scholarly journals Holistic Type Extension for Classical Logic via Toffoli Quantum Gate

Entropy ◽  
2019 ◽  
Vol 21 (7) ◽  
pp. 636 ◽  
Author(s):  
Hector Freytes ◽  
Roberto Giuntini ◽  
Giuseppe Sergioli
Keyword(s):  
Quantum Computation ◽  
Classical Logic ◽  
Propositional Logic ◽  
Quantum States ◽  
Quantum Gate ◽  
Logical System ◽  
Classical Propositional Logic ◽  
Mixed States ◽  
Werner States ◽  
To Receive

A holistic extension of classical propositional logic is introduced via Toffoli quantum gate. This extension is based on the framework of the so-called “quantum computation with mixed states”, where also irreversible transformations are taken into account. Formal aspects of this new logical system are detailed: in particular, the concepts of tautology and contradiction are investigated in this extension. These concepts turn out to receive substantial changes due to the non-separability of some quantum states; as an example, Werner states emerge as particular cases of “holistic" contradiction.

scholarly journals An Holistic Extension for Classical Logic via Quantum Fredkin Gate

Entropy ◽  
2021 ◽  
Vol 23 (9) ◽  
pp. 1178
Author(s):  
Hector Freytes ◽  
Giuseppe Sergioli
Keyword(s):  
Quantum Computation ◽  
Classical Logic ◽  
Propositional Logic ◽  
Classical Propositional Logic ◽  
Mixed States ◽  
Fredkin Gate ◽  
Bipartite States ◽  
Extended Notion

An holistic extension for classical propositional logic is introduced in the framework of quantum computation with mixed states. The mentioned extension is obtained by applying the quantum Fredkin gate to non-factorizable bipartite states. In particular, an extended notion of classical contradiction is studied in this holistic framework.

scholarly journals Hybrid Deduction–Refutation Systems

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 118 ◽  
Author(s):  
Valentin Goranko
Keyword(s):  
Propositional Logic ◽  
Proof Theory ◽  
Natural Deduction ◽  
Basic Theory ◽  
Logical System ◽  
Classical Propositional Logic ◽  
Deductive Systems ◽  
Refutation Systems ◽  
Soundness And Completeness

Hybrid deduction–refutation systems are deductive systems intended to derive both valid and non-valid, i.e., semantically refutable, formulae of a given logical system, by employing together separate derivability operators for each of these and combining ‘hybrid derivation rules’ that involve both deduction and refutation. The goal of this paper is to develop a basic theory and ‘meta-proof’ theory of hybrid deduction–refutation systems. I then illustrate the concept on a hybrid derivation system of natural deduction for classical propositional logic, for which I show soundness and completeness for both deductions and refutations.

scholarly journals QUANTUM COMPUTATION TREE LOGIC — MODEL CHECKING AND COMPLETE CALCULUS

2008 ◽  
Vol 06 (02) ◽  
pp. 219-236 ◽  
Author(s):  
P. BALTAZAR ◽  
R. CHADHA ◽  
P. MATEUS
Keyword(s):  
Model Checking ◽  
Quantum Computation ◽  
Propositional Logic ◽  
Key Distribution ◽  
Logic Model ◽  
Quantum States ◽  
Computation Tree Logic ◽  
Computation Tree ◽  
Complete Axiomatization ◽  
Logic Model Checking

Logics for reasoning about quantum states and their evolution have been given in the literature. In this paper, we consider quantum computation tree logic (QCTL), which adds temporal modalities to exogenous quantum propositional logic. We give a sound and complete axiomatization of QCTL and combine the standard CTL model-checking algorithm with the dEQPL model-checking algorithm to obtain a model-checking algorithm for QCTL. Finally, we illustrate the use of the logic by reasoning about the BB84 key distribution protocol.

FRACTIONAL SEMANTICS FOR CLASSICAL LOGIC

2019 ◽  
Vol 13 (4) ◽  
pp. 810-828
Author(s):  
MARIO PIAZZA ◽  
GABRIELE PULCINI
Keyword(s):  
Classical Logic ◽  
Propositional Logic ◽  
Logical Formula ◽  
Classical Propositional Logic ◽  
Multivalued Semantics

AbstractThis article presents a new (multivalued) semantics for classical propositional logic. We begin by maximally extending the space of sequent proofs so as to admit proofs for any logical formula; then, we extract the new semantics by focusing on the axiomatic structure of proofs. In particular, the interpretation of a formula is given by the ratio between the number of identity axioms out of the total number of axioms occurring in any of its proofs. The outcome is an informational refinement of traditional Boolean semantics, obtained by breaking the symmetry between tautologies and contradictions.

scholarly journals A classical limit of Grover’s algorithm induced by dephasing: Coherence versus entanglement

2019 ◽  
Vol 34 (07n08) ◽  
pp. 1950146
Author(s):  
Kazuo Fujikawa ◽  
C. H. Oh ◽  
Koichiro Umetsu
Keyword(s):  
Quantum Computation ◽  
Classical Limit ◽  
Quantum Coherence ◽  
Quantum States ◽  
Mixed States ◽  
Grover’S Algorithm ◽  
Specific Element ◽  
New Approach ◽  
Mechanical Amplification ◽  
Grover's Algorithm

