Introduction to Artificial Intelligence

By:

Dr. Mohammad Shoab

Week 7

Knowledge-based agents

• Knowledge base (KB) = set of sentences in a formal language
• Declarative approach to building an agent (or other system):
• Tell it what it needs to know
• Then it can ask itself what to do - answers should follow from the KB
• Distinction between data and program
• Fullest realization of this philosophy was in the field of expert systems or knowledge-based systems in the 1970s and 1980s

Inference engine

Knowledge base

Domain-independent algorithms

Domain-specific content

Introduction to Artificial Intelligence

Department of Computer Science

What is logic?

• Logic is a formal system for manipulating facts so that true conclusions may be drawn
• “The tool for distinguishing between the true and the false” – Averroes (12^th cen.)
• Syntax: rules for constructing valid sentences
• E.g., x + 2 ≥ y is a valid arithmetic sentence, ≥x2y + is not
• Semantics: “meaning” of sentences, or relationship between logical sentences and the real world
• Specifically, semantics defines truth of sentences
• E.g., x + 2 ≥ y is true in a world where x = 5 and y = 7

Introduction to Artificial Intelligence

Department of Computer Science

Propositional logic: Syntax

• Atomic sentence:
• A proposition symbol representing a true or false statement
• Negation:
• If P is a sentence, ¬P is a sentence
• Conjunction:
• If P and Q are sentences, P ∧ Q is a sentence
• Disjunction:
• If P and Q are sentences, P ∨ Q is a sentence
• Implication:
• If P and Q are sentences, P ⇒ Q is a sentence
• Biconditional:
• If P and Q are sentences, P ⇔ Q is a sentence

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• ¬, ∧, ∨, ⇒, ⇔ are called logical connectives

Introduction to Artificial Intelligence

Department of Computer Science

Propositional logic: Semantics

• A model specifies the true/false status of each proposition symbol in the knowledge base
• E.g., P is true, Q is true, R is false
• With three symbols, there are 8 possible models, and they can be enumerated exhaustively�
• Rules for evaluating truth with respect to a model:

¬P is true iff P is false

P ∧ Q is true iff P is true and Q is true

P ∨ Q is true iff P is true or Q is true

P ⇒ Q is true iff P is false or Q is true

P ⇔ Q is true iff P ⇒ Q is true and Q ⇒ P is true�

Introduction to Artificial Intelligence

Department of Computer Science

Truth tables

• A truth table specifies the truth value of a composite sentence for each possible assignments of truth values to its atoms

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• The truth value of a more complex sentence can be evaluated recursively or compositionally

Introduction to Artificial Intelligence

Department of Computer Science

Logical equivalence

• Two sentences are logically equivalent iff they are true in same models

Introduction to Artificial Intelligence

Department of Computer Science

Validity, satisfiability

A sentence is valid if it is true in all models,

e.g., True, A ∨¬A, A ⇒ A, (A ∧ (A ⇒ B)) ⇒ B�

A sentence is satisfiable if it is true in some model

e.g., A∨B, C

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A sentence is unsatisfiable if it is true in no models

e.g., A∧¬A

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Introduction to Artificial Intelligence

Department of Computer Science

Entailment

• Entailment means that a sentence follows from the premises contained in the knowledge base:�

KB ╞ α

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• Knowledge base KB entails sentence α if and only if α is true in all models where KB is true
• E.g., x = 0 entails x * y = 0
• Can α be true when KB is false?
• KB ╞ α iff (KB ⇒ α) is valid
• KB ╞ α iff (KB ∧¬α) is unsatisfiable

Introduction to Artificial Intelligence

Department of Computer Science

Inference

• Logical inference: a procedure for generating sentences that follow from a knowledge base KB
• An inference procedure is sound if whenever it derives a sentence α, KB╞ α
• A sound inference procedure can derive only true sentences
• An inference procedure is complete if whenever �KB╞ α, α can be derived by the procedure
• A complete inference procedure can derive every entailed sentence

