离散数学实践:真值表与范式
预备知识
根据合式公式的真值表与主合取范式与主析取范式的关系来求。在命题逻辑中，合式公式的真值表的应用非常广泛。列合式公式真值表的步骤如下：
- 找出合式公式中出现的所有命题变项。
- 按照二进制的顺序给出命题公式的2n种赋值。
- 对每个赋值按照合式公式的层次求出它的值。
- 所有成真赋值的合取即为主合取范式，所有成假赋值的析取即为主析取范式
熟悉真值表定义，并列出合式公式的真值表，并根据真值表的结果来判断公式的类型。
源程序参考
PropostionalLogicHeader头文件
/************************************************************************** * (C) Copyright 2015-2018 by Gavin Y. Liu All Rights Reserved. * * * * DISCLAIMER: The authors and publisher shall not be liable in any event * * for incidental or consequential damages in connection with, or arising * * out of, the furnishing, performance, or use of these programs. * **************************************************************************/ /* File: PropostionalLogicHeader.h *------------------------------- * This interface exports a simple symboll table abstraction * */ #ifndef _PROPOSTIONALLOGICHEADER_H_ #define _PROPOSTIONALLOGICHEADER_H_ /* * Constants *------------------ * LengthMaxLimit - Length Max value for the tables */ #define LengthMaxLimit 100 /*Private function prototypes*/ /* * Function: negation * Usage: negation(p); *------------------- * This funtion is an operation that takes a proposition p to another proposition "not p", * written ¬p, which is interpreted intuitively as being true when p is false and false when p is true. */ static int negation(const int p); /* * Function: conjunction * Usage: conjunction(p,q); * --------------------------------- * This function is an operation on two logical values, typically the values of two propositions, that * produces a value of true if and only if both of its operands are true.The conjunctive identity is 1, * which is to say that AND-ing an expression with 1 will never change the value of the expression. In * keeping with the concept of vacuous truth, when conjunction is defined as an operator or function of * arbitrary arity, the empty conjunction (AND-ing over an empty set of operands) is often defined as * having the result 1. */ static int conjunction(const int p,const int q); /* * Function: disjunction * Usage: disjunction(p,q); *---------------------------------------- * The Function is the values of two propositions, that has a value of false if and only if both of its * operands are false. More generally, a disjunction is a logical formula that can have one or more literals * separated only by ORs. A single literal is often considered to be a degenerate disjunction. * * The disjunctive identity is false, which is to say that the or of an expression with false has the same * value as the original expression. In keeping with the concept of vacuous truth, when disjunction is defined * as an operator or function of arbitrary arity, the empty disjunction (OR-ing over an empty set of operands) * is generally defined as false. */ static int disjunction(const int p, const int q); /* * Function: conditional * Usage: conditional(p,q) * ---------------------------------------- * The function is The material conditional is used to form statements of the form "p→q" (termed a conditional * statement) which is read as "if p then q" and conventionally compared to the English construction "If... * then...". But unlike as the English construction may, the conditional statement "p→q" does not specify a * causal relationship between p and q and is to be understood to mean "if p is true, then q is also true" such * that the statement "p→q" is false only when p is true and q is false.