Truncating and Shifting Weights for Max-Plus Automata

In this paper, for any real number $\lambdaup$, we transform the complete max-plus semiring $\mathbb{R}_\infty$ into a commutative, complete, additively idempotent semiring $\mathbb{R}_\infty^\lambdaup$, called the lower $\lambdaup$-truncation~of~$\mathbb{R}_\infty$. It is obtained by removing from $\mathbb{R}_\infty$ all real numbers smaller than $\lambdaup$, inheriting the addition operation, shifting the original products by $-\lambdaup$, and appropriately modifying the residuum operation. The purpose of lower truncations is to transfer the iterative procedures for computing the greatest presimulations and prebisimulations between max-plus automata, in cases where they cannot be completed in a finite number of iterations over $\mathbb{R}_\infty$, to $\mathbb{R}_\infty^\lambdaup$, where they could terminate in a finite number of iterations. For instance, we prove that this necessarily happens when working with max-plus automata with integer weights. We also show how presimulations and prebisimulations computed over $\mathbb{R}_\infty^\lambdaup$ can be transformed into presimulations and prebisimulations between the original automata over $\mathbb{R}_\infty$. Although they do not play a significant role from the standpoint of computing presimulations and prebisimulations, for theoretical reasons we also introduce two types of upper truncations of the complete max-plus semiring $\mathbb{R}_\infty$.