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Reasoning with Classical Propositional Logic

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Implicative Normal Form CPL (INFCPL) Horn CPL (HCPL) Semantics. Cognitive and Herbrand interpretations, ... f valid (or tautology) iff true in all Hi(f), ex, a ... – PowerPoint PPT presentation

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Title: Reasoning with Classical Propositional Logic


1
Reasoning with Classical Propositional Logic
  • Jacques Robin

2
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

3
Full Classical Propositional Logic (FCPL) syntax
Syntax
?(a ? (b ? ((?c ? d) ? a) ? b))
FCPLFormula
4
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
5
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
6
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
7
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
8
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
9
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

10
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

11
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

12
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

13
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)

14
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

15
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

16
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

17
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 ?,?,?,?,?
18
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

19
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

20
Direct x indirect use of search for agent
reasoning
21
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

22
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

23
Propositional forward chaining example
24
Propositional forward chaining example
25
Propositional forward chaining example
26
Propositional forward chaining example
27
Propositional forward chaining example
28
Propositional forward chaining example
29
Propositional forward chaining example
30
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

31
Propositional backward chaining example
Goal Stack Q
Alternative Stack ?
32
Propositional backward chaining example
Goal Stack P
Alternative Stack ?
33
Propositional backward chaining example
Goal Stack L M
Alternative Stack ?
34
Propositional backward chaining example
Goal Stack A P M
Alternative Stack A B
35
Propositional backward chaining example
Goal Stack P M
Alternative Stack A B
36
Propositional backward chaining example
Goal Stack A B M
Alternative Stack ?
37
Propositional backward chaining example
Goal Stack M
Alternative Stack ?
38
Propositional backward chaining example
Goal Stack B L
Alternative Stack ?
39
Propositional backward chaining example
Goal Stack ?
Alternative Stack ?
40
Propositional backward chaining example
Goal Stack ?
Alternative Stack ?
41
Propositional backward chaining example
Goal Stack ?
Alternative Stack ?
42
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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