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Automatic Proofs and Counterexamples for Some Ortholattice Identities
 Information Processing Letters
, 1998
"... This note answers questions on whether three identities known to hold for orthomodular lattices are true also for ortholattices. One identity is shown to fail by MACE, a program that searches for counterexamples, an the other two are proved to hold by EQP, an equational theorem prover. The problems, ..."
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Cited by 22 (2 self)
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This note answers questions on whether three identities known to hold for orthomodular lattices are true also for ortholattices. One identity is shown to fail by MACE, a program that searches for counterexamples, an the other two are proved to hold by EQP, an equational theorem prover. The problems, from work in quantum logic, were given to us by Norman Megill. Keywords: Automatic theorem proving, ortholattice, quantum logic, theory of computation. 1 Introduction An ortholattice is an algebra with a binary operation (join) and a unary operation 0 (complement) satisfying the following (independent) set of identities. x y = (x 0 y 0 ) 0 (definition of meet) x y = y x (x y) z = x (y z) x (x y) = x x 00 = x x (y y 0 ) = y y 0 Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W31109Eng38. From these identities one can...
Linear logic for generalized quantum mechanics
 In Proc. Workshop on Physics and Computation (PhysComp'92
, 1993
"... Quantum logic is static, describing automata having uncertain states but no state transitions and no Heisenberg uncertainty tradeoff. We cast Girard’s linear logic in the role of a dynamic quantum logic, regarded as an extension of quantum logic with time nonstandardly interpreted over a domain of l ..."
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Cited by 16 (2 self)
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Quantum logic is static, describing automata having uncertain states but no state transitions and no Heisenberg uncertainty tradeoff. We cast Girard’s linear logic in the role of a dynamic quantum logic, regarded as an extension of quantum logic with time nonstandardly interpreted over a domain of linear automata and their dual linear schedules. In this extension the uncertainty tradeoff emerges via the “structure veil. ” When VLSI shrinks to where quantum effects are felt, their computeraided design systems may benefit from such logics of computational behavior having a strong connection to quantum mechanics. 1
Representation Theorems and the Semantics of NonClassical Logics , and Applications to Automated Theorem Proving
, 2002
"... We give a uniform presentation of representation and decidability results related to the Kripkestyle semantics of several nonclassical logics. We show that a general representation theorem (which has as particular instances the representation theorems as algebras of sets for Boolean algebras, d ..."
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Cited by 4 (2 self)
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We give a uniform presentation of representation and decidability results related to the Kripkestyle semantics of several nonclassical logics. We show that a general representation theorem (which has as particular instances the representation theorems as algebras of sets for Boolean algebras, distributive lattices and semilattices) extends in a natural way to several classes of operators and allows to establish a relationship between algebraic and Kripkestyle models. We illustrate the ideas on several examples. We conclude by showing how the Kripkestyle models thus obtained can be used (if rstorder axiomatizable) for automated theorem proving by resolution for some nonclassical logics.
A new axiomatization of unified quantum logic
 International Journal of Theoretical Physics
, 1992
"... Summary. Unified quantum logic which is a propositional logic underlying quantum formalism is given a new much simplified axiomatization. A statistical basis for this propositional logical system is given so as to interpret unified quantum logic as a system of deduction. The soundness and completene ..."
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Cited by 1 (1 self)
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Summary. Unified quantum logic which is a propositional logic underlying quantum formalism is given a new much simplified axiomatization. A statistical basis for this propositional logical system is given so as to interpret unified quantum logic as a system of deduction. The soundness and completeness of algebraic semantics are proved. Kripkean and probabilistic semantics are discussed. PACS numbers: 03.65.Bz — Quantum logic, quantum measurements. 1.
Complete Axiomatizations for Quantum Actions
 International Journal of Theoretical Physics
"... We present two equivalent axiomatizations for a logic of quantum actions: one in terms of quantum transition systems, and the other in terms of quantum dynamic algebras. The main contribution of the paper is conceptual, offering a new view of quantum structures in terms of their underlying logical d ..."
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We present two equivalent axiomatizations for a logic of quantum actions: one in terms of quantum transition systems, and the other in terms of quantum dynamic algebras. The main contribution of the paper is conceptual, offering a new view of quantum structures in terms of their underlying logical dynamics. We also prove Representation Theorems, showing these axiomatizations to be complete with respect to the natural Hilbertspace semantics. The advantages of this setting are many: 1) it provides a clear and intuitive dynamicoperational meaning to key postulates (e.g. Orthomodularity, Covering Law); 2) it reduces the complexity of the SolèrMayet axiomatization by replacing some of their key higherorder concepts (e.g. “automorphisms of the ortholattice”) by firstorder objects (“actions”) in our structure; 3) it provides a link between traditional quantum logic and the needs of quantum computation.
QUANTUM LOGICS
"... 2 Orthomodular quantum logic and orthologic 11 3 The implication problem 22 4 Metalogical properties and anomalies 28 ..."
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2 Orthomodular quantum logic and orthologic 11 3 The implication problem 22 4 Metalogical properties and anomalies 28