CS 371: Introduction to Artificial Intelligence

Propositional Logic

Review/Introduction

Review: goal-based agent, simple problem-solving agent

How would we design a Minesweeper-playing agent?

Capacity for simple logical reasoning useful for some problem solving.

Examples of KB Agents

Toy example: Minesweeper Agent

Real-world example: Reaction Control System (RCS)

Propositional Logic Syntax

propositions - T/F atomic sentences about the world

propositional symbols (a.k.a. variables) P, Q, R, …, and constants True and False

e.g. P: “There is a mine at (1,2).”; Q: “There is a mine at (2,2).”

not (Ø) - negates meaning of a sentence

P, ØP, Q, ØQ, … are called literals

e.g. ØQ: “There is not a mine at (2,2)

Propositional Logic Syntax (cont.)

connectives (Ù,Ú,Þ,Û) - logically combine sentences

PÙQ, PÚQ, PÞQ, PÛQ are complex sentences

e.g. PÞØQ: “If there is a mine at (1,2), then there is not a mine at (2,2).”

parentheses - disambiguates grammar

e.g. PÞØQ ÙQ

PÞ(ØQ ÙQ) vs. (PÞØQ) ÙQ

Propositional Logic Semantics

commitments to the meanings of propositions

(“P” without an interpretation is meaningless.)

propositional symbol interpretations

commitments to the meanings of connectives, etc.

(“Þ” without an interpretation is meaningless.)

truth tables

Truth Tables

truth table - table of truth assignments to propositions and resulting truth values for complex sentences

truth tables for defining connectives

Truth Tables (cont.)

tables can easily become large!

Exponential growth with increase in propositional symbols

Validity, Satisfiability, Models

A sentence is valid if it is true for all possible truth assignments (a.k.a. a tautology) e.g. (P ÚØP)

A sentence is satisfiable if there exists a “satisfying” truth assignment (i.e. a truth assignment for which the sentence is true).

A truth assignment which satisfies a sentence is a model of that sentence.

Entailment

entailment - whether a sentences follows from other sentences

KB Ö f means knowledge base KB entails formula f

If a truth assignment that satisfies all sentences in KB also satisfies formula f, f is entailed by KB

Inference

inference rules - deduce new sentences from old sentences

KB | f means knowledge base KB infers or proves formula f

If there are a set of inference actions which derive the truth of formula f from the sentence in KB, f is inferred by KB

Semantics: Relating the Representation to the World

Relationships Between Entailment and Inference

soundness - all inferences correct?

completeness - all conclusions reachable?

Knowledge Representation and Reasoning

A logic consists of

a formal language consisting of

syntax

semantics

a proof theory

terms: sentences, KB, valid, satisfiable, model, entail, inference rule, sound, complete

Knowledge Representation and Reasoning (cont.)

All KR&R languages have syntax, semantics, and ontological and epistemological commitments.

Knowledge-Based Agent Review

Advantages

KB-agents can achieve competence quickly, by learning or being told facts

KB-agents can adapt. Changes in the world cause updates to KB

Flexible; no reprogramming!

Difficulties

expressivity - limits of what knowledge a logic can express

tractability - limits of what can be reasoned within computational constraints

fundamental tradeoff between expressivity and tractability

knowledge acquisition - the difficulty of expressing and codifying human knowledge

Example: SSA Circuit Fault Testing

Problem Overview

Combinational circuits

Single-Stuck-At (SSA) faults

Automatic Test Pattern Generation

Notation translation for Larrabee paper

Normal circuit example

Conversion to Conjunctive Normal Form (CNF)

Problem Overview

Suppose:

You’re working in circuit testing

You test a great many combinational circuit designs (many quite complex).

You wish to find a way to automate testing of circuits for Single Stuck-At (SSA) faults:

A single circuit wire “stuck at” a constant value (e.g. 0 = false=grounded)

For any given fault, it would be nice to have a circuit input which would give a different result if fault / no fault.

Combinational Circuits

Larrabee Notation

In the Larrabee paper, a different logical notation is used:

Conversion to CNF

Eliminate Þ and Û

Move negations in (form literals) using deMorgan’s Law

Distribute Ù and Ú to form conjunction of disjunctions

Worked example: X Û (Y Ù ØZ)

Eliminate Þ and Û

Replace XÛY with (XÞY) Ù (YÞX)

Replace XÞY with ØXÚY

Worked example:

X Û (YÙØZ) =

(X Þ (YÙØZ)) Ù ((YÙØZ) Þ X) =

(ØX Ú (YÙØZ)) Ù (Ø(YÙØZ) Ú X)

Move Negations In

deMorgan’s Law:

Replace Ø(XÚY) with (ØX ÙØY)

Replace Ø(XÙY) with (ØX Ú ØY)

Worked example:

(ØX Ú (YÙØZ)) Ù (Ø(YÙØZ) Ú X) =

(ØX Ú (YÙØZ)) Ù ((ØY Ú ØØZ) Ú X) =

(ØX Ú (YÙØZ)) Ù ((ØY Ú Z) Ú X) =

(ØX Ú (YÙØZ)) Ù (ØY Ú Z Ú X)

At this point, you should only have complex sentences made up of Ù, Ú, and literals

Form Conjunction of Disjunctions (CNF) with Distribution

Replace (f₀ Ú (f ₁ Ù … Ù f _n))

with ((f₀ Ú f₁) Ù … Ù (f₀ Ú f_n)) where f_i is a formula

Continue until you have one conjunction (and) of disjunctions (or’s).

