Reasoning with Classical Propositional Logic

OntologiesReasoningComponentsAgentsSimulations

Reasoning with Classical Reasoning with Classical Propositional LogicPropositional Logic

Jacques Robin

OutlineOutline

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 Full Classical Propositional Logic (FCPL): syntax(FCPL): syntax

FCPLFormula

Syntax

FCPLUnaryConnective

Connective: enum{}

FCPLBinaryConnective

Connective: enum{, , , }

FCPLConnectiveFunctor

ConstantSymbolArg1..2

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

CPL Normal FormsCPL Normal Forms

ConstantSymbol

INFCPLFormula

Functor =

INFCPLClause

Functor =

INFCLPLHS

Functor =

INFCLPRHS

Functor =

Premisse

Conclusion

Implicative Normal Form (INF)

Conjunctive Normal Form (CNF)

CNFCPLFormula

Functor =

CNFCPLClause

Functor =

NegativeLiteral

Functor =

Literal

ConstantSymbol

Semantic equivalence: a b c d (a b) c d a b c d

**

*

* *

Horn CLPHorn CLP

ConstantSymbolINFCPLFormula

Functor =

INFCPLClause

Functor =

INFCLPLHS

Functor =

INFCLPRHS

Functor =

Premisse

Conclusion

Implicative Normal Form (INF)

Conjunctive Normal Form (CNF)

CNFCPLFormula

Functor =

CNFCPLClause

Functor =

NegativeLiteral

Functor =

Literal ConstantSymbol

*

*

* *

DefiniteClause

IntegrityConstraint

Fact

DefiniteClause

IntegrityConstraint

Fact

context IntegrityConstraint inv IC: Conclusion.ConstantSymbol = false

context DefiniteClause inv DC: Conclusion.ConstantSymbol false

context Fact inv Fact: Premisse -> size() = 1 and Premisse -> ConstantSymbol = true

context IntegrityConstraint inv IC: Literal->forAll(oclIsKindOf(NegativeLiteral))

context DefiniteClause inv DC: Literal.oclIsKindOf(ConstantSymbol)->size() = 1

context Fact inv Fact: Literal->forAll(oclIsKindOf(ConstantSymbol))

a b c false

a b c d

true d

a b c

a b c d

d

FCPL semantics: cognitive FCPL semantics: cognitive interpretationinterpretation

FCPLFormula ConstantSymbolFCPLConnective

FCPLUnaryConnective

Connective: enum{}



TruthValue

Value: enum{true,false}

FCLPCognitiveInterpretation

Syntax

Semantics

ArgFunctor

CompoundDomainProperty

FormulaMapping

AtomicDomainProperty

ConstantMapping

1..2

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

csm1(pitIn12) = agent knows there is a pit in coordinates (1,2)csm2(pitIn12) = John is the King of England

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

FCPL semantics: FCPL semantics: Herbrand interpretationHerbrand interpretation


FCPLUnaryConnective

Connective: enum{}



TruthValue


FCLPHerbrandInterpretation

Syntax

Semantics

ArgFunctor

ConstantValuationFormulaValuation

FCLPHerbrandModel

1..2

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

{cv1(pitIn12) = true, cv1(pitIn11) = true, ...}{cv2(pitIn12) = true, cv2(pitIn11) = false, ...}

{fv1(pitIn12 pitIn11) = true, fv1(pitIn12 pitIn11) = true, ...}{fv2(pitIn12 pitIn11) = true, fv2(pitIn12 pitIn11) = false, ...}

FCPL semanticsFCPL semantics


CompoundDomainProperty

FCPLUnaryConnective

Connective: enum{}



AtomicDomainProperty TruthValue


FCLPCognitiveInterpretation

FCLPHerbrandInterpretation

Syntax

Semantics

ArgFunctor

ConstantValuationFormulaValuation

FCLPHerbrandModel

FormulaMapping

ConstantMapping

1..2

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

Entailment and modelsEntailment 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 iff

its 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

Valid formulas

Satisfiable formulas

Unsatisfiable formulas

Logic-Based AgentLogic-Based Agent

Ask

Tell

Retract

En

viro

nm

en

t

Actuators

KnowledgeBase B:

Domain Modelin Logic L

InferenceEngine:

TheoremProver

for Logic L

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)Sensors

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 checkingTruth-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 rulesFCLP 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

ponensR15: 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 applicationMultiple inference rule application

Idea: KB |= f ? KB0 = KB Apply inference rule: KBi |= g Update KBi+1 = KBi g Iterate until f KBk

or until f KBn and KBn+1 = 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 factoringResolution 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 strategiesResolution 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 (b) an ancestor of g in the proof tree

Input resolution: Specialization of linear resolution excluding (b) case Generates spine-looking proofs trees

FCPL theorem proving as boolean FCPL theorem proving as boolean CSP exhaustive global backtracking CSP exhaustive global backtracking

searchsearch 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 Li+1

if Li was last literal in f, fail, KB Query satisfiable, and thus KB | Query

FCPL theorem proving as boolean FCPL theorem proving as boolean CSP CSP

backtracking search: examplebacktracking 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 = [?,?,?,?,?]

V = [0,?,?]C = [1,?,?,?,?]

V = [1,?,?]C = [0,?,?,?,?]

V = [0,0,?]C = [1,0,?,?,0]

V = [0,1,?]C = [1,0,?,?,1]

V = [1,0,?]C = [0,1,?,?,0]

V = [1,1,?]C = [0,0,?,?,1]

V = [1,1,1]C = [0,0,0,0,1]

V = [1,1,0]C = [0,0,1,0,1]

V = [1,0,1]C = [0,1,0,0,0]

V = [1,0,0]C = [0,1,1,0,0]

V = [0,1,1]C = [1,0,0,1,1]

V = [0,1,0]C = [1,0,0,0,1]

V = [0,0,1]C = [1,0,0,0,0]

V = [0,0,0]C = [1,0,0,0,0]

DPLL algorithmDPLL 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 Satisfiability of formula as boolean CSP heuristic local stochastic searchCSP 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 Direct x indirect use of search for agent reasoninguse of search for agent reasoning

AgentDecisionProblem

Domain SpecificAgent Decision ProblemSearch Model:• State data structure• Successor function• Goal function• Heuristic function

Domain SpecificKnowledge Base Model:• Logic formulas

true df g h c

...

Domain Independent Inference Engine Search Model• State data structure• Successor function• Goal function• Heuristic function

Domain Independe

ntSearch

AlgorithmAgent

Application

Developer

Reasoning

Component

Developer

Horn CPL reasoningHorn 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 chainingPropositional 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 KB i

Add their respective conclusions to KB i+1

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: Propositional forward chaining: exampleexample

Propositional backward chainingPropositional 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: Propositional backward chaining: exampleexample

Goal Stack

Q

AlternativeStack


Goal Stack

P

AlternativeStack


Goal Stack

LM

AlternativeStack


Goal Stack

APM

AlternativeStack

AB


Goal Stack

PM

AlternativeStack

AB


Goal Stack

ABM

AlternativeStack


Goal Stack

M

AlternativeStack


Goal Stack

BL

AlternativeStack


Goal Stack

AlternativeStack

Limitations of propositional logicLimitations 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 rule:

“at 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)

Reasoning with Classical Propositional Logic

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