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Reasoning with Classical Propositional Logic

- Jacques Robin

Outline

- Syntax
- Full CPL
- Implicative Normal Form CPL (INFCPL)
- Horn CPL (HCPL)
- Semantics
- Cognitive and Herbrand interpretations, models
- Reasoning
- FCPL Reasoning
- Truth-tabel based model checking
- Multiple inference rules
- INFCPL Reasoning
- Resolution and factoring
- DPLL
- WalkSat
- HCPL Reasoning
- Forward chaining
- Backward chaining

Full Classical Propositional Logic (FCPL) syntax

Syntax

?(a ? (b ? ((?c ? d) ? a) ? b))

FCPLFormula

CPL Normal Forms

Implicative Normal Form (INF)

Premisse

ConstantSymbol

Conclusion

- Semantic equivalence
- a ? b ? c ? d
- ?(a ? b) ? c ? d
- ?a ? ?b ? c ? d

Conjunctive Normal Form (CNF)

Literal

ConstantSymbol

Horn CLP

Implicative Normal Form (INF)

Premisse

ConstantSymbol

Conclusion

IntegrityConstraint

context IntegrityConstraint inv IC

Conclusion.ConstantSymbol false

a ? b ? c ? false

DefiniteClause

context DefiniteClause inv DC Conclusion.Constant

Symbol ? false

a ? b ? c ? d

Fact

context Fact inv Fact Premisse -gt size() 1 and

Premisse -gt ConstantSymbol true

true ? d

Conjunctive Normal Form (CNF)

Literal

ConstantSymbol

IntegrityConstraint

context IntegrityConstraint inv IC

Literal-gtforAll(oclIsKindOf(NegativeLiteral))

?a ? ?b ? ?c

DefiniteClause

context DefiniteClause inv DC Literal.oclIsKindOf

(ConstantSymbol)-gtsize() 1

?a ? ?b ? ?c ? d

Fact

context Fact inv Fact Literal-gtforAll(oclIsKindOf

(ConstantSymbol))

d

FCPL semantics cognitive interpretation

Syntax

?(a ? (b ? ((?c ? d) ? a) ? b))

Arg

Functor

FCPLFormula

ConstantSymbol

FCPLConnective

1..2

fm1(pitIn12 ? ? pitIn11) agent knows there is a

pit in coordinates (1,2) and no pit in

coordinates (1,1) fm1(pitIn12 ? ? pitIn11) John

is the Kind of England and John is not the King

of France

csm1(pitIn12) agent knows there is a pit in

coordinates (1,2) csm2(pitIn12) John is the

King of England

FCLPCognitiveInterpretation

Semantics

FCPL semantics Herbrand interpretation

Syntax

?(a ? (b ? ((?c ? d) ? a) ? b))

cv1(pitIn12) true, cv1(pitIn11) true,

... cv2(pitIn12) true, cv2(pitIn11) false,

...

Arg

Functor

FCPLFormula

ConstantSymbol

FCPLConnective

1..2

FCLPHerbrandInterpretation

fv1(pitIn12 ? ? pitIn11) true, fv1(pitIn12 ?

pitIn11) true, ... fv2(pitIn12 ? ? pitIn11)

true, fv2(pitIn12 ? pitIn11) false, ...

Semantics

FCPL semantics

Syntax

?(a ? (b ? ((?c ? d) ? a) ? b))

Arg

Functor

FCPLFormula

ConstantSymbol

FCPLConnective

1..2

ConstantValuation

FormulaValuation

FormulaMapping

FCLPHerbrandInterpretation

FCLPCognitiveInterpretation

FCLPHerbrandModel

ConstantMapping

CompoundDomainProperty

AtomicDomainProperty

Semantics

Entailment and models

- Entailment
- f f iff ?Hi, Hi(f) true ? Hi(f) true
- Logical equivalence ?
- f ? f iff f f and f f
- Herbrand model
- An Herbrand interpretation Hi is a (Herbrand)

model of formula f iffits truth value

corresponds to the application of the truth-table

definition of the FCPL connectives to the truth

value in Hi of the constant symbols that compose

f - f valid (or tautology) iff true in all Hi(f), ex,

a ? ?a - f satisfiable iff true in at least one Hi(f)
- f unsatisfiable (or contradiction) iff false in

all Hi(f), ex, a ? ?a

Logic-Based Agent

Given B as axiom, formula f is a theorem of L? B

L f ? B ? f is valid in L? (Boolean CSP search

proof) B ? ?f is unsatisfiable in L? (Refutation

proof)

