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Materials: AIMA, Ch. 7
Basics for Logic
Syntax vs. Semantic
Syntax : well formed representation
Semantic : sentences' specific meaning in a "possible world"
possible world: "model", such as "a world where x=1 and y=2".
Satisfaction vs. Entailment
Satisfaction : if sentence (alpha) is true in model (m) , we say :
- (m) satisfies (alpha),
- (m) is a model of (alpha),
- (M(alpha)={m|m ext{ is a model of }alpha}).
e.g. "(x=2, y=2)" is a model of "(x+y=4)".
Entailment : A sentence follows logically from another sentence.
- (alphavDasheta Leftrightarrow M(alpha)subseteq M(eta)), which means (alpha) entails (beta) .
e.g. (x=0) entails (xy=0) .
Inference vs. Entailment
We use INFERENCE to find ENTAILMENT .
If we can use inference algorithm (i) to derive (alpha) from (KB) , we say:
- (KBvdash_ialpha), as (i) derives (alpha) from (KB) .
Soundness vs. Completeness
Soundness : An inference algorithm that derives only entailed sentences is called sound .
Model-checking ( To prove (KBvdash_{mc}alpha), check every model to prove (M(alpha)subseteq M(KB)) ) is a sound algorithm when possible.
e.g.: only find exist needles in a haystack.
Completeness : An inference algorithm that can derive any sentence that is entailed is called complete .
e.g.: find every exists needle in a haystack.
Representation vs. Real World
Corresponding relation between "x entails x" in logical representation and xx follows xx in real world .
Grounding issues. "Sensors" guarantees (KB) is true in Real World.
Propositional Logic
Syntax
- Atomic sentences: capitalized letters
- True/False
- Logical connectives: connect atomic sentences to make complex sentences
- ( eg) neg , negation
- (land) land , conjuction
- (lor) lor , disjunction
- (Rightarrow) Rightarrow , imply
- (Leftrightarrow) Leftrightarrow IFF , biconditional
- also (equiv) equiv
Semantic
Rules for Atomic sentences are easy:
• True is true in every model and False is false in every model.
• The truth value of every other proposition symbol must be specified directly in the model.
For complex sentences, we have five rules, which hold for any subsentences (P) and (Q) in any model m (here “iff” means “if and only if”):
• (
eg P) is true iff (P) is false in (m).
• (P land Q) is true iff both (P) and (Q) are true in (m).
• (P lor Q) is true iff either (P) or (Q) is true in (m).
• * (P Rightarrow Q) is true unless (P) is true and (Q) is false in (m).
• (P Leftrightarrow Q) is true iff (P) and (Q) are both true or both false in (m).
A tricky point: (PRightarrow Q) is true in model (M) EXCEPT (P) is true but (Q) is false.
You can refer to this: ((PRightarrow Q) Leftrightarrow ( eg P lor Q))
btw, Truth Table solves all.