A new approach to the classical limit of Grover’s algorithm is discussed by assuming a very rapid dephasing of a system between consecutive Grover’s unitary operations, which drives pure quantum states to decohered mixed states. One can identify a specific element among [Formula: see text] unsorted elements by a probability of the order of unity after [Formula: see text] steps of classical amplification, which is realized by a combination of Grover’s unitary operation and rapid dephasing, in contrast to [Formula: see text] steps in quantum mechanical amplification. The initial two-state system with enormously unbalanced existence probabilities, which is realized by a chosen specific state and a superposition of all the rest of the states among [Formula: see text] unsorted states, is crucial in the present analysis of classical amplification. This analysis illustrates Grover’s algorithm in extremely noisy circumstances. A similar increase from [Formula: see text] to [Formula: see text] steps due to the loss of quantum coherence takes place in the analog model of Farhi and Gutmann where the entanglement does not play an obvious role. This supports a view that entanglement is crucial in quantum computation to describe quantum states by a set of qubits, but the actual speedup of the quantum computation is based on quantum coherence.

scholarly journals Control effects as a modality

2009 ◽  
Vol 19 (1) ◽  
pp. 17-26 ◽  
Author(s):  
HAYO THIELECKE
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Propositional Logic ◽  
The Other ◽  
Classical Propositional Logic ◽  
Double Negation ◽  
Other Hand

AbstractWe combine ideas from types for continuations, effect systems and monads in a very simple setting by defining a version of classical propositional logic in which double-negation elimination is combined with a modality. The modality corresponds to control effects, and it includes a form of effect masking. Erasing the modality from formulas gives classical logic. On the other hand, the logic is conservative over intuitionistic logic.

scholarly journals Many-valued logics—implications and semantic consequences

2013 ◽  
Vol 5 (2) ◽  
pp. 145-166
Author(s):  
Katalin Pásztor Varga ◽  
Gábor Alagi
Keyword(s):  
Classical Logic ◽  
Propositional Logic ◽  
Matrix Method ◽  
Decision Problem ◽  
Modus Ponens ◽  
Deduction Theorem ◽  
Classical Propositional Logic ◽  
Consequence Operation ◽  
Semantic Consequence ◽  
Logical Calculi

Abstract In this paper an application of the well-known matrix method to an extension of the classical logic to many-valued logic is discussed: we consider an n-valued propositional logic as a propositional logic language with a logical matrix over n truth-values. The algebra of the logical matrix has operations expanding the operations of the classical propositional logic. Therefore we look over the Łukasiewicz, Post, Heyting and Rosser style expansions of the operations negation, conjunction, disjunction and with a special emphasis on implication. In the frame of consequence operation, some notions of semantic consequence are examined. Then we continue with the decision problem and the logical calculi. We show that the cause of difficulties with the notions of semantic consequence is the weakness of the reviewed expansions of negation and implication. Finally, we introduce an approach to finding implications that preserve both the modus ponens and the deduction theorem with respect to our definitions of consequence.

scholarly journals Natural three-valued logics and classical logic

2013 ◽  
Vol 19 ◽  
pp. 344-352 ◽  
Author(s):  
Н.Е. Томова
Keyword(s):  
Classical Logic ◽  
Propositional Logic ◽  
Classical Propositional Logic

In this paper implicative fragments of natural three- valued logic are investigated. It is proved that some fragments are equivalent by set of tautologies to implicative fragment of classical logic. It is also shown that some natural three-valued logics verify all tautologies of classical propositional logic.

scholarly journals Deduction in Non-Fregean Propositional Logic SCI

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 115 ◽  
Author(s):  
Joanna Golińska-Pilarek ◽  
Magdalena Welle
Keyword(s):  
Propositional Logic ◽  
Sequent Calculus ◽  
Classical Propositional Logic ◽  
Tableau System ◽  
Soundness And Completeness

We study deduction systems for the weakest, extensional and two-valued non-Fregean propositional logic SCI . The language of SCI is obtained by expanding the language of classical propositional logic with a new binary connective ≡ that expresses the identity of two statements; that is, it connects two statements and forms a new one, which is true whenever the semantic correlates of the arguments are the same. On the formal side, SCI is an extension of classical propositional logic with axioms characterizing the identity connective, postulating that identity must be an equivalence and obey an extensionality principle. First, we present and discuss two types of systems for SCI known from the literature, namely sequent calculus and a dual tableau-like system. Then, we present a new dual tableau system for SCI and prove its soundness and completeness. Finally, we discuss and compare the systems presented in the paper.

RELEVANCE LOGIC AND THE CALCULUS OF RELATIONS

2010 ◽  
Vol 3 (1) ◽  
pp. 41-70 ◽  
Author(s):  
ROGER D. MADDUX
Keyword(s):  
Propositional Logic ◽  
Relevance Logic ◽  
Binary Relations ◽  
Classical Propositional Logic

Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are representable as algebras of binary relations.

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