EFFICIENT KNOWLEDGE BASE MANAGEMENT IN DCSP

International Journal of Ad hoc, Sensor & Ubiquitous Computing (IJASUC) Vol.1, No.3, September 2010
DOI : 10.5121/ijasuc.2010.1302 21
EFFICIENT KNOWLEDGE BASE MANAGEMENT IN
DCSP
Hong Jiang
Mathematics & Computer Science Department, Benedict College, USA
jiangh@benedict.edu
ABSTRACT
DCSP (Distributed Constraint Satisfaction Problem) has been a very important research area in AI
(Artificial Intelligence). There are many application problems in distributed AI that can be formalized as
DSCPs. With the increasing complexity and problem size of the application problems in AI, the required
storage place in searching and the average searching time are increasing too. Thus, to use a limited
storage place efficiently in solving DCSP becomes a very important problem, and it can help to reduce
searching time as well. This paper provides an efficient knowledge base management approach based on
general usage of hyper-resolution-rule in consistence algorithm. The approach minimizes the increasing of
the knowledge base by eliminate sufficient constraint and false nogood. These eliminations do not change
the completeness of the original knowledge base increased. The proofs are given as well. The example
shows that this approach decrease both the new nogoods generated and the knowledge base greatly. Thus
it decreases the required storage place and simplify the searching process.
KEYWORDS
Distributed Constraint Satisfaction Problem, hyper-resolution-rule, knowledge base management
1. INTRODUCTION
Many problem solving techniques in AI (Artificial Intelligence) are searching based. And the
problems that have been addressed upon search algorithms can be categorized as three classes as
mentioned in [1]: (1) Constraint satisfaction problems (CSP), (2) path-finding problems, and (3)
two-player games. Among them, CSP and distributed version of CSP become a very important
research area, because many problems that arise in multiagent systems can be reduced to a
distributed constraint satisfaction problem(DCSP) and this approach has led to many successful
multiagent applications.
Hyper-resolution-rule [2, 3] is a basic unit used in many famous algorithms used in solving
DCSPs, such as asynchronous backtracking algorithm(ABT) [4–8] and asynchronous weak-
commitment search algorithm [9–11]. However the detailed usage of the hyper-resolution-rule in
those algorithms is not clearly defined, and the general usage of the hyper-resolution-rule for
consistence algorithm produces polynomial space usage. This is still fordable for some small size
problems, but for bigger size problems polynomial space usage will become a big problem.
Practically, most of the problems in multiagent system become bigger and bigger. Then the space
usage problem can not be ignored, and research on how to efficiently use limited storage place in
solving DCSP becomes important.
This paper provides an efficient knowledge base management approach based on general usage of
hyper-resolution-rule in consistency algorithm [12]. This approach minimizes the increasing of
the knowledge base by eliminate sufficient constraint and false nogood, which are defined in
section 3.1. These eliminations do not change the completeness of the original knowledge base
increased. The proofs are given in section 3.1.
In this paper, section 2.1 gives description about CSP and DCSP; section 2.2 describes the
general Hyper-Resolution-Based Consistency Algorithm and the related Polynomial Space
Problem; section 3.1 derives some theorems from hyper-resolution-rule, which are the foundation
of our approach based on; section 3.2 describes our approach; example to compare the general

22
approach with our approach is given in section 4; finally, evaluation and conclusion are given in
section 5 and 6.
2. HYPER-RESOLUTION BASED POLYNOMIAL SPACE PROBLEM IN DCSP
2.1. CSP and Distributed CSP
In a Constraint Satisfaction Problem (CSP) the goal is to find a consistent assignment of values
for a set of variables [1]. In general, there is no restriction about the form of the predicate for
constraint. It can be a logical or mathematical formula, or any arbitrary relation defined by a tuple
of variable values. We will sometimes also refer to these constraints as nogoods.
[4–6] Solving a distributed CSP can be considered as achieving coherence among the agents.
Many application problems in DAI, such as interpretation problems, assignment problems, and
multiagent truth maintenance tasks, can be formalized as distributed CSP.
For the agents, it assumes the following communication model:
- Agents communicate by sending messages. An agent can send messages to other agents
if and only if the agent knows the addresses of the agents.
- The delay in delivering a message is finite, though random. For the transmission between any
pair of agents, messages are received in the order in which they were sent.
In this model, the physical communication network may not be fully connected. In other words,
the topology of the physical communication network does not play an important role here, it
assumes the existence of a reliable underlying communication structure among the agents and
ignores the implementation of the physical communication network.
Every agent owns some variables and it tries to determine their values. However, there exist inter-
agent constraints which must be satisfied. Formally, there exist m agents 1, 2, ..., m. Each variable
xj belongs to one agent i, which could be represented as belongs(xj , i). Constraints are also
distributed among agents. The fact that an agent l knows a constraint predicate pk is represented
as known(pk , l).

