Logical Agents

Chapter 7

Outline

Knowledge-based agents

Wumpus world

Logic in general - models and entailment

Propositional (Boolean) logic

Equivalence, validity, satisfiability

Inference rules and theorem proving

forward chaining

backward chaining

resolution

Knowledge bases

Knowledge base = 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

Agents can be viewed at the knowledge level

i.e., what they know, regardless of how implemented

Or at the implementation level

i.e., data structures in KB and algorithms that manipulate them

A simple knowledge-based agent

The agent must be able to:

Represent states, actions, etc.

Incorporate new percepts

Update internal representations of the world

Deduce hidden properties of the world

Deduce appropriate actions

Wumpus World PEAS description

Performance measure

gold +1000, death -1000

-1 per step, -10 for using the arrow

Environment

Squares adjacent to wumpus are smelly

Squares adjacent to pit are breezy

Glitter iff gold is in the same square

Shooting kills wumpus if you are facing it

Shooting uses up the only arrow

Grabbing picks up gold if in same square

Releasing drops the gold in same square

Sensors: Stench, Breeze, Glitter, Bump, Scream

Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot

Wumpus world characterization

Fully Observable No – only local perception

Deterministic Yes – outcomes exactly specified

Episodic No – sequential at the level of actions

Static Yes – Wumpus and Pits do not move

Discrete Yes

Single-agent? Yes – Wumpus is essentially a natural feature

Exploring a wumpus world

Logic in general

Logics are formal languages for representing information such that conclusions can be drawn

Syntax defines the sentences in the language

Semantics define the "meaning" of sentences;

i.e., define truth of a sentence in a world

E.g., the language of arithmetic

x+2 ≥ y is a sentence; x2+y > {} is not a sentence

x+2 ≥ y is true iff the number x+2 is no less than the number y

x+2 ≥ y is true in a world where x = 7, y = 1

x+2 ≥ y is false in a world where x = 0, y = 6

Entailment

Entailment means that one thing follows from another:

KB ╞ α

Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true

E.g., the KB containing “the Giants won” and “the Reds won” entails “Either the Giants won or the Reds won”

E.g., x+y = 4 entails 4 = x+y

Entailment is a relationship between sentences (i.e., syntax) that is based on semantics

Models

Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated

We say m is a model of a sentence α if α is true in m

M(α) is the set of all models of α

Then KB ╞ α iff M(KB) Í M(α)

E.g. KB = Giants won and Reds
won α = Giants won

Entailment in the wumpus world

Situation after detecting nothing in [1,1], moving right, breeze in [2,1]

Consider possible models for KB assuming only pits

3 Boolean choices Þ 8 possible models

Wumpus models

Wumpus models

KB = wumpus-world rules + observations

Wumpus models

KB = wumpus-world rules + observations

α₁ = "[1,2] is safe", KB ╞ α₁, proved by model checking

Wumpus models

KB = wumpus-world rules + observations

Wumpus models

KB = wumpus-world rules + observations

α₂ = "[2,2] is safe", KB ╞ α₂

Inference

Define: KB ├_iα = sentence α can be derived from KB by procedure i

Soundness: i is sound if whenever KB ├_iα, it is also true that KB╞ α

Completeness: i is complete if whenever KB╞ α, it is also true that KB ├_iα

Preview: we will define a logic (first-order logic) which is expressive enough to say almost anything of interest, and for which there exists a sound and complete inference procedure.

That is, the procedure will answer any question whose answer follows from what is known by the KB.

Propositional logic: Syntax

Propositional logic is the simplest logic – illustrates basic ideas

The proposition symbols P₁, P₂ etc are sentences

If S is a sentence, ØS is a sentence (negation)

If S₁ and S₂ are sentences, S₁ Ù S₂ is a sentence (conjunction)

If S₁ and S₂ are sentences, S₁ Ú S₂ is a sentence (disjunction)

If S₁ and S₂ are sentences, S₁ Þ S₂ is a sentence (implication)

If S₁ and S₂ are sentences, S₁ Û S₂ is a sentence (biconditional)

Propositional logic: Semantics

Each model specifies true/false for each proposition symbol

E.g. P_1,2 P_2,2 P_3,1

false true false

With these symbols, 8 possible models, can be enumerated automatically.