Environment

Sensors

Ask

Knowledge Base BDomain Model in Logic L

Inference Engine Theorem Prover for Logic L

Tell

Retract

Actuators

- Strenghts
- Reuse results and insights about correct

reasoning that matured over 23 centuries - Semantics (meaning) of a knowledge base can be

represented formally as syntax, a key step

towards automating reasoning

Truth-table based model checking

- To answer Ask(?)
- Enumerate all His from domain proposition

alphabet - Use truth-table to compute Mh(KB) and Mh(?)
- If Mh(KB) ? Mh(?), then answer yes, else answer

no - Example
- KB ?pit11 ? ?breeze11 ? ?pit12 ? breeze12
- ?1 ?pit21
- ?2 ?pit22

FCLP inference rules

- Bi-directional (logical equivalences)
- R1 f ? g ? g ? f
- R2 f ? g ? g ? f
- R3 (f ? g) ? h ? f ? (g ? h)
- R4 (f ? g) ? h ? f ? (g ? h)
- R5 ??f ? f
- R6 f ? g ? ?g ? ?f
- R7 f ? g ? ?f ? g
- R8 f ? g ? (f ? g) ? (g ? f)
- R9 ?(f ? g) ? ?f ? ?g
- R10 ?(f ? g) ? ?f ? ?g
- R11 f ? (g ? h) ? (f ? g) ? (f ? h)
- R12 f ? (g ? h) ? (f ? g) ? (f ? h)
- R13 f ? f ? f factoring

- Directed (logical entailments)
- R14 f ? g, f g modus ponens
- R15 f ? g, ?g ?f modus tollens
- R16 f ? g f and-elimination
- R17 l1 ? ... ? li ? ... lk, m1 ? ... ?

mj-1 ? ?li ? mj-1... mk l1 ? ... ? li-1 ?

li-1... lk ? m1 ? ... ? mj-1 ? mj-1... mk - resolution

Multiple inference rule application

- Idea
- KB f ?
- KB0 KB
- Apply inference rule KBi g
- Update KBi1 KBi ? g
- Iterate until f ? KBk or until f ? KBn and KBn1

KBn - Transforms proving KB f into search problem
- At each step
- Which inference rule to apply?
- To which sub-formula of f?

- Example proof
- KB0 ?P1,1 ? (B1,1 ? P1,2 ? P2,1) ?

(B2,1 ? P1,1 ? P2,2 ? P3,1) ? ?B1,1 ?

B2,1 - Query ?(P1,2 ? P2,1)
- Cognitive interpretation
- BX,Y agent felt breeze in coordinate (X,Y)
- PX,Y agent knows there is a pit in coordinate

(X,Y) - Apply R8 to B1,1 ? P1,2 ? P2,1 KB1 KB0 ? (B1,1

? (P1,2 ? P2,1)) ? ((P1,2 ? P2,1) ?

B1,1) - Apply R6 to last sub-formula KB2 KB1 ? (?B1,1

? ?(P1,2 ? P2,1)) - Apply R14 to ?B1,1 and last sub-formula KB3

KB2 ? ?(P1,2 ? P2,1)

Resolution and factoring

- Repeated application of only two inference rules

- resolution and factoring
- More efficient than using multiple inference

rules - search space with far smaller branching factor
- Refutation proof
- Derive false from KB ? ?Query
- Requires both in normal form (conjunctive or

implicative) - Example proof in conjunctive normal form

Resolution strategies

- Search heuristics for resolution-based theorem

proving - Two heuristic classes
- Choice of clause pair to resolve inside current

