We define a formal model of dynamic programming algorithms which we 
call Prioritized Branching Programs (pBP). Our model is a generalization 
of the BT model of Alechmovich et al (AlekhnovichBBIMP05), which is in 
turn a generalization of the priority algorithms model of Borodin, Nielson 
and Rackoff.  One of the distinguishing features of these models is that they 
not only capture large classes of algorithms generally considered to be 
greedy, backtracking or dynamic programming algorithms, but they also allow 
characterizations of their limitations. Hence they give meaning to the 
statement that a given problem can or cannot be solved by dynamic programming.  
After defining the model, we prove three main results: 
(i) that certain types of natural restrictions of our seemingly more powerful 
model can be simulated by the BT model; 
(ii) that while the pBP model is capable of implementing the classic 
Bellman-Ford algorithm for finding the shortest path in a graph without 
negative cycles, the pBT model is incapable of solving the promise version of 
this problem efficiently (that is, every such algorithm must have high 
complexity on some instance, although it may be an instance that does not 
satisfy the promise of no negative cycles); 
(iii) that our model has very real limitations, namely that maximum unweighted 
bipartite matching cannot be efficiently computed in it without high-precision 
numbers. Since using subproblems of numeric complexity well beyond that of the 
input problem seems unnatural for dynamic programming, this suggests that there 
are problems that can be solved efficiently by network flow algorithms and by 
simple linear programming that cannot be solved by natural dynamic programming 
approaches.