We investigate the computational complexity of various satisfiability problems in Łukasiewicz logic, restricting attention to valuations in the standard MV-algebra [0,1]. Specifically, we focus on maximal r -satisfiability – the task of maximizing the number of formulas whose valuation is at least a given rational r ∈ (0, 1]. We also consider the decisional and weighted versions of this problem, as well as the partial (weighted) r -satisfiability problem.

A proof for the NP-containment for the probabilistic coherence problem over events represented by formulas of the infinite-valued Łukasiewicz logic was proposed in [1] . The geometric and combinatorial argument to prove that complexity bound contains a mistake that is fixed in the present paper. Actually we present two ways to restore that imprecise claim and, by doing so, we show that the main result of that paper is indeed valid.

Abstract In this paper we aim at completing the study of $\sigma $-complete Riesz MV-algebras that started in Di Nola et al. (2018, J. Logic Comput., 28, 1275–1292). To do so, we discuss polynomials, algebraic geometry and dualities in the infinitary variety of such algebras. In particular, we characterize the free objects as algebras of Baire-measurable functions and we generalize two dualities, namely the Marra–Spada duality and the Gelfand duality, obtaining a duality with basically disconnected compact Hausdorff spaces and an equivalence with Rickart $C^*$-algebras.

We extend Łukasiewicz logic obtaining the infinitary logic Infinitary Riesz Logic (⁠|$\mathcal{IR}$|Ł) whose models are algebras C(X, [0, 1]), where X is a basically disconnected compact Hausdorff space. Equivalently, our models are unit intervals in Dedekind |$\sigma $|-complete Riesz spaces with strong unit. The Lindenbaum–Tarski algebra of |$\mathcal{IR}$|Ł is, up to isomorphism, an algebra of [0, 1]-valued Borel functions. Finally, our system enjoys standard completeness with respect to the real interval [0, 1].