logical inference - university of texas at san antoniobylander/cs3793/notes/logic.pdfcs 3793/5233...

CS 3793/5233 Artificial Intelligence Logical Inference – 1

Logical Inference

Logic

⊲ Logic

Propositional Logic

Examples

Proof Procedures

Consistency-BasedDiagnosis

The Limitations ofLogic


We want to tell our computers facts that aretrue of the world.“It is raining.”

Some of these facts specify how one thing isrelated to another.“It is raining implies it is wet.”

We want our computers to be able to inferwhat else must be true of the world.“It is wet.”

A logic is a system for inference from facts.

Syntax

Logic

Propositional Logic

⊲ Syntax

Informal Semantics

Informal Example

Formal Semantics 1

Formal Semantics 2

Examples

Proof Procedures




A proposition is something that is true or false. An atomic proposition or atom consists of a

single symbol. (≈ boolean variable) A compound proposition is constructed from

simpler propositions p and q using logical

operators (≈ boolean expression):

– ¬p (read “not p”)–negation

– p ∧ q (read “p and q”)–conjunction

– p ∨ q (read “p or q”)–disjunction

– p→ q (read “p implies q”)–implication

– q ← p (read “q if p”)–implication

– p↔ q (read “p iff q”)–equivalence

[Note: I prefer using → to ←.]

Informal Semantics

Logic

Propositional Logic

Syntax

⊲InformalSemantics

Informal Example

Formal Semantics 1

Formal Semantics 2

Examples

Proof Procedures




Semantics maps between symbols and the world.

Begin with a task domain. Choose symbols in the computer to denote

propositions.Symbol ≈ variable name

Tell the system knowledge about the domain.Knowledge ≈ code and inputs

Ask the system true/false questions.Ask questions ≈ run a function

The system should answer true, false orunknown as appropriate.

You can interpret the answer because youknow the meaning of the symbols.

Informal Semantics Example

Logic

Propositional Logic

Syntax

Informal Semantics

⊲InformalExample

Formal Semantics 1

Formal Semantics 2

Examples

Proof Procedures




In computer:sw up ∧ power ∧ unlit l1→ l1 broken

In user’s mind: sw up = switch is up,power = there is power in,unlit l1 = light #1 isn’t lit,l1 broken = light #1 is broken

The computer doesn’t know the meaning ofthe symbols.

The user can interpret the symbols using theirmeaning.

Formal Semantics 1

Logic

Propositional Logic

Syntax

Informal Semantics

Informal Example

⊲FormalSemantics 1

Formal Semantics 2

Examples

Proof Procedures




An interpretation I maps atoms to true orfalse.

Based on how logical operators work, aninterpretation maps each proposition to a truthvalue.

Propositions may have different truth values indifferent interpretations.

p q ¬p p ∧ q p ∨ q p→ q q ← p p↔ q

true true false true true true true truetrue false false false true false false falsefalse true true false true true true falsefalse false true false false true true true

Formal Semantics 2

Logic

Propositional Logic

Syntax

Informal Semantics

Informal Example

Formal Semantics 1

⊲FormalSemantics 2

Examples

Proof Procedures




A knowledge base is a set of propositions thatthe agent is given as being true.

A model of knowledge base is aninterpretation in which all the propositions inthe knowledge base are true.

If KB is a knowledge base and p is aproposition, KB entails p (written KB |= p)if p is true in every model of KB.

KB |= p means that no interpretation exists inwhich KB is true and p is false.

If KB |= p we also say p logically follows fromKB, or p is a logical consequence of KB.

Simple Example

Logic

Propositional Logic

Examples

⊲ Simple Example

Simple Example

ElectricalEnvironment

Representation

Proof Procedures




KB = p→ q, p, s→ r

p q r s model?I1 true true true trueI2 false false false falseI3 true true false falseI4 true true true falseI5 true true false true

Which of p, q, r, s are entailed by KB?

