perform one calculation at a time whereas a QTM can perform multiple calculations at a time. Normal computer works by manipulating bits in which there exists two states (0 or 1), but Quantum Computers are not limited to two states. They encode information as Quantum Bits (Qubit) which exist in superposition. Qubit represent atoms, ions, photons, electrons. The quantumTuringmachine (QTM) has been introduced by Deutsch as the very first model of quantum computation. A quantumTuringmachine can be seen as a generalisation of a probabilistic Turingmachine where the probabilities which are associated with any transition are replaced by amplitudes i.e., complex numbers. The use of complex numbers leads to model fundamental phenomena of quantum mechanics like interferences and entanglement. A probabilistic Turingmachine is submitted to well-formed ness conditions ensuring that for any configuration, the probabilities of all the possible evolutions sums to one a quantumTuringmachine is submitted to similar well-formed ness conditions. These well-formed ness conditions ensures that the evolution of a quantumTuringmachine (i) does not violate the law of quantum mechanics; (ii) is reversible. This later condition can be rephrased in more physical terms as an isolation of the quantumTuringmachine from its environment during the computation. The reversibility assumption of quantumTuring machines is questionable for several reasons, including for instance the emergence of quantum computing models based on non reversible evolutions, like the one-way model or more generally the measurement-based quantum computations, which point out that a quantum computation is not necessary reversible. Moreover, the isolation assumption leads to technical issues like the impossibility to know whether a running computation is terminated or not i.e., whether the halting state is reached or not. Finally, due to the isolation assumption, quantumTuring machines are the natural quantum versions of reversible Turing machines. But the natural embedding of any reversible Turingmachine into a quantumTuringmachine cannot be extended to non- reversible and probabilistic Turing machines.
We can go on increasing the number of Registers and the number of Cohorts in each register (to compute and communicate various values). This will enable us to communicate and execute any complex operation. In this manner, we can also program K-Devices to interpret and execute smart contracts on a Quantum Blockchain (K-Chain). Such an idea can be extended to facilitate the construction of a QuantumTuringMachine (QTM), also known as a Universal Quantum Computer.
Before examining a QuantumTuringMachine for the Standard Model we will look at the features of “normal” Turing machines. A normal Turingmachine consists of a finitely describable black box (its features are describable in a finite number of statements) and an infinite tape. The tape plays the role of computer memory. The tape is divided into squares. Each square contains a symbol or character. The character can be the “blank” character or a symbol. A tape contains blank characters followed by a finite string of input symbols followed by blank characters.
In the present work, we introduce E-infinity theory in which we can, in a manner of speech, “differentiate” and “integrate” the non-differentiable and non-integrible, namely geometrical structures made up entirely from transfinite point set of higher dimensional Cantor sets -. At a minimum, this is shockingly unconven- tional at least in the first few moments. However noting the use of non-standard analysis in the context of Not- tale’s theory of scale relativity as well as similar tools used in the work of the great French mathematical phy- sicist A. Connes, the inventor of noncommutative geometry  , the situation may start appearing slowly but surely in a different light. On this optimistic note, we will give in the present paper some more detail and explanation of the essential tools of the E-infinity Cantorian spacetime trade - before embarking upon applying our theory to the major problem of the discrepancy between the measured energy density of the cosmos as compared to the theoretical expectations. We are of course not divulging prematurely any secret when we say that one of our main tools in reaching our exact result is the fractal version of Witten’s eleven dimensional theory as it is clear from the title of the present work and it goes without saying that this fractal M-theory   is as much the central piece of the work as it is the accurate determination of the ordinary and the dark energy density of this cosmos which turns out to be quite a surprise connected to what is probably the most famous equation in the history of theoretical physics, namely Einstein’s E = mc 2 where E is the energy, m is the mass and c is the speed of light -. The basic idea is that the fundamentally classical relativity theory E = mc 2 consists of two fundamentally non-classical quantum components that when added together, produce Eins tein’s beauty in the following manner: E ≅ mc 2 22 + mc 2 ( 21 22 ) = E O ( ) + E D ( ) = mc 2 where E(O) is the or- dinary energy and E(D) is the dark energy density of the cosmos -.
Differing-input obfuscation for Turing Machines: We define the notion of differing- input obfuscator for Turing machines and give a construction for Turing machines with bounded length inputs (without converting it to a circuit), assuming the existence of a differing-input obfuscator for circuits and SNARGs for P [BCCT13]. Additionally, assuming SNARKs for P [BCCT13], we can construct a differing-input obfuscator even for the setting where the length of the input is not bounded. (We stress that it is only for this extension that we need to assume SNARKs.) Moreover, our construction achieves input-specific running times (explained below). This means that evaluating the obfuscated machine on input x does not depend on the worst- case running time of the machine but just on the running time of the unobfuscated machine on input x.
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In this section, first, we expressed intuitively how to create an interpreter machine that can receive and simulate all possible gFSA. The notion is put forward intuitively in the sense that a Turingmachine is created without introducing gFSA encoding into an alphabet and then fed into a Turingmachine. Our goal is to make it easier to introduce this Turingmachine. Secondly, we wrote it in a formal form with the input encoding of the Turingmachine so that it meets a formal definition. The construction of a Turingmachine that can accept all of the gFSA is expressed intuitively as follows:
It has a one-way infinite read/write tape, a head that it uses for reading and writing on the cells of the tape and a register that holds the state of the machine. A single computation step is described by one application of the transition function: it reads a symbol in a specific state, and then writes down a symbol on that location, gets into some state and the head moves to the left, right or stays on that location.