Rules for evaluating truth with respect to a model m:

ØS is true iff S is false

S₁ Ù S₂ is true iff S₁ is true and S₂ is true

S₁ Ú S₂ is true iff S₁is true or S₂ is true

S₁ Þ S₂ is true iff S₁ is false or S₂ is true

i.e., is false iff S₁ is true and S₂is false

S₁ Û S₂is true iff S₁ÞS₂ is true andS₂ÞS₁ is true

Simple recursive process evaluates an arbitrary sentence, e.g.,

ØP_1,2Ù (P_2,2ÚP_3,1) = true Ù (true Ú false) = true Ù true = true

Truth tables for connectives

Wumpus world sentences

Let P_i,j be true if there is a pit in [i, j].

Let B_i,j be true if there is a breeze in [i, j].

ØP_1,1

ØB_1,1

B_2,1

"Pits cause breezes in adjacent squares"

B_1,1Û(P_1,2 Ú P_2,1)

B_2,1Û (P_1,1 Ú P_2,2Ú P_3,1)

Truth tables for inference

Inference by enumeration

Depth-first enumeration of all models is sound and complete

For n symbols, time complexity is O(2ⁿ), space complexity is O(n)

Logical equivalence

Two sentences are logically equivalent iff true in same models: α ≡ ß iff α╞ β and β╞ α

Validity and satisfiability

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

e.g., True, A ÚØA, A Þ A, (A Ù (A Þ B)) Þ B

Validity is connected to inference via the Deduction Theorem:

KB ╞ α if and only if (KB Þ α) is valid

A sentence is satisfiable if it is true in some model

e.g., AÚ B, C

A sentence is unsatisfiable if it is true in no models

e.g., AÙØA

Satisfiability is connected to inference via the following:

KB ╞ α if and only if (KB ÙØα) is unsatisfiable

Proof methods

Proof methods divide into (roughly) two kinds:

Application of inference rules

Legitimate (sound) generation of new sentences from old

Proof = a sequence of inference rule applications
Can use inference rules as operators in a standard search algorithm

Typically require transformation of sentences into a normal form

Model checking

truth table enumeration (always exponential in n)

improved backtracking, e.g., Davis--Putnam-Logemann-Loveland (DPLL)

heuristic search in model space (sound but incomplete)

e.g., min-conflicts like hill-climbing algorithms

Resolution

Conjunctive Normal Form (CNF)

conjunction of disjunctions of literals

clauses

E.g., (A Ú ØB) Ù (B Ú ØC Ú ØD)

Resolution inference rule (for CNF):

l_i Ú… Ú l_k, m₁ Ú … Ú m_n

l_i Ú … Ú l_i-1Úl_i+1Ú … Ú l_k Ú m₁ Ú … Ú m_j-1Ú m_j+1 Ú... Ú m_n

where l_i and m_j are complementary literals.

E.g., P_1,3 Ú P_2,2, ØP_2,2

P_1,3

Resolution is sound and complete
for propositional logic

Resolution

Soundness of resolution inference rule:

Ø(l_i Ú … Ú l_i-1Úl_i+1Ú … Ú l_k) Þ l_i

Øm_j Þ (m₁ Ú … Ú m_j-1Ú m_j+1 Ú... Ú m_n)

Ø(l_i Ú … Ú l_i-1Úl_i+1Ú … Ú l_k) Þ (m₁ Ú … Ú m_j-1Ú m_j+1 Ú... Ú m_n)

where l_i and m_j are complementary literals.

Conversion to CNF

B_1,1 Û (P_1,2 Ú P_2,1)

Eliminate Û, replacing α Û β with (α Þ β)Ù(β Þ α).

