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### Propositional Logic

Reading: C. 7.4-7.8, C. 8

Logic: Outline

- Propositional Logic
- Inference in Propositional Logic
- First-order logic

Agents that reason logically A Knowledge base agent

- A logic is a:
- Formal language in which knowledge can be expressed
- A means of carrying out reasoning in the language

- Tell: add facts to the KB
- Ask: query the KB

Towards General-Purpose AI General –purpose AI (e.g., logic-based)

- Problem-specific AI (e.g., Roomba)
- Specific data structure
- Need special implementation
- Can be fast

- Flexible and expressive
- Generic implementation possible
- Can be slow

Language Examples Natural Language Good representation language

- Programming languages
- Formal, not ambiguous
- Lacks expressivity (e.g., partial information)

- Very expressive, but ambiguous:
- Flying planes can be dangerous.
- The teacher gave the boys an apple.

- Inference possible, but hard to automate

- Both formal and can express partial information
- Can accommodate inference

Components of a Formal Logic

- Syntax: symbols and rules for combining themWhat you can say
- Semantics: Specification of the way symbols (and sentences) relate to the worldWhat it means
- Inference Procedures: Rules for deriving new sentences (and therefore, new semantics) from existing sentencesReasoning

Semantics

- A possible world (also called a model) is an assignment of truth values to each propositional symbol
- The semantics of a logic defines the truth of each sentence with respect to each possible world
- A model of a sentence is an interpretation in which the sentence evaluates to True
- E.g., TodayIsTuesday -> ClassAI is true in model {TodayIsTuesday=True, ClassAI=True}
- We say {TodayIsTuesday=True, ClassAI=True} is a model of the sentence

Exercise: Semantics

What is the meaning of these two sentences?

- If Shakespeare ate Crunchy-Wunchies for breakfast, then Sally will go to Harvard
- If Shakespeare ate Cocoa-Puffs for breakfast, then Sally will go to Columbia

Examples

- What are the models of the following sentences?
- KB1: TodayIsTuesday -> ClassAI
- KB2: TodayIsTuesday -> ClassAI, TodayIsTuesday

Proof by refutation

- A complete inference procedure
- A single inference rule, resolution
- A conjunctive normal form for the logic

Example: Wumpus World

- Agent in [1,1] has no breeze
- KB = R2Λ R4 = (B1,1<->(P1,2 V P2,1)) Λ⌐B1,1
- Goal: show ⌐P1,2

Inference Properties

- Inference method A is sound (or truth-preserving) if it only derives entailed sentences
- Inference method A is complete if it can derive any sentence that is entailed
- A proof is a record of the progress of a sound inference algorithm.

Other Types of Inference

- Model Checking
- Forward chaining with modus ponens
- Backward chaining with modus ponens

Model Checking

- Enumerate all possible worlds
- Restrict to possible worlds in which the KB is true
- Check whether the goal is true in those worlds or not

Wumpus Reasoning

- Percepts: {nothing in 1,1; breeze in 2,1}
- Assume agent has moved to [2,1]
- Goal: where are the pits?
- Construct the models of KB based on rules of world
- Use entailment to determine knowledge about pits

Properties of Model Checking

- Sound because it directly implements entailment
- Complete because it works for any KB and sentence to prove αand always terminates
- Problem: there can be way too many worlds to check
- O(2n) when KB and α have n variables in total

Inference as Search

- State: current set of sentences
- Operator: sound inference rules to derive new entailed sentences from a set of sentences
- Can be goal directed if there is a particular goal sentence we have in mind
- Can also try to enumerateevery entailed sentence

Complexity Forward chaining with Modus Ponens is complete for Horn logic

- N propositions; M rules
- Every possible fact can be establisehd with at most N linear passes over the database
- Complexity O(NM)

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