Eigenlogic: Interpretable Quantum Observables with applications to Fuzzy Behavior of Vehicular Robots

This work proposes a formulation of propositional logic, named Eigenlogic, using quantum observables as propositions. The eigenvalues of these operators are the truth-values and the associated eigenvectors the interpretations of the propositional system. Fuzzy logic arises naturally when considering vectors outside the eigensystem, the fuzzy membership function is obtained by the Born rule of the logical observable.This approach is then applied in the context of quantum robots using simple behavioral agents represented by Braitenberg vehicles. Processing with non-classical logic such as multivalued logic, fuzzy logic and the quantum Eigenlogic permits to enlarge the behavior possibilities and the associated decisions of these simple agents

Eigenlogic: a Quantum View for Multiple-Valued and Fuzzy Systems

We propose a matrix model for two- and many-valued logic using families of observables in Hilbert space, the eigenvalues give the truth values of logical propositions where the atomic input proposition cases are represented by the respective eigenvectors. For binary logic using the truth values {0,1} logical observables are pairwise commuting projectors. For the truth values {+1,-1} the operator system is formally equivalent to that of a composite spin 1/2 system, the logical observables being isometries belonging to the Pauli group. Also in this approach fuzzy logic arises naturally when considering non-eigenvectors. The fuzzy membership function is obtained by the quantum mean value of the logical projector observable and turns out to be a probability measure in agreement with recent quantum cognition models. The analogy of many-valued logic with quantum angular momentum is then established. Logical observables for three-value logic are formulated as functions of the Lz observable of the orbital angular momentum l=1. The representative 3-valued 2-argument logical observables for the Min and Max connectives are explicitly obtained.Comment: 11 pages, 2 table

A logical approach to fuzzy truth hedges

The starting point of this paper are the works of Hájek and Vychodil on the axiomatization of truth-stressing and-depressing hedges as expansions of Hájek's BL logic by new unary connectives. They showed that their logics are chain-complete, but standard completeness was only proved for the expansions over Gödel logic. We propose weaker axiomatizations over an arbitrary core fuzzy logic which have two main advantages: (i) they preserve the standard completeness properties of the original logic and (ii) any subdiagonal (resp. superdiagonal) non-decreasing function on [0, 1] preserving 0 and 1 is a sound interpretation of the truth-stresser (resp. depresser) connectives. Hence, these logics accommodate most of the truth hedge functions used in the literature about of fuzzy logic in a broader sense. © 2013 Elsevier Inc. All rights reserved.The authors acknowledge partial support of the MICINN projects TASSAT (TIN2010-20967-C04-01) and ARINF (TIN2009-14704-C03-03), and the FP7-PEOPLE-2009-IRSES project MaToMUVI (PIRSES-GA-2009-247584). Carles Noguera also acknowledges support of the research contract “Juan de la Cierva” JCI-2009-05453.Peer Reviewe

A formal approach to vague expressions with indexicals

In this paper, we offer a formal approach to the scantily investigated problem of vague expressions with indexicals, in particular including the spatial indexical `here' and the temporal indexical `now'. We present two versions of an adaptive fuzzy logic extended with an indexical, formally expressed by a modifier as a function that applies to predicative formulas. In the first version, such an operator is applied to non-vague predicates. The modified formulas may have a fuzzy truth value and fit into a Sorites paradox. We use adaptive fuzzy logics as a reasoning tool to address such a paradox. The modifier enables us to offer an adequate explication of the dynamic reasoning process. In the second version, a different result is obtained for an indexical applied to a formula with a possibly vague predicate, where the resulting modified formula has a crisp value and does not add up to a Sorites paradox

Fuzzy Logic Function as a Post-hoc Explanator of the Nonlinear Classifier

Pattern recognition systems implemented using deep neural networks achieve better results than linear models. However, their drawback is the black box property. This property means that one with no experience utilising nonlinear systems may need help understanding the outcome of the decision. Such a solution is unacceptable to the user responsible for the final decision. He must not only believe in the decision but also understand it. Therefore, recognisers must have an architecture that allows interpreters to interpret the findings. The idea of post-hoc explainable classifiers is to design an interpretable classifier parallel to the black box classifier, giving the same decisions as the black box classifier. This paper shows that the explainable classifier completes matching classification decisions with the black box classifier on the MNIST and FashionMNIST databases if Zadeh`s fuzzy logic function forms the classifier and DeconvNet importance gives the truth values. Since the other tested significance measures achieved lower performance than DeconvNet, it is the optimal transformation of the feature values to their truth values as inputs to the fuzzy logic function for the databases and recogniser architecture used

Uncertainty in the conjunctive approach to fuzzy inference

Fuzzy inference using the conjunctive approach is very popular in many practical applications. It is intuitive for engineers, simple to understand, and characterized by the lowest computational complexity. However, it leads to incorrect results in the cases when the relationship between a fact and a premise is undefined. This article analyses the problem thoroughly and provides several possible solutions. The drawbacks of uncertainty in the conjunctive approach are presented using fuzzy inference based on a fuzzy truth value, first introduced by Baldwin (1979c). The theory of inference is completed with a new truth function named 0-undefined for two-valued logic, which is further generalized into fuzzy logic as α-undefined. Eventually, the proposed modifications allow altering existing implementations of conjunctive fuzzy systems to interpret the undefined state, giving adequate results

Fuzzy logic is a helpful conceptual and operational tool for modelling the geography of ecological interactions

Fuzzy logic is a form of many-valued logic whose variables have a truth value that varies in degree. Spatial favourability for species occurrence may be considered a fuzzy concept, as historical, geographical, human, and environmental conditions make locations more or less favourable for the occurrence of particular species. The favourability function was conceptually conceived to define spatial favourability in a fuzzy gradient from 0 to 1, so facilitating the application of fuzzy logic to spatial modelling. Favourability values derived from the favourability function have the same meaning and the same mathematical value regardless the prevalence of the species, so enabling direct comparison of models built for different species and their combination using fuzzy logic operators. This characteristics make the favourability function particularly useful in the spatial modelling of ecologically interacting species. In particular, the fuzzy intersection of favourability for different species is useful to model the biogeographical consequences of different degrees of competition between species. Fuzzy logic operations allow also to combine autoecological and sinecological responses in a way that may account for the existence of parapatric distributions in current and future environments, as exemplified by hare species in Europe. Fuzzy logic may provide biogeographical modellers with the necessary flexibility in concepts and operational tools to deal with a highly unstable and intertwined biogeographical world.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

A Modern Syllogistic Method in Intuitionistic Fuzzy Logic with Realistic Tautology

The Modern Syllogistic Method (MSM) of propositional logic ferrets out from a set of premises all that can be concluded from it in the most compact form. The MSM combines the premises into a single function equated to 1 and then produces the complete product of this function. Two fuzzy versions of MSM are developed in Ordinary Fuzzy Logic (OFL) and in Intuitionistic Fuzzy Logic (IFL) with these logics augmented by the concept of Realistic Fuzzy Tautology (RFT) which is a variable whose truth exceeds 0.5. The paper formally proves each of the steps needed in the conversion of the ordinary MSM into a fuzzy one. The proofs rely mainly on the successful replacement of logic 1 (or ordinary tautology) by an RFT. An improved version of Blake-Tison algorithm for generating the complete product of a logical function is also presented and shown to be applicable to both crisp and fuzzy versions of the MSM. The fuzzy MSM methodology is illustrated by three specific examples, which delineate differences with the crisp MSM, address the question of validity values of consequences, tackle the problem of inconsistency when it arises, and demonstrate the utility of the concept of Realistic Fuzzy Tautology