GenB: A General Solver for AGM Revision
Aaron Hunter(B) and Eric Tsang
British Columbia Institute of Technology, Burnaby, Canada
aaron [email protected]
Abstract. We describe a general tool for solving belief revision problems with a range of different operators. Our tool allows a user to flexibly
specify a total pre-order over states, using simple selection boxes in a
graphic user interface. In this manner, we are able to calculate the result
of any AGM revision operator. The user is also able to specify so-called
trust partitions to calculate the result of trust-sensitive revision. The
overall goal is to provide users with a simple tool that can be used in
applications involving AGM-style revision. While the tool can be demonstrated and tested as a standalone application with a fixed user interface,
what we have actually developed is a set of libraries and functions that
can flexibly be incorporprated in other systems. It is anticipated that
this tool will be useful for experimentation, education, and prototyping
to solve problems in formal reasoning.
1
Introduction
We describe GenB, a general tool for solving belief revision problems. While the
theory of belief revision has been well-studied, there has been comparatively little
work on the development of tools to calculate the result of revision. Moreover,
the belief revision solvers that have appeared in the literature have commonly
focused on implementing a specific revision operator as effeciently as possible.
Our approach differs in that we develop a general tool that can capture any
AGM revision operator.
Due to well-known results on the complexity of revision [4], it is certainly
not possible to develop a solver that runs quickly for all instances of any given
AGM revision operator. Nevertheless, we suggest that a general tool that can
calculate the result of AGM revision would be useful in the development of
prototype reasoning systems. For instance, many reasoning problems involve
selective revision, where the input is pre-processed in some way [5]. Our tool can
be extended to solve such problems. GenB can also be useful for solving inverse
belief revision problems. For example, there are problems in which we have data
about the revision that has occurred, and we want to determine what sort of
plausibility ordering was used [9]. There are also problems where we know the
revision operator, but we want to find a formula to announce in order to bring
about a certain result [7]. Our tool can be useful for addressing this kind of
problem.
c Springer International Publishing AG 2016
L. Michael and A. Kakas (Eds.): JELIA 2016, LNAI 10021, pp. 564–569, 2016.
DOI: 10.1007/978-3-319-48758-8 40
GenB: A General Solver for AGM Revision
565
This paper makes several contributions to existing work. First, we present
a fully general solver that can capture all AGM revision operators. The development of such a tool should, in principle, facilitate prototyping for systems
that involve belief revision. Second, implementing a general solver allows us to
explore a new approach to revision that can be used in cases where no faithful
assignment is forthcoming. Finally, to the best of our knowledge, this is the first
implemented system for solving problems involving selective revision and trust.
2
Preliminaries
Belief revision is the process in which an agent’s beliefs change to incorporate new
information. One of the most influential models of belief revision has been the
AGM approach [1]. In AGM belief revision, the beliefs of an agent are represented
by a belief set, which is just a set of formulas K that is closed under consequence.
An AGM belief revision operator ∗ is a function that takes a belief set K and
a new formula φ as input, and it returns a new belief set K ∗ φ. Moreover, an
AGM revision operator must satisfy the so-called AGM postulates for revision.
We use the term state to refer to a propositional interpretation over the
underlying signature, and we use the term belief state to refer to a set of states.
A faithful assignment is a function that maps every belief set K to a total preorder ≺K over states, where the models of K are minimal. A representation result
has been proved to show that every AGM revision operator is characterized by
a faithful assignment [8]. To be slightly more precise, for every AGM revision
operator ∗, there is a faithful assignment such that K ∗ φ is the set of formulas
true in the ≺K -minimal models of φ. The converse also holds.
3
Implementation
We provide the user with a mechanism for entering a belief set and a new formula
for revision. Each of these is entered as a propositional formula; the underlying
vocabulary is just the set of proprositional variables that occur in the input.
In addition to these inputs, the user must also specify a total pre-order over
possible states. The basic revision algorithm operates as follows:
1.
2.
3.
4.
5.
Input: sentence φ, comparator Set sentenceM odels = {M | M |= φ}
Set nearestM odel = M where M is -minimal in sentenceM odels
Set nearestM odels = {M | M M M }
Return nearestM odels
GenB is implemented in Kotlin, which is essentially a variant of Java1 . Internally, we use a Comparator object to capture the ordering over possible states.
However, in order to facilitate the use of our program, we give users a few basic
revision operators to choose from. The GenB interface is actually inspired by the
interface of COBA 2.0, a belief revision system described in [3]. The interface is
displayed in Fig. 1.
1
Technical documentation and download available at http://kotlinlang.org.
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A. Hunter and E. Tsang
Fig. 1. GenB interface
The components identified in the interface above are as follows:
1. File Menu: Allows inputs to be loaded or saved to a file.
2. Initial Belief State: The initial beliefs of the agent can be entered and
displayed as a set of models or a set of formulas.
3. Sentences for Revision: Entry and display can be through sets of models
or formulas.