Simple Example

Logic

Propositional Logic

Examples

Simple Example

⊲ Simple Example


Representation

Proof Procedures




KB = p→ q, p, s→ r

p q r s model of KB?I1 true true true true yesI2 false false false false noI3 true true false false yesI4 true true true false yesI5 true true false true no

Which of p, q, r, s are entailed by KB?p and q

Electrical Environment

Logic

Propositional Logic

Examples

Simple Example

Simple Example

⊲ElectricalEnvironment

Representation

Proof Procedures




Representation

Logic

Propositional Logic

Examples

Simple Example

Simple Example


⊲ Representation

Proof Procedures




light l1 live w0 ∧ ok l1→ lit l1light l2 live w1 ∧ up s2→ live w0down s1 live w2 ∧ down s2→ live w0up s2 live w3 ∧ up s1→ live w1up s3 live w3 ∧ down s1→ live w2ok l1 live w4 ∧ ok l2→ lit l2ok l2 live w3 ∧ up s3→ live w4ok cb1 live w3→ live p1ok cb2 live w5 ∧ ok cb1→ live w3live outside live w6→ live p2

live w5 ∧ ok cb2→ live w6live outside→ live w5

Proofs

Logic

Propositional Logic

Examples

Proof Procedures

⊲ Proofs

Brute Force

CSP Inference

Definite Clauses

Example

Contradiction

Inference Rules

Resolution

Example




A proof is a derivation that a propositionlogically follows from a knowledge base.

Given a proof procedure, KB ⊢ p means p canbe derived or proved from KB.

Recall KB |= p means KB entails p, that p istrue in all models of KB.

A proof procedure is sound if KB ⊢ p only ifKB |= p. Anything that is proved is alsoentailed.

A proof procedure is complete if KB |= p thenalso KB ⊢ p. Everything that is entailed canbe proved.

Brute Force Inference

Logic

Propositional Logic

Examples

Proof Procedures

Proofs

⊲ Brute Force

CSP Inference

Definite Clauses

Example

Contradiction

Inference Rules

Resolution

Example




Enumerate all interpretations. Determine which interpretations are models of

the KB. Determine which atoms (and any other

propositions of interest) are true in all models(or false in all models).

This is Ω(2n) where n is the number of atoms.

CSP Inference

Logic

Propositional Logic

Examples

Proof Procedures

Proofs

Brute Force

⊲ CSP Inference

Definite Clauses

Example

Contradiction

Inference Rules

Resolution

Example




Set up KB as a CSP. Each atom is a variablewith two possible values. Each proposition inthe KB is a constraint.

Solutions of CSP = models of KB. Run arc consistency/domain splitting. Don’t stop after finding one CSP solution (KB

model). Find them all. Determine which atoms are true in all models

(or false in all models). This is still potentially exponential, but more

efficient than brute force. See Section 4.6.1.

Definite Clause Inference

Logic

Propositional Logic

Examples

Proof Procedures

Proofs

Brute Force

CSP Inference

⊲ Definite Clauses

Example

Contradiction

Inference Rules

Resolution

Example




Suppose all propositions in KB are definite

clauses, either:

– an atom (e.g., an observation), or– of the form p→ q, where p and q are

atoms (e.g., a rule about the behavior ofthe world)

– of the form p1 ∧ . . . ∧ pk → q, where q andeach pi are atoms

Running CSP inference is efficient (linear inthe length of the KB).

See Section 5.2.

Definite Clause Example

Logic

Propositional Logic

Examples

Proof Procedures

Proofs

Brute Force

CSP Inference

Definite Clauses

⊲ Example

Contradiction

Inference Rules

Resolution

Example




KB = a, b, a→ c, b ∧ c→ d, d ∧ e→ f

Know a and b.

Derive c from a and a→ c.

Derive d from b and c and b ∧ c→ d

Cannot derive e or f .

