The matrix connection method (MCM) is an alternative procedure for theorem proving than the usual resolution technique. We already have used the MCM for finding models in a real-time knowledge-based system generator. In this paper, we adapt the MCM to the particular case of sorne annotated propositional paraconsistent logics. Further developments related to these ideas are also outlined.Eje: 2do. Workshop sobre aspectos teóricos de la inteligencia artificialRed de Universidades con Carreras en Informática (RedUNCI

Kaestner, Celso A. A.

English

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416 2do. Workshop sobre Aspectos Teóricos de la Inteligencia Artificial 
MATRIX PROOF METHOD IN 
ANNOTATED PARACONSISTENT 
/ LOGIC 
Celso A.A. KAESTNER 
Departamento Academico de Informática - Centro Federal de Educac;ao Tecnológica do Paraná 
Departamento de Informática - Pontíficia Univcrsidadc Católica do Paraná 
CEFET/PR - DAINF - Av. Sctc de Seternbro, 3165 - 80.230-901 Curitiba - PR - BRAZIL 
e-mail: kaestncr@dainf.ccfctpr.br 
Décio KRAUSE 
Departamento de Matemática - Universidade Federal do Paraná 
Centro Politécnico - Caixa Postal 19.081 - 81.531-910 Curitiba - PR - BRAZIL 
e-mail: dkrause@mat.ufpr.br 
Abstract 
The matrix connection method (MCM) is an alternative procedure for the-
orem proving than the usual resolution technique. We already have used the 
MCM for finding models in a real-time knowledge-based system generator. In 
this paper, we adapt the MCM to the particular case of sorne annotated propo-
sitional paraconsistent logics. Further developments related to these ideas are 
also outlined. 
Keywords: 
Paraconsistent Logics, Annotated Logics, Thcorem Proving, 
Matrix Connection Method. 
2do. Worksbop sobre Aspectos Teóricos de la Inteligencia Artmcial 417 
1 Introduction 
Automatic proof methods are actually widcly used in Artificial Intelligence appli-
cations. In the particular case oC the Real~Time Knowledge-Based Systems do-
maín, knowledge-based embedded programs have the convenience of reacting conve-
niently stimulus within imposed time restrictions to the environment. In [KAE 92a], 
[KAE 92b], [KAE 9380], [KAE 93b], (KAE 95], it was presented the RETIKS system, 
which is a real-time knowlé'dge-based system generator based on a annotated p~ra­
consistent propositionallogic, which makes use oC the synchronous approach for time 
thodeling. 
In order to attain its main purposes, RETIKS exhaustively calculates its outputs 
for each possible input. This compilation procedure is equivalent to that one of finding 
a11 models oí the theory obtained from the rules that constitute the Knowledge-Base of 
the system [KAE 93b]. If it is desirable that each input determines an unique output, 
then the paraconsistent treatment should be used. The final execution structure 
obtained by the compilation ptocedure is a finite automataj this fact grants good 
performance and permits the verification oC the desired time requirements. 
In this paper, we describe the method used for finding models in the RETIKS sys-
temo Although theorem provers for similar paraconsistent logic8 based on resolution 
were already proposed, e.g. (SUB 87][BLA 88], thc RETIKS system uses a method 
based on a variation oí the Wallen and Bibel matriz connection method [BIB 82], 
[WAL 87], [WAL 90], [GOC 90], suited for these particular logics. 
We begin by adapting the basic concepts oí the matrix connection method in order 
to consider a peculiar paraconsistent case. We restrict ourselves to the propositional 
case, which is the base of the RETIKS compilation procedure. Meanwhile, the main 
ideas discussed here can be suitable adapted for first-order logics. 
The paper is organized as follows: in the next section the fundamental concepts 
of the matrix coimection method are given and the ground formalism is briefiy intro-
ducedj in the section 3 the semantics for this formalism is out,linedj in the section 4 
the concepts of path, connections, and a characterization of validity are introducedj in 
the section 5 the case of theorem proving and the technics for finding modela for the 
characterized systems are al so considered. FinalIy, further research and applications 
are also suggested. 
