Knowledge-Based -5YSTEMs

ELSEVIER Knowledge-Based Systems 10 (1998) 359-361

Finding relevant knowledge in a knowledge base of linear logic

Dong-Tsan Lee*, C.P. Tsang

Lqic und AI l*lhorafory, Department of Computer Science, The University of Western Australia, Nedlands. Perth, Western Australia. .4ustralirr

Received 25 September 1996; accepted 20 October 1997

Abstract

In terms of knowledge science, there clearly has been a trend toward the development of large-scale knowledge bases. In such knowledge bases, not all knowledge is closely related to each other and it is important to be able to determine which parts of knowledge are relevant to the current problem. In this article, we discuss theformula interconnectivity graphs but also its use for selecting relevant knowledge from a knowledge base that is based on linear logic. 0 1998 Elsevier Science B.V. All rights reserved.

Keywords: Knowledge bases; Knowledge representation; Linear logic

1. Introduction

Linear logic, introduced by Girard [3,4] in 1987 as a state based, resource-sensitive logic, is an important knowledge representation language that can model more real-world semantics. Linear logic specifically does not have two struc- tural rules, contraction and weakening. Removing the two rules gives a linear system in which each resource (formula) must be used exactly once. Once the two rules are dropped, the two possible traditions for the right A -rule in the Gentzen classical sequent calculus are no longer equivalent. The removal of contraction and weakening leads to two forms of conjunctions, namely @ (times) and & (with) and similarly to two forms of disjunctions, @ (par) and @ (plus). @ can be seen as a multiset constructor, with neutral 1. p is a way to split multisets into pieces, with neutral I. & corresponds to outer non-determinism, with neutral T. @ corresponds to inner non-determinism, with neutral 0. Girard [3] discovered that in coherent spaces, the function space A * B can be split into (!A)-0 B, where ! is the exponential operator and -0 is the linear implication. -0 can be thought as a method of rewriting multisets. ! can be interpreted as a persistent member of a multiset. Linear negation is denoted by (.)I. Unlike the negation as failure paradigm, linear negation is an involution satisfying De Morgan-like properties. Thus, it is possible to describe lin- ear logic in terms of one-sided sequents, transforming AFZ to FA’, C. Yet unlike classical negation, linear negation has a simple constructive meaning. Moreover, if a linear logic

* Corresponding author. e-mail: dtlee@cs.uwa.edu.au

0950.7051/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved PI/ SO950-705 I (97)00048-8

formula is prefixed with the exponential operator ! (of course) or ? (why not), it is called an exponential formula. Otherwise, a linear logic formula without ! and ? is called a non-exponential formula. Knowledge bases of linear logic are suited for applications in which a large amount of knowledge must be stored and complex queries must be supported. In such large-scale knowledge bases, it is neces- sary to control and restrict the search space of deduction. A natural way to reduce the search space is to determine which knowledge is relevant to the current problem. This idea has been discussed in refs [ 1,5,8].

A query within knowledge-bases of first-order linear logic always has the linear logic form: Q @ T A fragment of the deduction for I-A, Q @ T proceeds as:

. . _

A crucial problem arises in partitioning the knowledge base A into 2 parts and selecting facts and rules related to the formula Q. Also, the ( @ R) rule is a worst rule for proof search. The goal (sequent) FA,Q @ T can be replaced by 2” possibilities (n being the number of formulas in A) of the subgoals FA,,Q and FA2,T. where {A,, A,) is a partition of A. Here the neutral T can erase unused formulas.

2. Formula interconnectivity graphs

2.1. De$nition

Suppose A is a set of linear logic formulas. The formula

interconnectivity graph for A is a graph whose nodes are the formulas of A, and which has an undirected edge between formula F, and formula Fl, labelled with a most general unifier of two literals LI and L$, if there are literals L , in F, and Lz in F? such that L, and Li are unifiable.

Two literals, L1 and Lz, are uniliable if there exists a substitution 0 such that LIB = L20. Such graphs are similar to the connection graph introduced in [7].

2.2. Dejinition

A chain joining Fi and F, is a sequence F,, F2 ,..., F,, of nodes in the formula interconnectivity graph of A such that there is an edge between F, and F,,, in the graph, for 1 5 i < n. The length of the chain is n - 1. A cycle is a finite chain F,, . . . , F,, for which node F, coincides with F,,, and no other node is traversed more than once.

