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Polynomizing Lukasiewicz's Many-Valued Logics by Maple

Kahtan H. Alzubaidy

Introduction

Polynomizing of Lukasiewicz's Many-Valued Logics

by certain reduced polynomials of several variables over the ring

This method is applicable when n is a prime or a power of a prime. We shall restrict ourselves to n = 2, 3, 4 only.

Numerical representations

Propositional Lukasiewicz's many valued logics can be introduced by the matrix

Suppose that n is a prime.Then connectives are expressed by the numeric functions.

negation(ne) wx = n-1- x

implication(im) x

disjunction(di) x

conjunction(co) x

biconditional(bi) x

"We consider L[2] , L[3] and L[4] only. The connectives are given by the following numerical functions as follows"

For

(1)

(2)

(3)

(4)

(5)

For

(6)

(7)

(8)

(9)

(10)

Representations by polynomials

Let n be a prime.

"R[n]=`&Zopf;`[n][x[1],x[2],..,x[n]: (x[i])^(n)=x[i]] is a polynomial ring in x[1],x[2],..,x[n] over a field `&Zopf;`[n] subject to the relations (x[i])^(n)=x[i ] ."

Any polynomial p in
$"R[n] has the reduced form p(x[1],x[2],..,x[n])=(∑)a(x[1])^(i[1])...(x[n])^(i[n]) ; a in {0,...,n-1} ."$

Special cases

p(x)=a

p(x,y)=axy+bx+cy+d

p(x)=a

p(x,y)=a

Atomic proposition in
"L[n] is represented by a variable x in { x[1],x[2],..,x[n]}. The connectives are given as follows "

the unary onnective x= ax+b ; a,b

the binary connectives
"->, or , and ,↔ are given by by axy+bx+cy ; a,b,c in `&Zopf;`[2] In L[3]"

the unary onnective x= a ; a,b

the binary connectives a ;

To find now the connectives

(11)

>	conn2:=proc(ff)

>	local DS2,Eq2;

>	DS2 := [seq(seq([i, j], i = 0 .. 1), j = 0 .. 1)];

>	Eq2 := [seq(p2(u[])-ff(u[]), `in`(u, DS2))];

>	modp(Linear(Eq2, [a, b, c, d]),2);

end proc;

proc (ff) local DS2, Eq2; DS2 := [seq(seq([i, j], i = 0 .. 1), j = 0 .. 1)]; Eq2 := [seq(p2(u[])-ff(u[]), `in`(u, DS2))]; modp(SolveTools:-Linear(Eq2, [a, b, c, d]), 2) end proc

(12)

(13)

(14)

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(16)

Thus the connectives in

wx = 1+x

In terms of polynomial functions

(17)

(18)

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(22)

>	conn3:=proc(ff)

>	local DS3,Eq3;

>	DS3 := [seq(seq([i, j], i = 0 .. 2), j = 0 .. 2)];

>	Eq3:= [seq(p3(u[])-ff(u[]), `in`(u, DS3))];

>	`mod`(Linear(Eq3, [a, b, c, d,e,f,g,h,k]), 3);

end proc;

proc (ff) local DS3, Eq3; DS3 := [seq(seq([i, j], i = 0 .. 2), j = 0 .. 2)]; Eq3 := [seq(p3(u[])-ff(u[]), `in`(u, DS3))]; `mod`(SolveTools:-Linear(Eq3, [a, b, c, d, e, f, g, h, k]), 3) end proc

(23)

(24)

(25)

(26)

(27)

Therefore the connectives in

wx =

In terms of polynomial functions

(28)

(29)

(30)

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Checking tautologies

The following procedure based on Groebner basis can compute compound proposition as a reduced polynomial in

The proposition is a tautolog if the result is 1 for

In the following procedure ev (evaluate).

p = 2, 3 primes

V is the list of variables x,y,z,...

q is the proposition as a polynomial in x, y, z, ...

>	ev:=proc(p,V,q)

>	local n,W,K,B;

>	n:=nops(V);

>	W:=[seq(V[i],i=1..n)];

>	K:=[seq(x^p-x,x in W)];

>	B:=Basis(K,tdeg(seq(V[i],i=1..n)));

>	modp(NormalForm(q,B,tdeg(seq(V[i],i=1..n))),p);

end proc;

proc (p, V, q) local n, W, K, B; n := nops(V); W := [seq(V[i], i = 1 .. n)]; K := [seq(x^p-x, `in`(x, W))]; B := Groebner:-Basis(K, tdeg(seq(V[i], i = 1 .. n))); modp(Groebner:-NormalForm(q, B, tdeg(seq(V[i], i = 1 .. n))), p) end proc

(33)

1) x→(y→x)

2) (x→(y→z)→((x→y)→(x→z))

3) (wx→wy)→(y→x)

To verify the second axiom for example. The other two axioms are similar.

(34)

(35)

(36)

1) x→(y→x)

2) (x→y)→((y→z)→(x→z))

3) (wx→wy)→(y→x)

4) ((x→wx )→x)→x

To verify the fourth axiom. The rest are similar.

(37)

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(39)

4-valued Logic

The truth values of propositional Lukasiewicz's 4- valued logic
"L[4]={0,1/(3),2/(3),1} can be represented by the elements of Galois field GF(4)."

GF(4)=
$"(`&Zopf;`[2][t])/(<t^(2)+t+1>). GF(4)={0,1,t,t+1} modulo 2, modulo(t^(2)+t+1) . We reserve the letter t for this purpose only."$

Minimum and maximum functions on GF(4) are given by the following two proceddures.

