xaddpy 0.2.8

XADD package in Python

Python Implementation of XADD

This repository implements the Python version of XADD (eXtended Algebraic Decision Diagrams) which was first introduced in Sanner et al. (2011); you can find the original Java implementation from here.

Our Python XADD code uses SymEngine for symbolically maintaining all variables and related operations, and PULP is used for pruning unreachable paths. Note that we only check linear conditionals. If you have Gurobi installed and configured in the conda environment, then PULP will use Gurobi for solving (MI)LPs; otherwise, the default solver (CBC) is going to be used. However, we do not actively support optimizers other than Gurobi for now.

Note that the implementation for EMSPO --- Exact symbolic reduction of linear Smart Predict+Optimize to MILP (Jeong et al., ICML-22) --- has been moved to the branch emspo.

You can find the implementation for the CPAIOR-23 work --- A Mixed-Integer Linear Programming Reduction of Disjoint Bilinear Programs via Symbolic Variable Elimination --- in examples/dblp.

Installation

Load your Python virtual environment then type the following commands for package installation

pip install xaddpy

# Optional: if you want to use Gurobi for the 'reduce_lp' method
# that prunes out unreachable partitions using LP solvers
pip install gurobipy    # If you have a license

Installing pygraphviz for visualization

With pygraphviz, you can visualize a given XADD in a graph format, which can be very useful. Here, we explain how to install the package.

To begin with, you need to install the following:

• graphviz
• pygraphviz

Make sure you have activated the right conda environment with conda activate YOUR_CONDA_ENVIRONMENT.

Step 1: Installing graphviz

1. For Ubuntu/Debian users, run the following command.

sudo apt-get install graphviz graphviz-dev

2. For Fedora and Red Hat systems, you can do as follows.

sudo dnf install graphviz graphviz-devel

3. For Mac users, you can use brew to install graphviz.

brew install graphviz

Unfortunately, we do not provide support for Windows systems, though you can refer to the pygraphviz documentation for information.

Step 2: Installing pygraphviz

1. Linux systems

pip install pygraphviz

2. MacOS

python -m pip install \
    --global-option=build_ext \
    --global-option="-I$(brew --prefix graphviz)/include/" \
    --global-option="-L$(brew --prefix graphviz)/lib/" \
    pygraphviz

Note that due to the default installation location by brew, you need to provide some additional options for pip installation.

Using xaddpy

You can find useful XADD usecases in the xaddpy/tests/test_bool_var.py and xaddpy/tests/test_xadd.py files. Here, we will first briefly discuss different ways to build an initial XADD that you want to work with.

Loading from a file

If you know the entire structure of an initial XADD, then you can create a text file specifying the XADD and load it using the XADD.import_xadd method. It's important that, when you manually write down the XADD you have, you follow the same syntax rule as in the example file shown below.

Below is a part of the XADD written in xaddpy/tests/ex/bool_cont_mixed.xadd:

...
        ( [x - y <= 0]
            ( [2 * x + y <= 0]
                ([x])
                ([y])
            )
            ( [b3]
                ([2 * x])
                ([2 * y])
            )
        )
...

Here, [x-y <= 0] defines a decision expression; its true branch is another node with the decision [2 * x + y <= 0], while the decision of the false branch is a Boolean variable b3. Similarly, if [2 * x + y <= 0] holds true, then we get the leaf value [x]; otherwise, we get [y]. All expressions should be wrapped with brackets, including Boolean variables. A SymEngine Symbol object will be created for each unique variable.

To load this XADD, you can do the following:

from xaddpy import XADD
context = XADD()
fname = 'xaddpy/tests/ex/bool_cont_mixed.xadd'

orig_xadd = context.import_xadd(fname)

Following the Java implementation, we call the instantiated XADD object context. This object maintains and manages all existing/new nodes and decision expressions. For example, context._id_to_node is a Python dictionary that stores mappings from node IDs (int) to the corresponding Node objects. For more information, please refer to the constructor of the XADD class.

To check whether you've got the right XADD imported, you can print it out.

print(f"Imported XADD: \n{context.get_repr(orig_xadd)}")

The XADD.get_repr method will return repr(node) and the string representation of each XADD node is implemented in xaddpy/xadd/node.py. Beware that the printing method can be slow for a large XADD.