2 Matrix concepts and basic syntax 
The terminology is based on [BIB 82] and [BLA 88]. Let A be a non-empty finite 
ordered set of propositional symbols. To each element a E A, a non-empty finite 
lattice Ta. is associated. The elements of Ta. will be named annotated constants and 
denoted by p., 11. 
Definition 2.1 A (ground) literal is a triple (a, 1', p) where a E A, l' E Ta and 
P E {O,l}. p is the polarity of the literal. Literals will be denoted by [(, L, M. We 
also use qL, q E {0,1} to denote the literal (a,p, (p + q) mod 2). 
418 2do. Workshop sobre Aspectos Teóricos de la Inteligencia Artificial 
{{L},PM}} {M, {}, {{L},I M}} 
/\ !I\ 
{L} M {} {{L},lM} 
1/ /\ 
L {L} 
I 
L 
Figure 1: 'frcc rcprescntation of matrices 
Let R be an alphabet of occurrences or posilions. 'fhe elements of R are denoted 
by r. 
Deflnition 2.2 By indudion, we define the concepts of (propositional) matrices over 
(A, R), denoted by D, E, F, so as their size O'(F), their positions n(F) e R and their 
depth h(r) of r in F for any r E n(F): 
• For any literal L and for any r E R, the pair (L, r) = Lr is a matrix with 
O'(Lr) = O, n(Lr) = {r} and h(U) = O .
• If Ft, ... Fn, n 2: O are matrices such that n(F¡) n n(Fj ) = 0 for i f. j and 
1 ~ i, j :5 n, then the set F = {F1, • •• Fn} is a matrix where: 
- 0'(0) = O for n = O and O'(F) = 1 + Li:l O'(F¡) for n > O; 
- n(F) = n(F1) U ... U n(Fn)¡ 
- h(r) = m + 1 for any r E n(I'i) , 1 $ i $ n, where m is the depth of r in 
Fi. 
According to this definition, the atomic parts of the matrices are ground literals, 
and in general a matrix is a nested set of occurrences of literals. 
Example 2.1 Let us consider A = (a, b, e, d) with Ta = 7í, = Te = 7d = 2{O,1} (the 
Boolean lattice of the power set of {O, 1} ordered by indusion) and R = {O,!, 2, 3} 
be an alphabet of positions. Then L = (a, {O},O) and M = (e, {O, 1}, 1) are ground 
literals, while {{L}, {1M}} and {M, D, {{L},lM}} are matrices over (A, R). 
A matrix can also be viewed as a tree, where sorne leaves are associated to litera]s. 
Figure 1 presents the trees corresponding to the matrices oí above example . 
. ¡ 
... 
2do. Workshop sobre Aspectos Teóricos de la Inteligencia Artificial 419 
Definition 2.3 Let F be a matrix and 1, m E {D,l}. Thc set oí formulas F repre-
sented by F with respect to (1, m) i8 indudivcly dcfined as follows: 
• if F is a literal F = Lr and 1 = O thcn F = L; 
• if F is a literal F = U and 1 = 1 then j.' =IL; 
• if F = {Ft, ... , Fn},ft ~ O and ir m = 1, then F = /\(FI , ... , Fn ), where the Pi 
are formulas represénted by Fi wilh re8pect to (1, O), i = 1, ... n; 
• if F = {F¡, ... , Fn}, n.~ O and if m = O then P = V(fr., ... , Fn), where Fi are 
formulas represen ted by Fi wi th respect to (/, 1), i = 1, ... n. 
Deflnition 2.4 A formula F is positively rcpresented by a matrix F ir it is represented 
by F with respect to 1 = m = O; fr i8 negativcly represented ir it is represented by 
F with respect to 1 = m = 1. A proposilional formula is any Cormula representcd by 
sorne matrix. Formulas are also dcnotcd by /), B, F. 
