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CS 4700:Foundations of Artificial Intelligence

Carla P. Gomesgomes@cs.cornell.edu

Module: Propositional Logic:

Inference (Reading R&N: Chapter 7)

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Proof Methods

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• Different types of proofs

– 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

• (including some inference rules)

we’ve talked about this approach

Nextmodule

Current module

Proof

The sequence of wffs (w1, w2, …, wn) is called a proof (or deduction) of wn from a set of wffs Δ iff each wi in the sequence is either in Δ or can be inferred from a wff (or wffs) earlier in the sequence by using a valid rule of inference.

If there is a proof of wn from Δ, we say that wn is a theorem of the set Δ.

Δ├ wn

(read: wn can be proved or inferred from Δ)

The concept of proof is relative to a particular set of inference rules used. If we denote the set of inference rules used by R, we can write the fact that wn can be derived from Δ using the set of inference rules in R:

Δ├ R wn

(read: wn can be proved from Δ using the inference rules in R)

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Propositional logic: Rules of Inference or Methods of Proof

How to produce additional wffs (sentences) from other ones? What steps can we perform to show that a conclusion follows logically from a set of hypotheses?

ExampleModus Ponens

PP Q______________ Q

The hypotheses (premises) are written in a column and the conclusions below the barThe symbol denotes “therefore”. Given the hypotheses, the conclusion follows.The basis for this rule of inference is the tautology (P (P Q)) Q)[aside: check tautology with truth table to make sure]

In words: when P and P Q are True, then Q must be True also. (meaning ofsecond implication)

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Propositional logic: Rules of Inference or Methods of Proof

ExampleModus Ponens

If you study the CS 4700 material You will passYou study the CS4700material ______________ you will pass

Nothing “deep”, but again remember the formal reason is that ((P ^ (P Q)) Q is a tautology.

Propositional logic: Rules of Inference or Method of Proof

Rule of Inference Tautology (Deduction Theorem) Name

P

P QP (P Q) Addition

P Q P

(P Q) P Simplification

P

Q

P Q

[(P) (Q)] (P Q) Conjunction

P

PQ

Q

[(P) (P Q)] (P Q) Modus Ponens

Q

P Q

P

[(Q) (P Q)] P Modus Tollens

P Q

Q R

P R

[(PQ) (Q R)] (PR) Hypothetical Syllogism

(“chaining”)

P Q P

Q

[(P Q) (P)] Q Disjunctive syllogism

P Q P R Q R

[(P Q) (P R)] (Q R) Resolution

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Valid Arguments

An argument is a sequence of propositions. The final proposition is called the conclusion of the argument while the other propositions are called the premises or hypotheses of the argument.

An argument is valid whenever the truth of all its premises implies the truth of its conclusion.

How to show that q logically follows from the hypotheses (p1 p2 …pn)?

Show that

(p1 p2 …pn) q is a tautology

One can use the rules of inference to show the validity of an argument.

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Proof Tree

Proofs can also be based on partial orders – we can represent them using a tree structure:– Each node in the proof tree is labeled by a wff, corresponding to a wff in

the original set of hypotheses or be inferable from its parents in the tree using one of the rules of inference;

– The labeled tree is a proof of the label of the root node.

Example:Given the set of wffs:

P, R, PQGive a proof of Q R

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Tree Proof

P PQ R

Q

Q R

Given: P, P Q, R;Prove: Q R

What rules of inference did we use?

MP

Conj.

Length of Proofs

Why bother with inference rules? We could always use a truth table

to check the validity of a conclusion from a set of premises.

But, resulting proof can be much shorter than truth table method.

Consider premises:p_1, p_1 p_2, p_2 p_3 … p_(n-1) p_n

To prove conclusion: p_n

Inference rules: Truth table: n-1 P steps 2n

Key open question: Is there always a short proof for any validconclusion? Probably not. The NP vs. co-NP question.