[1] The material conditional is also to * be distinguished from logical consequence. */ static int conditional(const int p, const int q); /* * Function:bicondtional * Usage: bincontional(p,q) * ---------------------------------------------- * The function is the logical connective of two statements asserting "p if and only if q", where q is an * antecedent and p is a consequent. This is often abbreviated p iff q. The operator is denoted using a * doubleheaded arrow (↔), a prefixed E (Epq), an equality sign (=), an equivalence sign (≡), or EQV. It is * logically equivalent to (p → q) ∧ (q → p), or the XNOR (exclusive nor) boolean operator. It is equivalent to * "(not p or q) and (not q or p)". It is also logically equivalent to "(p and q) or (not p and not q)", meaning * "both or neither". * * The only difference from material conditional is the case when the hypothesis is false but the conclusion is * true. In that case, in the conditional, the result is true, yet in the biconditional the result is false. * * In the conceptual interpretation, a = b means "All a 's are b 's and all b 's are a 's"; in other words, the * sets a and b coincide: they are identical. This does not mean that the concepts have the same meaning. * Examples: "triangle" and "trilateral", "equiangular trilateral" and "equilateral triangle". The antecedent is * the subject and the consequent is the predicate of a universal affirmative proposition. * * In the propositional interpretation, a ⇔ b means that a implies b and b implies a; in other words, that the * propositions are equivalent, that is to say, either true or false at the same time. This does not mean that * they have the same meaning. Example: "The triangle ABC has two equal sides", and "The triangle ABC has two * equal angles". The antecedent is the premise or the cause and the consequent is the consequence. When an * implication is translated by a hypothetical (or conditional) judgment the antecedent is called the hypothesis * (or the condition) and the consequent is called the thesis. */ static int biconditional(const int p, const int q); /* * Function:compute * Usage: compute(p,q ch) * ---------------------------------- */ static int compute(const int p, const int q, const char ch); /* * Function: is_proposition * Usage: is_propositon(ch) * ---------------------------------- */ static int is_proposition(const char ch); /* * Function:is_LogicalConnectives * Usage: is_LogicalConnectives(c) * ---------------------------------- */ static int is_LogicalConnectives(const char c); /* * Function:get_isp * Usage: get_isp(ch) * ---------------------------------- */ static int get_isp(const char ch); /* * Function:get_icp * Usage: get_icp(ch) * ---------------------------------- */ static int get_icp(const char ch); /* * Function:to_InersePolandT * Usage: to_InersePolandT(*last_exp, *pre_exp) * ------------------------------------------- */ static void to_InversePolandT(char *last_exp, const char *pre_exp); /* * Function: add_blackets * Usage: add_blackets(s) * ---------------------------- */ static void add_blackets(char* s); /* * Function: exp_resolve * Usage: exp_resolve(*exp,length,(*re_exp)[LengMaxLimit],k) * ---------------------------- */ static void exp_resolve(const char* exp, const int length, char (*re_exp)[LengthMaxLimit] , int k); static void binary_inc(int* a, int length); /* * Function: get_proposition * Usage: get_proposition(*exp,length,*p) * ------------------------------------------ * The propositions in these logics are more complex. First, terms must be defined. A term is (i) a variable or * (ii) a function symbol applied to the number of terms required by the function symbol's arity. For example, * if + is a binary function symbol and x, y, and z are variables, then x+(y+z) is a term, which might be written * with the symbols in various orders. A proposition is (i) a predicate symbol applied to the number of terms * required by its arity, (ii) an operator applied to the number of propositions required by its arity, or * (iii) a quantifier applied to a proposition. For example, if = is a binary predicate symbol and ∀ is a * quantifier, then ∀x,y,z [(x = y) → (x+z = y+z)] is a proposition. This more complex structure of propositions * allows these logics to make finer distinctions between inferences, i.e., to have greater expressive power. * * In this context, propositions are also called sentences, statements, statement forms, formulas, and well-formed * formulas, though