4. Resulting Belief State: Display can be through sets of models or formulas.
5. Belief Revision Strategy: This is a combo box that allows the user to
specify different belief revision strategies, to be discussed below.
6. Trust Partition: Allows a trust partition to be specified, for trust-sensitive
belief revision.
7. Revise Button: Pressing the revise button causes the revision to be performed, and displays the output.
8. Commit button: Pressing the commit button moves the new belief state to
the initial belief box.
9. Display Method: Combo boxes for toggling between different ways to show
information.
Users can toggle between the different display modes freely, switching from the
model display to several different formula options.
4
4.1
Revision Strategies
Hamming Distance
GenB supports several different revision strategies. For the purposes of this
paper, we focus primarily on the so-called Comparator-based revision strategies. These strategies correspond to AGM revision.
GenB: A General Solver for AGM Revision
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The first revision option is the Dalal revision operator based on the Hamming
distance [2]. In order to use this operator, the user simply selects Hamming
Distance from the combo box. No additional information is required for revision,
because GenB is able to automatically define the comparator by calculating
the minimal Hamming distance from each state to a model of the input sentence.
Example 1. Suppose that we initially believe both p and q are true, but we want
to revise by ¬p ∨ ¬q. We proceed as follows:
1.
2.
3.
4.
Click Add on the Initial Belief State panel, enter p and q in the pop-up.
Click Add on the Sentences for Revision panel, enter −p or −q in the pop-up.
Select the Hamming Distance option.
Click the Revise button.
The resulting beliefs will be ¬p ∨ ¬q, as expected.
4.2
Weighted Hamming Distance
In some cases, we want to use a variation of the Dalal operator where certain
variables are understood to carry greater “importance.” In GenB, this can be
done by selecting the Weighted Hamming Distance, where each propositional
formula has an associated priority. In this manner, the distance between two
states will vary depending on the ‘wieght’ of the variables where they differ.
Example 2. Modifying the previous example, if we select the Weighted Hamming
Distance at step 3, we will be prompted to list the propositional variables in order
of importance. If we enter p, q as the ordering, then states are considered more
plausible when the they differ on q as opposed to when they differ on p. As such,
in this case, GenB will return q as the resulting set of beliefs.
4.3
Ordered Sets
The third revision option is based on Ordered Sets. When this option is selected,
the user is prompted to provide an ordered list of formulas. Intuitively, the user
is actually specifying the most plausible states for the comparator. Hence, if a
user enters p, q in the corresponding dialog, they are saying that models of p are
more plausible than models of q. It is possible (though tedious) to specify any
total pre-order over states in this manner. The system also allows a randomized
ordering, for users that do not want to produce a list.
We suggest that Ordered Sets revision can be used for experimentation. For
example, we can iterated over all orderings and then look at credulous or skeptical reasoning. By using GenB as a library, we can address this kind of novel
approach to revision.
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4.4
A. Hunter and E. Tsang
Trust-Sensitive Revision
One issue with AGM revision for practical applications is the fact that the source
of new information is always trusted. This can be addressed by trust-sensitive
belief revision operators [6]. In trust-sensitive revision, we associate a partition
Π over states with each information source. When we are given a sentence φ for
revision, we first find {s | Π(s, s ) and s |= φ}. Rather than revising by φ, we
then revise by this set. In other words, we consider φ to be evidence for every
state that is Π-related to a model of φ. In this manner, the partition Π identifies
the states that we trust a particular source to be capable of distinguishing.
GenB allows the user to define a trust-partition, and then perform trustsensitive revision.
Example 3. Suppose that a particular agent is trusted on the value of the variable
S, but not on the variable D. Using GenB, we can model and solve this problem
as follows:
1. Click Add on the Initial Belief State panel, enter −S and D in the pop-up.
2. Click Add on the Sentences for Revision panel, enter S and −D in the pop-up.
3. Select the formula option on the Trust Partition combo box, enter S in the
pop-up.
4. Click the Revise button.
Note that, if step 3 is omitted, then normal revision is performed and the result
will include −D. However, using the trust-partition option, the result is not
forced to include this information.
5
Discussion
In this preliminary report, we have described the development of an automated
tool that can calculate the result of any AGM revision operator. To demonstrate
potential applications, we included an implementation of trust-sensitive revision
in our tool. We have addressed several interface design issues, such as providing
the user a means for specifying a total pre-order in a flexible manner. In practice,
however, we expect GenB to be deployed as a set of libraries that can easily be
integrated into more complex reasoning systems.
There are several directions for future research. First, although the practical
performance is strong, the effeciency of the system could be improved. Second,
we are interested in using GenB to find propositional announcements for belief
revision. Following the general approach in [7], we would like to use GenB to
implement a practical robot controller involving public announcements. Finally,
we would like to use GenB to address problems in security. To the best of our
knowledge, no existing protocol verification tools actually implement any approach to belief revision. By implementing a general solver, we hope to explore the
utlility of belief revision for security through exploration and experimentation.
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