KB - Choice of literals to resolve inside chosen

clause pair - Unit preference
- Prefer pairs with one unit clause (i.e.,

literals) - Rationale generates smaller clauses, eliminates

much literal choice in pair - Unit resolution turn preference into

requirement - Set of support
- Define small subset of initial clauses as

initial set of support - At each step
- Only consider clause pairs with one member from

current set of support - Add step result to set of support
- Efficiency depend on cleverness of initial set

of support - Common domain-independent initial set of

support negated query - Beyond efficiency, results in easier to

understand, goal-directed proofs - Linear resolution
- At each step only consider pairs (f,g) where f

is either - (a) in KB0, or

FCPL theorem proving as boolean CSP exhaustive

global backtracking search

- Put f KB ? ?Query in conjunctive normal form
- Try to prove it unsatisfiable
- Consider each literal in f as a boolean variable
- Consider each clause in f as a constraint on

these variables - Solve the underlying boolean CSP problem by

using - Exhaustive global backtracking search
- of all complete variable assignments
- showing none satisfies all constraint in f
- Initial state empty assignment of pre-ordered

variables - Search operator
- Tentative assignment of next yet unassigned

variable Li (ith literal in f) - Apply truth table definitions to propagate

constraints in which Li appears (clauses of f

involving L) - If propagation violates one constraint,

backtrack on Li - If propagation satisfies all constraints
- iterate on Li1
- if Li was last literal in f, fail, KB ? ?Query

satisfiable, and thus KB ? Query

FCPL theorem proving as boolean CSP backtracking

search example

- Variables B1,1 , P1,2, P2,1
- Constraints ?B1,1 , ?P1,2 ? B1,1 , ?P2,1 ?

B1,1, ?B1,1 ? P1,2 ? P2,1 , P1,2

V ?,?,? C ?,?,?,?,?

DPLL algorithm

- General purpose CSP backtracking search very

inefficient for proving large CFPL theorems - Davis, Putnam, Logemann Loveland algorithm

(DPPL) - Specialization of CSP backtracking search
- Exploiting specificity of CFPL theorem proving

recast as CSP search - To apply search completeness preserving

heuristics - Concepts
- Pure symbol S yet unassigned variable positive

in all clauses or negated in all clauses - Unit clause C clause with all but one literal

already assigned to false - Heuristics
- Pure symbol heuristic assign pure symbols first
- Unit propagation
- Assign unit clause literals first
- Recursively generate new ones
- Early termination heuristic
- After assigning Li true, propagate Cj true

?Cj Li ? Cj (avoiding truth-table look-ups) - Prune sub-tree below any node where ?Cj Cj

false - Clause learning

Satisfiability of formula as boolean CSP

heuristic local stochastic search

- DPLL is not restricted to proving entailment by

proving unsatisfiability - It can also prove satisfiability of a FCPL

formula - Many problems in computer science and AI can be

recast as a satisfiability problem - Heuristic local stochastic boolean CSP search

more space scalable than DPLL for satisfiability - However since it is not exhaustive search, it

cannot prove unsatisfiability (and thus

entailment), only strongly suspect it - WalkSAT
- Initial state random assignment of pre-ordered

variables - Search operator
- Pick a yet unsatisfied clause and one literal in

it - Flip the literal assignment
- At each step, randomly chose between to picking

strategies - Pick literal which flip results in steepest

decrease in number of yet unsatisfied clauses - Random pick

Direct x indirect use of search for agent

reasoning

Horn CPL reasoning

- Practical limitations of FCPL reasoning
- For experts in most application domain

(medicine, law, business, design,

troubleshooting) - Non-intuitiveness of FCPL formulas for knowledge

acquisition - Non-intuitiveness of proofs generated by FCPL

algorithms for knowledge validation - Theoretical limitation of FCPL reasoning
- exponential in the size of the KB
- Syntactic limitation to Horn clauses overcome

both limitations - KB becomes base of simple rules If p1 and ...

and pn then c, with logical semantics p1 ? ... ?

pn ? c - Two algorithms are available, rule forward

chaining and rule backward chaining, that are - Intuitive
- Sound and complete for HCPL
- Linear in the size of the KB
- For most application domains, loss of

expressiveness can be overcome by addition of new

symbols and clauses - ex, FCPL KB1 p ? q ? c ? d has no logical

equivalent in HCPL in terms of alphabet

p,q,c,d - However KB2 (p ? q ? notd ? c) ? (p ? q ? notc

? d) ? (c ? notc ? false) ?