these terms are usually not synonymous within a single text. This definition treats propositions * as syntactic objects, as opposed to semantic or mental objects. That is, propositions in this sense are meaningless, * formal, abstract objects. They are assigned meaning and truth-values by mappings called interpretations and * valuations, respectively. */ static int get_proposition(const char* exp, const int length, char* p); static void proposition_ass(const char* pre_exp, int length, char *p, int* v, int n, char* last_exp); static int bin2dec(int* v, int n); /* Public function prototyes*/ /* * Function:is_wellformed * Usage:is_wellformed(s,length) * ----------------------------------- * The formulas are inductively defined as follows: * Each propositional variable is, on its own, a formula. * If φ is a formula, then \lnotφ is a formula. * If φ and ψ are formulas, and • is any binary connective, then ( φ • ψ) is a formula. * Here • could be (but is not limited to) the usual operators ∨, ∧, →, or ↔. * This definition can also be written as a formal grammar in Backus–Naur form, provided the set * of variables is finite. */ int is_wellformed(const char *s,const int length); /* * Function: compute_wellformed * Usage: compute_wellformed(exp,length) * ------------------------------------------- */ int compute_wellformed(const char* exp, const int length); /* * Function: truth_table * Usage: truth_table(exp,length) * ---------------------------------- * a truth table is composed of one column for each input variable (for example, A and B), and * one final column for all of the possible results of the logical operation that the table is * meant to represent (for example, A XOR B). Each row of the truth table therefore contains one * possible configuration of the input variables (for instance, A=true B=false), and the result of * the operation for those values. See the examples below for further clarification. */ void truth_table(const char* exp, const int length); int get_main_disjunction(const char* exp, char* resu); int get_main_conjunction( char* exp, char* resu); void help(); #endif // end of _PROPOSTIONALLOGICHEADER_H_
PropostionalLogicSource源文件
/**************************************************************************
* (C) Copyright 2015-2018 by Gavin Y. Liu All Rights Reserved. *
* *
* DISCLAIMER: The authors and publisher shall not be liable in any event *
* for incidental or consequential damages in connection with, or arising *
* out of, the furnishing, performance, or use of these programs. *
**************************************************************************/
#include <stdio.h>
#include <stdlib.h>
#include <ctype.h>
#include <string.h>
#include <math.h>
#include "PropostionalLogicHeader.h"
/*Private functions */
//Compute testing
static int compute(const int p, const int q, const char ch){
switch(ch){
case '~':
return negation(q);
case '+':
return disjunction(p, q);
case '*':
return conjunction(p, q);
case '>':
return conditional(p, q);
case '=':
return biconditional(p, q);
default:
return -1;
}
}
//Negation
static int negation(const int p){
return p == 0 ? 1 : 0;
}
//Conjunction
static int conjunction(const int p, const int q){
return (p != 0 && q != 0) ? 1 : 0;
}
//Disjunction
static int disjunction(const int p, const int q){
return (p == 0 && q == 0) ? 0 : 1;
}
//Conditional
static int conditional(const int p, const int q){
return (p != 0 && q == 0) ? 0 : 1;
}
//Biconditional
static int biconditional(const int p, const int q){
return ((p == 0 && q == 0) || (p != 0 && q != 0)) ? 1 : 0;
}
//Whether or not is Propostion?
static int is_proposition(const char ch){
return isalpha(ch) || isalnum(ch);
}
//Whether or not is Logical Connnectives?
static int is_LogicalConnectives(const char c){
switch(c){
case '+':
case '*':
case '>':
case '~':
case '=':
return 1;
default:
return 0;
}
}
//Stack Priority
static int get_isp(const char ch){
switch(ch){
case '#':
return 0;
case '+':
case '*':
