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The inference algorithm is sound if it derives only sentences that are entailed by KB.

The inference algorithm is complete if it can derive any sentence that is entailed by KB

A literal is a propositional variable (called a positive literal) or its negation (called a negative literal).

ground term: They behave just like propositional symbols in automated reasoning. They can unify with literals containing only universally quantified(al) variables(al* *). They may contain terms that are functions.

(•Ground term: A term that does not contain a variable. A constant symbol•A function applies to some ground term)

Horn Clauses

The definite clause language does not allow a contradiction to be stated. However, a simple expansion of the language can allow proof by contradiction.

A definite clause is a Horn clause that has exactly one positive literal. (~p v q v ~r)

A Horn clause is a clause (a disjunction of literals) with at most one positive unnegated, literal.

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An integrity constraint is a clause of the form. A Horn clause without a positive literal is called a [goal]

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BC: DFS L-R Incomplete due to infinite loops(prolog), sound in FOL

**FC：sound in FOL sound and complete in Definite clauses **

In Propositional LOGIC unification algo need one or more substitution that causes two literals like P and ~P to match so that they cancel out each other. Every FOL KB can be propositionalized . Propositional logic does not scale to the environments of unbounded size

Propositional logical is monotonic. Propositional logic has two quantifiers: for universal and exists.

• Propositional Logic converts a complete sentence into a symbol and makes it logical whereas in First-Order Logic relation of a particular sentence will be made that involves relations, constants, functions, and constants.
• The limitation of PL is that it does not represent any individual entities whereas FOL can easily represent the individual establishment that means if you are writing a single sentence then it can be easily represented in FOL.

PL<FOL(In propositional logic you are allowed to use the usual propositional connectives (and, or, …). In first order logic you are allowed to quantify (forall, exist) over variables, which makes it strictly more expressive.)

Every first-order logic knowledge base can be translated into a propositional one(WRONG IN COMPLEX SI)

One predicate that takes one argument. In such cases, we can automatically convert our first-order logic knowledge base to an equivalent propositional logic knowledgebase. For each possible combination of arguments of each predicate, we must define a symbol in the propositional logic version.

α |= β α entails sentence β if and only if β is true in all worlds where α is true(KB ⊨ α, if and only if KB ⇒ α)

The inference using Modus Ponens is incomplete (MP cannot use on ~HF)

GMP

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PDDL(planing domain definite language) avoids the frame problem by using add and delete lists.

Using clever indexing, can reduce number of calls to unification

∀x Libra(x) ⇒ BornIn( x ,September) ∨ BornIn( x ,October)
B.（a小范围，b大范围）******

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Satisfiable sentence: there exists a truth value assignment for the variables that makes the sentence true (truth value = t).

Try all the possible assignments to see if one works.

Valid sentence: all truth value assignments for the variables make the sentence true.

• Try all possible assignments and check that they all work.

De Morgan’s Laws：（negation in）

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Distributed Law

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Logical reasoning systems

•Use hash table to avoid looping over entire KB for each TELL or FETCH

Predicates (relations/functions).

To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. The inverse of “If it rains, then they cancel school” is “If it does not rain, then they do not cancel school.”

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If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.

The goal of the unification procedure is to look for at least one substitution that causes two literals to match.

an atomic sentence is a type of declarative sentence which is either true or false.

complex sentences cannot be created by using a single propositional symbol

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Skolemization can Introduce new functions and Remove all existential quantifiers

commutative operation: 可交换的

universal instantiation/elimination

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Resolution is a sound and complete inference procedure for both Propositional Logic and First-Order Logic that amounts to proof by contradiction.

truth table checking requires exponential time and the decidability of a logic does not necessarily impact its tractability.

vice versa 反过来也一样

sentence is sentence =》

entailment

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计算器要算fuzzy logic，center of mass（可能要算质心）。

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CNF 没有quantify

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In plain first-order logic without equality, every formula is monotonic（KB的无关sentence增加，不会导致之前的entailed的结果变化）FOL不一定是

To be Reflexive you should have (0,0) (1,1) (2,2) (3,3)

To be symmetric if you have (0,1), then you should have (1,0)

To be transitive if you have (1,2) (2,4), then you should have (1,4)

ordered plans are created by a search through a space of plans. (rather than the state space)

referential transparency is generally defined as the fact that an expression, in a program, may be replaced by its value (or anything having the same value) without changing the result of the program.

Predicate symbol must be same, atoms or expression with different predicate symbol can never be unified

Following are some basic conditions for unification:

• Predicate symbol must be same, atoms or expression with different predicate symbol can never be unified.
• Number of Arguments in both expressions must be identical.
• Unification will fail if there are two similar variables present in the same expression.

unified.

• Number of Arguments in both expressions must be identical.
• Unification will fail if there are two similar variables present in the same expression.

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来源： https://blog.csdn.net/qq_35651140/article/details/119041414

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