Tri-state: from Shannon to Galois Fields in Communication Systems

: Communication, compression of information, transmission of information through noisy 1 channels, interconnecting different information systems, cryptography, gate construction —- these 2 areas all depend on classical information theory. We show that, in classical terms, semantic aspects 3 of communication are not at all irrelevant to the engineering45problem, contrary to Shannon, and 4 affect the message intended to be transmitted. This is revisited and captured by an analogy to trust, 5 in that they are essential to the channel (for proper use),47but cannot be transferred (under risk of 6 ﬂaws) through that same channel. Information is also described by, at least, a tri-state system — 7 not by a binary logic. The trust analogy semantics can be coded as the Curry-Howard relationship, 8 connecting computer code with structural logic, by way of different categories. Two-state and 9 Boolean logic (aka Shannon semantics) was used classically before, with Shannon theory, but 10 without trust analogy semantics – found to be a sine qua non condition. This is now familiar 11 in classical gate construction with physical systems with, e.g., Verilog and SystemVerilog. The 12 applications to computation and quantum theory are further explored, at least dismissing qubits 13 and explaining its difﬁculties in representation. The most fundamental entity in today‘s theory 14 of information is proposed to use at least three logical states, not bits, and in all applications, 15 including: cyber-physical systems, devices, in computation, and in quantum theory.


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In 1948, Claude Shannon published "A Mathematical Theory of Communication" 20 [1], where the once fuzzy concept of "information" was proposed in a precise way to 21 quantify the fundamental unit of classical information, the bit. This is a binary logic level 22 system, following Boolean or classical logic, which carries two possible values, "0" and 23 "1".

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In its classical realization as a realizable physical system, using early bipolar junction 26 transistors representing the bit, which, for example could be imagined to be just a relay, a 27 transistor gate, or a mechanical switch, as pioneered by Shannon [1], one builds a system 28 which is designed to have two distinguishable states only 1 . The Law of the Excluded 29 Middle (LEM) applies naturally to such a system, although it does no apply in a logic 30 system with three or more logic levels 2 31 32 Shannon did not consider semantics part of the design, writing: "Frequently the mes- 33 1 There should be a sufficiently large energy barrier between them to create a "don't care" region -a region with no spontaneous transition, which would be evidently detrimental -occurring between the two logical states "0' and "1", allow for variations in the power supply, and noise pick-up in the lower level as well as the upper level. 2 LEM says that "For every proposition 'p', either 'p' or 'not p' holds". To many this is a self-evident truth, but only works in a binary logical system, such as Boolean or classical logic [2]. 48 problem, of the message intended to be transmitted, and are captured by an analogy to 49 trust (as discussed in Section 2), in that they are essential to the channel (for proper use), 50 but cannot be transferred (under risk of flaws) through that same channel [4].

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The significant aspect is still that the actual message is one selected from a set of possible 53 messages, but that it also is qualified to have the proper meaning. The discussed framework allows one to use the concept of trust in a common heteroge- on factors independent of that information. One realizes that trust is essentially com-103 municable, and that is why it can interconnect directly. But trust, as qualified reliance 104 on information, needs multiple, independent channels to be communicated. If one has 105 two entities (e.g., a client and a server) talking to one another, one has only one channel 106 of communication. Clearly, one needs more than two entities. It seems unreasonable to 107 require a hundred entities. Looking into millennia of human uses of trust, one realizes 108 that one needs no more than four parties, in general, to induce trust (i.e., to communicate    As this Section will discuss, tri-state offers more discriminating channels than binary 159 logic, allowing a much better resolution of indeterminate contributions, allowing them 160 to be much better discriminated for and filtered. In this paradigm, everything helps -161 even noise. The noise is used to highlight the signal, which noise is considered to be too 162 valuable to discard, bearing the result of previous processing (and cost).

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Tri-state here is understood as:

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Contrast with classical sentential or Boolean logic, that we consider has only one "true 175 or false" ontic level, leading to two fixed epistemic levels.

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To those who question that tri-state would be somehow "illogical" to consider, something   1, as the, at-first, ignored but necessary "don't care" region.

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The two-state logic levels are given below:   But this has also replaced in the field the Boolean algebra of a two-state system in-222 dicated by Shannon [1], mostly in gate construction with physical systems, with a trust 223 analogy semantics (see Section 2) that allows a common bus to carry varying signals, in 224 a coherent time. The primary reason, emerging from this Section, seems to be able to 225 deal more effectively with noise.  Fig.(2), not two as in Fig.(1), for communication.

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The Internet has given us much more than a simple means of exchanging information The tri-state logic, and each implementation have benefits/drawbacks, however, has 250 a more popular application in everyday devices than binary logic, or Shannon mode,

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where one can better see speed plus power savings, whereas with two-states we would 252 have one model but sacrifice those benefits. As explained above, any three-valued logic 253 system can be embedded in a two-valued logical system. An important feature of most tri-state gates is that the output enable delay is longer than 294 the output disable delay. If a control circuit enables one gate and disables another at 295 the same time, the disabled gate enters the high-impedance state before the other gate 296 is enabled. This eliminates the situation of both gates being active at the same time.

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There is a very small leakage current associated with the high-impedance condition in a 298 tri-state gate. Nevertheless, this current is so small that as many as 100 tri-state outputs 299 can be connected together to form a common-bus line.

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Three states are needed also in election information systems, to represent the anal- meaning to rely on, e.g., that modern robots are represented as possible, without dis-311 continuity, by humans, and vice versa. This is accomplished by sharing the same ADT, 312 not the same information (information is measured by surprise, not by knowledge; false 313 information exists [1]). Since the system's election purpose is often to define who will hold power, it must be impervious to its abuse.