In order to adequate our notation to the usual one, we will introduce the following 
notations: 
• if n = O /\(Ft , ••. , Fn) = /\0 is abbreviated by T, and VO by F¡ 
• if n = 1 both /\(F) and V(F) are abbreviated by F¡ 
• iI n ~ 2 /\(Ft, ... , Fn) is a conjunction and V(F., ... , Fn) is a disjunction¡ 
• for any literal L, the formula -.k L is called a hypcr-litcml¡ if L ia a hyper-literal, 
then ...,k L = (a, ...,k(¡.t),p), where..., : Ta. ~ Ta. denotes sorne fixed function (that 
gives the meaning of the negation), and k ia a multiplicity factor (a natural 
number)¡ 
• if F is a formula which is not a hyper-literal, then -.F is defined by: 
- if F = /\(F., ... ,Fn ), n ~ O then -.F = V(-.F., ... ,-.Fn )¡ 
- if F = V(Ft, ... ,Fn ), n ~ O then -./i' = /\(-.F¡, ... ,...,Fn ). 
• any formula -.F V G may be writtcn as F -+ G¡ 
• any formula (F -+ G) /\ (G -+ F) may be written as F H G¡ 
• any formula F -+ ((F -+ F) 1\ -.(F -+ F)) may be written as '" F and it is 
called the strong negation of F¡ 
• parentheses are eliminated in the obvious way. 
According to the aboye conventions, every well-formed formula (defined in the 
standard way) determines an unique matrix; notwithstanding, a matrix may represent 
more than one formula. 
420 2do. Workshop sobre Aspectos Teóricos de la Inteligencia Artificial 
{ f(, { 1,,1 M}, {} } 
~O,I=O 
J( 
m = 1,1 = O 
{D,' M} {} m/\o m=l,/=O 
L 1M 
m = 0,1 = O m = 0,1 = O 
Figure 2: Positivc rcprcscntation of F == {J(, {L, M}, {}} 
Example 2.2 Let F = {I<, {D, M}, {}} be a rnat.rix. Thc trec in figure 2 is the 
positive representation of F. 
Example 2.3 LetbeP = J( --+ (LVI M); Fisanabbreviationof-.I<V(LV1M). This 
formula is represented by the matrix {-.I<, L, 1M}. Obviously the formula -.I<v Lv1M 
ha.s the same matrix. 
Results presented in [BIB 82], chaptet 2 remain applicable here, such as for in-
stance: 
• If a formula P is positively represented by a matrix F then rv P is negatively 
represented by Fj 
• If two formulas PI and F2 are positivcly rcpresented by the same matrix F, 
then P1 and F2 are logically equivalents in the sense of the annotated logics 
(BLA 88]. 
These results justify the use of matrices ¡nstead of formulas. 
The following example justify the name matrix employed firstly by Bibel [BIB 82] 
and used also here. 
Example 2.4 Let F be the formula: 
where I<, L, M and N are literalsj if we put F' in the disjunctive normal form (as 
usual), we will obtain: 
2do. Workshop sobre Aspectos Teóricos de la Inteligencia Artificial 421 
This formula may be prescnted in a bidimensional arrangemcnt, where thc literals 
placcd in a fixed column are conncctcd by "1\", al\(l tlae columns are connected by 
"v", as íollows: 
/ 
.3 Semantics 
As usual, we admit that thc truth of a ccrlain knowledge depends on the truth values 
oí its atomic constituents. 
Definition 3.1 An interpretation M for our formalism is a fundion which associates 
an element of the lattice to every propositional symbol. By denoting M(a) = /la, we 
may write (a, b, e, . .. ) H (/la, /lb, ¡le' .. ). 
Let us consider the two "spedal" matrices inlroduced earlicr: 
vO = F = 0 = {} and 1\0 = T = {0} = {{}} .. 
Now we will define the "truth value" M(F) of a matrix Fas follows: 
• if F is a literal (a, /l,p), then M(F) = T = {0} iff M(a) ~ /l when p = O and 
M(a) 'l/l when p = 1. Otherwise M(F) = F = 0 . 
• if F is a matrix F = {FI!"" Fn}, n ~ o, then M(F) = Uk=t M(Fk) when 
m = O and M(F) = nk=t M(Fk) whcn m = 1. 
We will write M sat F (and also M sat F) iff M(F) = T for a matrix F which 
represents F. 
Definition 3.2 A matrix F is valid iff M(F) = T for every possible interpretation. 
