完整定义一套严格的三值代数系统，仿照布尔代数的结构但扩展到三个值。
1. 三值代数系统定义
基本集合和值
"""
三值代数系统定义
仿照布尔代数结构，但值域扩展到三个值
"""
from enum import Enum
from typing import List, Tuple
class TernaryValue(Enum):
"""三值系统的三个基本值"""
FALSE = 0 # 假
UNKNOWN = 1 # 未知/不确定
TRUE = 2 # 真
def __str__(self):
return ["F", "U", "T"][self.value]
基本运算定义
class TernaryAlgebra:
"""三值代数系统，包含基本运算和定律"""
@staticmethod
def ternary_not(a: TernaryValue) -> TernaryValue:
"""非运算（否定）"""
truth_table = {
TernaryValue.FALSE: TernaryValue.TRUE,
TernaryValue.UNKNOWN: TernaryValue.UNKNOWN, # 未知的否定仍是未知
TernaryValue.TRUE: TernaryValue.FALSE
}
return truth_table[a]
@staticmethod
def ternary_and(a: TernaryValue, b: TernaryValue) -> TernaryValue:
"""与运算（合取）- 使用Kleene三值逻辑定义"""
# 真值表实现
if a == TernaryValue.FALSE or b == TernaryValue.FALSE:
return TernaryValue.FALSE
elif a == TernaryValue.TRUE and b == TernaryValue.TRUE:
return TernaryValue.TRUE
else:
return TernaryValue.UNKNOWN
@staticmethod
def ternary_or(a: TernaryValue, b: TernaryValue) -> TernaryValue:
"""或运算（析取）- 使用Kleene三值逻辑定义"""
if a == TernaryValue.TRUE or b == TernaryValue.TRUE:
return TernaryValue.TRUE
elif a == TernaryValue.FALSE and b == TernaryValue.FALSE:
return TernaryValue.FALSE
else:
return TernaryValue.UNKNOWN
@staticmethod
def ternary_xor(a: TernaryValue, b: TernaryValue) -> TernaryValue:
"""异或运算"""
and_result = TernaryAlgebra.ternary_and(a, b)
or_result = TernaryAlgebra.ternary_or(a, b)
not_and = TernaryAlgebra.ternary_not(and_result)
return TernaryAlgebra.ternary_and(or_result, not_and)
@staticmethod
def ternary_imply(a: TernaryValue, b: TernaryValue) -> TernaryValue:
"""蕴含运算 (a → b) = ¬a ∨ b"""
not_a = TernaryAlgebra.ternary_not(a)
return TernaryAlgebra.ternary_or(not_a, b)
@staticmethod
def ternary_equiv(a: TernaryValue, b: TernaryValue) -> TernaryValue:
"""等价运算 (a ≡ b) = (a → b) ∧ (b → a)"""
a_implies_b = TernaryAlgebra.ternary_imply(a, b)
b_implies_a = TernaryAlgebra.ternary_imply(b, a)
return TernaryAlgebra.ternary_and(a_implies_b, b_implies_a)
2. 完整真值表
class TruthTableGenerator:
"""生成和显示三值逻辑的真值表"""
@staticmethod
def generate_truth_table(operation, operation_name: str):
"""生成指定运算的真值表"""
values = [TernaryValue.FALSE, TernaryValue.UNKNOWN, TernaryValue.TRUE]
print(f"\n{operation_name} 真值表:")
print("a\\b | F U T")
print("----+------------")
for a in values:
row = f" {a} |"
for b in values:
result = operation(a, b)
row += f" {result} "
print(row)
@staticmethod
def generate_unary_truth_table(operation, operation_name: str):
"""生成一元运算的真值表"""
values = [TernaryValue.FALSE, TernaryValue.UNKNOWN, TernaryValue.TRUE]
print(f"\n{operation_name} 真值表:")
print("a | result")
print("--+-------")
for a in values:
result = operation(a)
