A toolkit for equivalence-based belief change

## Project description

Equibel is a Python package for working with consistency-based belief change in a graph-oriented setting.

## Currently Supported Platforms

Mac OS X (tested on OS X 10.10.5) with Python 2.7.x

64-bit Linux (tested on Ubuntu 14.04) with Python 2.7.x

Note that while Equibel is distributed as a Python package, the core of the system is implemented using Answer Set Programming (ASP), and relies on an underlying ASP solver called clingo, which is part of the Potsdam Answer Set Solving Collection (Potassco).

In particular, Equibel has two ASP-related dependencies: the Python gringo module, which provides an interface to an ASP solver from within Python, and asprin.parser, which is a component of the asprin preference-handling framework. asprin is described in more detail here, and can be downloaded from here.

The *Python* component of Equibel is highly portable across platforms; however, the gringo
and asprin.parser dependencies must be compiled for specific system configurations,
producing system-specific binaries. In order to simplify usage for some *common* system
configurations, Equibel includes pre-compiled binaries of these dependencies for
64-bit Ubuntu Linux and Mac OS.

*If Equibel does not function correctly once it is installed, this is likely due to the fact that the pre-compiled binaries are not compatible with your system.*
In this case, you must compile the dependencies manually, by downloading the required
components directly from Potassco.
In order to use your custom binaries with Equibel, it is recommended that you follow the
installation instructions given on the
Github project page, which involve downloading
the source code of Equibel to provide access to directories in which you can overwrite the
included binaries with your own.

## Installation

The following steps assume that you have the pip Python package manager installed. If you don’t have pip, you can get it here.

The pre-compiled gringo modules included with Equibel for either 64-bit Linux or Mac OS require a dependency called Threading Building Blocks (tbb).

The easiest way to install the tbb library on

**Mac OS**is to use Homebrew:$ brew install tbb

On

**Ubuntu Linux**, the tbb library can be installed using the apt package manager:$ sudo apt-get install libtbb-dev

Install Equibel using pip:

$ pip install equibel

If you are installing Equibel system-wide, you may need to use sudo:

$ sudo pip install equibel

## Quickstart

To use Equibel within a Python program, you need to import the equibel package. The statement import equibel as eb imports this package, and gives it a shorter alias eb. The following Python script creates a path graph, assigns formulas to nodes, finds the global completion, and prints the resulting formulas at each node:

```
import equibel as eb
if __name__ == '__main__':
# Create an EquibelGraph object, which represents a graph and
# associated scenario.
G = eb.EquibelGraph()
# Create nodes:
G.add_nodes_from([1, 2, 3, 4])
# Create edges:
G.add_edges_from([(1,2), (2,3), (3,4)])
# Add formulas to nodes:
G.add_formula(1, "p & q & r")
G.add_formula(4, "~p & ~q")
# Find the global completion of the G-scenario:
R = eb.global_completion(G, simplify=True)
# Pretty-print the resulting formulas at each node:
eb.print_formulas(R)
```

If the above code is saved in a file called completion.py, then it can be run by typing python completion.py at the command line, as follows:

$ python completion.py Node 1: p ∧ q ∧ r Node 2: r Node 3: r Node 4: r ∧ ¬p ∧ ¬q

If you get an error running this example, it is most likely due to the *dependencies* of
Equibel not being compatible with your platform. As noted above, Equibel includes
pre-compiled binaries of the Python gringo module, as well as of asprin.parser,
for 64-bit Linux distributions (tested on Ubuntu 14.04) and for Mac OS (tested on OSX 10.10.5).
If you are not using one of these systems, you will need to manually compile the gringo
and asprin.parser dependencies (see the Github page).

## Implemented Approaches

Equibel allows for experimentation with several different approaches to consistency-based belief change in a graph-oriented setting, namely:

Global completion

Simple iteration

Expanding iteration

Augmenting iteration

Ring iteration

The global completion operation is performed on an EquibelGraph G by
eb.global_completion(G); this performs a “one-shot” procedure to update
the information at every node in the graph, and thus is not an iterative approach. All
of the other approaches—*simple*, *expanding*, *augmenting*, and *ring*—can be performed
iteratively, and each one iterates to a *fixpoint*. The table below summarizes the Equibel
functions used to perform single iterations of each approach, as well as to find the fixpoints
reached by each approach:

Method |
Single Iteration |
Iterate to Fixpoint |
---|---|---|

Simple Iteration |
eb.iterate_simple(G) |
eb.iterate_simple_fixpoint(G) |

Expanding Iteration |
eb.iterate_expanding(G) |
eb.iterate_expanding_fixpoint(G) |

Augmenting Iteration |
eb.iterate_augmenting(G) |
eb.iterate_augmenting_fixpoint(G) |

Ring Iteration |
eb.iterate_ring(G) |
eb.iterate_ring_fixpoint(G) |

Each of the approaches has two separate implementations, corresponding to its equivalent *semantic*
and *syntactic* characterizations. In addition, there are two ways of performing the core optimization
procedure over equivalences, involving either *inclusion-based* or *cardinality-based* maximization.