(B_1,1 Þ (P_1,2 Ú P_2,1)) Ù ((P_1,2 Ú P_2,1) Þ B_1,1)

2. Eliminate Þ, replacing α Þ β with ØαÚ β.

(ØB_1,1 Ú P_1,2 Ú P_2,1) Ù (Ø(P_1,2 Ú P_2,1) Ú B_1,1)

3. Move Ø inwards using de Morgan's rules and double-negation:

(ØB_1,1Ú P_1,2 Ú P_2,1) Ù ((ØP_1,2Ú ØP_2,1) Ú B_1,1)

4. Apply distributivity law (Ù over Ú) and flatten:

(ØB_1,1 Ú P_1,2 Ú P_2,1) Ù (ØP_1,2Ú B_1,1) Ù (ØP_2,1 Ú B_1,1)

Resolution algorithm

Proof by contradiction, i.e., show KBÙØα unsatisfiable

Resolution example

KB = (B_1,1 Û (P_1,2Ú P_2,1)) ÙØ B_1,1

α = ØP_1,2(negate the premise for proof by refutation)

Forward and backward chaining

Horn Form (restricted)

KB = conjunction of Horn clauses

Horn clause =

proposition symbol; or

(conjunction of symbols) Þ symbol

E.g., C Ù (B Þ A) Ù (C Ù D Þ B)

Modus Ponens (for Horn Form): complete for Horn KBs

α₁, … ,α_n, α₁ Ù … Ù α_n Þ β

β

Can be used with forward chaining or backward chaining.

These algorithms are very natural and run in linear time

Forward chaining

Idea: fire any rule whose premises are satisfied in the KB,

add its conclusion to the KB, until query is found

Forward chaining algorithm

Forward chaining is sound and complete for Horn KB

Forward chaining example

Proof of completeness

FC derives every atomic sentence that is entailed by KB

FC reaches a fixed point where no new atomic sentences are derived

Consider the final state as a model m, assigning true/false to symbols

Every clause in the original KB is true in m

a₁Ù … Ù a_k_Þb

Hence m is a model of KB

If KB╞ q, q is true in every model of KB, including m

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

Avoid loops: check if new subgoal is already on the goal stack

Avoid repeated work: check if new subgoal

has already been proved true, or

has already failed

Backward chaining example

Forward vs. backward chaining

FC is data-driven, automatic, unconscious processing,

e.g., object recognition, routine decisions

May do lots of work that is irrelevant to the goal

BC is goal-driven, appropriate for problem-solving,

e.g., Where are my keys? How do I get into a PhD program?

Complexity of BC can be much less than linear in size of KB

Efficient propositional inference

Two families of efficient algorithms for propositional inference:

Complete backtracking search algorithms

DPLL algorithm (Davis, Putnam, Logemann, Loveland)

Incomplete local search algorithms

WalkSAT algorithm

The DPLL algorithm

Determine if an input propositional logic sentence (in CNF) is satisfiable.

Improvements over truth table enumeration:

Early termination

A clause is true if any literal is true.

A sentence is false if any clause is false.

Pure symbol heuristic

Pure symbol: always appears with the same "sign" in all clauses.

e.g., In the three clauses (A Ú ØB), (ØB Ú ØC), (C Ú A), A and B are pure, C is impure.

Make a pure symbol literal true.

Unit clause heuristic

Unit clause: only one literal in the clause

The only literal in a unit clause must be true.

The DPLL algorithm

The WalkSAT algorithm

Incomplete, local search algorithm

Evaluation function: The min-conflict heuristic of minimizing the number of unsatisfied clauses

Balance between greediness and randomness

The WalkSAT algorithm

Hard satisfiability problems

Consider random 3-CNF sentences. e.g.,

(ØD Ú ØB Ú C) Ù (B Ú ØA Ú ØC) Ù (ØC Ú ØB Ú E) Ù (E Ú ØD Ú B) Ù (B Ú E Ú ØC)

m = number of clauses

n = number of symbols

Hard problems seem to cluster near m/n = 4.3 (critical point)

Hard satisfiability problems

Hard satisfiability problems

Median runtime for 100 satisfiable random 3-CNF sentences, n = 50

Inference-based agents in the wumpus world

A wumpus-world agent using propositional logic:

