Three-valued derived logics for classical phase spaces

Reichenbach proposed a three-valued logic to describe quantum mechanics. In his development, Reichenbach presented three different negation operators without providing any criteria for choosing among them. In this paper we develop two three-valuedderived logics for classical systems. These logics are derived in that they are based on a theory of physical measurement. In this regard they have some of the characteristics of the quantum logic developed by Birkhoff and von Neumann. The theory of measurement used in the present development is the one used previously in developingbivalent derived logics for classical systems. As these systems are derived logics, many of the ambiguities possessed by systems such as Reichenbach's are avoided.     

Automaton partition logic versus quantum logic

The propositional system of a general class of discrete deterministic systems is formally characterized. We find that any finite prime orthomodular lattice allowing two-valued states can be represented by an automaton logic.     

Automaton logic

The experimental logic of Moore and Mealy-type automata is investigated.     

Conservative logic

Conservative logic is a comprehensive model of computation which explicitly reflects a number of fundamental principles of physics, such as the reversibility of the dynamical laws and the conservation of certainadditive quantities (among which energy plays a distinguished role). Because it more closely mirrors physics than traditional models of computation, conservative logic is in a better position to provide indications concerning the realization of high-performance computing systems, i.e., of systems that make very efficient use of the computing resources actually offered by nature. In particular, conservative logic shows that it is ideally possible to build sequential circuits with zero internal power dissipation. After establishing a general framework, we discuss two specific models of computation. The first uses binary variables and is the conservative-logic counterpart of switching theory; this model proves that universal computing capabilities are compatible with the reversibility and conservation constraints. The second model, which is a refinement of the first, constitutes a substantial breakthrough in establishing a correspondence between computation and physics. In fact, this model is based on elastic collisions of identical balls, and thus is formally identical with the atomic model that underlies the (classical) kinetic theory of perfect gases. Quite literally, the functional behavior of a general-purpose digital computer can be reproduced by a perfect gas placed in a suitably shaped container and given appropriate initial conditions.This research was supported by the Defense Advanced Research Projects Agency and was monitored by the Office of Naval Research under Contract No. N000 14-75-C-0661.     

Quantum logic revisited

An adequate conjunction-implication pair is given for complete orthomodular lattices. The resulting conjunction is noncommutative in nature. We use the well-known lattice of closed subspaces of a Hilbert space, to give physical meaning to the given lattice operation.To the memory of Thomas A. Brody.     

Unified quantum logic

Unified quantum logic based on unified operations of implication is formulated as an axiomatic calculus. Soundness and completeness are demonstrated using standard algebraic techniques. An embedding of quantum logic into a new modal system is carried out and discussed.On leave of absence from Department of Mathematics, University of Zagreb, Pot. Pret. 165, YU-41001 Zagreb, Yugoslavia.     

Optical logic redux

Twenty years ago IBM physicist Robert Keyes published a paper entitled “Optical Logic—in the light of computer technology.” It caused an instant furor in the fledgling optical logic community. Now, 20 years after that devastating critique, the field of optical logic has grown enormously. There are literally thousands of papers. Many of them are collected in a bibliography given here. Was Keyes’ critique wrong? Have opticists simply ignored what Keyes pointed out? Have new developments made some of his remarks not quite so relevant? We argue here that

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Keyes was and still is mostly correct, but that may change in a few years

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Many researchers have indeed simply ignored what he said

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New developments in both optical logic and its applications open niches for optical logic that Keyes did not (and probably could not) anticipate

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New and anticipated developments in electronics may increase the role for optics

     

Coherent quantum logic

The von Neumann quantum logic lacks two basic symmetries of classical logic, that between sets and classes, and that between lower and higher order predicates. Similarly, the structural parallel between the set algebra and linear algebra of Grassmann and Peano was left incomplete by them in two respects. In this work a linear algebra is constructed that completes this correspondence and is interpreted as a new quantum logic that restores these invariances, and as a quantum set theory. It applies to experiments with coherent quantum phase relations between the quantum and the apparatus. The quantum set theory is applied to model a Lorentz-invariant quantum time-space complex.     