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A Distributed CSP is solved if and only if the following conditions are satisfied: for any i and xj
where belongs(xj , i), the value of xj is dj , and for any l and pk where known(pk , l), pk is true under
the assignment xj = dj .
2.2. Hyper-Resolution-Based Consistency Algorithm and Polynomial Space Problem
Hyper-resolution-based consistency algorithm is a basic algorithm used to solve CSP problem.
Some other famous algorithms such as Asynchronous Backtracking algorithm [6] and
Asynchronous Weak-Commitment Search algorithm [9] can be considered as an extension of this
algorithm. The core of this hyper-resolution-based consistency algorithm is using hyper-
resolution rule, which is described as follows [1]:
We can use hyper-resolution rule to solve DCSP problem. First, we map the elements of DCSP
problem to above hyper-resolution rule, then the domain information of one variable can be
represented by the first row of equation (3), where each proposition Ai represents a possible
domain value. Initial constraints related to this variable can be represented by the nogoods as in
the next rows of equation (3). For each variable, we could represent the domain information and
constraints similarly, and then use above hyper-resolution rule to generate new nogoods. These
new nogoods could be communicated to related variables. Once the related variables get the new
nogoods, they can update their constraint database, and use above rule to generate new nogoods
based on the communicated nogood. Continuing above operations, if an empty nogood is
generated, then the problem is over-constraint and has no solution. The hyper-resolution- based
consistency algorithm is also described as in [1].
For example, for the graph coloring problem with 3 nodes connected to each other, and with color
limited to 2 kinds, the table 1 lists domain value and initial constraints for each variable, where
we use value 1 and 2 to represent the 2 different colors.
We assume x1 generates a new nogood ¬(x2 = 1 x3 = 2) using nogood ¬(x1 = 1 x2 = 1) and
nogood ¬(x1 = 2 x3 = 2) by hyper-resolution rule. This nogood is communicated to x2 and x3.x2
generates a new nogood ¬(x3 = 2) using this communicated nogood and nogood ¬(x2 = 2 x3 =
2). Similarly, x1 generates a new nogood ¬(x2= 2 x3= 1) from nogood ¬(x1 = 2 x2 = 2) and
nogood ¬(x1 = 1 x3 = 1). x2 generates a new nogood ¬(x3 = 1) using this nogood and nogood
¬(x2 = 1 x3 = 1). Then x3 generates an empty nogood from nogood ¬(x3 = 2) and nogood
¬(x3 = 1). In other words, an empty nogood means nothing is good, which represents that the
problem is over-constrained and has no solution. In this case, it is easy to understand for the last
usage of the hyper-resolution rule – the domain information for x3 says x3 could be 1 or 2,
however one nogood says x3 can not be 1, another says x3 can not be 2, so that it is not possible to
find a value for x3 , which means no solution.

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Above example gives us a clear idea about how the algorithm solves the DCSP problem.
However, above example starts from variable x1 only and ignores other possible nogoods.
Practically, if we use 3 agents work for each variable, and work asynchronously, we can generate
a very large number of nogoods, which is in polynomial. In this case, for each variable, such as
for x1 , we can choose any one of the two nogoods with x1 = 1, and any one of two nogoods with
x1 = 2 to generate new nogoods, so that we can generate 4 nogoods from the initial nogood set.
For 3 variables, we can then generate 12 nogoods totally. Once these 12 nogoods communicate to
each other, we can imagine how the knowledge base is increased.
3. HYPER-RESOLUTION-BASED CONSISTENCY ALGORITHM WITH E CIENT
KNOWLEDGE BASE MANAGEMENT
3.1. Theorems Derived from Hyper-Resolution Rule
Before describing our management idea, we give several definitions and theorems as follows.
For example, assume x1 has domain information A = (x1 = 1 x1 = 2). Correspondingly, A =
{A1 , A2 }, where A1 = (x1 = 1) and A2 = (x1 = 2). Assume x1 has nogoods set B = {¬(x1 = 1 x3
= 1), ¬(x1 = 1 x3 = 2), ¬(x1 = 2 x2 = 2)}, then we have B1 = {x3 = 1, x3 = 2} and B2 = {x2 =
2}. By using above theorem 2, we can get new nogoods set C = {(x3 = 1 x2 = 2), (x3 = 2 x2 =
2)}.

