FOL消解法步骤. 判断公式 α 是否为永真式等价于判断 \neg α 不可满足. 步骤1: 求 α 的前束范式 β (等值变换 α \Leftrightarrowβ) 步骤2: 求 \neg β 的Skolem范式的CNF β′, 如 β′ = ∀x_1∀x_2··· ∀x_n(A_1∧A_2∧···∧A_m) ( \neg β “等可满足”变换, \neg β ≈ β′)

Using FOL The set（集合）domain: 集合就是空集或通过将一些元素添加到一个集合而构成 8s Set(s),(s = {} ) _(9x,s2 Set(s2) ^s = {x|s2}) 空集没有任何元素，也就是说，空集无法再分解为更小的集合和 元素:9x,s {x|s} = {} 将已经存在于集合中的元素添加到该集合，无任何变化 8x,s x ...

There exist complete and sound proof procedures for propositional and FOL. Use the definition of entailment directly. Proof procedure is exponential in n, the number of symbols. In practice, can be much faster... where the Pi and Q are non-negated atoms. Godel’s completeness theorem showed that a proof procedure exists...

一阶逻辑(first order logic, FOL)也叫一阶谓词演算，允许量化陈述的公式，是使用于数学、哲学、语言学及计算机科学中的一种形式系统。一阶逻辑是区别于高阶逻辑的数理逻辑，它不允许量化性质。

• In FOL form: ∃x.Older(Lulu, x) • Denial: ¬∃x.Older(Lulu, x) ∀x.¬Older(Lulu, x) in clause form: ¬Older(Lulu, x) • Successful proof gives {x/Fifi} [Verify!!] Example 2: “What is older than what?” • In FOL form: ∃x ∃y.Older(x, y) • Denial: ¬∃x∃y.Older(x, y) in clause form: ¬Older(x, y) • …

FOL的真值 语句是在给定的模型和解释下为真 模型包含对象(领域元素) 和它们之间的关系 解释指定了指示物： constant symbols → objects predicate symbols → relations function symbols → functional relations 一个原子语句predicate(term 1,...,term n) 为真...

Having discussed the syntactic and semantic framework of FOL in Chapter 9. FOL, we’ll now turn to methods for step-wise inference in FOL. As you’ll see, FOL inference packs a bit more of a punch than propositional inference: not only are the rules more complicated, but we have to be careful about how we apply them.

Every FOL formula F can be transformed to formula F′ in PNF s.t. F′ ⇔ F. Example: Find equivalent PNF of F : ∀x. ¬(∃y. p(x,y) ∧ p(x,z)) ∨ ∃y. p(x,y) ↑ to the end of the formula 1. Write F in NNF F1: ∀x. (∀y. ¬p(x,y) ∨ ¬p(x,z)) ∨ ∃y. p(x,y) 2- 21 2. Rename quantified variables to fresh names F2: ∀x. (∀y. ¬p(x ...

Test cases can only describe behavior in concrete instances. Specifications, written as first order formulas, are much richer! Introduction: To prove a forall goal ∀x : T, G(x): Suppose you have a (new, freshly named) x : T in your context, and prove G(x) for that new x.

First-order logic (FOL) is a richer language than propositional logic. Its lexicon contains not only the symbols ^, _, ¬, and ! (and parentheses) from propositional logic, but also the symbols 9 and 8 for “there exists” and “for all”, along with various symbols to represent variables, constants, functions, and relations.