It is called contradictory iff M(F) = F for evcry interpretation. 
We can note that if F is valid then ,..., F is contradictory, and the converse is also 
true. 
Other semantical concepts can be introduced in such a way so that the standard 
semantical results can be obtained, but we will not present such details here. 
4 Paths, Connections, and a Characterization of 
Validity 
Definition 4.1 A path through a matrix F is a set of occurrences oí literals, defined 
as follows: 
422 2do. Workshop sobre Aspectos Teóricos de la Inteligencia Artificial 
Figure 3: The ¡>áth {J<,-,M, -,2 L,t/<} through the matrix F 
• if F = 0 then the only path through F is 0; 
• if F = Lr then the only path through F is the set {U}; 
• if F = {F¡, ... , Fm, Fm+1, •.. , Fm+n}, m, n 2:: o, m + n 2:: 1 for m literals 
FI!"" Fm and for n matrices which are not literals Fm+1 , ••• , Fm+n, then for 
any matrix E, E Fm+i and ror any palh Pi Lhrough Ei, 1 ~ i ~ n, the set 
Uf=1 {Fi} U Ui:1 Pi is a path through F. 
Example 4.1 Let F be the matrix in example 2.4; a path through F is a crossing 
from left to right,constrained to pass by the literals (to be interpreted as "gates") as 
shown in figure~. 
Definition 4.2 We caH complementary literals a pair of literals L = (a,¡.t,p), 
M = (a,v,q) such that: 
• (p+q) mod 2 = 1, and 
• U(J.t) U n(v) = 7;,., where we denote U(J.t) = {r E 7;,. : J.t ~ r} and 
n(r) = 7;,. \ U(r). 
Definition 4.3 Paths which have complementary literala as elements are called con-
nections. 
Example 4.2 If 7;,. = 2{O,1} then (a, {ll, O) and (a, {O}, 1) are complementary literals. 
Example 4.3 Let F be the matrix in the example 2A¡ 
the paths are {K, -,M,-,2 L,INl, {J<, -,M, -,2 L,t/<}, pL, -,M, -,2 L,lN}, PL,-,M, -,2 L,lI<}, 
{-,lN, -,M, -,2 L,IN}, and {..,lN,..,M,..,2 L,lK}. 
The path {K, ..,M,..,2 L,lK} is obviously a connection, since !( and tK are com-
plementary literats. In order to others paths be connections, it is necessary to analyze 
their compounding literals, the negation function definition, and so on. 
Proposition 4.1 (Soundness and Completeness): A matrix F is valid iff every path 
through F is a connection. 
Proof: Adapted from [BIB 82, pp. 30-31], by using induction over the size of u in 
the matrix F. 
2do. Worksbop sobre Aspectos Te6ricos de la Inteligencia Artificial 423 
5 Theorem proving technics and models 
In order to adapt the usual proof theorem procedures to our case, we should consider 
the following situation. Suppose that we have a set of formulas 
r = {Ft, ... , F,,} and a query G. Then, in order to investigate if Gis a semantical 
consequence of the set r, we should verify if the matrix provided by (Af=l F¡) V f'V G 
is contradictory. , 
The paraconsistent céÍse is includcd in this procedure due to our definition of 
~ complementary literals. In fact, in this case the existence of a literal 
L = (a, ¡.L,p) and its 'negation' ...,L = (a, ""(¡.L),p) is not a sufficient condition 
to assure compIementary literals in the path. This ,exemplifies perfectly wetl the 
underlying ideas of the general,paraconsistence program [COS 74]. We note that 
in order to obtain a proof of the query, it is necessary that 8011 paths of the matrix 
f'V (Af=l F¡) V f'V G) have connections, which imply the existence of complementary 
literals in every path. 
If there are no complemenfary literals in the paths, then the set of paths represents 
the set of models of r U {f'V G} as in usual tableaux semantic method¡ this was the 
methodology used in [KAE 93b). 
6 Further Developments 
The methodoIogy presented in this paper were implemented in an prototype version 
o~ the system written in CommomLisp [KAE 93b]. It is interesting to note that 
this method is adequate also for the paraconsistent treatment, so that it provides an 
alternative way for theorem proving than the classical resolution procedures presented 
in the papers described in the Introdudion. 