case '=':
case '>':
return 5;
case '~':
return 7;
case '(':
return 1;
case ')':
return 8;
default:
return -1;
}
}
//Stack Priority
static int get_icp(const char ch){
switch(ch)
{
case '#':
return 0;
case '+':
case '*':
case '=':
case '>':
return 4;
case '~':
return 6;
case '(':
return 8;
case ')':
return 1;
default:
return -1;
}
}
//Inverse Poland Type
static void to_InversePolandT(char* last_exp, const char* pre_exp){
char sign_stack[LengthMaxLimit] = {'\0'};
int sp = -1;
int pp = 0;
int lp = 0;
sign_stack[++sp] = '#';
while(pre_exp[pp] != '\0'){
if(is_proposition(pre_exp[pp])) {
last_exp[lp++] = pre_exp[pp];
}
else{
while(get_icp(pre_exp[pp]) < get_isp(sign_stack[sp])){
last_exp[lp++] = sign_stack[sp];
sign_stack[sp--] = '\0';
}
if(get_icp(pre_exp[pp]) == get_isp(sign_stack[sp]))
sign_stack[sp--] = '\0';
else
sign_stack[++sp] = pre_exp[pp];
}
pp++;
}
while(sp != 0){
last_exp[lp++] = sign_stack[sp--];
}
}
static void add_blackets(char* s){
int len = strlen(s);
memcpy(s + 1, s, len);
s[0] = '(';
s[len + 1] = ')';
}
static void exp_resolve(const char* exp, const int length, char (*re_exp)[LengthMaxLimit] , int k){
char stack[LengthMaxLimit][LengthMaxLimit] = {"\0"};
char rpn_exp[LengthMaxLimit] = "\0";
char tmp_exp[LengthMaxLimit] = "\0";
int sp = -1;
int len = 0;
int i = 0;
int j = 0;
to_InversePolandT(rpn_exp, exp);
for(i = 0; i < length; i++){
if(j > k){
break;
}
else if(rpn_exp[i] == '~'){
memset(tmp_exp, '\0', sizeof(char) * LengthMaxLimit);
strncpy(tmp_exp, &rpn_exp[i], sizeof(char));
strcpy(tmp_exp + strlen(tmp_exp), stack[sp]);
add_blackets(tmp_exp);
memset(stack[sp], '\0', sizeof(char) * LengthMaxLimit);
sp--;
strcpy(stack[++sp], tmp_exp);
j++;
}
else if(is_LogicalConnectives(rpn_exp[i])){
memset(tmp_exp, '\0', sizeof(char) * LengthMaxLimit);
strcpy(tmp_exp, stack[sp-1]);
strncpy(tmp_exp + strlen(tmp_exp), &rpn_exp[i], 1);
strcpy(tmp_exp + strlen(tmp_exp), stack[sp]);
add_blackets(tmp_exp);
memset(stack[sp], '\0', sizeof(char) * LengthMaxLimit);
memset(stack[sp - 1], '\0', sizeof(char) *LengthMaxLimit);
sp -= 2;
strcpy(stack[++sp], tmp_exp);
j++;
}
else if(is_proposition(rpn_exp[i])){
strncpy(stack[++sp], &rpn_exp[i], 1);
}
}
strcpy(re_exp[k], tmp_exp);
}
static void binary_inc(int* a, int length){
int i = 0, c = 0;
a[length - 1] += 1;
for(i = length - 1; i > 0; i--) {
if(a[i] > 1){
a[i] = 0;
a[i - 1] += 1;
}
else
break;
}
}
static int get_proposition(const char* exp, const int length, char* p){
int i = 0;
int j = 0;
int k = 0;
int n = 0;
for(i = 0; i < length; i++){
if(is_proposition(exp[i])){
j = 0;
k = 0;
while(p[j] != '\0') {
if(exp[i] == p[j]){
k = 1;
break;
}
j++;
}
if(k == 0){
p[j] = exp[i];
n++;
}
}
}
return n;
}
static void proposition_ass(const char* pre_exp, int length, char *p, int* v, int n, char* last_exp)
{
int j = 0, k = 0;
for(j = 0; j < length; j++) {
if(is_proposition(pre_exp[j])){
for(k = 0; k < n; k++){
if(pre_exp[j] == p[k])
last_exp[j] = v[k] + '0';
}
}
else
last_exp[j] = pre_exp[j];
}
}
//Binary to Dec
static int bin2dec(int* v, int n)
{
int num = 0, i = 0;
for(i = n - 1; i > -1; i--)
{
if(v[i] == 0)
continue;
num += (int)pow(2.0, n - i - 1);
}
return num;
}
/*Public functions */
// Compute for well formed formula Values
int compute_wellformed(const char* exp, const int length)
{
if(!is_wellformed(exp, length)){
printf("The expression is not a well formed\n");
return -1;
}
char last_exp[LengthMaxLimit] = {'\0'};
int stack[LengthMaxLimit] = {0};
int lp = 0, sp = -1, r, i;
for(i = 0; i < length; i++){
if(isalpha(exp[i])){
printf("does not assigment");
return -1;
}
}
to_InversePolandT(last_exp, exp);
while(last_exp[lp] != '\0')
{
if(isalnum(last_exp[lp])){
stack[++sp] = last_exp[lp] - '0';
}
else if(last_exp[lp] == '~'){
r = compute(0, stack[sp--], last_exp[lp]);
stack[++sp] = r;
}
else if(is_LogicalConnectives(last_exp[lp])){
r = compute(stack[sp-1], stack[sp], last_exp[lp]);
sp -= 2;
stack[++sp] = r;
}
lp++;
}
return stack[sp];
}
//Whether or not is well-formed?