print(f"{a} | {result}")
# 显示所有基本运算的真值表
def display_all_truth_tables():
"""显示完整的真值表集合"""
print("=== 三值代数系统完整真值表 ===")
TruthTableGenerator.generate_unary_truth_table(
TernaryAlgebra.ternary_not, "NOT"
)
TruthTableGenerator.generate_truth_table(
TernaryAlgebra.ternary_and, "AND"
)
TruthTableGenerator.generate_truth_table(
TernaryAlgebra.ternary_or, "OR"
)
TruthTableGenerator.generate_truth_table(
TernaryAlgebra.ternary_xor, "XOR"
)
TruthTableGenerator.generate_truth_table(
TernaryAlgebra.ternary_imply, "IMPLY"
)
TruthTableGenerator.generate_truth_table(
TernaryAlgebra.ternary_equiv, "EQUIV"
)
3. 代数定律验证
class TernaryLaws:
"""验证三值代数系统的代数定律"""
@staticmethod
def test_identity_laws():
"""恒等律验证"""
print("\n=== 恒等律验证 ===")
values = [TernaryValue.FALSE, TernaryValue.UNKNOWN, TernaryValue.TRUE]
for a in values:
# a ∧ T = a
and_true = TernaryAlgebra.ternary_and(a, TernaryValue.TRUE)
# a ∨ F = a
or_false = TernaryAlgebra.ternary_or(a, TernaryValue.FALSE)
print(f"a={a}: a∧T={and_true}, a∨F={or_false}")
@staticmethod
def test_complement_laws():
"""补元律验证"""
print("\n=== 补元律验证 ===")
values = [TernaryValue.FALSE, TernaryValue.UNKNOWN, TernaryValue.TRUE]
for a in values:
# a ∧ ¬a = F
and_complement = TernaryAlgebra.ternary_and(a, TernaryAlgebra.ternary_not(a))
# a ∨ ¬a = T
or_complement = TernaryAlgebra.ternary_or(a, TernaryAlgebra.ternary_not(a))
print(f"a={a}: a∧¬a={and_complement}, a∨¬a={or_complement}")
@staticmethod
def test_commutative_laws():
"""交换律验证"""
print("\n=== 交换律验证 ===")
test_pairs = [
(TernaryValue.FALSE, TernaryValue.TRUE),
(TernaryValue.UNKNOWN, TernaryValue.TRUE),
(TernaryValue.FALSE, TernaryValue.UNKNOWN)
]
for a, b in test_pairs:
and_ab = TernaryAlgebra.ternary_and(a, b)
and_ba = TernaryAlgebra.ternary_and(b, a)
or_ab = TernaryAlgebra.ternary_or(a, b)
or_ba = TernaryAlgebra.ternary_or(b, a)
print(f"a={a}, b={b}: AND交换: {and_ab}={'=' if and_ab == and_ba else '!'}{and_ba}")
print(f"a={a}, b={b}: OR交换: {or_ab}={'=' if or_ab == or_ba else '!'}{or_ba}")
@staticmethod
def test_associative_laws():
"""结合律验证"""
print("\n=== 结合律验证 ===")
a, b, c = TernaryValue.FALSE, TernaryValue.UNKNOWN, TernaryValue.TRUE
# (a ∧ b) ∧ c = a ∧ (b ∧ c)
left_and = TernaryAlgebra.ternary_and(
TernaryAlgebra.ternary_and(a, b), c
)
right_and = TernaryAlgebra.ternary_and(
a, TernaryAlgebra.ternary_and(b, c)
)
# (a ∨ b) ∨ c = a ∨ (b ∨ c)
left_or = TernaryAlgebra.ternary_or(
TernaryAlgebra.ternary_or(a, b), c
)
right_or = TernaryAlgebra.ternary_or(
a, TernaryAlgebra.ternary_or(b, c)
)
print(f"AND结合律: ({a}∧{b})∧{c} = {left_and}")
print(f" {a}∧({b}∧{c}) = {right_and}")
print(f" 结果: {'相等' if left_and == right_and else '不相等'}")
print(f"OR结合律: ({a}∨{b})∨{c} = {left_or}")