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Proof. The theorem 2 generates all possible nogoods by hyper-resolution rule. Since the new
nogoods set is the collection of all the nogoods generated, obviously it is complete.
Proof. When predicates bi and bj that can not be satisfied at the same time, bi bj will be always
false, and it can be looked as a predicate also. According to the definition 4, since bi bj is
always false, we have B to be a false nogood.
Proof. Let’s consider about the two conditions separately, if bj is a sufficient constraint of
bi , then according to the definition 3, we can write bj and bi to be the format:
where m ≤ n and set {ak | k {1, 2, · · · , m}} {a k | k {1, 2, · · · , n}}. So that we
can rewrite bi to be: bi = ¬(a1 a2 · · · am a k · · ·)
If bj is satisfied, that is, when bj is true, then a1 a2 · · · am is false. Because 0 x = 0,
where 0 donates false and x donates any proposition, a1 a2 · · · am a k · · · will
also be false, so bi is true. In other words, when bj is satisfied, bi will also be satisﬁed. So
by removing bi from the nogood set B, we don’t lose any constraint information.For the
second condition, if bi is a false nogood, according to the definition 4, there is a

26
proposition always false. Since 0 x = 0, a 1 a 2 · · · a n will be always false, so
that bi will be always true. Since bi will be always satisfied, it does not supply any
constraint information, so we can simply remove it from without change any thing.
3.2. Hyper-Resolution-Based Consistency Algorithm with Efficient Knowledge Base
Management
Based on above derivations, we generate following algorithm:
1. Initialize the knowledge base with domain information and nogoods;
2. Generate new nogoods with false nogoods eliminated;
3. Communicate new nogoods to related variables;
4. Once the related variables receive the new nogoods, they update their knowledge base by
remove all sufficient nogoods of the new added nogoods, and eliminate the new added
nogoods if the new added nogoods are sufficient nogoods of some nogoods in the knowledge
base.
5. If the new nogood is not added, do nothing, otherwise generate new nogoods based on the
new added nogoods.
6. If empty nogoods is generated, report no solution and stop, otherwise, repeat 3 to 6.
Assume initial domain information is represented as A = A1 A2 A3 · · · Am, and let set A =
{Ai | i = 1, 2, · · · , m, and m Z +
}, where Z +
donates the set of positive integers. Assume
nogoods set B = {¬(Ai Aij 1 Aij 2 · · · Aij k ) | Ai A }, and we let Bi ={Aij 1 Aij 2 · · · Aij k
|¬(Ai Aij 1 Aij 2 · · · Aij k ) B, Ai A and j, k Z +
}. Specially, if ¬Ai B, then we have
Bi , where donates empty symbol, which means the rest part without Ai is empty.
For example, according to the initial domain information and nogoods as in Table 1, for x1 we can
get A = {x1 = 1, x1 = 2}, B1 = {x2 = 1, x3 = 1}, B2 = {x2 = 2, x3 = 2}.
Then the details of generate new nogoods could be described as follows:
Above procedure can be used for step 2 and 5. For step 4, instead of simply add the new nogoods
in, based on above assumptions, the details of the procedure can be described as follows:

27
By eliminating false nogoods in NOGOODS−GENERATION , we reduce the number of the
nogoods to be communicated; by eliminating sufficient nogoods in KNOWLEDGE−BASE−UP
DATE, we decrease the size of the knowledge base.
4. EXAMPLE
This section will give an example based on Table 1, and show how our approach decreases the
knowledge base with comparison to the general one.
Example given in Section 2.2 just shows how the algorithm works to solve the problem, which
focus on one variable, and ignore messages that are not used. Here, we show all messages
produced and added to knowledge base, and compare the result with our approach.
Following table 2 shows the knowledge base changes based on the general hyper-resolution-
based consistency algorithm. In this table the constraints are proceeded by a number which
indicated the time at which they were added.
By comparing above two approaches, we see that each variable initially has 4 nogoods, based on
which each can generate 4 new nogoods, because we have 2 elements in B1 and 2 in B2 . However
in the second approach, by removing false nogoods, such as ¬(x2 = 1 x2 = 2) and ¬(x3 = 1 x3
= 2) for x1 , we get only 2 new nogoods for each variable to communicate. After ﬁrst
communication, there are 5 nogoods added to each variable for the first approach, and 4 added to
each variable for the second approach, as in Table 2 and 3.
Then based on the new nogoods, the ﬁrst approach can generate 21 new nogoods for each
variable. For example, for variable x1 , since there are 3 with x1 = 1 and 3 with x2 = 2 in the new
received nogood, then they could generate 3 × 2 + 3 × 2 nogoods with old nogoods in the
knowledge base, and they could generate 3 × 3 nogoods among the new nogoods. By
communicating these nogoods to related variables, there are only 2 that can be really added to the
knowledge base for each variable, since most of them are redundant (already in the knowledge
base). For the second approach, based on the 4 new received nogoods, each variable can generate
10 new nogoods. For example, for x1 , since there are 2 with x1 = 1 and 2 with x2 = 2 in the new
received nogood, then they could generate 2 × 2 + 2 × 2 nogoods with old nogoods in the
knowledge base, and they could generate 2 × 2 nogoods among the new nogoods, and by
removing false nogoods ¬(x2 = 1 x2 = 2) and ¬(x3 = 1 x3 = 2), we have 10 new nogoods left.
By communicating these nogoods to related variables, there are only 2 added to knowledge base,

