Reasoning with Classical Propositional Logic

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Reasoning with Classical Propositional Logic

• Jacques Robin

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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

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Full Classical Propositional Logic (FCPL) syntax
Syntax
?(a ? (b ? ((?c ? d) ? a) ? b))
FCPLFormula
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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
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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
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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
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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
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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
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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

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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

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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

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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

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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)

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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

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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

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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

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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 ?,?,?,?,?
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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

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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

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Direct x indirect use of search for agent
reasoning
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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

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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

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Propositional forward chaining example
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Propositional forward chaining example
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Propositional forward chaining example
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Propositional forward chaining example
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Propositional forward chaining example
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Propositional forward chaining example
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Propositional forward chaining example
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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

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Propositional backward chaining example
Goal Stack Q
Alternative Stack ?
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Propositional backward chaining example
Goal Stack P
Alternative Stack ?
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Propositional backward chaining example
Goal Stack L M
Alternative Stack ?
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Propositional backward chaining example
Goal Stack A P M
Alternative Stack A B
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Propositional backward chaining example
Goal Stack P M
Alternative Stack A B
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Propositional backward chaining example
Goal Stack A B M
Alternative Stack ?
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Propositional backward chaining example
Goal Stack M
Alternative Stack ?
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Propositional backward chaining example
Goal Stack B L
Alternative Stack ?
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Propositional backward chaining example
Goal Stack ?
Alternative Stack ?
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Propositional backward chaining example
Goal Stack ?
Alternative Stack ?
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Propositional backward chaining example
Goal Stack ?
Alternative Stack ?
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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)