ØP_1,1

ØW_1,1

B_x,y Û (P_x,y+1 Ú P_x,y-1 Ú P_x+1,y Ú P_x-1,y)

S_x,y Û (W_x,y+1 Ú W_x,y-1 Ú W_x+1,y Ú W_x-1,y)

W_1,1 Ú W_1,2 Ú … Ú W_4,4

ØW_1,1 Ú ØW_1,2

ØW_1,1 Ú ØW_1,3

…

Þ 64 distinct proposition symbols, 155 sentences

Slide 72

Expressiveness limitation of propositional logic

KB contains "physics" sentences for every single square

For every time t and every location [x,y],

L_x,y Ù FacingRight^t Ù Forward^t Þ L_x+1,y

Rapid proliferation of clauses

Summary

Logical agents apply inference to a knowledge base to derive new information and make decisions

Basic concepts of logic:

syntax: formal structure of sentences

semantics: truth of sentences wrt models

entailment: necessary truth of one sentence given another

inference: deriving sentences from other sentences

soundness: derivations produce only entailed sentences

completeness: derivations can produce all entailed sentences

Wumpus world requires the ability to represent partial and negated information, reason by cases, etc.

Resolution is complete for propositional logic
Forward, backward chaining are linear-time, complete for Horn clauses

Propositional logic lacks expressive power


	Knowledge-based agents
	Wumpus world
	Logic in general - models and entailment
	Propositional (Boolean) logic
	Equivalence, validity, satisfiability
	Inference rules and theorem proving
		forward chaining
		backward chaining
		resolution


	Knowledge base = 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
	Agents can be viewed at the knowledge level
		i.e., what they know, regardless of how implemented
	Or at the implementation level
		i.e., data structures in KB and algorithms that manipulate them


	The agent must be able to:
		Represent states, actions, etc.
		Incorporate new percepts
		Update internal representations of the world
		Deduce hidden properties of the world
		Deduce appropriate actions


	Performance measure
		gold +1000, death -1000
		-1 per step, -10 for using the arrow

	Environment
		Squares adjacent to wumpus are smelly
		Squares adjacent to pit are breezy
		Glitter iff gold is in the same square
		Shooting kills wumpus if you are facing it
		Shooting uses up the only arrow
		Grabbing picks up gold if in same square
		Releasing drops the gold in same square

	Sensors: Stench, Breeze, Glitter, Bump, Scream
	Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot


	Fully Observable No – only local perception
	Deterministic Yes – outcomes exactly specified
	Episodic No – sequential at the level of actions
	Static Yes – Wumpus and Pits do not move
	Discrete Yes
	Single-agent? Yes – Wumpus is essentially a natural feature


	Logics are formal languages for representing information such that conclusions can be drawn
	Syntax defines the sentences in the language
	Semantics define the "meaning" of sentences;
		i.e., define truth of a sentence in a world

	E.g., the language of arithmetic
		x+2 ≥ y is a sentence; x2+y > {} is not a sentence
		x+2 ≥ y is true iff the number x+2 is no less than the number y
		x+2 ≥ y is true in a world where x = 7, y = 1
		x+2 ≥ y is false in a world where x = 0, y = 6


	Entailment means that one thing follows from another:
	KB ╞ α
	Knowledge base KB entails sentence α if and only if α is true in all worlds where KB is true

		E.g., the KB containing “the Giants won” and “the Reds won” entails “Either the Giants won or the Reds won”
		E.g., x+y = 4 entails 4 = x+y
		Entailment is a relationship between sentences (i.e., syntax) that is based on semantics


	Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated

	We say m is a model of a sentence α if α is true in m

	M(α) is the set of all models of α

	Then KB ╞ α iff M(KB) Í M(α)
		E.g. KB = Giants won and Reds won α = Giants won


	Situation after detecting nothing in [1,1], moving right, breeze in [2,1]

	Consider possible models for KB assuming only pits

	3 Boolean choices Þ 8 possible models


	KB = wumpus-world rules + observations
	α₁ = "[1,2] is safe", KB ╞ α₁, proved by model checking


	KB = wumpus-world rules + observations
	α₂ = "[2,2] is safe", KB ╞ α₂


	Define: KB ├_iα = sentence α can be derived from KB by procedure i
	Soundness: i is sound if whenever KB ├_iα, it is also true that KB╞ α
	Completeness: i is complete if whenever KB╞ α, it is also true that KB ├_iα
	Preview: we will define a logic (first-order logic) which is expressive enough to say almost anything of interest, and for which there exists a sound and complete inference procedure.
	That is, the procedure will answer any question whose answer follows from what is known by the KB.