(ØX Ú (YÙØZ)) Ù (ØY Ú Z Ú X) =

((ØXÚY) Ù (ØXÚØZ)) Ù (ØY Ú Z Ú X) =

(ØXÚY) Ù (ØXÚØZ) Ù (ØY Ú Z Ú X)

Conjunctive Normal Form

This is called Conjunctive Normal Form (CNF).

Any CNF form can be converted to 3CNF or k-CNF by introducing variables and splitting disjunctions:

(PÚQÚRÚSÚT) is equivalent to (PÚQÚU) Ù (ØUÚRÚV) Ù (ØVÚSÚT)

SSA Circuit Fault Testing Example

Suppose we wish to test the following circuit for a SSA fault at D with D stuck at true:

Fault Model

We model the fault by introducing D’, which is stuck at true and affects the value of X which we rename X’:

Test Pattern Generation

We want to find a test pattern (inputs resulting in different outputs for correct/faulty cases)

This is accomplished by first creating a circuit with both our correct and fault models, which evaluates to true when the correct and fault circuits have different outputs.

Test Pattern Generation (cont.)

Then we fix the output of this combined circuit to true, and treat this as a boolean satisfiability problem, seeking a satisfying truth assignment to the inputs.

Such an assignment will be a test pattern suitable for testing for the given fault.

SSA Test Pattern Generation Summary

Given: combinational circuit, possible fault

Build fault model

Build “test circuit” combining normal and fault model, evaluating true iff outputs of each are different

Fix output of test circuit to true

Convert circuit logic to CNF form

Use GSAT to find a satisfying truth assignment to inputs.

Theorem Proving

Proof by Contradiction:

Given KB, want to show KB Ö f (KB entails f)

Let KB’ = KB È {Øf}

If we can prove that KB’ Ö false (inconsistent, contradiction), then we’ve proven that KB Ö f

If KB’ | false and our inference rules are sound then, KB’ Ö false

Propositional Logic Inference

Can use propositional logic inference rules of Figure 6.13, or…

Can convert to CNF and do resolution theorem proving

Example:

KB = {P, PÞQ, QÞR}

Does this infer R?

Resolution Theorem Proving Example

KB’ = {P, PÞQ, QÞR, ØR} =

= (P Ù (ØP Ú Q) Ù (ØQ Ú R) Ù ØR)

{P} (KB)

{ØP Q} (KB)

{ØQ R} (KB)

{ØR} (negated goal)

{Q} (1,2)

{R} (3,5)

{} (4,6) à Contradiction! Q.E.D.

Summary

KB-agent: “Knowledge is power” à “Ability to derive new knowledge is ability to increase power.”

Overview of Propositional Logic as a language for a KB-agent

Tradeoff: expressivity vs. tractability

Practical Example: circuit test pattern generation

Automatic theorem proving approaches


	Review: goal-based agent, simple problem-solving agent
	How would we design a Minesweeper-playing agent?
	Capacity for simple logical reasoning useful for some problem solving.


	Toy example: Minesweeper Agent
	Real-world example: Reaction Control System (RCS)


	propositions - T/F atomic sentences about the world
		propositional symbols (a.k.a. variables) P, Q, R, …, and constants True and False
		e.g. P: “There is a mine at (1,2).”; Q: “There is a mine at (2,2).”
	not (Ø) - negates meaning of a sentence
		P, ØP, Q, ØQ, … are called literals
		e.g. ØQ: “There is not a mine at (2,2)


	connectives (Ù,Ú,Þ,Û) - logically combine sentences
		PÙQ, PÚQ, PÞQ, PÛQ are complex sentences
		e.g. PÞØQ: “If there is a mine at (1,2), then there is not a mine at (2,2).”
	parentheses - disambiguates grammar
		e.g. PÞØQ ÙQ
		PÞ(ØQ ÙQ) vs. (PÞØQ) ÙQ


	commitments to the meanings of propositions
		(“P” without an interpretation is meaningless.)
		propositional symbol interpretations
	commitments to the meanings of connectives, etc.
		(“Þ” without an interpretation is meaningless.)
		truth tables


	truth table - table of truth assignments to propositions and resulting truth values for complex sentences
	truth tables for defining connectives


	tables can easily become large!
		Exponential growth with increase in propositional symbols


	A sentence is valid if it is true for all possible truth assignments (a.k.a. a tautology) e.g. (P ÚØP)
	A sentence is satisfiable if there exists a “satisfying” truth assignment (i.e. a truth assignment for which the sentence is true).
	A truth assignment which satisfies a sentence is a model of that sentence.


	entailment - whether a sentences follows from other sentences
		KB Ö f means knowledge base KB entails formula f
		If a truth assignment that satisfies all sentences in KB also satisfies formula f, f is entailed by KB


	inference rules - deduce new sentences from old sentences
		KB \| f means knowledge base KB infers or proves formula f
		If there are a set of inference actions which derive the truth of formula f from the sentence in KB, f is inferred by KB


	soundness - all inferences correct?
	completeness - all conclusions reachable?