print(f" {a}∨({b}∨{c}) = {right_or}")
print(f" 结果: {'相等' if left_or == right_or else '不相等'}")
@staticmethod
def test_distributive_laws():
"""分配律验证"""
print("\n=== 分配律验证 ===")
a, b, c = TernaryValue.FALSE, TernaryValue.UNKNOWN, TernaryValue.TRUE
# a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
left_dist = TernaryAlgebra.ternary_and(
a, TernaryAlgebra.ternary_or(b, c)
)
right_dist = TernaryAlgebra.ternary_or(
TernaryAlgebra.ternary_and(a, b),
TernaryAlgebra.ternary_and(a, c)
)
print(f"分配律: {a}∧({b}∨{c}) = {left_dist}")
print(f" ({a}∧{b})∨({a}∧{c}) = {right_dist}")
print(f" 结果: {'成立' if left_dist == right_dist else '不成立'}")
@staticmethod
def test_de_morgans_laws():
"""德摩根定律验证"""
print("\n=== 德摩根定律验证 ===")
test_pairs = [
(TernaryValue.FALSE, TernaryValue.TRUE),
(TernaryValue.UNKNOWN, TernaryValue.UNKNOWN),
]
for a, b in test_pairs:
# ¬(a ∧ b) = ¬a ∨ ¬b
left_dm1 = TernaryAlgebra.ternary_not(
TernaryAlgebra.ternary_and(a, b)
)
right_dm1 = TernaryAlgebra.ternary_or(
TernaryAlgebra.ternary_not(a),
TernaryAlgebra.ternary_not(b)
)
# ¬(a ∨ b) = ¬a ∧ ¬b
left_dm2 = TernaryAlgebra.ternary_not(
TernaryAlgebra.ternary_or(a, b)
)
right_dm2 = TernaryAlgebra.ternary_and(
TernaryAlgebra.ternary_not(a),
TernaryAlgebra.ternary_not(b)
)
print(f"a={a}, b={b}:")
print(f" ¬({a}∧{b}) = {left_dm1}, ¬{a}∨¬{b} = {right_dm1} {'✓' if left_dm1 == right_dm1 else '✗'}")
print(f" ¬({a}∨{b}) = {left_dm2}, ¬{a}∧¬{b} = {right_dm2} {'✓' if left_dm2 == right_dm2 else '✗'}")
4. 三值逻辑应用示例
class TernaryApplication:
"""三值代数的实际应用示例"""
@staticmethod
def database_query_example():
"""数据库三值逻辑查询示例"""
print("\n=== 数据库三值逻辑应用 ===")
# 模拟数据库记录：age字段可能有NULL值
records = [
{"name": "Alice", "age": TernaryValue.TRUE}, # 年龄已知
{"name": "Bob", "age": TernaryValue.UNKNOWN}, # 年龄未知
{"name": "Charlie", "age": TernaryValue.FALSE} # 无年龄信息
]
# 查询：年龄已知或未知的记录
query_condition = lambda age: TernaryAlgebra.ternary_or(
age, TernaryValue.UNKNOWN
)
print("查询结果:")
for record in records:
satisfies = query_condition(record["age"])
print(f" {record['name']}: 满足条件 = {satisfies}")
@staticmethod
def circuit_design_example():
"""三值电路设计示例"""
print("\n=== 三值电路设计 ===")
# 三值信号的逻辑组合
input_a = TernaryValue.TRUE
input_b = TernaryValue.UNKNOWN
input_c = TernaryValue.FALSE
# 复杂逻辑表达式: (A ∧ B) ∨ ¬C
circuit_output = TernaryAlgebra.ternary_or(
TernaryAlgebra.ternary_and(input_a, input_b),
TernaryAlgebra.ternary_not(input_c)
)
print(f"电路输入: A={input_a}, B={input_b}, C={input_c}")
print(f"电路输出: (A∧B)∨¬C = {circuit_output}")
@staticmethod
def uncertainty_reasoning():
"""不确定性推理示例"""
print("\n=== 不确定性推理 ===")
# 知识库中的事实（可能不确定）
fact_rain = TernaryValue.UNKNOWN # 可能下雨
fact_wet_ground = TernaryValue.TRUE # 地面是湿的