Resolution in Propositional Logic

Resolution (for CNF)

P Q P R Q R

Very important inference rule – several other inference rulescan be seen as special cases of resolution.

Resolution for CNF – applied to a special type of wffs: conjunction of clauses.

Literal – either an atom (e.g., P) or its negation (P).Clause – disjunction of of literals (e.g., (P Q R)).

Note: Sometimes we use the notation of a set for a clause: e.g. {P,Q,R} correspondsto the clause (PQ R); the empty clause (sometimes written as Nil or {}) is equivalentto False;

Soundness of rule (validity of rule): [(P Q) (P R)] (Q R) is valid

Soundness of Resolution:Validity of the Resolution Inference Rule

P Q R (PQ) (PR) (PQ)(PR) (QR) (P Q) (P R) (Q R)

0 0 0 0 1 0 0 1

0 0 1 0 1 0 1 1

0 1 0 1 1 1 1 1

1 0 0 1 0 0 0 1

1 1 0 1 0 0 1 1

1 0 1 1 1 1 1 1

0 1 1 1 1 1 1 1

0 0 0 0 1 0 0 1

P Q P R Q R

resolving on P

Validity (Tautology): (P Q) (P R) (Q R) ;

Resolution:Special Cases

1 – Rule of Inference: Chaining

R P P Q R Q

R P P Q R Q

can be re-written

Rule of Inference Chaining

2 – Rule of Inference: Modus Ponens

P P Q Q

P P Q Q

can be re-written

Rule of Inference: Modus Ponens

Resolution:Special Cases

3 – Unit Resolution

P P Q Q

P

P Q Q

Resolution:Caution!

No duplications in the resolvent set

P Q R S P Q W Q R S W

Resolving one pair at a time

only one instance of Qappears in the resolvent,which is a set!

Resolving on Q

Resolving on R

P Q R P W Q R P R R W

P Q R P W Q R P Q Q W

True

P Q R P W Q R P W

DO NOT Resolve on Q and R

CNF

Conjunctive Normal Form (CNF)

A wff is in CNF format when it is a conjunction of disjunctions of literals.

Resolution for CNF – applied to wffs in CNF format.

(P Q R) (S P T R) (Q S)

{λ} Σ1

{ λ} Σ2

Σ1 Σ2

Σi- sets of literals i =1 ,2λ – atom;

atom resolved upon

Resolvent of thetwo clauses

Resolution

Conversion to CNF

P (Q R)

1.Eliminate , replacing α β with (α β)(β α).(P (Q R)) ((Q R) P)

2. Eliminate , replacing α β with α β.(P Q R) ((Q R) P)

3. Move inwards using de Morgan's rules and double-negation:(P Q R) ((Q R) P)

4. Apply distributivity law ( over ) and flatten:(P Q R) (Q P) (R P)–

Converting DNF (Disjunctions of conjunctions) into CNF

1 – create a table – each row corresponds to the literals in each conjunct;

2 - Select a literal in each row and make a disjunction of these literals;

P Q R

S R P

Q S P

Example:

(PQ R ) (S R P) (Q S P)

(P S Q) (P R Q) (P P Q) (P S S)(P R S) (P P S) (P P Q)…

How many clauses?

Resolution:Wumpus World

P31 P2,2, P2,2

P31

P?

P?

Resolution Refutation

Resolution is sound – but resolution is not complete – e.g., (P R) ╞ (P R) but we cannot infer (P R) using resolution

we cannot use resolution directly to decide all logical entailments.

Resolution is Refutation Complete:We can show that a particular wff W is entailed from a given KB, how?

Proof by contradiction:Write the negation of what we are trying to prove (W) as a conjunction of clauses;Add those clauses (W) to the KB (also a set of clauses), obtaining KB’; prove

inconsistency for KB’, i.e.,

Apply resolution to the KB’ until:• No more resolvents can be added• Empty clause is obtained

To show that (P R) ╞Res (P R) do: (1) negate (P R), i.e.: (P) (R) ; (2) prove that (P R) (P) (R) is inconsistent

!

