计量逻辑学及其随机化研究
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摘要
经典的命题逻辑中最基本的推理模式为{A_1,…,A_n}|-A~*,从语法的角度看,它表明A_1→(A_2→…→(A_n→A~*)…)是定理,而从语义的角度看,它表明如果任一赋值v使前提A_1,…,A_n都为真,则v也使结论A~*为真。值得注意的是,这一推理的前提是否可靠并未考虑,因而从实际应用的角度看,这种单纯的形式推理似有不足之处。正是基于这种考虑,从20世纪70年代以来,逐渐兴起了概率逻辑学的研究。在概率逻辑学中,对推理的前提集中的各公式,要考虑其“不确定性”,不确定性是通过一个数值表征的,这个数值是由1减去该命题为真的概率而得的。通过Kolmogorov公理将概率方法与逻辑推理相结合,通过诸前提的不可靠度可以估计出其结论不可靠度的变化范围。但概率推理模式是“一事一议”型的,即,对于不同的有效推理,前提中同一公式的概率不必相同。这固然有其方便的一面,但从其理论的完整性来看,似乎只具有局部性而缺乏整体性。这不能不说是一种局限。
另一方面,为了把逻辑概念程度化而提出的计量逻辑学理论,其目的是为了尝试在人工智能科学与数值计算理论之间架起沟通的桥梁。计量逻辑学一方面具有整体性的优点,但同时又有缺少随机性的不足。事实上,在计量逻辑学中,每个公式都被赋予了一个真度,但在该真度意义下,每个原子公式都有相同的的真度,用概率的观点来考察,即每个原子公式为真的概率均相等。事实上各简单命题是否为真以及在多大的程度上为真是不确定的、随机的。所以赋予不同原子公式以不同的概率,可以使由此产生的公式的真度更具实用性,这种基于随机性的逻辑概念的程度化方法已经成为当前概率化人工智能研究的一个热点课题,从而展示了更为广阔的应用前景。本文正是以此为出发点,着力于将逻辑概念程度化与随机化相结合,从而把计量逻辑学中的程度化研究及近似推理模式纳入于更为宽泛的研究体系之中。
本文第一章首先通过引入生成状态集和生成概率给出了概率逻辑学基本定理的简捷证明,并进一步通过引入自然合并概率将概率逻辑学的基本定理推广到了更一般的形式,改进了对推理结论的不可靠度上界的估计。然后将概率逻辑学的基本方法引入计量逻辑学,得出了带参数(?)的有限逻辑理论相容度概念,是δ-相容度的推广。
第二章论证了有限多个公式的概率分布与生成它的原子公式集的概率分布之间的关系,然后把计量逻辑学与概率逻辑学相结合,在二值逻辑中提出了概率真度、概率逻辑伪度量空间。指出当取均匀概率分布时,概率真度就转化为计量逻辑学中的真度,同时两公式间的概率逻辑伪距离就转化为计量逻辑学中的伪距离。从而在有限理论中建立了一种更具一般性的概率逻辑伪度量空间理论。
第三章利用赋值集的随机化方法,在二值逻辑中首先提出了公式的D-随机真度概念,证明了全体公式的D-随机真度之集在[0,1]中没有孤立点。接着给出了D-逻辑伪距离和D-逻辑度量空间,证明了该空间中没有孤立点。指出当取均匀概率测度,且各概率测度均为1/2时,D-随机真度就转化为计量逻辑学中的真度,同时两公式间的D-逻辑伪距离就转化为计量逻辑学中的伪距离,从而建立了更具一般性的D-逻辑度量空间。
通过概率逻辑学基本定理,证明了D-逻辑度量空间中逻辑运算的连续性,从而实现了概率逻辑学与计量逻辑学的融合。在D-逻辑度量空间中提出了公式之间的3种不同类型的近似推理模式。证明了D-逻辑度量空间中三种近似推理模式是等价的;指出了全体原子公式之集在D-逻辑度量空间中未必是全发散的。
在D-逻辑度量空间中提出了理论的D-开放度,得出一个理论的D-开放度与它的D-发散度取值相等。提出了理论的D-相容度,得出D-相容度在D-逻辑度量空间中能保持相容度在逻辑度量空间中的基本性质。
最后,在三值R_0命题逻辑系统,三值Lukasiewicz命题逻辑系统,三值Goguen命题逻辑系统和三值G(o|¨)del命题逻辑系统中提出了公式的随机真度和随机距离,建立了随机逻辑度量空间。指出当取均匀概率测度,且各概率测度均为1/3时,随机真度就转化为计量逻辑学中的真度,同时两公式间的随机距离就转化为计量逻辑学中的伪距离,从而在三值逻辑中建立了更具一般性的随机逻辑度量空间。
The most fundamental inference pattern in propositional logic is {A_1,…,A_n}|-A~*. It means,from syntactic viewpoint,A_1→(A_2→…→(A_n→A~*)…) is a theorem, while it means,from semantic viewpoint,A~* is true under any valuation v whenever A_1,…,A_n are true under v.Notice that if the premises are sound or not is not taken into account,and hence this formalized effective inference seems not to be suitable for practical reasoning.In view of this situation,the probability logic emerged from the 70's of the 20~(th) century,where uncertainty of premises were considered,and uncertainty degree of the conclusion had been deducted by using the Kolmogorov axioms.It is remarkable that the theory is developed by means of individual cases while the probability of one and the same formula varies in different effective inferences,and only a few(mostly two or three) formulas are involved in premises of effective inferences,therefore the theory seems to be locally but not globally.
