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Łukasiewicz μ-calculus

The paper explores properties of the Łukasiewicz μ-calculus, or Łμ for short, an extension of Łukasiewicz logic with scalar multiplication and least and greatest fixed-point operators (for monotone formulas). We observe that Łμ terms, with $n$ variables, define monotone piecewise linear functions from $[0, 1]^n$ to $[0, 1]$. Two effective procedures for calculating the output of Łμ terms on rational inputs are presented. We then consider the Łukasiewicz modal μ-calculus, which is obtained by adding box and diamond modalities to Łμ. Alternatively, it can be viewed as a generalization of Kozen's modal μ-calculus adapted to probabilistic nondeterministic transition systems (PNTS's). We show how properties expressible in the well-known logic PCTL can be encoded as Łukasiewicz modal μ-calculus formulas. We also show that the algorithms for computing values of Łukasiewicz μ-calculus terms provide automatic (albeit impractical) methods for verifying Łukasiewicz modal μ-calculus properties of finite rational PNTS's.