int is_wellformed(const char* s, const int length){
int l_bracket = 0;
int r_bracket = 0;
int i = 0;
if(length == 1 && is_proposition(s[i]))
return 1;
while(i < length){
if(is_proposition(s[i])){
if(!is_LogicalConnectives(s[i + 1]) && s[i + 1] != ')' && s[i + 1] != '\0' || s[i + 1] == '~')
return 0;
}
else if(is_LogicalConnectives(s[i])){
if(!is_proposition(s[i + 1]) && s[i + 1] != '~' && s[i + 1] != '(')
return 0;
}
else if(s[i] == '(')
{
if(s[i + 1] != '~' && !is_proposition(s[i + 1]) && s[i + 1] != '(')
return 0;
l_bracket++;
}
else if(s[i] == ')')
{
if(s[i + 1] != '\0' && !is_LogicalConnectives(s[i + 1]) && s[i + 1] != ')' || s[i + 1] == '~')
return 0;
r_bracket++;
}
i++;
}
if(l_bracket == r_bracket)
return 1;
else
return 0;
}
//print truth table
void truth_table(const char* exp, const int length)
{
if(!is_wellformed(exp, length)) {
printf("The expression is not a wff\n");
return;
}
int i, j, k, r, n = 0, m = 0;
char s[LengthMaxLimit] = "\0";
char p[LengthMaxLimit] = "\0";
char ss[LengthMaxLimit][LengthMaxLimit] = {'\0'};
n = get_proposition(exp, length, p);
m = (int)pow(2.0, n);
int* v = new int[m];
memset(v, 0, sizeof(int) * m);
memset(ss, '\0', sizeof(char) * LengthMaxLimit * LengthMaxLimit);
for(i = 0, k = 0; i < length; i++){
if(is_LogicalConnectives(exp[i])){
exp_resolve(exp, length, ss, k);
k++;
}
}
for(j = 0; j < n; j++)
printf("%c ", p[j]);//print propostional
for(j = 0; j < k; j++)
printf("%s ", ss[j]);
printf("\n");
for(i = 0; i < m; i++){
for(j = 0; j < n; j++)
printf("%d ", v[j]);
for(j = 0; j < k; j++){
memset(s, '\0', sizeof(char) * LengthMaxLimit);
proposition_ass(ss[j], strlen(ss[j]), p, v, n, s);
r = compute_wellformed(s, strlen(s));
printf("%d ", r);
}
binary_inc(v, n);
printf("\n");
}
delete[] v;
}
//Main Disjunction
int get_main_disjunction(const char* exp, char* resu)
{
int len = strlen(exp);
if(!is_wellformed(exp, len))
return 0;
memset(resu, '\0', sizeof(char) * LengthMaxLimit);
char tmp_exp[LengthMaxLimit] = "\0";
char p[LengthMaxLimit] = "\0";
int i = 0, r = 0;
int n = get_proposition(exp, len, p);
int m = (int)pow(2.0, n);
int* v = new int[m];
memset(v, 0, sizeof(int) * m);
int r_index = 0;
for(i = 0; i < m; i++)
{
proposition_ass(exp, len, p, v, n, tmp_exp);
r = compute_wellformed(tmp_exp, strlen(tmp_exp)); //compute to well-fomred
if(r == 1)
{
if(r_index - 1 > -1 && resu[r_index - 1] != '\0')
resu[r_index++] = '+';
char strn[10] = "\0", num = 0;
num = bin2dec(v, n);
resu[r_index++] = 'm';
_itoa(num, strn, 10);
memcpy(resu + r_index, strn, sizeof(char) * strlen(strn));
resu += strlen(strn); //以上几行为添加范式
}
binary_inc(v, n);
}
delete[] v;
return 1;
}
//Main conjunction
int get_main_conjunction( char* exp, char* resu)
