A toolkit for equivalence-based belief change

`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.

- The default
`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.

- The default
`simplify`, which is a Boolean flag specifying whether to simplify the final formulas at each node.- The default value for
`simplify`is`False`

- The default value for

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)`

- This function call computes the global completion of
`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.

- This finds the global completion of
`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.

- This finds the global completion of
`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.

- This finds the global completion of

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.

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

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

- This computes the fixpoint reached by a sequence of
`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.

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

- This computes the fixpoint reached by a sequence of

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