2.3. Dejnition

The length CI(Fi,F*) between F, and F? is the shortest chain joining F, and Fz in the graph of A and x if no such chain exists. Intuitively, the length d(Fi, F*)determines the closeness between formulas F, and Fz.

2.4. Example I

Fig. 1 illustrates a formula interconnectivity graph for the set of formulas { BIG(Electorate, Kalgoorlie) ‘, BIG(Jetty ‘, Busselton-Jetty) ‘, BIG(Tree, Karri)’ @ BIG(Tree, Mountain- Ash)l, BIG(Tree, Karri)’ @ ?BIG(Tree, Red-Wood)l, LOC( Karri, WA) ‘, LOC(Mountain-Ash, VIC)‘, (BIG (Tree, X) @ LOC(x, y)) @ LARGEST(Tree, X, y)‘]. A node in the graph corresponds to a linear logic formula. Edges are labeled with most general unifiers.

3. Selecting a relevant subset of knowledge

Let Q @ T be a query for a knowledge base A; that is, we consider the sequent AtQ @ T. We propose the following method to select a subset of knowledge that is relevant to Q:

1. 2.

3. 4. 5.

convert A into A’; simplify A’ using some known equivalents (see Ref. [3]): forinstance,A-OB=A’@B,(A’)‘=A,l’=I,etc. Note that the simplified formulas do not contain the con- nective -0; construct formula interconnectivity graph Gl for A’; add Q @ T to Gl, and then obtain a new graph G2; construct a solution graph from G2.

We now present and discuss an algorithm for the con- struction of a solution graph containing relevant knowledge. Let Q @ T be a query. We begin the construction by select- ing a connected component G’ containing Q @ T from a given formula interconnectivity graph G. Then, we select a

node from G’ and mark the node. Initially. we select the node Q @ T. Later. we check each literal in this selected node. For each literal of this node. we only select one edge incident on the literal of this node. If the selected edge is in a cycle, then all nodes in the cycle will be selected. Otherwise, we choose another node connected by this selected edge. At the same time, we remove other edges incident on this lit- eral, which are not selected. Moreover, if the resulting graph contains more than one component, remove components that do not contain Q @ T. In general, nodes without unlinked literals are selected earlier in time than those with unlinked literals. We repeat the same procedure on other nodes having chains associated with the node Q @ T.

Let us consider a special knowledge base consisting of the following linear logic formulas: nonexponential fact L or non-exponential rule Le(L, @ Ll@ . . @ L,,)a. Here L, L,, Lz,. . L,,, are literals that are not prefixed with the con- nective ! or ?. If a node in a solution graph contains unlinked literals, then the query related to the graph will fail.

Converting A into A’, simplifying A’, and adding Q @ T to the graph can be done in time proportional to the number of formulas in a knowledge base. Constructing formula inter- connectivity graph for A’ can be done in O(n’) time, where n is the number of nodes in the graph. Also, a solution graph can be constructed in O(n) time. The total time to select a relevant subset of knowledge is therefore O(n’).

3.1. Example 2

Consider a knowledge base A of the following linear logic formulas:

BIG(Electorate, Kalgoorlie). BIG(Jetty, Busselton-Jetty). BIG(Tree. Karri) & BIG(Tree, Mountain-Ash). BIG(Tree, Karri) & !BIG(Tree, Red-Wood). LOC(Karri, WA). LOC(Mountain-Ash, VIC). (BIG(Tree,x) @ LOC(x,y))*LARGEST(Tree, X, y).

We will use just right-handed sequents. First, we must move the formulas from the left-hand side to the right-hand side in a sequent and simplify these formulas. That is, A can be converted into A’, and A’ can be simplified using some known equivalents (see Ref. [3]). For this example, the

BIG(Electorate, Kalgoorlie)’ BIG(Jetty. Busselton-Jetty)’

BIG(Tree, Karri)- Q ?BIG(Tree. Red-Wood)’

BIG(Tree, Karri)’ 8 BIG(Tree, Mountain-Ash)’

kwood \ 1 x/~Mounmin-Ash

( BIG(Tree, x) 0 LOC(x, y))@LARGEST(Tree, x. y)’

x/Karri, y/ WA

/I

x/Mountain-Ash, y / V/C

LOC(Karri, WA)’ LOC(Mountain-Ash. VIC)’

Fig. 1. The initial formula interconnecttvity graph.