!

Satisfiability is connected to inference via the following:

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

One assumes α and shows that this leads to a contradiction with the facts in KB

Propositional Logic:Proof by refutation or contradiction:

Resolution:Robot Domain

Example:

BatIsOk

RobotMoves

BatIsOk BlockLiftable RobotMoves KBShow that KB ╞ BlockLiftable

BatIsOk RobotMovesBatIsOk BlockLiftable RobotMoves BlockLiftable

KB’

BlockLiftableBatIsOk BlockLiftable RobotMoves

BatIsOk RobotMovesRobotMoves

BatIsOk BatIsOk

Nil

Resolution

Resolution is refutation complete (Completeness of resolution refutation):

If KB ╞ W, the resolution refutation procedure, i.e., applying resolution on KB’, will produce the empty clause.

Decidability of propositional calculus by resolution refutation:

If KB is a set of finite clauses and if KB ╞ W, then the resolution refutation procedure will terminate without producing the empty clause.

Ground Resolution Theorem– If a set of clauses is not satisfiable, then resolution closure of those

clauses contains the empty clause.

In general, resolution for propositional logic is exponential !

The resolution closure of a set of clauses W in CNF, RC(W), is the set of all clauses derivable by repeated applicationof the resolution rule to clauses in W or their derivatives.

Resolution example:Wumpus World

KB = (B1,1 (P1,2 P2,1)) B1,1 α = P1,2

Resolution example:Wumpus World

KB = (B1,1 (P1,2 P2,1)) B1,1 α = P1,2

KB = (B11 (P1,2 P2,1)) ^ ((P1,2 P2,1) B11) B1,1

=(B11 P1,2 P2,1) ^ ((P1,2 P2,1) B11) B1,1 =(B11 P1,2 P2,1) ^(( P1,2 ^ P2,1) B11)) B1,1 =(B11 P1,2 P2,1) ^( P1,2 B11) ^ ( P2,1 B11) B1,1

Resolution example:Wumpus World

KB = (B1,1 (P1,2 P2,1)) B1,1 α = P1,2

KB = (B11 (P1,2 P2,1)) ^ ((P1,2 P2,1) B11) B1,1

=(B11 P1,2 P2,1) ^ ((P1,2 P2,1) B11) B1,1 =(B11 P1,2 P2,1) ^(( P1,2 ^ P2,1) B11)) B1,1 =(B11 P1,2 P2,1) ^( P1,2 B11) ^ ( P2,1 B11) B1,1

Resolution algorithm

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

New resolvents added at the end

Resolution algorithm

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

Any complete search algorithm applying only the resolution rule, can derive any conclusion entailed by any knowledge base in propositional logic – resolution can always be used to either confirm or refute a sentence – refutation completeness (Given A, it’s true we cannot use resolution to derive A OR B; butwe can use resolution to answer the question of whether A OR B is true.)

New resolvents added at the end

Resolution Closure

Definition

The resolution closure RC(f) of a formula f in CNF is the set of

all clauses derivable by repeated application of the resolution rule

to clauses in f or their derivatives.

Resolution

Theorem (Ground Resolution Theorem)– If f ( in CNF) is unsatisfiable then RC(f) contains the empty clause (that

evolves from P¬P for some variable P).

Proof (by contraposition)

If RC(f) does not contain the empty clause then it is satisfiable.

Let the variables in f be P1,..,Pn, and let us assume that RC(f) does not contain the empty clause.

Therefore we can construct a model for m for f with suitable truth values for P1,..,Pn.

Resolution (contd.)

Construction procedure: For i=1,..,n, if there exists a clause in RC(f) containing the

literal ¬Pi such that all its other literals are false under the assignment chosen for P1,..,Pi-1, then assign false to Pi. . Otherwise we set Pi=True.