On quantum logic

The status and justification of quantum logic are reviewed. On the basis of several independent arguments it is concluded that it cannot be a logic in the philosophical sense of a general theory concerning the structure of valid inferences. Taken as a calculus for combining quantum mechanical propositions, it leaves a number of significant aspects of quantum physics unaccounted for. It is shown, moreover, that quantum logic, far from being more general than Boolean logic, forms a subset of a slight and natural extension of Boolean logic, a subset which corresponds to incomplete statements. The philosophical background of this unsatisfactory state of affairs is briefly explored.     

Infinitesimals without logic

We introduce a ring of the so-called Fermat reals, which is an extension of the real field containing nilpotent infinitesimals. The construction is inspired by Smooth Infinitesimal Analysis (SIA) and provides a powerful theory of actual infinitesimals without any background in mathematical logic. In particular, in contrast to SIA, which admits models in intuitionistic logic only, the theory of Fermat reals is consistent with the classical logic. We face the problem of deciding whether or not a product of powers of nilpotent infinitesimals vanishes, study the identity principle for polynomials, and discuss the definition and properties of the total order relation. The construction is highly constructive, and every Fermat real admits a clear and order-preserving geometrical representation. Using nilpotent infinitesimals, every smooth function becomes a polynomial because the remainder in Taylor’s formulas is now zero. Finally, we present several applications to informal classical calculations used in physics, and all these calculations now become rigorous, and at the same time, formally equal to the informal ones. In particular, an interesting rigorous deduction of the wave equation is given, which clarifies how to formalize the approximations tied with Hooke’s law using the language of nilpotent infinitesimals.     

Relativistic quantum logic

On the basis of the well-known quantum logic and quantum probability a formal language of relativistic quantum physics is developed. This language incorporates quantum logical as well as relativistic restrictions. It is shown that relativity imposes serious restrictions on the validity regions of propositions in space-time. By an additional postulate this relativistic quantum logic can be made consistent. The results of this paper are derived exclusively within the formal quantum language; they are, however, in accordance with well-known facts of relativistic quantum physics in Hilbert space.     

Magnetic-bubble conservative logic

Among integrated-circuit devices, magnetic bubbles are a particularly interesting candidate to implement the Fredkin gate and conservative logic. The magnetostatic repulsion of magnetic bubbles simulates the bouncing-ball model of conservative logic.     

Reflections on quantum logic

This is a short, self-contained summary of problems connected with the interpretation of state vectors in quantum mechanics. We discuss the reconstruction of the “ψ function” from statistical data, some related mathematical questions, the classical “paradoxes,” the probability interpretation of the state vectors, and, finally, quantum logic in relation to hidden variable theories and Hilbert space formalism, to build up a consistent framework for the indeterministic quantum picture of nature.     

Fuzzy quantum logic I

The paper gives a review of the application of fuzzy set ideas in quantum logics. After a brief introduction to the fuzzy set theory, the historical development of the main attempts to utilize fuzzy set ideas in quantum logics are presented. Results of investigations of all major researchers (except the Italian group discussed elsewhere), who work or worked in the field, are discussed.     

Fuzziness in abacus logic

The concept of abacus logic has recently been developed by the author (Malhas, n.d.). In this paper the relation of abacus logic to the concept of fuzziness is explored. It is shown that if a certain regularity condition is met, concepts from fuzzy set theory arise naturally within abacus logics. In particular it is shown that every abacus logic then has a pre-Zadeh orthocomplementation. It is also shown that it is then possible to associate a fuzzy set with every proposition of abacus logic and that the collection of all such sets satisfies natural conditions expected in systems of fuzzy logic. Finally, the relevance to quantum mechanics is discussed.     

Intramolecular Hamiltonian logic gates

An intramolecular computing model is presented that is based on the quantum time evolution of a single molecule driven by the preparation of a non-stationary single-electron state. In our scheme, the input bits are encoded into the coupling constants of the Hamiltonian that governs the molecular quantum dynamics. The results of the computation are obtained by carrying out a quantum measurement on the molecule. We design reliable , , and logic gates. This opens new avenues for the design of more complex logic circuits at a single-molecular scale.     

States on causal logic

Conserved currents are shown to define states on causal logic of Minkowski space.