	Propositional logic is the simplest logic – illustrates basic ideas

	The proposition symbols P₁, P₂ etc are sentences

		If S is a sentence, ØS is a sentence (negation)
		If S₁ and S₂ are sentences, S₁ Ù S₂ is a sentence (conjunction)
		If S₁ and S₂ are sentences, S₁ Ú S₂ is a sentence (disjunction)
		If S₁ and S₂ are sentences, S₁ Þ S₂ is a sentence (implication)
		If S₁ and S₂ are sentences, S₁ Û S₂ is a sentence (biconditional)


	Each model specifies true/false for each proposition symbol
		E.g. P_1,2 P_2,2 P_3,1
		false true false

	With these symbols, 8 possible models, can be enumerated automatically.
	Rules for evaluating truth with respect to a model m:
	ØS is true iff S is false
	S₁ Ù S₂ is true iff S₁ is true and S₂ is true
	S₁ Ú S₂ is true iff S₁is true or S₂ is true
	S₁ Þ S₂ is true iff S₁ is false or S₂ is true
	i.e., is false iff S₁ is true and S₂is false
	S₁ Û S₂is true iff S₁ÞS₂ is true andS₂ÞS₁ is true

	Simple recursive process evaluates an arbitrary sentence, e.g.,

	ØP_1,2Ù (P_2,2ÚP_3,1) = true Ù (true Ú false) = true Ù true = true


	Let P_i,j be true if there is a pit in [i, j].
	Let B_i,j be true if there is a breeze in [i, j].
		ØP_1,1
		ØB_1,1
		B_2,1

	"Pits cause breezes in adjacent squares"
		B_1,1Û(P_1,2 Ú P_2,1)
		B_2,1Û (P_1,1 Ú P_2,2Ú P_3,1)


	Depth-first enumeration of all models is sound and complete









	For n symbols, time complexity is O(2ⁿ), space complexity is O(n)


	Two sentences are logically equivalent iff true in same models: α ≡ ß iff α╞ β and β╞ α


	A sentence is valid if it is true in all models,
		e.g., True, A ÚØA, A Þ A, (A Ù (A Þ B)) Þ B

	Validity is connected to inference via the Deduction Theorem:
		KB ╞ α if and only if (KB Þ α) is valid

	A sentence is satisfiable if it is true in some model
		e.g., AÚ B, C

	A sentence is unsatisfiable if it is true in no models
		e.g., AÙØA

	Satisfiability is connected to inference via the following:
		KB ╞ α if and only if (KB ÙØα) is unsatisfiable


Proof methods divide into (roughly) two kinds:

	Application of inference rules
		Legitimate (sound) generation of new sentences from old
		Proof = a sequence of inference rule applications Can use inference rules as operators in a standard search algorithm
		Typically require transformation of sentences into a normal form

	Model checking
		truth table enumeration (always exponential in n)
		improved backtracking, e.g., Davis--Putnam-Logemann-Loveland (DPLL)
		heuristic search in model space (sound but incomplete)
		e.g., min-conflicts like hill-climbing algorithms


	Conjunctive Normal Form (CNF)
		conjunction of disjunctions of literals
		clauses
		E.g., (A Ú ØB) Ù (B Ú ØC Ú ØD)

	Resolution inference rule (for CNF):
	l_i Ú… Ú l_k, m₁ Ú … Ú m_n
	l_i Ú … Ú l_i-1Úl_i+1Ú … Ú l_k Ú m₁ Ú … Ú m_j-1Ú m_j+1 Ú... Ú m_n

	where l_i and m_j are complementary literals.
	E.g., P_1,3 Ú P_2,2, ØP_2,2
	P_1,3

