# Generalization of ALMM Based Learning Method for Planning and Scheduling
## 2 Algebraic-Logical Meta-Model of Multistage Decision Processes
As a result, all limitations concerning the decisions in a given state \(s\) can be defined in a convenient way by means of so-called sets of possible decisions \(U_{p}(s)\), and defined as: \(U_{p}(s)=\{u\in U:(u,s)\in Domf\}\)._  
_The cMDP is obtained by reducing a generalized state to a proper state with a transition function \(f=f_{x}\). For both defined types of the multistage decision processes, in the most general case, sets \(U\) and \(X\) may be presented as a Cartesian product \(U=U^{1}\times U^{2}\times...\times U^{m}\), \(X=X^{1}\times X^{2}\times...\times X^{n}\), i.e., \(u=(u^{1},u^{2},...,u^{m})\), \(x=(x^{1},x^{2},...,x^{n})\)._  
_In particular, \(u^{i}\), \(i=1,2,...,m\) represent separate decisions that must or may be taken simultaneously and relate to particular objects. Values of particular coordinates of a state or a decision may be names of elements (symbols) as well as some objects (e.g., finite set, sequence etc. )._  
_There are no limitations imposed on a type of elements of the sets; in particular they do not have to be numerical. Thus, values of particular co-ordinates of a state or a decision may be names of elements (symbols) as well as some objects (e.g., finite set, sequence etc.). The sets \(X_{N},S_{N},X_{G},S_{G}\) and \(U_{p}\) are formally defined with the use of both algebraic and logical formulae, hence the algebraic-logic model descriptor._  
The most significant characteristic, unique for the proposed ALMM of MDP paradigm, is the fact that:  
* Proper state coordinates can be higher order variables;
* Decision \(u\) can take a complex form, consisting of individual decisions related to various issues/objects; these individual decisions may or have to be taken or executed at the same time;
* The transition function is defined as a partial one, which allows taking into account various different restrictions on sensible decisions in different states. Based on the meta-model recalled herein, AL models may be created for individual problems consisting of seeking admissible or optimal solutions. In the case of an admissible solution, an AL model is equivalent to a suitable multistage decision process, hence it is denoted as process \(\mathbf{P}\). An optimization problem is then denoted as a \((\mathbf{P},\mathbf{Q})\) pair, where \(\mathbf{Q}\) is a criterion. An optimization task (instance of the problem) is denoted as a \((P,Q)\), where \(P\) is an instance of the process \(\mathbf{P}\) and is named an individual process. At the same time, an individual process \(P\) is represented by a set of its trajectories. A finite trajectory is a sequence of consecutive states from the initial state to a final state (goal, not admissible or blind one), computed by the transition function. Though trajectories may be finite or infinite, for further consideration we assume only finite ones. For \(s_{0}\), it is assumed that no state of a trajectory, apart from the last one, may belong to the set \(S_{N}\). Only a trajectory that ends in the set of goal states is admissible (feasible). The decision sequence determining an admissible trajectory is an admissible decision sequence. Obviously, the set of finite trajectories corresponds to the state graph of the process \(P\).