We note that the measure of width complexity in the case of circuits is related to the measure of space complexity in the case of Turing machines. Indeed, we can transform a Turingmachine M that requires space s on inputs of length n into a circuit of width O(n +s); similarly, a circuit of width w can be simulated by a Turingmachine which takes space at most O(w). Moreover, there are actually major similarities between the security proofs of [HJO + 16] (for their width- dependent adaptive garbling scheme) and [BGL + 15] (for their space-dependent succinct garbling
Unfortunately the depth of this idea has been obscured by the interpretation of the test given by John Searle in his mental experiment of the Chinese Room. Searle’s challenge to artifi- cial intelligence was exactly a critique of the concept of “sym- bol manipulation”, considered literally as working with sym- bols detached from any real interaction with the environment. In Searle’s mental experiment an English speaker has some instructions in English to take Chinese symbols as input and is to give some other symbols as output; the rules in English per- mit the English speaker to produce as output reasonable an- swers to the questions in Chinese. A Chinese speaker therefore would understand the answers to her questions produced by this procedure, thinking that whoever is inside the room under- stands Chinese. It is apparently a rhetorical presentation of the Turing test with the aim of depriving the test of its significance. In fact, Searle asks whether we can say that the man in the room understands Chinese. Certainly not! He understands Eng- lish, and is able to use rules (formal or not) to give as output some patterns of Chinese symbols as answers to other patterns of Chinese symbols taken as input, without having any idea of what those symbols may mean. Symbol manipulation is not understanding language! What is missing is the understanding on the meaning of the Chinese symbols and the intentionality, that is the ability to understand what a symbol refers to. The English speaker inside the room has no idea what the Chinese symbols refer to; he only knows how to manipulate symbols, he is only using a syntactic ability without semantics.
Indistinguishability Obfuscation. Constructing iO for TMs given miFE for TM is straightforward, and adapts the miFE to iO circuit compiler by [GGG + 14] to the TM setting. As in the circuit case, an miFE for TM that supports two ciphertext queries and single key query suffices for this transformation. Please see Section 5 for details. Since our security proof for miFE for TM is tight, this compiler yields iO for TM from sub-exponentially secure FE for circuits rather than sub-exponentially secure iO for circuits. Organization of the paper. The paper is organized as follows. In Section 2 we provide the definitions and preliminaries used by our constructions. In Section 3, we provide our construction for single input FE for Turing machines. In Section 4, we provide our construction for multi-input FE for Turing machines for any fixed arity k and in Section 5 we describe the construction of iO for Turing machines for bounded inputs. Our constructions use constrained PRFs which are instantiated in Appendix D and decomposable FE which is constructed in Appendix F.
The Turing test extends this polite convention to machines: don’t ask the question "can M think?" If a machine acts as intelligently as a human being, then it is intelligent and it should lay aside the mystery about consciousness for the moment. Turing declared in  that "I do not wish to give the impression that I think there is no mystery about consciousness.""But I do not think these mysteries necessarily need to be solved before we can answer the question (Can machines think?) with which we are concerned in this paper." Turing test's idea is the correct basic concepts of AI and is consistent with the approximate separability hypothesis of the whole and the definition of AI. On today's vibrant internet, chatbots, which serve customers, have exceeded Turing's expectations in their ability to talk to people.
I would like to warmly thank Joel Hamkins for suggesting to me the idea of doing Kleene’s O in the infinite time Turingmachine setting; for providing guidance and ideas – many of the results reported below were anticipated by him; for reading and commenting extensively on several earlier drafts of this thesis; finally, but not least, his great interest in my work all along the way, and his vitality and passion for mathematics, was an indispensable source of inspiration. Further, I would like to thank Benedikt L¨ owe for letting me change thesis topic at a late stage; for providing guidance in my individual study projects during my time at the ILLC; for spawning my interest in infinite time Turing machines, and for convincing me to come to Amsterdam. I would also like to thank Dick de Jongh for wanting to sit on my thesis committee, and for providing support as my academic mentor. Finally, I want to thank Peter Koepke and the Hausdorff Center for Mathematics in Bonn for inviting me to Bonn International Workshop for Ordinal Computability.
The Robotic Turing Test. But was the language-‐only Turing Test, T2, really the one Turing intended (or should have intended)? After all, if the essence of Turing’s “cognition is as cognition does” criterion is Turing-‐indistinguishability from what a human cognizer can do, then a human cognizer can certainly do a lot more than just produce and understand language. A human cognizer can do countless other kinds of things in the real world of objects, actions, events, states and traits, and if the T2 candidate could not do all those kinds of things too, then that incapacity would be immediately detectable, and the candidate would fail the test. To be able to do all those things successfully, the T2 candidate would have to be a lot more than just a computer: It would have to be a sensorimotor robot, capable of sensing and acting upon all those objects, etc. -‐-‐ again Turing-‐indistinguishably from the way real human cognizers do.
In 1936, Turing developed his theoretical computational model . The deterministic and nondeterministic Turing machines have become in two of the most important definitions related to this theoretical model for computation . A deterministic Turingmachine has only one next action for each step defined in its program or transition function . A nondeterministic Turingmachine could contain more than one action defined for each step of its program, where this one is no longer a function, but a relation .
Another interesting application of the algorithmic the- ory of randomness is the following explanation why the bulk of work in machine learning under the gen- eral iid assumption (such as statistical learning theory and PAC theory) has been done in the batch setting. (The on-line setting has also been very popular, eg, in the theory of prediction with expert advice, but it uses dierent assumptions.) The algorithmic theory of ran- domness implies that on-line prediction under the iid