Recursively building an XADD

Another way of creating an initial XADD node is by recursively building it with the apply method. A very simple example would be something like this:

from xaddpy import XADD
import symengine.lib.symengine_wrapper as core

context = XADD()

x_id = context.convert_to_xadd(core.Symbol('x'))
y_id = context.convert_to_xadd(core.Symbol('y'))

sum_node_id = context.apply(x_id, y_id, op='add')
comp_node_id = context.apply(sum_node_id, y_id, op='min')

print(f"Sum node:\n{context.get_repr(sum_node_id)}\n")
print(f"Comparison node:\n{context.get_repr(comp_node_id)}")

You can check that the print output shows

Sum node:
( [x + y] ) node_id: 9

Comparison node:
( [x <= 0] (dec, id): 10001, 10
         ( [x + y] ) node_id: 9 
         ( [y] ) node_id: 8 
)

which is the expected outcome!

Check out a much more comprehensive example demonstrating the recursive construction of a nontrivial XADD from here: pyRDDLGym/XADD/RDDLModelXADD.py.

Directly creating an XADD node

Finally, you might want to build a constant node, an arbitrary decision expression, and a Boolean decision directly. To this end, let's consider building the following XADD:

([b]
    ([1])
    ([x + y <= 0]
        ([0])
        ([2])
    )
)

To do this, we will first create an internal node whose decision is [x + y <= 0], the low and the high branches are [0] and [2] (respectively). Using SymEngine's S function (or you can use sympify), you can turn an algebraic expression involving variables and numerics into a symbolic expression. Given this decision expression, you can get its unique index using XADD.get_dec_expr_index method. You can use the decision ID along with the ID of the low and high nodes connected to the decision to create the corresponding decision node, using XADD.get_internal_node.

import symengine.lib.symengine_wrapper as core
from xaddpy import XADD

context = XADD()

# Get the unique ID of the decision expression
dec_expr: core.Basic = core.S('x + y <= 0')
dec_id, is_reversed = context.get_dec_expr_index(dec_expr, create=True)

# Get the IDs of the high and low branches: [0] and [2], respectively
high: int = context.get_leaf_node(core.S(0))
low: int = context.get_leaf_node(core.S(2))
if is_reversed:
    low, high = high, low

# Create the decision node with the IDs
dec_node_id: int = context.get_internal_node(dec_id, low=low, high=high)
print(f"Node created:\n{context.get_repr(dec_node_id)}")

Note that XADD.get_dec_expr_index returns a boolean variable is_reversed which is False if the canonical decision expression of the given decision has the same inequality direction. If the direction has changed, then is_reversed=True; in this case, low and high branches should be swapped.

Another way of creating this node is to use the XADD.get_dec_node method. This method can only be used when the low and high nodes are terminal nodes containing leaf expressions.

dec_node_id = context.get_dec_node(dec_expr, low_val=core.S(2), high_val=core.S(0))

Note also that you need to wrap constants with the core.S function to turn them into core.Basic objects.

Now, it remains to create a decision node with the Boolean variable b and connect it to its low and high branches.

from xaddpy.utils.symengine import BooleanVar

b = BooleanVar(core.Symbol('b'))
dec_b_id, _ = context.get_dec_expr_index(b, create=True)

First of all, you need to import and instantiate a BooleanVar object for a Boolean variable. Otherwise, the variable won't be recognized as a Boolean variable in XADD operations!

Once you have the decision ID, we can finally link this decision node with the node created earlier.

high: int = context.get_leaf_node(core.S(1))
node_id: int = context.get_internal_node(dec_b_id, low=dec_node_id, high=high)
print(f"Node created:\n{context.get_repr(node_id)}")

And we get the following print outputs.

Output:
Node created:
( [b]   (dec, id): 2, 9
        ( [1] ) node_id: 1 
        ( [x + y <= 0]  (dec, id): 10001, 8
                ( [0] ) node_id: 0 
                ( [2] ) node_id: 7 
        )  
) 

XADD Operations

XADD.apply(id1: int, id2: int, op: str)

You can perform the apply operation to two XADD nodes with IDs id1 and id2. Below is the list of the supported operators (op):

Non-Boolean operations

• 'max', 'min'
• 'add', 'subtract'
• 'prod', 'div'

Boolean operations

• 'and'
• 'or'

Relational operations

• '!=', '=='
• '>', '>='
• '<', '<='

XADD.unary_op(node_id: int, op: str) (unary operations)

You can also apply the following unary operators to a single XADD node recursively (also check UNARY_OP in xaddpy/utils/global_vars.py). In this case, an operator will be applied to each and every leaf value of a given node. Hence, the decision expressions will remain unchanged.

• 'sin, 'cos', 'tan'
• 'sinh', 'cosh', 'tanh'
• 'exp', 'log', 'log2', 'log10', 'log1p'
• 'floor', 'ceil'
• 'sqrt', 'pow'
• '-', '+'
• 'sgn' (sign function... sgn(x) = 1 if x > 0; 0 if x == 0; -1 otherwise)
• 'abs'
• 'float', 'int'
• '~' (negation)

The pow operation requires an additional argument specifying the exponent.

