Sunday, August 21, 2011

Is quantal Boolean reverse engineering possible?

The quantal version of Boolean algebra means that the basic logical functions have quantum inverses. The inverse of C=A ∧ B represents the quantum superposition of all pairs A and B for which A∧ B=C hols true. Same is true for ∨. How could these additional quantum logical functions with no classical counterparts extend the capacities of logician?

What comes in mind is logical reverse engineering. Consider the standard problem solving situation repeatedly encountered by my hero Hercule Poirot. Someone has been murdered. Who could have done it? Who did it? Actually scientists who want to explain instead of just applying the method to get additional items to the CVC, meet this kind of problem repeatedly. One has something which looks like an experimental anomaly and one has to explain it. Is this anomaly genuine or is it due to a systematic error in the information processing? Could the interpretation of data be somehow wrong? Is the model behind experiments based on existing theory really correct or has something very delicate been neglected? If a genuine anomaly is in question (someone has been really murdered- this is always obvious in the tales about the deeds of Hercule Poirot since the mere presence of Hercule guarantees the murder unless it has been already done) one encounters what might be called Poirot problem in honor of my hero. As a matter fact, from the point of view of Boolean algebra, one has the same reverse Boolean engineering problem irrespective of whether it was a genuine anomaly or not.

This brings in my mind the enormously simplified problem. The logical statement C is found to be true. Which pairs A,B could have implied C as C=A∧ B (or A∨ B). Of course, much more complex situations can be considered where C corresponds to some logical function C=f(A1,A2,...,An). Quantum Poirot could use quantum computer able to realize the co-gates for gates AND and OR (essentially time reversals) and write a quantum computer program solving the problem by constructing the Boolean co-function of Boolean function f.

What would happen in TGD Universe obeying zero energy ontology is following.

  1. The statement C is represented as as positive energy part of zero energy state (analogous to initial state of physical event) and A1,..An is represented as one state in the quantum superposition of final states representing various value combinations for A1,...,An. Zero energy states (rather than only their evolution) represets the arrow of time. The M-matrix characterizing time-like entanglement between positive and negative energy states generalizes generalizes S-matrix. S-matrix is such that initial states have well defined particle numbers and other quantum numbers whereas final states do not. They are analogous to the outcomes of quantum measurement in particle physics.

  2. Negentropy Maximization Principle maximizing the information contents of conscious experience (sic!) forces state function reduction to one particular A1,...,An and one particular value combination consistent with C is found in each state function reduction. At the ensemble level one obtains probabilities for various outcomes and the most probable combination might represent the most plausible candidate for the murderer in quantum Poirot problem. Also in particle physics one can only speak about plausibility of the explanation and this leads to the endless n sigma talk. Note that it is absolutely essential that state function reduction occurs. Ironically, quantum problem solving causes dissipation at the level of ensemble but the ensemble probabilities carry actually information! Second law of thermodynamics tells us that Nature is a pathological problem solver- just like my hero!

  3. In TGD framework basic logical binary operations have a representation at the level of Boolean algebra realized in terms of fermionic oscillator operators. They have also space-time correlates realized topologically. ∧ has a representation as the analog of three-vertex of Feynman graph for partonic 2-surfaces: partonic 2-surfaces are glued along the ends to form outgoing partonic 2-surface. ∨ has a representation as the analog of stringy trouser vertex in which partonic surfaces fuse together. Here TGD differs from string models in a profound manner.

To conclude, I am a Boolean dilettante and know practically nothing about what quantum computer theorists have done- in particular I do not know whether they have considered quantum inverse gages. My feeling is that only the gates with bits replaced with qubits are considered: very natural when one thinks in terms of Boolean logic. If this is really the case, quantal co-AND and co-OR having no classical counterparts would bring a totally new aspect to quantum computation in solving problems in which one cannot do without (quantum) Poirot and his little gray (quantum) brain cells.

For details and background the reader can consult either to the chapter Physics as Generalized Number Theory: Infinite Primes of "Physics as Generalized Number Theory" or to the article How infinite primes relate to other views about mathematical infinity?.

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