>	m:=proc(u,v)

local r;

>	r:=min(u,v);

>	r:=eval(subs(t=2,r));

>	subs(2=t,3=t+1,r);

end proc;

proc (u, v) local r; r := min(u, v); r := eval(subs(t = 2, r)); subs(2 = t, 3 = t+1, r) end proc

(40)

>	M:=proc(u,v)

local r;

>	r:=max(u,v);

>	r:=eval(subs(t=2,r));

>	subs(2=t,3=t+1,r);

end proc;

proc (u, v) local r; r := max(u, v); r := eval(subs(t = 2, r)); subs(2 = t, 3 = t+1, r) end proc

(41)

(42)

(43)

The order on GF(4) is therefore given by 0 < 1 < t < t+1 .

The connectives are given by the following functions

(44)

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Consider the following lists

(49)

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(51)

The reduced polynomial when n = 4

p4 := proc (x, y) options operator, arrow; a[1]*x^3*y^3+a[2]*x^3*y^2+a[3]*x^2*y^3+a[4]*x^3*y+a[5]*x^2*y^2+a[6]*x*y^3+a[7]*x^3+a[8]*x^2*y+a[9]*x*y^2+a[10]*y^3+a[11]*x^2+a[12]*x*y+a[13]*y^2+a[14]*x+a[15]*y+a[16] end proc

proc (x, y) options operator, arrow; a[1]*x^3*y^3+a[2]*x^3*y^2+a[3]*x^2*y^3+a[4]*x^3*y+a[5]*x^2*y^2+a[6]*x*y^3+a[7]*x^3+a[8]*x^2*y+a[9]*x*y^2+a[10]*y^3+a[11]*x^2+a[12]*x*y+a[13]*y^2+a[14]*x+a[15]*y+a[16] end proc

(52)

>	conn4:=proc(ff)

>	local DS4,Eq4;

>	DS4 := [seq(seq([i, j], `in`(i, L4)), `in`(j, L4))];

>	Eq4:= [seq(p4(u[])-ff(u[]), `in`(u, DS4))];

>	`mod`(Linear(Eq4, CC), 2);

end proc;

proc (ff) local DS4, Eq4; DS4 := [seq(seq([i, j], `in`(i, L4)), `in`(j, L4))]; Eq4 := [seq(p4(u[])-ff(u[]), `in`(u, DS4))]; `mod`(SolveTools:-Linear(Eq4, CC), 2) end proc

(53)

The connectives

{a[1] = t, a[2] = 1, a[3] = 0, a[4] = 0, a[5] = t, a[6] = t+1, a[7] = 0, a[8] = 1, a[9] = t+1, a[10] = 0, a[11] = 0, a[12] = 1, a[13] = 0, a[14] = 1, a[15] = 0, a[16] = t+1}

(54)

{a[1] = 0, a[2] = t+1, a[3] = t+1, a[4] = t, a[5] = 1, a[6] = t, a[7] = 0, a[8] = t, a[9] = t, a[10] = 0, a[11] = 0, a[12] = 0, a[13] = 0, a[14] = 1, a[15] = 1, a[16] = 0}

(55)

{a[1] = 0, a[2] = t+1, a[3] = t+1, a[4] = t, a[5] = 1, a[6] = t, a[7] = 0, a[8] = t, a[9] = t, a[10] = 0, a[11] = 0, a[12] = 0, a[13] = 0, a[14] = 0, a[15] = 0, a[16] = 0}

(56)

{a[1] = 0, a[2] = 1, a[3] = 1, a[4] = t+1, a[5] = 0, a[6] = t+1, a[7] = 0, a[8] = t, a[9] = t, a[10] = 0, a[11] = 0, a[12] = 0, a[13] = 0, a[14] = 1, a[15] = 1, a[16] = t+1}

(57)

Connectives in terms of polynomials

proc (x) options operator, arrow; y = tx^3*y^3+x^3*y^2+tx^2*y^2+(t+1)*xy^3+x^2*y+(t+1)*xy^2+xy+x+t+1 end proc

x or y = (t+1)*x^3*y^2+(t+1)*x^2*y^3+tx^3*y+x^2*y^2+txy^3+tx^2*y+txy^2+x+y

`↔`(x, y) and y = x^3*y^2+x^2*y^3+(t+1)*x^3*y+(t+1)*xy^3+tx^2*y+txy^2+x+y+t+1

(58)

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(62)

The following procedure evaluate compound propositions in terms of reduced polynomials in

p = 2 and V, q as above in ev.

>	ev4:=proc(p,V,q)

>	local n,W,K,B;

>	n:=nops(V);

>	W:=[seq(V[i],i=1..n)];

>	K:=[seq(x^(2*p)-x,x in W)];

>	B:=Basis(K,tdeg(seq(V[i],i=1..n)));

>	modp(NormalForm(q,B,tdeg(seq(V[i],i=1..n))),p);

end proc;

proc (p, V, q) local n, W, K, B; n := nops(V); W := [seq(V[i], i = 1 .. n)]; K := [seq(x^(2*p)-x, `in`(x, W))]; B := Groebner:-Basis(K, tdeg(seq(V[i], i = 1 .. n))); modp(Groebner:-NormalForm(q, B, tdeg(seq(V[i], i = 1 .. n))), p) end proc