(d ? notd ? false) is an HCPL formula

logically equivalent to KB1

Propositional forward chaining

- Repeated application of modus ponens until

reaching a fixed point - At each step i
- Fire all rules (i.e., Horn clauses with at least

one positive and one negative literal) with all

premises already in KBi - Add their respective conclusions to KBi1
- Fixed point k reached when KBk KBk-1
- KBk f KB0 f, i.e., all logical

conclusions of KB0 - If f ? KBk, then KB0 f, otherwise, KB0 ? f
- Naturally data-driven reasoning
- Guided by fact (axioms) in KB0
- Allows intuitive, direct implementation of

reactive agents - Generally inefficient for
- Inefficient for specific entailment query
- Cumbersome for deliberative agent

implementations - Builds and-or proof graph bottom-up

Propositional forward chaining example

Propositional forward chaining example

Propositional forward chaining example

Propositional forward chaining example

Propositional forward chaining example

Propositional forward chaining example

Propositional forward chaining example

Propositional backward chaining

- Repeated application of resolution using
- Unit input resolution strategy with negated query

as initial set of support - At each step i
- Search KB0 for clause of the form p1 ?...? pn ?

g to resolve with clause g popped from the goal

stack - If there are several ones, pick one, push p1

?...? pn on goal stack, and push other ones

alternative stack to consider upon backtracking - If there are none, backtrack (i.e., pop

alternative stack) - Terminates
- Successfully when goal stack is empty
- As failure when goal stack is non empty but

alternative stack is - Naturally goal-driven reasoning
- Guided by goal (theorem to prove)
- Allows intuitive, direct implementation of

deliberative agents - Generally
- Inefficient for deriving all logical conclusions

from KB - Cumbersome implementation of reactive agents
- Builds and-or proof graph top-down

Propositional backward chaining example

Goal Stack Q

Alternative Stack ?

Propositional backward chaining example

Goal Stack P

Alternative Stack ?

Propositional backward chaining example

Goal Stack L M

Alternative Stack ?

Propositional backward chaining example

Goal Stack A P M

Alternative Stack A B

Propositional backward chaining example

Goal Stack P M

Alternative Stack A B

Propositional backward chaining example

Goal Stack A B M

Alternative Stack ?

Propositional backward chaining example

Goal Stack M

Alternative Stack ?

Propositional backward chaining example

Goal Stack B L

Alternative Stack ?

Propositional backward chaining example

Goal Stack ?

Alternative Stack ?

Propositional backward chaining example

Goal Stack ?

Alternative Stack ?

Propositional backward chaining example

Goal Stack ?

Alternative Stack ?

Limitations of propositional logic

- Ontological
- Cannot represent knowledge intentionally
- No concise representation of generic relations

(generic in terms of categories, space, time,

etc.) - ex, no way to concisely formalize the Wumpus

world ruleat any step during the exploration,

the agent perceiving a stench makes him knows

that there is a Wumpus in a location adjacent to

his - Propositional logic
- Requires conjunction of 100,000 equivalences to

represent this rule for an exploration of at most

1000 steps of a cavern size 10x10 - (stench1_1_1 ? wumpus1_1_2 ? wumpus1_2_1) ?

... ... ? (stench1000_1_1 ? wumpus100_1_2 ?

wumpus1000_2_1) ? ...... ? (stench1_10_10 ?

wumpus1_9_10 ? wumpus1_10_9) ? ... ... ?

(stench1000_10_10 ? wumpus100_9_10 ?

wumpus1000_9_10) - Epistemological
- Agent always completely confident of its

positive or negative beliefs - No explicit representation of ignorance (missing

knowledge) - Only way to represent uncertainty is disjunction
- Once held, agent belief cannot be questioned by

new evidence (ex, from sensors)

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