human
I am working on the file "Dynamic programming class.ipynb".
The first task is:
$$
v(s) = \min_x(cost(x,s) + v(new state(x,s)))
$$
Can you write Python code for it?
assistant
#Here is the Python code you requested.

class DynamicProgram(object):
    """
    Generate a dynamic program to find a set of optimal descissions using the HJB.
    
    define the program by:
    
    Setting intial states via:            set_inital_state(list or int)
    Setting the number of steps via:      set_step_number(int) 
    
    Add a set of descissions:             add_decisions_set(set)
    Add a cost function:                  add_cost_function(function in terms of state )
    
    Add a state change equation:          add_state_eq(function(state))
    Add an expression for the last value: add_final_value_expression(function(state,settings))
    Add limits on the states:             add_state_limits(lower=list or int,upper = list or int)

    See below for examples:
    
    """
    def __init__(self):
        self.settings = {
                        'Lower state limits' : [],
                        'Upper state limits' : [],    
                        'x_set' : set(),
                        'cache' : {},}
        
    def add_state_eq(self,function):
        """Returns a tuple describing the states. 
        Remember to increment the first state, b convention the number of steps.
        Load additional parameters (usually needed for cost and state value with global variables)
        
        Example of a function that changes the state by the decission:
        def new_state(x,s):
            return (s[0]+1,s[1]+x) #Return a tuple, use (s[:-1]+(5,)) to slice tuples.  
                
        """
        self.settings['State eq.'] = function
        
    def add_cost_function(self,function):
        """
        Returns a float or integer describing the cost value.
        Load additional parameters (usually needed for cost and state value with global variables)
        
        Example is a function that simply returns the decision as cost:
        def cost(x,s):
            return x
            
        """
        self.settings['Cost eq.'] = function
    
    def add_final_value_expression(self, function,):
        """
     ...