Claim: RC(f)|m is true. I.e., m is a model for f.

Resolution (contd.)

Claim, RC(f)|m is true. I.e., m is a model for f.

Assume the opposite. Then, there exists a clause that evaluates to false under m.

Consider a non-satisfied clause in RC(f) over variables P1,..Pi that minimizes i.

Wlog, we may write this clause as (ab), where b is a literal over Pi and a is a formula over variables P1,.., Pi-1.

Note that there cannot exist another such clause where b is negated (c¬b) as otherwise i was not minimal [consider (ac) RC(f)].

Resolution (contd.)

So we write the clause as (ab), with b a literal over Pi

However, then Pi is set such that b evaluates to True, which contradicts the assumption that the clause was not satisfied.

Therefore all the clauses are satisfied and m is indeed a model for RC(f).

QED

Resolution Refutation – Ordering Search Strategies

Original clauses – 0th level resolvents– Breadth first strategy

• Generate all 1st level resolvents, then all 2nd level resolvents, etc.

– Depth first strategy • Produce a 1st level resolvent;

• Resolve the 1st level resolvent with a 0th level resolvent to produce a 2nd level resolvent, etc.

• With a depth bound, we can use a backtrack search strategy;

BlockLiftableBatIsOk BlockLiftable RobotMoves

BatIsOk RobotMovesRobotMoves

BatIsOk BatIsOk

Nil

BatIsOk RobotMovesBatIsOk BlockLiftable RobotMoves BlockLiftable

0th level resolvents

Depth first strategy

Refinement Resolution Strategies

Definitions:

A clause γ2 is a descendant of a clause γ1 iif:– Is a resolvent of γ1 with some other clause – Or is a resolvent of a descendant of γ1 with some

other clause;

If γ2 is a descendant of γ1, γ1 is an ancestor of γ2;

Set-of-support – set of clauses that are either clauses coming from the negation of the theorem to be proved or descendants of those clauses.

Set-of-support Strategy – it allows only refutations in which one of the clauses being resolved is in the set of support.

Set-of-support Strategy is refutation complete.

Set-of-support Resolution Strategy

BlockLiftableBatIsOk BlockLiftable RobotMoves

BatIsOk RobotMovesRobotMoves

BatIsOk BatIsOk

Nil

Set-of-support Strategy

Refinement Strategies

Ancestry-filtered strategy – allows only resolutions in which at least one member of the clauses being resolved either is a member of the original set of clauses or is an ancestor of the other clause being resolved;

The ancestry-filtered strategy is refutation complete.

Refinement Strategies

Linear Input Resolution Strategy – at least one of the clauses being resolved is a member of the original set of clauses (including the theorem being proved).

Linear Input Resolution Strategy is not refutation complete.

Example:

(P Q) (P Q) (P Q) (P Q)

This set of clauses is inconsistent; but there is no linear-input refutation strategy; but there is a resolution refutation strategy;

(P Q) (P Q)

Q

(P Q) (P Q)

Q

Nil

This is NOT Linear Input

Resolution Strategy

Consider a formula to represent the Pigeon Hole principle: (n+1) pigeons

in n holes.

Pij – pigeon i goes in hole j Pij {0,1} i= 1, 2, …, n+1; j = 1, 2, …n

Each pigeon has to go in a hole:

Starting with pigeon 1

P11 P12 … P1n

…

Pn+1,1 Pn+1,2 … Pn+1,n

.

: Resolution and Pigeon Hole Principle

Resolution Proofs of PH

And?

Two pigeons cannot go in the same hole

P11 ~P21; P11 ~P31; … P11 ~Pn+1,1;

~P11 ~P21; ~P11 ~P31; … ~P11 ~Pn+1,1;

….