XADD.evaluate(node_id: int, bool_assign: dict, cont_assign: bool, primitive_type: bool)

When you want to assign concrete values to Boolean and continuous variables, you can use this method. An example is provided in the test_mixed_eval function defined in xaddpy/tests/test_bool_var.py.

As another example, let's say we want to evaluate the XADD node defined a few lines above.

x, y = core.symbols('x y')

bool_assign = {b: True}
cont_assign = {x: 2, y: -1}

res = context.evaluate(node_id, bool_assign=bool_assign, cont_assign=cont_assign)
print(f"Result: {res}")

In this case, b=True will directly leads to the leaf value of 1 regardless of the assignment given to x and y variables.

bool_assign = {b: False}
res = context.evaluate(node_id, bool_assign=bool_assign, cont_assign=cont_assign)
print(f"Result: {res}")

If we change the value of b, we can see that we get 2. Note that you have to make sure that all symbolic variables get assigned specific values; otherwise, the function will return None.

XADD.substitute(node_id: int, subst_dict: dict)

If instead you want to assign values to a subset of symbolic variables while leaving the other variables as-is, you can use the substitute method. Similar to evaluate, you need to pass in a dictionary mapping SymEngine Symbols to their concrete values.

For example,

subst_dict = {x: 1}
node_id_after_subs = context.substitute(node_id, subst_dict)
print(f"Result:\n{context.get_repr(node_id_after_subs)}")

which outputs

Result:
( [b]   (dec, id): 2, 16
        ( [1] ) node_id: 1 
        ( [y + 1 <= 0]  (dec, id): 10003, 12
                ( [0] ) node_id: 0 
                ( [2] ) node_id: 7 
        )  
) 

as expected.

XADD.collect_vars(node_id: int)

If you want to extract all Boolean and continuous symbols existing in an XADD node, you can use this method.

var_set = context.collect_vars(node_id)
print(f"var_set: {var_set}")

Output:
var_set: {y, b, x}

This method can be useful to figure out which variables need to have values assigned in order to evaluate a given XADD node.

XADD.make_canonical(node_id: int)

This method gives a canonical order to an XADD that is potentially unordered. Note that the apply method already calls make_canonical when the op is one of ('min', 'max', '!=', '==', '>', '>=', '<', '<=', 'or', 'and').

Variable Elimination

1. Sum out: XADD.op_out(node_id: int, dec_id: int, op: str = 'add')

Let's say we have a joint probability distribution function over Boolean variables b1, b2, i.e., P(b1, b2) as in the following example. P(b1, b2)=

( [b1]
    ( [b2] 
        ( [0.25] )
        ( [0.3] )
    )
    ( [b2]
        ( [0.1] )
        ( [0.35] )
    )
)

Notice that the values are non-negative and sum up to one, making this a valid probability distribution. Now, one may be interested in marginalizing out a variable b2 to get P(b1) = \sum_{b2} P(b1, b2). This can be done in XADD by using the op_out method.

Let's directly dive into an example:

# Load the joint probability as XADD
p_b1b2 = context.import_xadd('xaddpy/tests/ex/bool_prob.xadd')

# Get the decision index of `b2`
b2 = BooleanVar(core.Symbol('b2'))
b2_dec_id, _ = context.get_dec_expr_index(b2, create=False)

# Marginalize out `b2`
p_b1 = context.op_out(node_id=p_b1b2, dec_id=b2_dec_id, op='add')
print(f"P(b1): \n{context.get_repr(p_b1)}")

Output: 
P(b1): 
( [b1]  (dec, id): 1, 26
        ( [0.55] ) node_id: 25 
        ( [0.45] ) node_id: 24 
)

As expected, the obtained P(b1) is a function of only b1 variable, and the probabilities sum up to 1.

2. Prod out

Similarly, if we specify op='prod', we can 'prod out' a Boolean variable from a given XADD.

3. Max out (or min out) continuous variables: XADD.min_or_max_var(node_id: int, var: VAR_TYPE, is_min: bool)

One of the most interesting and useful applications of symbolic variable elimination is 'maxing out' or 'minning out' continuous variable(s) from a symbolic function. See Jeong et al. (2023) and Jeong et al. (2022) for more detailed discussions. Look up the min_or_max_var method in the xadd.py file. For now, we only support optimizing a linear or disjointly bilinear expressions at the leaf values and decision expressions.

