Narrow Results

We investigate formal properties, mainly issues connected with propositional logic, of reaction systems introduced by Ehrenfeucht and Rozenberg. We are concerned only with the most simple variant of the systems. Basic properties of propositional formulas are expressed in terms of reaction systems. This leads to NP-completeness (resp. co-NP-completeness) of many problems concerning reaction systems. Among such problems are: (i) deciding whether the function defined by the system is total, (ii) determining the inverse of the function, (iii) deciding whether state sequences always end with a loop. Propositional formulas with monotonic truth-functions yield a particularly simple representation in terms of reaction systems.

In this paper, we first propose two Chebyshev-type inequalities associated with the general fractional-order (Yang–Abdel–Aty–Cattani) integrals with the Rabotnov fractional-exponential kernel under the condition that $\mu$ and $\nu$ are synchronous functions. What is more, by the mathematical induction, we prove a new Chebyshev-type inequality in the case that ${({\mu }_i)}_{i=1,\dots ,n}$ be $n$ positive increasing functions. Finally, we introduce a novel Chebyshev-type inequality via the general fractional-order integrals with the Rabotnov fractional-exponential kernel under the condition that $\mu$ and $\nu$ are monotonic functions.