As mentioned in [BIB 82, p. 45ff] and repeated by [WAL 87], [WAL 90], [GOC 90], 
we guess that the method presented here is more suitable for finding modela as re-
quired by the RETIKS system. In fad, as the mentioned authors sustain, the con-
nection method seems to be algorithmicaly more efficient than resolution in most 
cases. 
It would be also interesting to ask for the algebra of connections, which apparentIy 
might differ from the classical case, as presented in [BIB 82]. 
The first-order case can be obtained without difficulty by adapting the procedure 
sketched in this paper. 
One couId a1so to investigate the possibility of extending the method presented 
here to the so called multideduclive logics leos 95]. In short, multideductive logics 
are defined so that several deduction symbols, say h, 1-2 , ••• can be introduced in 
such a way so that the notions of h-dedudion (i = 1,2, ... ) are defined as usual by 
means of the stated 1-¡-axioms. Then, if r is a set of formulas oC the language and if F 
is a particular formula, the concept of r h F as well as the concept of an i-theorem 
are well defined. Since the set oC i-theorems and the set of j-theorems (i i: j) may 
contain contradictory formulas, the general underlying logic must be a paraconsistent 
one, as shown in just mentioned papero 
424 2do. Workshop sobre Aspectos Teóricos de la Inteligencia Artificial 
Then, if it is done a query G and a set oí formulas r, one could ask if the query is 
a consequence of r by one of the concepts f-¡. In fad, in a most general idealized case, 
the 'proof' of the query might require more that one deductive notion. To investigate 
the applicability of the connection method in this situation could be useful in several 
cases, such as for instance when a certain knowledge may dependa on different canons 
of inference. 
In the particular case gf Artificial Intelligence applications, the aboye discussion 
appears both in the conteit of knowledge-based systems obtained from several experts 
~ an.d in the context of cooperati ve multiagent systems. In the first case, the obtained 
date-base may be inconsistent in several ways, and the matrix connection approach 
could be done in a rather different way than that one provided by [COS 89], [COS 90]. 
In the context of cooperative multiagent systems, the use of multideductive logics 
seems to be more evident, since to each one of the agents it could be associated a 
different f- ¡ notion. 
References 
[BIB 82] W. Bibel. Automated Theorcm Proving. Friedr. Vieweg & Sohn, Braun-
schweig, Wiesbaden, 1982. 
[BLA 88] R.A. Blair; V.S. Subrahmanian. 'Paraconsistent Logic Programming', 7 th. 
Int. Con/o on Foundations 01 Software Tech. & Theoret. Computer Science, 1988. 
[COS 74] N.C.A. da Costa. IOn the Theory of Inconsistent Formal Systems', Notre 
Dame Journal o/ Formal Logic, 15, 497-510, 1974. 
[COS 89] N.C.A. da Costa; V.S. Subrahmanian. 'Paraconsistent Logics as a Formal-
ism for Reasoning About Inconsistent Knowledge Bases', Artificial Intelligence 
in Medicine, 1, 167-174, 1989. 
[COS 90] N.C.A. da Costa, et al.. 'Automatic theorem proving in paraconsistent log-
ics: theory and implementation', Proceendings o/ the 10th International Con/er·· 
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[COS 91] N.C.A. da Costa; V.S. Subrahmanian; C. Vago. 'The Paraconsistent Logics 
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[COS 95] N. C. A. da Costa et al.. 'Paraconsistent logic: an application to physics', 
preprint. 
[GOC 90] P. Gochet; P. Gribomont. Logic 1: Methods lor Fundamental In.formatics 
(in french). Rermes, Paris, 1990. 
[KAE 92a] C.A.A. Kaestner. 'Real-Time Al Systems: the Synchronous Approach 
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[KAE 92b] C.A.A. Kaestner. 'Artificial Intclligcncc Real-Time Systems: Synchronous 
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Matrix proof method in annotated paraconsistent logic

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Matrix proof method in annotated paraconsistent logic

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