerase enzymes, is striking and has recently motivated researchers to start programs on the development of biomolecular computers”.
 Since at least the 1990s, there has been an intense study of the biochemistry of life processes that are deemed to be driven by a selfassembly of biotic macromolecules with mechanistic features [86,97]. The shift in focus from the autonomous nature of this selfassembly to a program for selfassembly in the key transcription and translation processes [85], such that a gene code is viewed as a program that can both build the machine that can be instructed by the same program to read and run/execute it is also present in [5].
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 Note that a machine listable set or recursively enumerable set ${\mathit{W}}_{x}$ , is the domain of the function, viz. the y as inputs, on which ${\varphi}_{x}\left(y\right)$↓ for $y\text{}\u03f5\text{}\aleph $, where ℵ is the set of integers. Thus, ${\mathit{W}}_{x}=\left\{\mathrm{y}\text{}\text{}{\varphi}_{x}\left(y\right)\text{}\downarrow ;{\mathrm{TM}}_{\mathrm{x}}\left(\mathrm{y}\right)\text{}\mathrm{halts};\text{}\mathrm{y}\text{}\u03f5\text{}\aleph \right\}$. The archetypical set C contains the domain of halting functions ${\varphi}_{x}\left(x\right)$of Diag(x) operations and hence set C contains the g.ns of those recursively enumerable sets, W_{x}, that contain their own indexes (see [17] (p. 123) and [16] (p.62)): C = $\left\{x\text{}\text{}{\varphi}_{x}\left(x\right)\text{}\downarrow ;{\mathrm{TM}}_{\mathrm{x}}\left(\mathrm{x}\right)\text{}\mathrm{halts};\text{}\mathrm{x}\text{}\u03f5\text{}{\mathit{W}}_{x}\text{}\right\}$ . The complement of C, denoted by C^{¬}, represents the “antidiagonal” set, which is different from every listable set W_{x} for all $x\text{}\u03f5\aleph $: C^{¬} = $\left\{x\text{}\text{}{\varphi}_{x}\left(x\right)\uparrow ;{\mathrm{TM}}_{\mathrm{x}}\left(\mathrm{x}\right)\text{}\mathrm{does}\text{}\mathrm{not}\text{}\mathrm{halt};\text{}\mathrm{x}\text{}\notin \text{}{\mathit{W}}_{x}\text{}\right\}$. A set A is creative if it is recursively enumerable and its complement ℵ–A is not recursively enumerable such that for any W_{n} ⊂ ℵ–A, there is a recursive function f(n)$\u03f5\text{}$ ℵ–A–W_{n}. The complement set of A is said to be productive. Thus, set C is creative and C^{¬} is productive, see also [45].
 Note when focusing exclusively on nonself antigens, a subscript p is used in f_{p}^{¬!}.
 It is well known that, by what is called the SMN Theorem or the Parameterization Theorem [16,17]), new g.ns for recursive operations on extant g.ns can be mechanically generated.
 However, it is well known that only V(D)J motifs that the mTEC selection process can deal with as those selfrepresented in the Thymic MHC viz. g$\u03f5$ G*, G* ⊂ G as given in Equation (7). Hence, at the negative selection process TCRs, especially, for example, when there is deficiency of the autoimmune regulator AIRE, restricted gene codes do not get “SelfRepped” in the mTECs for the training of the Tcell receptors [60]. So, TCRs with motifs σ(f◦g, g) for g ∉ G* in (7) can cause autoimmune disease.
 Michael Lotz, in his BioInspired ICT (BICT) 2019 Keynote, apocryphally called “Know yourself, know the enemy”, stated that, while almost 99.9% of genes are the same for humans, only 6% of Tcell repertoires of different humans are the same. Hence, while the V(D)J realizations of some can combat new pathogens because they succeeded in cloning the code for this in the Tcells, others may not due to the random generation of codes in Tcell receptors and hence, without the anticipatory capacity to identify the novel nonself antigen when it strikes a tissue online, they succumb to them.
 Rogers Fixed point Theorem [16] (Section 11.2) states that any total computable function, f_{p}^{¬!}, for the case in question, has as its fixed point an index given by an integer v such that ${\varphi}_{{f}_{p}^{\neg}!\left(v\right)}$(s) $\cong $ ${\varphi}_{v}$(s), viz. either both sides are defined and are equal or else both sides are undefined. This index for the fixed point for a total computable function f can be obtained constructively. The first part of the proof is already satisfied with the index function σ(.) in the SelfRep Theorem in (5) for the program for the partial computable function Diag(g_{n}) = ${\varphi}_{{g}_{n}}\left({g}_{n}\right)$. The next step is to assign the index g_{n}^{¬} to f_{p}^{¬!}Diag(g_{n}) which leads to the function ${\varphi}_{{\varphi}_{{g}_{n}^{\neg}}\left({g}_{n}\right)}$ with a TCR motif of σ(g_{n}^{¬}, g_{n}) as in Equations (10) and (11). The final step is to substitute g_{n}^{¬} into f_{p}^{¬!} Diag(g_{n}) to get f_{p}^{¬!} Diag(g_{n}^{¬}), which is the function ${\varphi}_{{g}_{n}^{\neg}}\left({g}_{n}^{\neg}\right)$. Then, assign v as the index for ${\varphi}_{{g}_{n}^{\neg}}\left({g}_{n}^{\neg}\right)$ = Diag(g_{n}^{¬}) = σ(g_{n}^{¬}, g_{n}), which yields ${\varphi}_{{f}_{p}^{\neg}!\left(v\right)}$(s) $\cong $ ${\varphi}_{v}$(s).
 On updating Equation (11) with the Liar/Malware strategy g_{n}^{¬} in the place holder for the other, and assuming v = ${\varphi}_{{g}_{n}^{\neg}}\left({g}_{n}^{\neg}\right)$ = Diag(g_{n}^{¬}) = σ(g_{n}^{¬}, g_{n}) is a halting computation then we have a contradiction: ${\varphi}_{{\varphi}_{{g}_{n}^{\neg}}\left({g}_{n}^{\neg}\right)}\left(s\right)$ = ${\varphi}_{\sigma \left({g}_{n}^{\neg},{g}_{n}^{\neg}\right)}\left(s\right)$ = ${\varphi}_{{f}_{p}\neg !Diag\left({g}_{n}^{\neg}\right)}\left(s\right)$ = ¬${\varphi}_{{\varphi}_{{g}_{n}^{\neg}}\left({g}_{n}^{\neg}\right)}\left(s\right)$. The last two terms follow from Equation (11) when g_{n}^{¬} is substituted for ${g}_{n}$ in ${\varphi}_{{f}_{p}\neg !Diag\left({g}_{n}^{\neg}\right)}\left(s\right)=$ ¬${\varphi}_{{\varphi}_{{g}_{n}^{\neg}}\left({g}_{n}^{\neg}\right)}\left(s\right)$.
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Markose, S.M. Genomic Intelligence as Über BioCybersecurity: The Gödel Sentence in ImmunoCognitive Systems. Entropy 2021, 23, 405. https://doi.org/10.3390/e23040405
Markose SM. Genomic Intelligence as Über BioCybersecurity: The Gödel Sentence in ImmunoCognitive Systems. Entropy. 2021; 23(4):405. https://doi.org/10.3390/e23040405
Chicago/Turabian StyleMarkose, Sheri M. 2021. "Genomic Intelligence as Über BioCybersecurity: The Gödel Sentence in ImmunoCognitive Systems" Entropy 23, no. 4: 405. https://doi.org/10.3390/e23040405