It is known that a simple numeric planning problem (SNP) with one numeric variable is decidable but undecidable with three (Helmert 2002). A more recent result (Gnad et. al 2023) showed undecidability for two numeric and one propositional variable. In this paper, we show the decidability of SNP with exactly two numeric variables. For this, we first partition the state space into a finite number of regions and demonstrate the decidability of SNP when restricted to any of these regions. Afterwards, we develop a correct search algorithm that abstracts from these regions by tracking an infinite number of states following an arithmetic progression pattern. Finally, we prove termination of the search and draw conclusions about the reasons for undecidability for general SNP.

In this paper, we first define numeric additive planning (NAP), a planning formulation equivalent to Hoffmann's Restricted Tasks over Integers. Then, we analyze the minimal number of action repetitions required for a solution, since planning turns out to be decidable as long as such numbers can be calculated for all actions. We differentiate between two kinds of repetitions and solve for one by integer linear programming and the other by search. Additionally, we characterize the differences between propositional planning and NAP regarding these two kinds. To achieve this, we define so-called multi-valued partial order plans, a novel compact plan representation. Finally, we consider decidable fragments of NAP and their complexity.

Semiring semantics evaluates logical statements by values in some commutative semiring (K, +, ·, 0, 1). Random semiring interpretations, induced by a probability distribution on K, generalise random structures, and we investigate here the question of how classical results on first-order logic on random structures, most importantly the 0-1 laws of Glebskii et al. and Fagin, generalise to semiring semantics. For positive semirings, the classical 0-1 law implies that every first-order sentence is, asymptotically, either almost surely evaluated to 0 by random semiring interpretations, or almost surely takes only values different from 0. However, by means of a more sophisticated analysis, based on appropriate extension properties and on algebraic representations of first-order formulae, we can prove much stronger results.