PASTA: Probabilistic Answer Set programming for STAtistical probabilities

This software allows to perform inference in probabilistic answer set programs under the credal semantics (PASPs) and with statistical statements. Currently it supports:

• exact inference
• approximate inference via sampling
• abduction
• MAP/MPE inference
• parameter learning
• lifted inference
• decision theory

Installation

git clone https://github.com/damianoazzolini/pasta
cd pasta
pip install .

or

pip install git+https://github.com/damianoazzolini/pasta

You can also use Docker (image on Dockerhub) to test the application (it is not always updated):

docker container run -it damianodamianodamiano/pasta bash

then you are ready to go (follows the next instructions to run an example).

How to Use

After the installation, use

pastasolver --help

to see the available options.

Exact inference

The probability of a query in a PASP is given by a range $P(q) = [\underline{P}(q),\overline{P}(q)]$ where $\underline{P}(q) = \sum_{w_i \mid |AS(w_i)| > 0 \ \land \ m \in AS(w_i), \ m \models q} P(w_i)$ and $\overline{P}(q) = \sum_{w_i \mid \exists m \in AS(w_i), \ m \models q} P(w_i)$ where $P(w)$ is the probability of the world $w$ computed as $P(w) = \prod_{i \mid f_i = \top} \Pi_i \cdot \prod_{i \mid f_i = \bot} (1 - \Pi_i)$ and $AS(w)$ is the set of answer sets for a world $w$. The current algorithm adopts projected answer set enumeration to solve the task.

Example:

pastasolver examples/conditionals/bird_4_cond.lp --query="fly"

Asks the query fly for the program stored in examples/conditionals/bird_4_cond.lp. Expected result:

Lower probability for the query: 0.7
Upper probability for the query: 1.0

You can specify evidence with --evidence.

Abduction

This is still experimental and some features might not work as expected.

pastasolver examples/abduction/bird_4_abd_prob.lp --query="fly(1)" --abduction

MAP/MPE inference

pastasolver examples/map/color_map.lp --query="win" --map

Approximate inference

Available: sampling (--approximate), gibbs sampling (--gibbs), metropolis hastings sampling (--mh), rejection sampling (--rejection).

pastasolver examples/inference/bird_4.lp --query="fly(1)" --approximate

Use the flag --samples to set the number of samples (1000 by default), for example --samples=2000.

Use the flag --processes to set the number of processes (1 by default), for example --processes=8. The maximum number is 16.

Parameter Learning

pastasolver examples/learning/background_bayesian_network.lp --pl

Decision Theory

pastasolver examples/decision_theory/dummy.lp -dt

For normalization, you should use -dtn instead of -dt.

Handling Inconsistent Programs

The credal semantics requires that every world has at least one answer set. These programs are called consistent. Here, we make the same assumption.

If you ask a query on a program that is not consistent, you should get an error. You can normalize the probability with the flag --normalize.

Use PASTA as a Library

You can also use it as a library

from pasta.pasta_solver import Pasta

filename = "examples/inference/bird_4.lp"
query = "fly(1)"
solver = Pasta(filename, query)
lp, up = solver.inference()

print("Lower probability for the query " + query + ": " + str(lp))
print("Upper probability for the query " + query + ": " + str(up))

where filename and query are the name of the file and the query to ask.

You can also pass the file as a string, in this way:

program = """
0.5::bird(1).
0.5::bird(2).
0.5::bird(3).
0.5::bird(4).

% A bird can fly or not fly
0{fly(X)}1 :- bird(X).

% Constraint: at least 60% of the birds fly
:- #count{X:fly(X),bird(X)} = FB, #count{X:bird(X)} = B, 10*FB<6*B.
"""

query = "fly(1)"

solver = Pasta("", query)  # leave the filename as ""
lp, up = solver.inference(program)

print(f"Lower probability: {lp}")
print(f"Upper probability: {up}")

where program is a string containing your program.

All the above tasks can be used with the Python interface.

Caveat

Make sure to not write clauses with the same functor of probabilistic facts. For example, you should not write:

0.4::c(1).
c(X):- a(X).

In other words, probabilistic facts cannot appear as head atoms of any rule.

Syntax

Basically, PASTA (PASP) programs are ASP programs plus probabilistic facts. Probabilistic facts can be added with the syntax: prob::atom. where prob is a floating point number (0 < number <= 1) and atom is a standard ASP fact. For example, 0.5::a. states that a has probability 0.5.

For more examples, see the examples folder.

Using exact inference, you can also express statistical statements (x% of the y elements share the same behavior) with the syntax: (A | B)[LP,UP]. For example, "at least 60% of the birds fly" can be expressed with

(fly(X) | bird(X))[0.6,1].

See examples/conditionals/bird_4_cond.lp for an example.

Note: be super careful when using rules with disjunction in the head. You should replace them with choice rules.

Issues

Open an issue.

Description and How to Cite

The system and the various types of inferences are currently described in:

• Exact inference and statistical staments: Damiano Azzolini, Elena Bellodi, and Fabrizio Riguzzi. Statistical statements in probabilistic logic programming. In Georg Gottlob, Daniela Inclezan, and Marco Maratea, editors, Logic Programming and Nonmonotonic Reasoning, pages 43--55, Cham, 2022. Springer International Publishing.
• Abduction (preliminary): Damiano Azzolini, Elena Bellodi, and Fabrizio Riguzzi. Abduction in (probabilistic) answer set programming. In Roberta Calegari, Giovanni Ciatto, and Andrea Omicini, editors, Proceedings of the 36th Italian Conference on Computational Logic, volume 3204 of CEUR Workshop Proceedings, pages 90--103, Aachen, Germany, 2022. Sun SITE Central Europe.
• MAP/MPE inference: Damiano Azzolini, Elena Bellodi, and Fabrizio Riguzzi. Map inference in probabilistic answer set programs. In Agostino Dovier, Angelo Montanari, and Andrea Orlandini, editors, AIxIA 2022 -- Advances in Artificial Intelligence, pages 413--426, Cham, 2023. Springer International Publishing.
• Approximate inference: Damiano Azzolini, Elena Bellodi, and Fabrizio Riguzzi. Approximate inference in probabilistic answer set programming for statistical probabilities. In Agostino Dovier, Angelo Montanari, and Andrea Orlandini, editors, AIxIA 2022 -- Advances in Artificial Intelligence, pages 33--46, Cham, 2023. Springer International Publishing.
• Lifted inference: Damiano Azzolini and Fabrizio Riguzzi. Lifted inference for statistical statements in probabilistic answer set programming. International Journal of Approximate Reasoning, 163:109040, 2023.
• Parameter learning: Damiano Azzolini, Elena Bellodi, and Fabrizio Riguzzi. Learning the parameters of probabilistic answer set programs. In Stephen H. Muggleton and Alireza Tamaddoni-Nezhad, editors, Inductive Logic Programming - ILP 2022, volume 14363 of Lecture Notes in Computer Science, pages 1--14, Cham, 2024. Springer Nature Switzerland.
• Decision theory: in press