Introduction to Artificial Intelligence

Department of Computer Science

Inference

• How can we check whether a sentence α is entailed by KB?
• How about we enumerate all possible models of the KB (truth assignments of all its symbols), and check that α is true in every model in which KB is true?
• Is this sound?
• Is this complete?
• Problem: if KB contains n symbols, the truth table will be of size 2ⁿ
• Better idea: use inference rules, or sound procedures to generate new sentences or conclusions given the premises in the KB

Introduction to Artificial Intelligence

Department of Computer Science

Inference rules

• Modus Ponens

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• And-elimination

premises

conclusion

Introduction to Artificial Intelligence

Department of Computer Science

Inference rules

• And-introduction

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• Or-introduction

Introduction to Artificial Intelligence

Department of Computer Science

Inference rules

• Double negative elimination

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• Unit resolution

Introduction to Artificial Intelligence

Department of Computer Science

Resolution

• Example:

α: “The weather is dry”

β: “The weather is rainy”

γ: “I carry an umbrella”

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or

Introduction to Artificial Intelligence

Department of Computer Science

Resolution is complete

• To prove KB╞ α, assume KB ∧ ¬ α and derive a contradiction
• Rewrite KB ∧ ¬ α as a conjunction of clauses, �or disjunctions of literals
• Conjunctive normal form (CNF)
• Keep applying resolution to clauses that contain complementary literals and adding resulting clauses �to the list
• If there are no new clauses to be added, then KB does not entail α
• If two clauses resolve to form an empty clause, we have a contradiction and KB╞ α

Introduction to Artificial Intelligence

Department of Computer Science

Complexity of inference

• Propositional inference is co-NP-complete
• Complement of the SAT problem: α ╞ β if and only if the sentence α ∧ ¬ β is unsatisfiable
• Every known inference algorithm has worst-case exponential running time
• Efficient inference possible for restricted cases

Introduction to Artificial Intelligence

Department of Computer Science

Definite clauses

• A definite clause is a disjunction with exactly one positive literal
• Equivalent to (P₁ ∧ … ∧ P_n) ⇒ Q

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• Basis of logic programming (Prolog)
• Efficient (linear-time) complete inference through forward chaining and backward chaining

premise or body

conclusion �or head

Introduction to Artificial Intelligence

Department of Computer Science

Forward chaining

• Idea: find any rule whose premises are satisfied in the KB, add its conclusion to the KB, and keep going until query is found

Introduction to Artificial Intelligence

Department of Computer Science

Forward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Forward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Forward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Forward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Forward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Forward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Forward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Forward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Backward chaining

Idea: work backwards from the query q:

to prove q by BC,

check if q is known already, or

prove by BC all premises of some rule concluding q�

Introduction to Artificial Intelligence

Department of Computer Science

Backward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Backward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Backward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Backward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Backward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Backward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Backward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Backward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Backward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Backward chaining example

Introduction to Artificial Intelligence

Department of Computer Science

Forward vs. backward chaining

• Forward chaining is data-driven, automatic processing
• May do lots of work that is irrelevant to the goal

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• Backward chaining is goal-driven, appropriate for problem-solving
• Complexity can be much less than linear in size of KB

Introduction to Artificial Intelligence

Department of Computer Science

The End

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Introduction to Artificial Intelligence

Department of Computer Science

Exercise

Q1. Explain knowledge based agents.

Q2. What is Logic?

Q3. Explain each syntax of propositional logic.

Q4. Explain truth table of propositional logic.

Q5. What is entailment?

Q6. What is inference?

Q7. Explain complexity of inference.

Q8. What is definite clause?

Q9. Write about Forward Chaining and Backward Chaining.

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Introduction to Artificial Intelligence

Department of Computer Science

Q10. Which is Domain-independent algorithm

1. Intelligence
2. Knowledge base
3. Inference engine
4. None of the above

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Q11. Which is the rule for constructing valid sentences

1. Logic
2. Syntax
3. Semantics
4. Non-observable

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42

Introduction to Artificial Intelligence

Department of Computer Science

Q12. Two sentences are logically equivalent iff they are true in

1. Same model
2. Different model
3. Selective model
4. Hybrid model

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Q13. A sound inference procedure can derive only

1. False sentences
2. True sentences
3. Discrete sentences
4. None of the above

43

Introduction to Artificial Intelligence

Department of Computer Science