28
with same reason in the first approach. However, when these 2 new nogoods are added, which are
sufficient nogoods for some nogoods in the knowledge base, so that those nogoods are
eliminated. For example, for x1 , once ¬(x1 = 1) is added, which is sufficient nogood of all other
nogoods with x1 = 1, so that all such nogoods with x1 = 1 such as 01, 02, 11 and 13 are
eliminated. Similarly, once ¬(x1 = 2) is added, 03, 04, 12, and 14 can be eliminated.
So far, at step 2, there are only 2 nogoods left for each variable in the knowledge base for the
second approach, while there are 4 + 5 + 2 nogoods left for each variable for the first approach.
Thus, in the last step, the first approach generate 11 nogoods (1 × 5 + 1 × 5 + 1 × 1), among them,
there is an empty nogood, which terminate the program by report no solution. For the second
approach, we only generate one empty nogood with the left nogood for each variable.

29
5. EVALUATION
From above example in Section 4, we can see that how our approach reduces the new nogoods
generated for communication, and how our approach decreases the knowledge base when
possible. While the general hyper-resolution-based approach never decreases the knowledge base.
The efficiency of our approach is obvious. The comparison of the two algorithms: the original
Hyper-resolution-based consistency algorithm and the one with Efficient Knowledge Base
Management (EKBM), based on example is given in Figure 3.
From the Figure, we can ﬁnd out that the algorithm with EKBM generates less nogoods for
communication than the original algorithm. And the knowledge space used for storing the
nogoods keeps increasing in the original algorithm, there is no deceasing process. However, in
the algorithm with EKBM, it involves a decreasing process. Thus, the provided EKMB method
helps to maintain a minimal required knowledge space. This will also speed up the searching
process by reducing the search space.
Other than the efficiency showed in above example, the correctness and completeness for the
EKBM approach are proved in Section 3.1 as well.
The Asynchronous Backtracking (ABT) algorithm and asynchronous weak-commitment search
can be considered as an extension of hyper-resolution based consistency algorithm. The
difference between ABT and hyper-resolution based consistency algorithm is that the former
communicates the new generated no-goods to higher priority variable only, while the later sends
the nogoods to all related variable; asynchronous weak-commitment search algorithm also
considers the priority in, however not like ABT in which the priority of variables are determined,
asynchronous weak-commitment search can change the priority of the variables. Disregarding the
priority of the variables, we can directly apply above NOGOODS− GENERATION and
KNOWLEDGE − BASE − UPDATE into ABT and asynchronous weak-commitment search,
which benefits the implement the same way as above.

30
6. CONCLUSION
This paper provides an efficient knowledge base management approach based on general usage of
hyper-resolution-rule in consistence algorithm. The approach minimizes the increasing of the
knowledge base by eliminating sufficient constraint and false nogood. These eliminations do not
change the correctness and completeness of the original knowledge base increased. The proofs
are given as well.
The given example shows how this new approach with EKMB reduces the new nogoods
generated for communication, and how it decreases the knowledge base during the searching
process. The comparison of the original algorithm and the new algorithm is given as well. It
shows that the algorithm with EKBM generates less nogoods for communication than the original
algorithm, and it involves a decreasing process. However the original hyper-resolution-based
approach never decreases the knowledge base.
Thus, the provided EKMB method helps to maintain a minimal required knowledge space, and
will potentially speeds up the searching process by reducing the search space and simplifying the
searching process. With the problem size increasing, the benefit of this approach will be more
obvious.
Considering that the provided EKBM approach is on a general base, all other algorithms adopting
hyper-resolution-rule can take the advantage of this approach, such as the Asynchronous
Backtracking (ABT) algorithm and asynchronous weak- commitment search algorithm in solving
DCSP.
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