D.-T. Lee, C. P. Tsang/Knowledge-Based Systems 10 ( 1998) 359-36 I

BIG(Electorate, Kalgoorlie)’ BIG(Jetty, Busselton-Jetty)’

BIG(Tree, Karri)’ 8 ?BIG(Tree. Red-Wood)’

\ BIG(Tree, Karri)’ Q BlG(Tree, Mountain-Ash)’

x/ Kurt \/

x / Karri

(BIG(Tree, x) @ LOC(x, y)) 8 LARGEST(Tree, x, y)’

x / Karri. y / WA

/ LOC(Karri. WA)’

LOC(Mountain-Ash. VIC)’

x/a. y/WA

LARGEST(Tree, a, WA)@T

Fig. 2. The new formula interconnectivity graph.

formulas C converted from A are:

BIG(Electorate, Kalgoorlie) ‘. BIG(Jetty, Busselton-Jetty) ‘. BIG(Tree, Karri) _L @ BIG(Tree, Mountain-Ash) I. BIG(Tree, Karri) _L @ ?BIG(Tree, Red-Wood)’ LOC(Karri, WA)‘. LOC(Mountain-Ash, VIC) ‘. (BIG(Tree, X) @ LOC(x,y)) @ LARGEST(Tree, x,g) ‘.

The formula interconnectivity graph for the above formulas is depicted in Fig. 1. Consider the query: LARGEST(Tree, a, WA) @ T. We can add the node LARGEST(Tree, a, WA) @ T to the graph in Fig. 1 and construct a new graph as illustrated in Fig. 2. The new graph in Fig. 2 con- tains 4 connected components. We only select one con- nected component related to LARGEST(Tree, a, WA) in the query. It follows from the algorithm above that a solu- tion graph G’ is illustrated in Fig. 3. The formulas in this solution graph are relevant to LARGEST(Tree, a, WA). Then, the sequent t--C,LARGEST(Tree, a, WA) @ T can be split into the sequent t-BIG(Tree, Karri)’ @ BIG(Tree, Mountain-Ash)‘, LOC(Karri, WA)‘, (BIG(Tree, X) @ LOC(x, y)) @ LARGEST(Tree, X, y)‘), LARGEST(Tree, a, WA) and the sequent k BIG(Electorate, Kalgoorlie)l. BIG(Jetty, Busselton-Jetty)' , LOC(Mountain-Ash, VIC)‘. BIG(Tree, Karri)- @ ?BIG(Tree. Red-Wood)‘,T.

4. Discussion

For knowledge bases and queries of full first-order linear logic it is sufficient to cope with a wide range of real world applications even though, in at least some cases, a proof search may not be complete [9,6]. It is known that reasoning in first-order linear logic, without ! and ?, is decidable [lo]. Our method can also be useful in knowledge-bases based on

361

BIG(Tree, Karri)’ $ BIG(Tree, Mountain-Ash)’

x / Karri

(BIG(Tree, x) @ LOC(x, y)) ~8 LARGEST(Tree. x. y:l’

x/Karri. y/ WA

I 1

x/a. y/WA

LOC(Karri. WA)’ LARGEST(Tree. a, WA)@T

Fig. 3. A solution graph for Sectton 3. I.

this linear logic fragment (i.e. first-order linear logic without ! and ?). Although our method cannot guarantee that every unused formula is eliminated, it can partition the knowledge base in a reasonable way, reduce the search space and make it easier to find a solution. Moreover, our method can be applied to the (sub)query form: Q, @ Qz. There are two partitions involved in this query. One partition is relevant to Q ,. The other is relevant to Q ?.

Further research will be concentrated on using the for- mula interconnectivity graphs to guide and facilitate reason- ing in knowledge bases of linear logic. By use of our method, we are interested in examining knowledge-bases, which are based on various fragments of full first-order linear logic such as the tensor-bang fragment [2]. Horn frag- ment [6], etc.

References

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[3] J.Y. Girard, Linear logic, Theoretical Computer Science 50 (1987) I-102.

[4] J.Y. Girard, Linear logic: its syntax and semantics. in: J.Y. Girard. Y. Lafont, L. Regnier (Eds.), Advances in Linear Logic. Cambridge University Press, 1995.

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