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Probably Partially True: Satisfiability for Łukasiewicz Infinitely-Valued Probabilistic Logic and Related Topics

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Abstract

We study probabilistic-logic reasoning in a context that allows for “partial truths”, focusing on computational and algorithmic properties of non-classical Łukasiewicz Infinitely-valued Probabilistic Logic. In particular, we study the satisfiability of joint probabilistic assignments, which we call ŁIPSAT. Although the search space is initially infinite, we provide linear algebraic methods that guarantee polynomial size witnesses, placing ŁIPSAT complexity in the NP-complete class. An exact satisfiability decision algorithm is presented which employs, as a subroutine, the decision problem for Łukasiewicz Infinitely-valued (non probabilistic) logic, that is also an NP-complete problem. We investigate efficient representation of rational McNaughton functions in Łukasiewicz Infinitely-valued Logic modulo satisfiability.

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Notes

  1. By the term “partial truth” we refer to the concept usually referred in the literature as “degree of truth”, not to be confused with partial valuations or models.

  2. Satisfiability problem in Łukasiewicz Infinitely-valued logic has been shown to be NP-complete [23] and there are some implementations discussed in the literature [3], but there are many implementation options with considerable efficiency differences which are analyzed in [15].

  3. An earlier version of this paper has appeared in [15]. In this work we present proofs of lemmas and theorems that were omitted. Section 5 on representation of rational McNaughton functions is totally new; on the other hand, due to space limitations, implementational and experimental issues have been omitted.

  4. Thus C is more restrictive than the full class of states of an MV-algebra, in the sense of [26], which will not be discussed here.

  5. We abuse the notation by using the same symbols for the propositional variables and for the metavariables in the functions description.

  6. These functions are only different from the Schauder hats in [24] on the values \(\beta _i\in {\mathbb {Q}}\).

  7. Of course, since such classes are identified with rational McNaughton functions, that was already a consequence of Theorem 5.

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Correspondence to Marcelo Finger.

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This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)— Finance Code 001; M. Finger—Partially supported by Fapesp Projects 2015/21880-4 and 2014/12236-1 and CNPq Grant PQ 303609/2018-4.

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Finger, M., Preto, S. Probably Partially True: Satisfiability for Łukasiewicz Infinitely-Valued Probabilistic Logic and Related Topics. J Autom Reasoning 64, 1269–1286 (2020). https://doi.org/10.1007/s10817-020-09558-9

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