# 推理：如果下雨，那么地面是湿的
implication = TernaryAlgebra.ternary_imply(fact_rain, fact_wet_ground)
print(f"事实: 下雨={fact_rain}, 地面湿={fact_wet_ground}")
print(f"推理: 下雨→地面湿 = {implication}")
if implication == TernaryValue.TRUE:
print("结论: 推理成立")
elif implication == TernaryValue.UNKNOWN:
print("结论: 推理不确定")
else:
print("结论: 推理不成立")
5. 主演示程序
def main():
"""主演示函数"""
print("三值代数系统完整演示")
print("=" * 50)
# 1. 显示真值表
display_all_truth_tables()
# 2. 验证代数定律
TernaryLaws.test_identity_laws()
TernaryLaws.test_complement_laws()
TernaryLaws.test_commutative_laws()
TernaryLaws.test_associative_laws()
TernaryLaws.test_distributive_laws()
TernaryLaws.test_de_morgans_laws()
# 3. 应用示例
TernaryApplication.database_query_example()
TernaryApplication.circuit_design_example()
TernaryApplication.uncertainty_reasoning()
print("\n" + "=" * 50)
print("三值代数系统演示完成")
if __name__ == "__main__":
main()
运行结果
三值代数系统完整演示
==================================================
=== 三值代数系统完整真值表 ===
NOT 真值表:
a | result
--+-------
F | T
U | U
T | F
AND 真值表:
a\b | F U T
----+------------
F | F F F
U | F U U
T | F U T
OR 真值表:
a\b | F U T
----+------------
F | F U T
U | U U T
T | T T T
XOR 真值表:
a\b | F U T
----+------------
F | F U T
U | U U U
T | T U F
IMPLY 真值表:
a\b | F U T
----+------------
F | T T T
U | U U T
T | F U T
EQUIV 真值表:
a\b | F U T
----+------------
F | T U F
U | U U U
T | F U T
=== 恒等律验证 ===
a=F: a∧T=F, a∨F=F
a=U: a∧T=U, a∨F=U
a=T: a∧T=T, a∨F=T
=== 补元律验证 ===
a=F: a∧¬a=F, a∨¬a=T
a=U: a∧¬a=U, a∨¬a=U
a=T: a∧¬a=F, a∨¬a=T
=== 交换律验证 ===
a=F, b=T: AND交换: F==F
a=F, b=T: OR交换: T==T
a=U, b=T: AND交换: U==U
a=U, b=T: OR交换: T==T
a=F, b=U: AND交换: F==F
a=F, b=U: OR交换: U==U
=== 结合律验证 ===
AND结合律: (F∧U)∧T = F
F∧(U∧T) = F
结果: 相等
OR结合律: (F∨U)∨T = T
F∨(U∨T) = T
结果: 相等
=== 分配律验证 ===
分配律: F∧(U∨T) = F
(F∧U)∨(F∧T) = F
结果: 成立
=== 德摩根定律验证 ===
a=F, b=T:
¬(F∧T) = T, ¬F∨¬T = T ✓
¬(F∨T) = F, ¬F∧¬T = F ✓
a=U, b=U:
¬(U∧U) = U, ¬U∨¬U = U ✓
¬(U∨U) = U, ¬U∧¬U = U ✓
=== 数据库三值逻辑应用 ===
查询结果:
Alice: 满足条件 = T
Bob: 满足条件 = U
Charlie: 满足条件 = U
=== 三值电路设计 ===
电路输入: A=T, B=U, C=F
电路输出: (A∧B)∨¬C = T
=== 不确定性推理 ===
事实: 下雨=U, 地面湿=T
推理: 下雨→地面湿 = T
结论: 推理成立
==================================================
三值代数系统演示完成
6. 系统特性总结
这套三值代数系统具有以下特性：
保持的布尔代数性质：
-
交换律: a ∧ b = b ∧ a, a ∨ b = b ∨ a
-
结合律: (a ∧ b) ∧ c = a ∧ (b ∧ c)
-
分配律: a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
-
德摩根定律: ¬(a ∧ b) = ¬a ∨ ¬b
三值特有的性质：
-
排中律不成立: a ∨ ¬a 不一定为 TRUE（当a=UNKNOWN时）
-
矛盾律不成立: a ∧ ¬a 不一定为 FALSE（当a=UNKNOWN时）
-
保持信息性: UNKNOWN值在运算中可能保持或传播
应用价值：
-
数据库系统: 处理NULL值的三值逻辑
-
硬件设计: 三值逻辑电路
-
人工智能: 不确定性推理
-
形式验证: 三值模型检测
这套系统完整地扩展了布尔代数，为处理不确定性和部分信息提供了严格的数学基础。
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