On the other hand,a global quantified logic theory is proposed,where logic concepts are graded into different levels so as to try to establish a bridge between artificial intelligence and numerical computation.In quantified logic every atomic formula has the same truth degree.This is not consistent with corresponding problems in the real world.In fact if a simple proposition in the real world is true or not,or in what extent it is true is uncertain,hence to follow the way of probabilistic AI and develop a probabilistic style quantified logic is certainly a beneficial task.In view of the above analysis the present paper focus on the alliance of quantified logic and probability logic so that quantified logic and most of its results can be considered special cases of the new setting.
In the first chapter,a simple and direct proof of the fundamental theorem of probability logic is proposed by introducing the concepts of generating states set and generating probability.Next,by introducing the concept of naturally merged probability the fundamental theorem has been generalized.Moreover,the present paper extends the concept ofδ-consistency degree of finite logic theories to be the concept of(?)-consistency degree and therefore certain relationship between probability logic and quantitative logic has been obtained.
In the second chapter,the relationship between the probability distribution of finite formulas and the probability distribution of atomic formulas generating them is discussed, and the concepts of probability truth degree and probability logic pseudo-metric space are proposed by combining quantitative logic and probability logic.It is proved that when the probability distribution is even,the value of probability truth degree is equal to the value of truth degree and the value of probability logic pseudo-metric is equal to the value of pseudo-metric in quantitative logic.Thus a more general pseudo-metric is established in finite theories.
In the third chapter,the concept of D- randomized truth degree of formulas in two-valued propositional logic is introduced by means of randomization,and it is proved that the set of values of D- randomized truth degree of formulas has no isolated point in[0,1]. The concepts of D- logic pseudo-metric and D- logic metric space are also introduced and it is proved that there is no isolated point in the space.The new built D- randomized concepts are extensions of the corresponding concepts in quantified logic.
Moreover,it is proved that the basic logic connectives are continuous operators in D-logic metric space.Three different types of approximate reasoning pattems are proposed and it is proved that the three different types of approximate reasoning pattems are equivalent to each other and the set of atomic formulas is not totally divergent in D-logic metric space.
After D- opening degree is introduced in D- logic metric space,it is proved that the D- opening degree of a theory is equivalent to it's D- divergence degree.Then the D- consistency degree is defined and it can maintain the basic properties in logic metric space.
By means of randomization,the concept of randomized truth degree and randomized logic pseudo-metric of formulas in R_0 three-valued propositional logic, Lukasiewicz three-valued propositional logic,Goguen three-valued propositional logic and G(o|¨)del three-valued propositional logic are introduced.The concept of randomized logic metric space is also introduced and it is proved that the new built randomized concepts are extensions of the corresponding concepts in quantified logic.
另一方面,为了把逻辑概念程度化而提出的计量逻辑学理论,其目的是为了尝试在人工智能科学与数值计算理论之间架起沟通的桥梁。计量逻辑学一方面具有整体性的优点,但同时又有缺少随机性的不足。事实上,在计量逻辑学中,每个公式都被赋予了一个真度,但在该真度意义下,每个原子公式都有相同的的真度,用概率的观点来考察,即每个原子公式为真的概率均相等。事实上各简单命题是否为真以及在多大的程度上为真是不确定的、随机的。所以赋予不同原子公式以不同的概率,可以使由此产生的公式的真度更具实用性,这种基于随机性的逻辑概念的程度化方法已经成为当前概率化人工智能研究的一个热点课题,从而展示了更为广阔的应用前景。本文正是以此为出发点,着力于将逻辑概念程度化与随机化相结合,从而把计量逻辑学中的程度化研究及近似推理模式纳入于更为宽泛的研究体系之中。
本文第一章首先通过引入生成状态集和生成概率给出了概率逻辑学基本定理的简捷证明,并进一步通过引入自然合并概率将概率逻辑学的基本定理推广到了更一般的形式,改进了对推理结论的不可靠度上界的估计。然后将概率逻辑学的基本方法引入计量逻辑学,得出了带参数(?)的有限逻辑理论相容度概念,是δ-相容度的推广。
第二章论证了有限多个公式的概率分布与生成它的原子公式集的概率分布之间的关系,然后把计量逻辑学与概率逻辑学相结合,在二值逻辑中提出了概率真度、概率逻辑伪度量空间。指出当取均匀概率分布时,概率真度就转化为计量逻辑学中的真度,同时两公式间的概率逻辑伪距离就转化为计量逻辑学中的伪距离。从而在有限理论中建立了一种更具一般性的概率逻辑伪度量空间理论。
第三章利用赋值集的随机化方法,在二值逻辑中首先提出了公式的D-随机真度概念,证明了全体公式的D-随机真度之集在[0,1]中没有孤立点。接着给出了D-逻辑伪距离和D-逻辑度量空间,证明了该空间中没有孤立点。指出当取均匀概率测度,且各概率测度均为1/2时,D-随机真度就转化为计量逻辑学中的真度,同时两公式间的D-逻辑伪距离就转化为计量逻辑学中的伪距离,从而建立了更具一般性的D-逻辑度量空间。
通过概率逻辑学基本定理,证明了D-逻辑度量空间中逻辑运算的连续性,从而实现了概率逻辑学与计量逻辑学的融合。在D-逻辑度量空间中提出了公式之间的3种不同类型的近似推理模式。证明了D-逻辑度量空间中三种近似推理模式是等价的;指出了全体原子公式之集在D-逻辑度量空间中未必是全发散的。
在D-逻辑度量空间中提出了理论的D-开放度,得出一个理论的D-开放度与它的D-发散度取值相等。提出了理论的D-相容度,得出D-相容度在D-逻辑度量空间中能保持相容度在逻辑度量空间中的基本性质。