Proof by Contradiction

Logic

Propositional Logic

Examples

Proof Procedures

Proofs

Brute Force

CSP Inference

Definite Clauses

Example

⊲ Contradiction

Inference Rules

Resolution

Example




Suppose we want to determine if KB |= p. Let KB′ = KB ∪ ¬p Determine that no model exists for KB′. Conclude that KB |= p. Should probably show that KB has at least

one model.

Inference Rules

Logic

Propositional Logic

Examples

Proof Procedures

Proofs

Brute Force

CSP Inference

Definite Clauses

Example

Contradiction

⊲ Inference Rules

Resolution

Example




Modus ponens is an inference rule. If p is true,and if p→ q is true, then q is true.

That is, if KB |= p and KB |= p→ q, thenKB |= q.

Resolution inference rule (really, two rules)

– If KB |= p ∨ q and KB |= ¬p, thenKB |= q.

– If KB |= p ∨ q and KB |= ¬p ∨ r, thenKB |= q ∨ r.

Remember p and q and r can be anypropositions, not just atoms.

Resolution Theorem Proving

Logic

Propositional Logic

Examples

Proof Procedures

Proofs

Brute Force

CSP Inference

Definite Clauses

Example

Contradiction

Inference Rules

⊲ Resolution

Example




Resolution theorem proving is a sound andcomplete inference procedure for propositionallogic.

Transform the KB to conjunctive normal

form, meaning each propositions in the KB isof the form l or l1 ∨ . . . ∨ lk, where each li is aliteral, an atom or the negation of an atom.

To show KB |= p, let KB′ = KB ∪ ¬p,and ensure KB′ is in CNF.

Proof is by deriving a contradiction, deriveboth a and ¬a for some atom a.

Worst-case exponential-time. Lots ofapproaches to reduce the exponential.

Resolution Example

Logic

Propositional Logic

Examples

Proof Procedures

Proofs

Brute Force

CSP Inference

Definite Clauses

Example

Contradiction

Inference Rules

Resolution

⊲ Example




KB = a∨b, b∨c, ¬a∨¬b, ¬a∨¬c, ¬b∨¬c

To prove KB |= b, add ¬b and prove acontradiction using the resolution inference rule.

~b V ~c~a V ~ca V b~b

a c

b V c

b

~a V ~b

~b ~c ~a

a V ~c~a V c b V ~c

Conflicts

Logic

Propositional Logic

Examples

Proof Procedures


⊲ Conflicts

Conflict Example

Conflict Example


Representation

Electrical Conflicts

Diagnosis

Diagnosis Examples

Diagnosis Examples



Assume components are working normally. Need a set of assumables A, e.g.,

A = ok l1, ok l2, ok s1, . . . Show that KB ∪A leads to a contradiction,

which implies A is false, that something’swrong. In this case, A is called a conflict.

Want to isolate the problem, e.g., is there asingle fault that is consistent with KB?

C is a minimal conflict if C ⊆ A, C is aconflict, and no proper subset of C is aconflict.

Conflict Example

Logic

Propositional Logic

Examples

Proof Procedures


Conflicts

⊲ Conflict Example

Conflict Example


Representation


Diagnosis

Diagnosis Examples

Diagnosis Examples



A = c, d, e, z

KB = ¬a ∨ ¬b, c→ a, d→ b, e→ b

A is a conflict.Assuming A implies ¬a ∨ ¬b is false.

What subsets of A are minimal conflicts?

Conflict Example

Logic

Propositional Logic

Examples

Proof Procedures


Conflicts

Conflict Example

⊲ Conflict Example


Representation


Diagnosis

Diagnosis Examples

Diagnosis Examples



A = c, d, e, z

KB = ¬a ∨ ¬b, c→ a, d→ b, e→ b

A is a conflict.Assuming A implies ¬a ∨ ¬b is false.

