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Popper

Popper is an inductive logic programming system. If you use Popper, please cite the paper Learning programs by learning from failures (MLJ 2021).

Requirements

Installation

Install Popper with the command pip install git+https://github.com/logic-and-learning-lab/Popper@main

Command line usage

Run Popper with the command python popper.py <input dir>. For instance, python popper.py examples/zendo1 produces:

11:11:14 Generating programs of size: 3
11:11:14 Generating programs of size: 4
11:11:14 Generating programs of size: 5
11:11:14 Generating programs of size: 6
11:11:14 ********************
11:11:14 New best hypothesis:
11:11:14 tp:19 fn:1 tn:20 fp:0 size:20
11:11:14 zendo(A):- piece(A,B),red(B),coord1(B,C),size(B,C).
11:11:14 zendo(A):- piece(A,C),contact(C,B),blue(B),rhs(C).
11:11:14 zendo(A):- piece(A,C),contact(C,B),red(B),upright(B).
11:11:14 zendo(A):- piece(A,C),contact(C,B),red(B),lhs(B).
11:11:14 ********************
********** SOLUTION **********
Precision:1.00 Recall:1.00 TP:20 FN:0 TN:20 FP:0 Size:6
zendo(A):- piece(A,D),red(D),contact(D,B),size(B,C),small(C).
******************************

Example problem

Popper requires three input files:

  • an examples file
  • a background knowledge (BK) file
  • a bias file

An examples file contains positive and negative examples of the relation you want to learn:

pos(grandparent(ann,amelia)).
pos(grandparent(steve,amelia)).
pos(grandparent(ann,spongebob)).
neg(grandparent(amy,amelia)).

A BK file contains other information about the problem:

mother(ann,amy).
mother(ann,andy).
mother(amy,amelia).
mother(linda,gavin).
father(steve,amy).
father(steve,andy).
father(gavin,amelia).
father(andy,spongebob).

A bias file defines the search space of Popper. Predicate declarations tell Popper which predicate symbols it can use in the head (head_pred) or body (body_pred) of a rule, such as:

head_pred(grandparent,2).
body_pred(mother,2).
body_pred(father,2).

These declarations say that Popper can use the symbol grandparent with two arguments in the head of a rule and mother or father in the body, also each with two arguments.

Noisy examples

Popper can learn from noisy data with the --noisy flag. Popper learns the minimal description length program.

Recursion

Popper can learn recursive programs where a predicate symbol appears in both the head and body of a rule, such as to find a duplicate element (python popper.py examples/find-dupl) in a list:

f(A,B):- tail(A,C),head(A,B),element(C,B).
f(A,B):- tail(A,C),f(C,B).

To enable recursion, add enable_recursion. to the bias file. However, recursion is expensive, so it is best to avoid it if possible.

Types

Popper supports type annotations in the bias file. A type annotation is of the form type(p,(t1,t2,...,tk) for a predicate symbol p with arity k, such as:

type(head,(list,element)).
type(tail,(list,list)).
type(length,(list,int,)).
type(empty,(list,)).
type(prepend,(element,list,list)).

Types are optional but can substantially reduce learning times.

Directions

Prolog often requires arguments to be ground. For instance, when asking Prolog to answer the query:

X is 3+K.

It throws an error:

ERROR: Arguments are not sufficiently instantiated

To avoid these issues, Popper supports optional direction annotations. A direction annotation is of the form direction(p,(d1,d2,...,dk) for a predicate symbol p with arity k, where each di is either in or out. An in variable must be ground when calling the relation. By contrast, an out variable need not be ground. Here are example directions:

direction(head,(in,out)).
direction(tail,(in,out)).
direction(length,(in,out)).
direction(prepend,(in,in,out)).
direction(geq,(in,in)).

Popper cannot learn with partial directions. If you provide them, you must provide them for all relations.

Bias

Popper has two important bias settings:

  • max_vars(N) sets the maximum number of variables in a rule to N (default: 6)
  • max_body(N) sets the maximum number of body literals in a rule to N (default: 6)

These settings greatly influence performance. If the values are too high, Popper might struggle to find a solution. If the settings are too low, the search space might be too small to contain a good solution.

Predicate invention

Popper supports automatic predicate invention (PI). With PI enabled, Popper (python popper.py examples/kinship-pi) learns the following program:

grandparent(A,B):-inv1(C,B),inv1(A,C).
inv1(A,B):-mother(A,B).
inv1(A,B):-father(A,B).
% Precision:1.00, Recall:1.00, TP:5, FN:0, TN:1, FP:0

To enable PI, add the setting enable_pi. to the bias file. However, predicate invention is currently very expensive so it is best to avoid it if possible.

Popper settings

  • --noisy (default: false) learn from noisy (misclassified examples)
  • --stats (default: false) shows runtime statistics
  • --debug (default: false) runs in debug mode
  • --quiet (default: False) runs in quiet mode
  • --timeout (default: 600 seconds) sets a maximum learning time
  • --eval-timeout (default: 0.001 seconds) sets a maximum example testing time. This flag only applies when learning recursive programs.
  • --solver {clingo,rc2,uwr,wmaxcdcl}(default: rc2) which exact solver to use
  • --anytime-solver {wmaxcdcl,nuwls}(default: None) which anytime solver to use
  • --anytime-timeout (default: 10 seconds) sets the maximum time allowed by the anytime solver

Solvers

Popper uses various MaxSAT solvers. By default, Popper uses the RC2 exact solver provided by PySAT. However, we have found that other solvers work much better. Popper supports these solvers:

  • UWrMaxSat (exact)
  • WMaxCDCL (exact)
  • NuWLS (anytime)

You can download and compile these solvers from the MaxSAT 2023 evaluation website. We strongly recommend using these solvers, especially NuWLS To use them, ensure that the solver is available on your path. See the install solvers file for help.

Performance tips

  • Transform your BK to Datalog, which allows Popper to perform preprocessing on the BK
  • Use one of the MaxSAT solvers, above, especially the NuWLS anytime solver.
  • Use 6 variables or fewer
  • Avoid recursion and predicate invention

Library usage

You can import Popper and use it in your Python code like so:

from popper.util import Settings, print_prog_score
from popper.loop import learn_solution

settings = Settings(kbpath='input_dir')
prog, score, stats = learn_solution(settings)
if prog != None:
    print_prog_score(prog, score)