Each function listed above can take three optional arguments:

method, which is a string that is either “semantic” or “syntactic”, representing the method to use when performing the approach; e.g. based on either the syntactic or semantic characterizations

The default method is

*semantic*To avoid typos when entering strings, Equibel has constants eb.SEMANTIC and eb.SYNTACTIC which equal the strings “semantic” and “syntactic”, respectively.

opt_type, which is a string that is either “inclusion” or “cardinality”, representing the type of maximization to be performed over equivalences

The default opt_type is

*inclusion*To avoid typos when entering strings, Equibel has constants eb.INCLUSION and eb.CARDINALITY which equal the strings “inclusion” and “cardinality”, respectively.

simplify, which is a Boolean flag specifying whether to simplify the final formulas at each node.

The default value for simplify is False

By definition, the semantic and syntactic characterizations of an approach yield
*equivalent results*; however, depending on the input scenario and type of approach, the
performance of the characterizations may differ significantly. A good example of this is
in the case of expanding iteration, where we have an *early-stoppping condition* over the
radius of the expanding neighbourhood when using the semantic characterization, but not when
using the syntactic characterization (causing the semantic characterization to be significantly
faster for large graphs in practice).

### Some Examples

To show how the method and opt_type arguments can be combined, we consider the following (by no means exhaustive) examples.

In the following example, we can see the difference between using inclusion-based optimization and cardinality-based optimization in the global completion:

```
import equibel as eb
if __name__ == '__main__':
# Creates a star graph with nodes [0, 1, 2, 3] and undirected edges [(0,1), (0,2), (0,3)]
G = eb.star_graph(3)
G.add_formula(1, 'p')
G.add_formula(2, 'p')
G.add_formula(3, '~p')
# Using inclusion-based maximization over equivalences
R_inclusion = eb.global_completion(G, method=eb.SEMANTIC, opt_type=eb.INCLUSION, simplify=False)
eb.print_formulas(R_inclusion)
# Using cardinality-based maximization over equivalences
R_cardinality = eb.global_completion(G, method=eb.SEMANTIC, opt_type=eb.CARDINALITY, simplify=False)
eb.print_formulas(R_cardinality)
```

Saving this code in a file inclusion_vs_cardinality.py and running it yields:

$ python inclusion_vs_cardinality.py Node 0: p ∨ ¬p Node 1: p Node 2: p Node 3: ¬p Node 0: p Node 1: p Node 2: p Node 3: ¬p

The following example function calls for the global completion operation show the flexible way in which options can be combined in Equibel:

R_semantic = eb.global_completion(G)

This function call computes the global completion of G. With no options explicitly specified, the defaults are used; thus, this call involves the

*semantic characterization*with*inclusion-based*optimization, and does not simplify the resultant formulas.With all options explicitly specified, the above function call is equivalent to R_semantic = eb.global_completion(G, method=eb.SEMANTIC, opt_type=eb.INCLUSION, simplify=False)

R_syntactic = eb.global_completion(G, method=eb.SYNTACTIC)

This finds the global completion of G, using the

*syntactic characterization*, the default*inclusion-based*optimization, and no simplification of formulas.

R_syntactic = eb.global_completion(G, method=eb.SYNTACTIC, opt_type=CARDINALITY)

This finds the global completion of G, using the

*syntactic characterization*,*cardinality-based*optimization, and no simplification of formulas.

R_syntactic = eb.global_completion(G, method=eb.SYNTACTIC, opt_type=CARDINALITY, simplify=True)

This finds the global completion of G, using the

*syntactic characterization*and*cardinality-based*optimization. With the simplify=True option, the resulting scenario will have simplified formulas for each node in the graph.

These options can be similarly combined for each of the iterative approaches, as shown in the following example calls:

R_semantic = eb.iterate_simple(G, method=eb.SEMANTIC, simplify=True)

This function call computes the graph and scenario that result from performing a single

*simple iteration*over G, using the*semantic characterization*with default*inclusion-based*optimization. With the simplify=True option, the resulting scenario will have simplified formulas for each node in the graph.

R_syntactic = eb.iterate_simple(G, method=eb.SYNTACTIC, simplify=True)

This call is similar to the previous call, except that it uses the

*syntactic characterization*of simple iteration, rather than the semantic characterization.

R_semantic_fixpoint = eb.iterate_simple_fixpoint(G, method=eb.SEMANTIC, opt_type=eb.CARDINALITY, simplify=True)

This computes the fixpoint reached by a sequence of

*simple iterations*starting from the graph and scenario represented by G, using the*semantic characterization*and*cardinality-based*optimization.

R_semantic = eb.iterate_expanding(G, simplify=True)

This function call computes the graph and scenario that result from performing a single

*expanding iteration*over G, using the default*semantic characterization*with default*inclusion-based*optimization. Since simplify=True, the resulting scenario will have simplified formulas for each node in the graph.

R_semantic = eb.iterate_augmenting_fixpoint(G, simplify=True)

This computes the fixpoint reached by a sequence of

*augmenting iterations*starting from the graph and scenario represented by G, using the default*semantic characterization*and*inclusion-based*optimization. Since simplify=True, the resulting scenario will have simplified formulas for each node in the graph.

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