As a concrete toy example, imagine the problem of inventory management. There is a Boolean variable d which denotes the level of demand (i.e., d=True if demand is high; otherwise d=False). Let's say the current inventory level of a product of interest is x \in [-1000, 1000]. Suppose we can place an order of amount a \in [0, 500] for this product. And we will have the following reward based on the current demand, inventory level, and the new order:

( [d]
    ( [x >= 150]
        ( [150 - 0.1 * a - 0.05 * x ] )
        ( [(x - 150) - 0.1 * a - 0.05 * x] )
    )
    ( [x >= 50]
        ( [50 - 0.1 * a - 0.05 * x] )
        ( [(x - 50) - 0.1 * a - 0.05 * x] )
    )
)

Though it is natural to consider multi-step decisions for this kind of problem, let's only focus on optimizing this reward for a single step, for the sake of simplicity and illustration.

So, given this reward, what we might be interested in is the maximum reward we can obtain, subject to the demand level and the current inventory level. That is, we want to compute max_a reward(a, x, d).

# Load the reward function as XADD
reward_dd = context.import_xadd('xaddpy/tests/ex/inventory.xadd')

# Update the bound information over variables of interest
a, x = core.Symbol('a'), core.Symbol('x')
context.update_bounds({a: (0, 500), x: (-1000, 1000)}) 

# Max out the order quantity
max_reward_d_x = context.min_or_max_var(reward_dd, a, is_min=False, annotate=True)
print(f"Maximize over a: \n{context.get_repr(max_reward_d_x)}")

Output:
Maximize over a: 
( [d]   (dec, id): 1, 82
        ( [-150 + x <= 0]       (dec, id): 10002, 81
                ( [-150 + 0.95*x] ) node_id: 72 anno: 0 
                ( [150 - 0.05*x] ) node_id: 58 anno: 0 
        )  
        ( [-50 + x <= 0]        (dec, id): 10003, 51
                ( [-50 + 0.95*x] ) node_id: 42 anno: 0 
                ( [50 - 0.05*x] ) node_id: 29 anno: 0 
        )  
)

To obtain this result, note that we should provide the bound information over the continuous variables. If not, then (-oo, oo) will be used as the bounds.

If we want to know which values of a will yield the optimal outcomes, we can apply the argmax operation. Specifically,

argmax_a_id = context.reduced_arg_min_or_max(max_reward_d_x, a)
print(f"Argmax over a: \n{context.get_repr(argmax_a_id)}")

Output:
Argmax over a: 
( [0] ) node_id: 0

Trivially in this case, not ordering any new products will maximize the one-step reward, which makes sense. A more interesting case would, of course, be when we have to make sequential decisions taking into account stochastic demands and the product level that changes according to the order amount and the demand. For this kind of problems, we suggest you take a look at Symbolic Dynamic Programming (SDP).

Now, if we further optimize the max_reward_d_x over x variable, we get the following:

# Max out the inventory level
max_reward_d = context.min_or_max_var(max_reward_d_x, x, is_min=False, annotate=True)
print(f"Maximize over x: \n{context.get_repr(max_reward_d)}")

# Get the argmax over x
argmax_x_id = context.reduced_arg_min_or_max(max_reward_d, x)
print(f"Argmax over x: \n{context.get_repr(argmax_x_id)}")

Output:
Maximize over x: 
( [d]   (dec, id): 1, 105
        ( [142.5] ) node_id: 102 anno: 99 
        ( [47.5] ) node_id: 89 anno: 85 
)
Argmax over x: 
( [d]   (dec, id): 1, 115
        ( [150] ) node_id: 99 
        ( [50] ) node_id: 85 
)

The results tells us that the maximum achievable reward is 142.5 when d=True, x=150 or 47.5 when d=False, x=50.

4. Max (min) out Boolean variables with XADD.min_or_max_var

Eliminating Boolean variables with max or min operations can be easily done by using the previously discussed min_or_max_var method. You just need to pass the Boolean variable to the method.

Definite Integral

Given an XADD node and a symbolic variable, you can integrate out the variable from the node. See test_def_int.py which provides examples of this operation.

Citation

Please use the following bibtex for citations:

@InProceedings{pmlr-v162-jeong22a,
  title = 	 {An Exact Symbolic Reduction of Linear Smart {P}redict+{O}ptimize to Mixed Integer Linear Programming},
  author =       {Jeong, Jihwan and Jaggi, Parth and Butler, Andrew and Sanner, Scott},
  booktitle = 	 {Proceedings of the 39th International Conference on Machine Learning},
  pages = 	 {10053--10067},
  year = 	 {2022},
  volume = 	 {162},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {17--23 Jul},
  publisher =    {PMLR},
}