~Pn+1,n ~P2,n; ~Pn+1,n ~P3n; … ~Pn+1,n ~Pn,n;

Resolution proofs of PH take exponentially many steps:Resolution proofs of inconsistency of PH require an exponential number of clauses, no matter in what order we resolve the clauses (Armin Haken 85).

Related to NP vs. Co-NP questions

Horn Clauses

Horn Clauses

Definition:

A Horn clause is a clause that has at most one positive literal.

Examples:

P; P Q; P Q; P Q R;

Types of Horn Clauses:Fact – single atom – e.g., P;Rule – implication, whose antecendent is a conjunction of positive literals and whose consequent consists of a single

positive literal – e.g., PQ R; Head is R; Tail is (PQ )Set of negative literals - in implication form, the antecedent is a conjunction of positive literals and the consequent is empty.

e.g., PQ ; equivalent to P Q.

Inference with propositional Horn clauses can be done in linear time !

Unit Resolution for Horn Clauses

Theorem– Unit resolution is refutation complete for HF, i.e. if kb¬a is a HF then

unit propagation shows that kb¬a is unsatisfiable iff kb╞ a.

Proof– If kb¬a is not satisfiable then kb¬a must contain a clause that contains

one non-negated variable only:This clause must be a unit clause. (otherwise consider the model that sets all variables to false).

– Using unit resolution of this variable with all other clauses essentially eliminates the variable from the formula while preserving the Horn property.

– Continuing this process, unit propagation hits P¬P iff the formula is not satisfiable.

P

P Q Q

P P Q Q

Forward chaining

HORN (Expert Systems and Logic Programming)

Horn Form (restricted)KB = conjunction of Horn clauses

– Horn clause = • proposition symbol; or• (conjunction of symbols) symbol

– E.g., C (B A) (C D B)

Modus Ponens (for Horn Form): complete for Horn KBsα1, … ,αn, α1 … αn β

β

–

–

Deciding entailment with Horn clauses can be done in linear time, in the size of the KB

!

Forward Chaining:Diagnosis systems

Example: diagnostic systemIF the engine is getting gas and the engine turns over

THEN the problem is spark plugs

IF the engine does not turn over and the lights do not come onTHEN the problem is battery or cables

IF the engine does not turn over and the lights come onTHEN the problem is starter motor

IF there is gas in the fuel tank and there is gas in the carburator

THEN the engine is getting gas

Forward chaining(Data driven reasoning)

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

AND-OR graph

Forward Chaining

Algorithm

Forward chaining is sound and complete for Horn KB

Count

P => Q 1L and M => P 2B and L => M 2A and P => L 2A and B => L 2

Inferred

P FL FM FB FA F

Agenda

AB

Forward chaining example

Forward chaining example

Forward chaining example

Forward chaining example

Forward chaining example

Forward chaining example

Forward chaining example

Forward chaining example

Forward Chaining

Algorithm

Forward chaining is sound and complete for Horn KB

Count

P => Q 1L and M => P 2B and L => M 2A and P => L 2A and B => L 2

Inferred

P FL FM FB FA F

Agenda

AB

Backward chaining

Idea: work backwards from the query q:to prove q by BC,

check if q is known already, orprove 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 subgoal1. has already been proved true, or2. has already failed

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 in pactice.

Prominent expert systems

• CADUCEUS (expert system)- Blood-borne infectious bacteria • Dendral- Analysis of mass spectra • Jess- Java Expert System Shell. A CLIPS engine implemented in Java

used in the development of expert systems • LogicNets- Web based expert system modeling environment to create

expert systems (in collaboration with NASA) • Mycin - Diagnose infectious blood diseases and recommend antibiotics

(by Stanford University) • NEXPERT Object- An early general-purpose commercial backwards-

chaining inference engine used in the development of expert systems • Prolog- Programming language used in the development of expert

systems • R1 (expert system)/XCon Order processing • STD Wizard - Expert system for recommending medical screening tests • PyKe- Pyke is a knowledge-based inference engine (expert system)