Logical Agents

Chapter 7 (continued)

Outline: Inference

Resolution in CNF

Sound and Complete

Forward and Backward Chaining using Modus Ponens in Horn Form

Sound and Complete

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

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)

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

Reasoning Patterns in Prop Logic

Given(s)

Conclusion

A Þ B, A

B

B Ù A

A

Rules that allow us to introduce new propositions while preserving truth values: logically equivalent

Two Examples:

Modus Ponens

And Elimination

Logical equivalence

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

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 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)

The power of false

Given: (P) Ù (ØP)

Prove: Z

Can we prove ØZ using the givens above?

Applying inference rules

Equivalent to a search problem

KB state = node

Inference rule application = edge

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.

Completeness

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

An incomplete inference algorithm cannot reach all possible conclusions

Equivalent to completeness in search (chapter 3)

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.

What if l_i and Øm_jare false?

What if l_i and Øm_j are true?

Completeness of Resolution

That is, that resolution can decide the truth value of S

S = set of clauses

RC(S) = Resolution closure of S = Set of all clauses that can be derived from S by the resolution inference rule.

RC(S) has finite cardinality (finite number of symbols P₁, P₂, … P_k), thus resolution refutation must terminate.

Completeness of Resolution (cont)

Ground resolution theorem = if S unsatisfiable, RC(S) contains empty clause.

Prove by proving contrapositive:

i.e., if RC(S) doesn’t contain empty clause, S is satisfiable

Do this by constructing a model:

For each P_i, if there is a clause in RC(S) containing ØP_iand all other literals in the clause are false, assign P_i = false

Otherwise P_i = true

This assignment of P_iis a model for S.

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 example

Proof of completeness

FC derives every atomic sentence that is entailed by KB (only for clauses in Horn form)

FC reaches a fixed point (the deductive closure) 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 example

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 38

Expressiveness limitation of propositional logic

We didn’t keep track of location and time in the KB. To do this we need more variables:

L_1,1to show that agent in L_1,1.Does this work?

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


	Resolution in CNF
		Sound and Complete
	Forward and Backward Chaining using Modus Ponens in Horn Form
		Sound and Complete


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


	Depth-first enumeration of all models is sound and complete









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


	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)


	Given(s)
	Conclusion


	A Þ B, A
	B

	B Ù A
	A

	Rules that allow us to introduce new propositions while preserving truth values: logically equivalent

	Two Examples:
	Modus Ponens

	And Elimination


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


	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


	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)


	Given: (P) Ù (ØP)
	Prove: Z



	Can we prove ØZ using the givens above?


	Equivalent to a search problem

	KB state = node
	Inference rule application = edge


	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.


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

	An incomplete inference algorithm cannot reach all possible conclusions
		Equivalent to completeness in search (chapter 3)


	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.

	What if l_i and Øm_jare false?
	What if l_i and Øm_j are true?


	That is, that resolution can decide the truth value of S

	S = set of clauses
	RC(S) = Resolution closure of S = Set of all clauses that can be derived from S by the resolution inference rule.
	RC(S) has finite cardinality (finite number of symbols P₁, P₂, … P_k), thus resolution refutation must terminate.


Ground resolution theorem = if S unsatisfiable, RC(S) contains empty clause.
Prove by proving contrapositive:
	i.e., if RC(S) doesn’t contain empty clause, S is satisfiable
	Do this by constructing a model:
		For each P_i, if there is a clause in RC(S) containing ØP_iand all other literals in the clause are false, assign P_i = false
		Otherwise P_i = true
	This assignment of P_iis a model for S.


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


FC derives every atomic sentence that is entailed by KB (only for clauses in Horn form)
	FC reaches a fixed point (the deductive closure) 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


	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.


	Incomplete, local search algorithm
	Evaluation function: The min-conflict heuristic of minimizing the number of unsatisfied clauses
	Balance between greediness and randomness


	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)


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


	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


	We didn’t keep track of location and time in the KB. To do this we need more variables:
		L_1,1to show that agent in L_1,1.Does this work?

	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


	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