{
int len = strlen(exp);
if(!is_wellformed(exp, len))
return 0;
memset(resu, '\0', sizeof(char) *LengthMaxLimit);
char tmp_exp[LengthMaxLimit] = "\0";
char p[LengthMaxLimit] = "\0";
int i = 0, r = 0;
int n = get_proposition(exp, len, p);
int m = (int)pow(2.0, n);
int* v = new int[m];
memset(v, 0, sizeof(int) * m);
int r_index = 0;
for(i = 0; i < m; i++)
{
proposition_ass(exp, len, p, v, n, tmp_exp);
r = compute_wellformed(tmp_exp, strlen(tmp_exp));
if(r == 0)
{
if(r_index - 1 > -1 && resu[r_index - 1] != '\0')
resu[r_index++] = '*';
char strn[10] = "\0", num = 0;
num = bin2dec(v, n);
resu[r_index++] = 'M';
_itoa(num, strn, 10);
memcpy(resu + r_index, strn, sizeof(char) * strlen(strn));
resu += strlen(strn);
}
binary_inc(v, n);
}
delete[] v;
return 1;
}
void help(){
printf("*********************************************************");
printf("\n*\t\t\t\t\t\t\t*");
printf("\n*\t\t Welcome to Propostion Set\t\t*");
printf("\n*\t\t\t\t\t\t\t*");
printf("\n*\t Usage:\t\t\t\t\t\t*");
printf("\n*\t\t'~': Negation\t\t\t\t*");
printf("\n*\t\t'+': Disjunction\t\t\t\*");
printf("\n*\t\t'*': Conjunction\t\t\t*");
printf("\n*\t\t'>': Conditional\t\t\t*");
printf("\n*\t\t'=': Biconditional\t\t\t*");
printf("\n*\t\t'default': -1\t\t\t\t*");
printf("\n*\t\t\t\t\t\t\t*");
printf("\n*********************************************************");
printf("\n");
}
Main源文件
/**************************************************************************
* (C) Copyright 2015-2018 by Gavin Y. Liu All Rights Reserved. *
* *
* DISCLAIMER: The authors and publisher shall not be liable in any event *
* for incidental or consequential damages in connection with, or arising *
* out of, the furnishing, performance, or use of these programs. *
**************************************************************************/
#include <stdio.h>
#include <string.h>
#include "PropostionalLogicHeader.h"
/*main program */
int main( ){
help();
char s[100];
int length, r = 0;
while(scanf("%s", s) && s[0] != '#'){
length = strlen(s);
printf("expression is: %s\n\n", s);
//Whether or not is well-formed?
printf("is_wellformed Formula? %d\n\n", is_wellformed(s, length));
printf("compute_wellfomred is: ");
r = compute_wellformed(s, length);
if(r > -1)
printf("%d", r);
printf("\n\n");
printf("Print True table\n");
truth_table(s,length);
char dis_form[LengthMaxLimit];
get_main_disjunction(s, dis_form);
printf("main disjunction normal form is: %s\n\n", dis_form);
char con_form[LengthMaxLimit];
get_main_conjunction(s, con_form);
printf("main conjunction normal form is: %s\n\n", con_form);
}
return 0;
}
测试结果
界面
测试:p蕴含q,然后蕴含r
测试: p析取q,然后蕴含r
测试:p蕴含q,然后析取r
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