bayesian statistics biologically inspired bounded rationality Control desirability dual probabilistic programming filtering Gaussian processes lexicographic machine learning Quantum mechanics radar tracking rationality set of probabilities SOS Sum-of-squares polynomials

## 2019 |

Benavoli, Alessio ; Facchini, Alessandro ; Zaffalon, Marco Computational Complexity and the Nature of Quantum Mechanics (Extended version) Technical Report 2019. Abstract | Links | BibTeX | Tags: Quantum mechanics, Sum-of-squares polynomials @techreport{Benavoli2019c, title = {Computational Complexity and the Nature of Quantum Mechanics (Extended version)}, author = {Benavoli, Alessio and Facchini, Alessandro and Zaffalon, Marco }, url = {https://arxiv.org/abs/1902.03513}, year = {2019}, date = {2019-02-11}, abstract = {Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by having it follow from two main postulates (i) the theory should be logically consistent; (ii) inferences in the theory should be computable in polynomial time. The first postulate is what we require to each well-founded mathematical theory. The computation postulate defines the physical component of the theory. We show that the computation postulate is the only true divide between QT, seen as a generalised theory of probability, and classical probability. All quantum paradoxes, and entanglement in particular, arise from the clash of trying to reconcile a computationally intractable, somewhat idealised, theory (classical physics) with a computationally tractable theory (QT) or, in other words, from regarding physics as fundamental rather than computation. }, keywords = {Quantum mechanics, Sum-of-squares polynomials}, pubstate = {published}, tppubtype = {techreport} } Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by having it follow from two main postulates (i) the theory should be logically consistent; (ii) inferences in the theory should be computable in polynomial time. The first postulate is what we require to each well-founded mathematical theory. The computation postulate defines the physical component of the theory. We show that the computation postulate is the only true divide between QT, seen as a generalised theory of probability, and classical probability. All quantum paradoxes, and entanglement in particular, arise from the clash of trying to reconcile a computationally intractable, somewhat idealised, theory (classical physics) with a computationally tractable theory (QT) or, in other words, from regarding physics as fundamental rather than computation. |

Benavoli, Alessio; Facchini, Alessandro; Piga, Dario; Zaffalon, Marco Sum-of-squares for bounded rationality Journal Article International Journal of Approximate Reasoning, 105 , pp. 130 - 152, 2019, ISSN: 0888-613X. Abstract | Links | BibTeX | Tags: Sum-of-squares polynomials @article{Benavoli2019b, title = {Sum-of-squares for bounded rationality}, author = {Alessio Benavoli and Alessandro Facchini and Dario Piga and Marco Zaffalon}, url = {https://arxiv.org/abs/1705.02663}, doi = {10.1016/j.ijar.2018.11.012}, issn = {0888-613X}, year = {2019}, date = {2019-01-01}, journal = {International Journal of Approximate Reasoning}, volume = {105}, pages = {130 - 152}, abstract = {In the gambling foundation of probability theory, rationality requires that a subject should always (never) find desirable all nonnegative (negative) gambles, because no matter the result of the experiment the subject never (always) decreases her money. Evaluating the nonnegativity of a gamble in infinite spaces is a difficult task. In fact, even if we restrict the gambles to be polynomials in Rn, the problem of determining nonnegativity is NP-hard. The aim of this paper is to develop a computable theory of desirable gambles. Instead of requiring the subject to desire all nonnegative gambles, we only require her to desire gambles for which she can efficiently determine the nonnegativity (in particular sum-of-squares polynomials). We refer to this new criterion as bounded rationality.}, keywords = {Sum-of-squares polynomials}, pubstate = {published}, tppubtype = {article} } In the gambling foundation of probability theory, rationality requires that a subject should always (never) find desirable all nonnegative (negative) gambles, because no matter the result of the experiment the subject never (always) decreases her money. Evaluating the nonnegativity of a gamble in infinite spaces is a difficult task. In fact, even if we restrict the gambles to be polynomials in Rn, the problem of determining nonnegativity is NP-hard. The aim of this paper is to develop a computable theory of desirable gambles. Instead of requiring the subject to desire all nonnegative gambles, we only require her to desire gambles for which she can efficiently determine the nonnegativity (in particular sum-of-squares polynomials). We refer to this new criterion as bounded rationality. |