	Resolution is sound and complete for propositional logic


	Soundness of resolution inference rule:

	Ø(l_i Ú … Ú l_i-1Úl_i+1Ú … Ú l_k) Þ l_i
	Øm_j Þ (m₁ Ú … Ú m_j-1Ú m_j+1 Ú... Ú m_n)
	Ø(l_i Ú … Ú l_i-1Úl_i+1Ú … Ú l_k) Þ (m₁ Ú … Ú m_j-1Ú m_j+1 Ú... Ú m_n)

	where l_i and m_j are complementary literals.


	B_1,1 Û (P_1,2 Ú P_2,1)

	Eliminate Û, replacing α Û β with (α Þ β)Ù(β Þ α).
		(B_1,1 Þ (P_1,2 Ú P_2,1)) Ù ((P_1,2 Ú P_2,1) Þ B_1,1)

	2. Eliminate Þ, replacing α Þ β with ØαÚ β.
		(ØB_1,1 Ú P_1,2 Ú P_2,1) Ù (Ø(P_1,2 Ú P_2,1) Ú B_1,1)

	3. Move Ø inwards using de Morgan's rules and double-negation:
		(ØB_1,1Ú P_1,2 Ú P_2,1) Ù ((ØP_1,2Ú ØP_2,1) Ú B_1,1)

	4. Apply distributivity law (Ù over Ú) and flatten:
		(ØB_1,1 Ú P_1,2 Ú P_2,1) Ù (ØP_1,2Ú B_1,1) Ù (ØP_2,1 Ú B_1,1)


	KB = (B_1,1 Û (P_1,2Ú P_2,1)) ÙØ B_1,1
	α = ØP_1,2(negate the premise for proof by refutation)


Horn Form (restricted)
	KB = conjunction of Horn clauses
	Horn clause =
		proposition symbol; or
		(conjunction of symbols) Þ symbol
	E.g., C Ù (B Þ A) Ù (C Ù D Þ B)
Modus Ponens (for Horn Form): complete for Horn KBs
α₁, … ,α_n, α₁ Ù … Ù α_n Þ β
β

Can be used with forward chaining or backward chaining.
These algorithms are very natural and run in linear time


	Idea: fire any rule whose premises are satisfied in the KB,
		add its conclusion to the KB, until query is found


FC derives every atomic sentence that is entailed by KB
	FC reaches a fixed point where no new atomic sentences are derived
	Consider the final state as a model m, assigning true/false to symbols
	Every clause in the original KB is true in m
		a₁Ù … Ù a_k_Þb
	Hence m is a model of KB
	If KB╞ q, q is true in every model of KB, including m


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

Avoid loops: check if new subgoal is already on the goal stack

Avoid repeated work: check if new subgoal
	has already been proved true, or
	has already failed


	FC is data-driven, automatic, unconscious processing,
		e.g., object recognition, routine decisions

	May do lots of work that is irrelevant to the goal

	BC is goal-driven, appropriate for problem-solving,
		e.g., Where are my keys? How do I get into a PhD program?

	Complexity of BC can be much less than linear in size of KB


	Two families of efficient algorithms for propositional inference:

	Complete backtracking search algorithms
	DPLL algorithm (Davis, Putnam, Logemann, Loveland)
	Incomplete local search algorithms
		WalkSAT algorithm


Determine if an input propositional logic sentence (in CNF) is satisfiable.

Improvements over truth table enumeration:
	Early termination
		A clause is true if any literal is true.
		A sentence is false if any clause is false.

	Pure symbol heuristic
		Pure symbol: always appears with the same "sign" in all clauses.
		e.g., In the three clauses (A Ú ØB), (ØB Ú ØC), (C Ú A), A and B are pure, C is impure.
		Make a pure symbol literal true.

	Unit clause heuristic
		Unit clause: only one literal in the clause
		The only literal in a unit clause must be true.