A logic consists of
	a formal language consisting of
		syntax
		semantics
	a proof theory
terms: sentences, KB, valid, satisfiable, model, entail, inference rule, sound, complete


	All KR&R languages have syntax, semantics, and ontological and epistemological commitments.


	KB-agents can achieve competence quickly, by learning or being told facts
	KB-agents can adapt. Changes in the world cause updates to KB
	Flexible; no reprogramming!


	expressivity - limits of what knowledge a logic can express
	tractability - limits of what can be reasoned within computational constraints
	fundamental tradeoff between expressivity and tractability
	knowledge acquisition - the difficulty of expressing and codifying human knowledge


	Problem Overview
		Combinational circuits
		Single-Stuck-At (SSA) faults
		Automatic Test Pattern Generation
	Notation translation for Larrabee paper
	Normal circuit example
	Conversion to Conjunctive Normal Form (CNF)


Suppose:
	You’re working in circuit testing
	You test a great many combinational circuit designs (many quite complex).
	You wish to find a way to automate testing of circuits for Single Stuck-At (SSA) faults:
		A single circuit wire “stuck at” a constant value (e.g. 0 = false=grounded)
		For any given fault, it would be nice to have a circuit input which would give a different result if fault / no fault.


	In the Larrabee paper, a different logical notation is used:


	Eliminate Þ and Û
	Move negations in (form literals) using deMorgan’s Law
	Distribute Ù and Ú to form conjunction of disjunctions
	Worked example: X Û (Y Ù ØZ)


	Replace XÛY with (XÞY) Ù (YÞX)
	Replace XÞY with ØXÚY
	Worked example:
	X Û (YÙØZ) =
	(X Þ (YÙØZ)) Ù ((YÙØZ) Þ X) =
	(ØX Ú (YÙØZ)) Ù (Ø(YÙØZ) Ú X)


	deMorgan’s Law:
		Replace Ø(XÚY) with (ØX ÙØY)
		Replace Ø(XÙY) with (ØX Ú ØY)
	Worked example:
	(ØX Ú (YÙØZ)) Ù (Ø(YÙØZ) Ú X) =
	(ØX Ú (YÙØZ)) Ù ((ØY Ú ØØZ) Ú X) =
	(ØX Ú (YÙØZ)) Ù ((ØY Ú Z) Ú X) =
	(ØX Ú (YÙØZ)) Ù (ØY Ú Z Ú X)
	At this point, you should only have complex sentences made up of Ù, Ú, and literals


	Replace (f₀ Ú (f ₁ Ù … Ù f _n))
		with ((f₀ Ú f₁) Ù … Ù (f₀ Ú f_n)) where f_i is a formula
	Continue until you have one conjunction (and) of disjunctions (or’s).
	(ØX Ú (YÙØZ)) Ù (ØY Ú Z Ú X) =
	((ØXÚY) Ù (ØXÚØZ)) Ù (ØY Ú Z Ú X) =
	(ØXÚY) Ù (ØXÚØZ) Ù (ØY Ú Z Ú X)


	This is called Conjunctive Normal Form (CNF).
	Any CNF form can be converted to 3CNF or k-CNF by introducing variables and splitting disjunctions:
		(PÚQÚRÚSÚT) is equivalent to (PÚQÚU) Ù (ØUÚRÚV) Ù (ØVÚSÚT)


	Suppose we wish to test the following circuit for a SSA fault at D with D stuck at true:


	We model the fault by introducing D’, which is stuck at true and affects the value of X which we rename X’:


	We want to find a test pattern (inputs resulting in different outputs for correct/faulty cases)
	This is accomplished by first creating a circuit with both our correct and fault models, which evaluates to true when the correct and fault circuits have different outputs.


	Then we fix the output of this combined circuit to true, and treat this as a boolean satisfiability problem, seeking a satisfying truth assignment to the inputs.
	Such an assignment will be a test pattern suitable for testing for the given fault.


	Given: combinational circuit, possible fault
	Build fault model
	Build “test circuit” combining normal and fault model, evaluating true iff outputs of each are different
	Fix output of test circuit to true
	Convert circuit logic to CNF form
	Use GSAT to find a satisfying truth assignment to inputs.


	Proof by Contradiction:
		Given KB, want to show KB Ö f (KB entails f)
		Let KB’ = KB È {Øf}
		If we can prove that KB’ Ö false (inconsistent, contradiction), then we’ve proven that KB Ö f
	If KB’ \| false and our inference rules are sound then, KB’ Ö false


	Can use propositional logic inference rules of Figure 6.13, or…
	Can convert to CNF and do resolution theorem proving
	Example:
		KB = {P, PÞQ, QÞR}
		Does this infer R?