最后,在三值R_0命题逻辑系统,三值Lukasiewicz命题逻辑系统,三值Goguen命题逻辑系统和三值G(o|¨)del命题逻辑系统中提出了公式的随机真度和随机距离,建立了随机逻辑度量空间。指出当取均匀概率测度,且各概率测度均为1/3时,随机真度就转化为计量逻辑学中的真度,同时两公式间的随机距离就转化为计量逻辑学中的伪距离,从而在三值逻辑中建立了更具一般性的随机逻辑度量空间。
The most fundamental inference pattern in propositional logic is {A_1,…,A_n}|-A~*. It means,from syntactic viewpoint,A_1→(A_2→…→(A_n→A~*)…) is a theorem, while it means,from semantic viewpoint,A~* is true under any valuation v whenever A_1,…,A_n are true under v.Notice that if the premises are sound or not is not taken into account,and hence this formalized effective inference seems not to be suitable for practical reasoning.In view of this situation,the probability logic emerged from the 70's of the 20~(th) century,where uncertainty of premises were considered,and uncertainty degree of the conclusion had been deducted by using the Kolmogorov axioms.It is remarkable that the theory is developed by means of individual cases while the probability of one and the same formula varies in different effective inferences,and only a few(mostly two or three) formulas are involved in premises of effective inferences,therefore the theory seems to be locally but not globally.
On the other hand,a global quantified logic theory is proposed,where logic concepts are graded into different levels so as to try to establish a bridge between artificial intelligence and numerical computation.In quantified logic every atomic formula has the same truth degree.This is not consistent with corresponding problems in the real world.In fact if a simple proposition in the real world is true or not,or in what extent it is true is uncertain,hence to follow the way of probabilistic AI and develop a probabilistic style quantified logic is certainly a beneficial task.In view of the above analysis the present paper focus on the alliance of quantified logic and probability logic so that quantified logic and most of its results can be considered special cases of the new setting.
In the first chapter,a simple and direct proof of the fundamental theorem of probability logic is proposed by introducing the concepts of generating states set and generating probability.Next,by introducing the concept of naturally merged probability the fundamental theorem has been generalized.Moreover,the present paper extends the concept ofδ-consistency degree of finite logic theories to be the concept of(?)-consistency degree and therefore certain relationship between probability logic and quantitative logic has been obtained.
In the second chapter,the relationship between the probability distribution of finite formulas and the probability distribution of atomic formulas generating them is discussed, and the concepts of probability truth degree and probability logic pseudo-metric space are proposed by combining quantitative logic and probability logic.It is proved that when the probability distribution is even,the value of probability truth degree is equal to the value of truth degree and the value of probability logic pseudo-metric is equal to the value of pseudo-metric in quantitative logic.Thus a more general pseudo-metric is established in finite theories.
In the third chapter,the concept of D- randomized truth degree of formulas in two-valued propositional logic is introduced by means of randomization,and it is proved that the set of values of D- randomized truth degree of formulas has no isolated point in[0,1]. The concepts of D- logic pseudo-metric and D- logic metric space are also introduced and it is proved that there is no isolated point in the space.The new built D- randomized concepts are extensions of the corresponding concepts in quantified logic.
Moreover,it is proved that the basic logic connectives are continuous operators in D-logic metric space.Three different types of approximate reasoning pattems are proposed and it is proved that the three different types of approximate reasoning pattems are equivalent to each other and the set of atomic formulas is not totally divergent in D-logic metric space.
After D- opening degree is introduced in D- logic metric space,it is proved that the D- opening degree of a theory is equivalent to it's D- divergence degree.Then the D- consistency degree is defined and it can maintain the basic properties in logic metric space.
By means of randomization,the concept of randomized truth degree and randomized logic pseudo-metric of formulas in R_0 three-valued propositional logic, Lukasiewicz three-valued propositional logic,Goguen three-valued propositional logic and G(o|¨)del three-valued propositional logic are introduced.The concept of randomized logic metric space is also introduced and it is proved that the new built randomized concepts are extensions of the corresponding concepts in quantified logic.
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