Abstract
How are abstract concepts and musical themes recognized on the basis of some previous experience? It is interesting to compare the different behaviors of human and of artificial intelligences with respect to this problem. Generally, a human mind that abstracts a concept (say, table) from a given set of known examples creates a table-Gestalt: a kind of vague and out of focus image that does not fully correspond to a particular table with well determined features. A similar situation arises in the case of musical themes. Can the construction of a gestaltic pattern, which is so natural for human minds, be taught to an intelligent machine? This problem can be successfully discussed in the framework of a quantum approach to pattern recognition and to machine learning. The basic idea is replacing classical data sets with quantum data sets, where either objects or musical themes can be formally represented as pieces of quantum information, involving the uncertainties and the ambiguities that characterize the quantum world. In this framework, the intuitive concept of Gestalt can be simulated by the mathematical concept of positive centroid of a given quantum data set. Accordingly, the crucial problem “how can we classify a new object or a new musical theme (we have listened to) on the basis of a previous experience?” can be dealt with in terms of some special quantum similarity-relations. Although recognition procedures are different for human and for artificial intelligences, there is a common method of “facing the problems” that seems to work in both cases.
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Notes
- 1.
See, for instance, Honing (2009).
- 2.
See, for instance, Ehrenstein et al. (2003).
- 3.
See, for instance, Dalla Chiara et al. (2018).
- 4.
Hilbert spaces are special examples of vector spaces that represent generalizations of geometric Euclidean spaces. A simple example of a Hilbert space is the geometric plane, whose set of points corresponds to the set of all possible ordered pairs of real numbers.
- 5.
More precisely, this probability-value is represented by the real number |ci|2 (the squared modulus of ci). Since the length of |ψ〉 is 1, we have: ∑i|ci|2 = 1.
- 6.
In this space (usually indicated by the symbol \(\mathbb {C}^2\)) the two classical bits |0〉 and |1〉 are identified with the two number-pairs (1, 0) and (0, 1), respectively.
- 7.
More precisely, the probability of the answer “No” is the number |c0|2, while the probability of the answer “Yes” is the number |c1|2.
- 8.
Any density operator ρ of a given Hilbert space can be represented (in a non-unique way) as a weighted sum of some projection-operators, having the form: \(\rho = \sum _i w_i P_{\vert {\psi _i} \rangle }, \) where the weights wi are positive real numbers such that ∑iwi = 1, while each \(P_{\vert {\psi _i} \rangle }\) is the projection operator that projects over the closed subspace determined by the vector |ψi〉. Thus, any pure state |ψ〉 corresponds to a special example of a density operator: the projection P|ψ〉. The physical interpretation of a mixed state \(\rho = \sum _i w_i P_{\vert {\psi _i} \rangle } \) is the following: a quantum system in state ρ might be in the pure state \(P_{\vert {\psi _i} \rangle } \) with probability-value wi.
- 9.
Technical details can be found in Dalla Chiara et al. (2018).
- 10.
- 11.
We recall that \(P_{\vert {\psi _i} \rangle }\) indicates the projection operator that projects over the closed subspace determined by the vector |ψi〉: a special example of a density operator that corresponds to the pure state represented by the vector |ψi〉. According to the canonical physical interpretation of mixtures, ρ+ represents a state that ambiguously describes a quantum system that might be in the pure state |ψi〉 with probability-value \(\frac {1}{n^+}\).
- 12.
See Dalla Chiara et al. (2012).
- 13.
The use of the squared brackets is suggested by a notation often used in mathematics, where operations involving an abstraction are frequently indicated by the brackets [……].
- 14.
Of course, we might use a “more mathematical” notation, indicating all melodic intervals by convenient arithmetical expressions. However, this kind of notation (which plays an important role in the framework of computer music) would be too heavy and hardly interesting for the aims of our semantic approach.
- 15.
See Schönberg (1995).
- 16.
See, for instance, Schenker (1935).
- 17.
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Dalla Chiara, M.L., Giuntini, R., Negri, E., Sergioli, G. (2023). Recognizing Concepts and Recognizing Musical Themes. In: Arenhart, J.R.B., Arroyo, R.W. (eds) Non-Reflexive Logics, Non-Individuals, and the Philosophy of Quantum Mechanics. Synthese Library, vol 476. Springer, Cham. https://doi.org/10.1007/978-3-031-31840-5_14
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