What subsets of A are minimal conflicts?c, d and c, e

Electrical Environment

Logic

Propositional Logic

Examples

Proof Procedures


Conflicts

Conflict Example

Conflict Example

⊲ElectricalEnvironment

Representation


Diagnosis

Diagnosis Examples

Diagnosis Examples



Representation

Logic

Propositional Logic

Examples

Proof Procedures


Conflicts

Conflict Example

Conflict Example


⊲ Representation


Diagnosis

Diagnosis Examples

Diagnosis Examples



up s1 live w0 ∧ ok l1→ lit l1up s2 live w1 ∧ up s2 ∧ ok s2→ live w0up s3 live w2 ∧ down s2 ∧ ok s2→ live w0live outside live w3 ∧ up s1 ∧ ok s1→ live w1¬lit l1 live w3 ∧ down s1 ∧ ok s1→ live w2¬lit l2 live w4 ∧ ok l2→ lit l2

live w3 ∧ up s3 ∧ ok s3→ live w4live w3→ live p1live w5 ∧ ok cb1→ live w3live w6→ live p2live w5 ∧ ok cb2→ live w6live outside→ live w5


Logic

Propositional Logic

Examples

Proof Procedures


Conflicts

Conflict Example

Conflict Example


Representation

⊲ElectricalConflicts

Diagnosis

Diagnosis Examples

Diagnosis Examples



Each ok atom is an assumable. This issimplified by no ok atoms for wires.

To infer lit l1 (which would contradict¬lit l1), need ok cb1, ok s1, ok s2 and ok l1.

ok cb1, ok s1, ok s2, ok l1 is a conflict. To infer lit l2 (which would contradict¬lit l2), need ok cb1, ok s3 and ok l2.

ok cb1, ok s3, ok l2 is a conflict.

Diagnosis

Logic

Propositional Logic

Examples

Proof Procedures


Conflicts

Conflict Example

Conflict Example


Representation


⊲ Diagnosis

Diagnosis Examples

Diagnosis Examples



If C is a minimal conflict, then some elementof C is false.

Explanation: C is a conjunction of assumablesa1 ∧ a2 ∧ . . ..If C is false, then ¬C = ¬a1 ∨ ¬a2 ∨ . . . istrue.

Some element of each minimal conflict mustbe false.

A diagnosis is a subset of assumables D ⊆ A

such that D includes at least one element fromeach minimal conflict.

A diagnosis D is minimal if no subset of D isa diagnosis.

Diagnosis Examples

Logic

Propositional Logic

Examples

Proof Procedures


Conflicts

Conflict Example

Conflict Example


Representation


Diagnosis

⊲DiagnosisExamples

Diagnosis Examples



In the first example, c, d and c.e wereminimal conflicts.

What are the minimal diagnoses?

In the second example, the minimal conflictswere ok cb1, ok s1, ok s2, ok l1 andok cb1, ok s3, ok l2.

What are the minimal diagnoses?

Diagnosis Examples

Logic

Propositional Logic

Examples

Proof Procedures


Conflicts

Conflict Example

Conflict Example


Representation


Diagnosis

Diagnosis Examples

⊲DiagnosisExamples



In the first example, c, d and c.e wereminimal conflicts.

What are the minimal diagnoses?c and d.e

In the second example, the minimal conflictswere ok cb1, ok s1, ok s2, ok l1 andok cb1, ok s3, ok l2.

What are the minimal diagnoses?ok cb1 and several double fault possibilities:ok s1, ok s3, ok s1, ok l2,ok s2, ok s3, ok s2, ok l2,ok l1, ok s3, ok l1, ok l2,

Limitations of Logic

Logic

Propositional Logic

Examples

Proof Procedures



⊲Limitations ofLogic


Assumes we can write down the truth.

– The world is round.– The sky is blue.– What goes up must come down.

Assumes our observations are always true.

– Optical illusions.– Photoshopping.– Referees and umpires.

Assumes our inferences are certain.

– Overturned convictions.– Car accidents.– Scams.

No real-world inference is truly logical.

logical inference - university of texas at san antoniobylander/cs3793/notes/logic.pdfcs 3793/5233...

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