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Transcript of 1 CS 4700: Foundations of Artificial Intelligence Carla P. Gomes [email protected] Module:...
1
CS 4700:Foundations of Artificial Intelligence
Carla P. [email protected]
Module: Propositional Logic:
Inference (Reading R&N: Chapter 7)
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)
6
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)
7
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
9
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.
10
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
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 (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: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 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
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
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
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)