Ontologies Reasoning Components Agents Simulations First-Order Classical Deduction Jacques Robin.

OntologiesReasoningComponentsAgentsSimulations

First-Order Classical DeductionFirst-Order Classical Deduction

Jacques Robin

OutlineOutline

Classical First-Order Logic (CFOL) Syntax

Full CFOL Implicative Normal Form CFOL (INFCFOL) Horn CFOL (HCFOL)

Semantics First-order interpretations and models

Reasoning Lifting propositional reasoning to first-order reasoning INFCFOL reasoning:

First-order resolution

An ontology of logics and engines Properties of logics

Commitments, complexity Properties of inference engines

Soundness, completeness, complexity

Full Classical First-Order Logic Full Classical First-Order Logic (FCFOL): syntax(FCFOL): syntax

Syntax

FCLUnaryConnective

Connective: enum{}

FCLBinaryConnective

Connective: enum{, , , }

FCLConnective FCFOLFormulaFunctor

QuantifierExpression

Quantifier: enum{,}

*

Arg1..2

FCFOLAtomicFormula

PredicateSymbol

FCFOLTerm

Arg1..*

FCFOLFunctionalTermFCFOLNonFunctionalTerm

Arg1..*

FunctionSymbolConstantSymbolFOLVariable

Arg1..*

X,Y (p(f(X),Y) q(g(a,b))) (U,V Z ((X = a) r(Z)) (U = h(V,Z))))

Functor

Functor

FCFOL Normal FormsFCFOL Normal Forms

FCFOLAtomicFormula

PredicateSymbol

FCFOLTerm

Arg1..*

FCFOLFunctionalTerm

FCFOLNonFunctionalTerm

Arg1..*


Functor

Functor

Conjunctive Normal Form (CNF)

CNFCFOLFormula

Functor =

CNFCFOLClause

Functor =

NegativeLiteral

Functor =

Literal* *

INFCFOLFormula

Functor =

INFCFOLClause

Functor =

INFCLPLHS

Functor =

INFCLPRHS

Functor =

Premisse

Conclusion

**

*

Implicative Normal Form (INF)

*

Horn CFOL (HCFOL)Horn CFOL (HCFOL)

INFCFOLFormula

Functor =

INFCFOLClause

Functor =

INFCFOLLHS

Functor = Premisse

Conclusion

Implicative Normal Form (INF)

Conjunctive Normal Form (CNF)

CNFCFOLFormula

Functor =

CNFCFOLClause

Functor =

NegativeLiteral

Functor =

Literal

*

*

* *

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))

FCFOLAtomicFormula

FCFOLNonGroundTerm

FCFOL semantics: FCFOL semantics: cognitive interpretationscognitive interpretations

Syntax

FCFOLFormulaArg1..2

FCFOLAtomicFormula

PredicateSymbol

FCFOLTerm

Arg1..*


Arg1..*

FunctionSymbolConstantSymbolFOLVariable FCFOLGroundTerm

SimpleEntityProperty

SimpleRelation

*

EntitySet *

ComplexEntityProperty

ComplexRelation *

*

EntityName

ConstantMapping FunctionMapping

EntityPropertyName

RelationName

PredicateMapping

Entity

*

Semantics

TruthValue

Value: enum{true,false}

EntitySet

FCFOLNonGroundTerm


Syntax

FCFOLFormulaArg1..2

FCFOLAtomicFormula

PredicateSymbol

FCFOLTerm

Arg1..*


Arg1..*


EntityPropertyName

RelationName EntityName

SimpleEntityProperty

SimpleRelation

Entity

FCFOLGroundTerm

ComplexEntityProperty

ComplexRelation

*

*

*

*

TruthValue


NounGroundTermMapping GroundTermMappingAtomMapping

*

Semantics

FormulaMapping

TruthMapping


FCFOLFormula

NounGroundTermMapping

AtomMapping

GroundTermMapping

FormulaMapping

TruthMapping

ConstantMapping

PredicateMapping

FunctionMapping

FCFOLCognitiveInterpretation

semantics

FCFOL semantics: FCFOL semantics: Herbrand interpretationsHerbrand interpretations

Herbrand universe Uh of FCFOL formula k: Set of all terms built from constant and function symbols appearing

in k Uh(k) = {t = f(t1,...,tn) | f functions(k), ti constants(k) Uh(k)}

ex: k = {parent(joe,broOf(dan)) parent(broOf(dan),pat) (A,D anc(A,D) (parent(A,D) (P anc(A,P) parent(P,D))))} Uh(k) = {joe,dan,pat,broOf(joe),broOf(dan),broOf(pat), broOf(broOf(joe), broOf(broOf(dan), broOf(broOf(pat), ...}

Herbrand base Bh of FCFOL formula k: Set of all atomic formulas built from predicate symbols appearing in

k and Herbrand universe elements as arguments Bh = {a = p(t1,...,tn) | p predicates(k), ti Uh(k)}

ex: Bh = {parent(joe,joe), parent(joe,dan),..., parent(broOf(pat),broOf(pat)),..., anc(joe,joe), anc(joe,dan),..., anc(broOf(pat),broOf(pat)},...}


Herbrand interpretation Ih of FCFOL formula k: Truth valuation of Herbrand base Ih(k): Bh(k) {true,false} ex: {parent(joe,joe) = false, ...parent(joe,dan) = true, ...

parent(broOf(pat),broOf(pat))= false, ... anc(joe,joe) = true, ..., anc(joe,dan) = true}

Herbrand modelmodel Mh of FCFOL formula k: Interpretation Ih(k) in which k is true

ex, Mh(k) = {parent(joe,broOf(dan)) = true, parent(broOf(dan),pat) = true, anc(joe,brofOf(dan)) = true, anc(joe,pat) = true, all others members of Bh(k) = false }

FCFOLNonGroundTerm


Syntax

FCFOLFormulaArg1..2

FCFOLAtomicFormula

PredicateSymbol

FCFOLTerm

Arg1..*


Arg1..*

FunctionSymbolConstantSymbolFOLVariable FCFOLGroundTerm

Semantics

HerbrandUniverse

HerbrandModel

TruthValue


AtomValuationHerbrandInterpretation

Herbrandsemantics

HerbrandBase

1..*

Reasoning in CFOLReasoning in CFOL

Key difference between CFOL and CPL? Quantified variables which extend expressive power of CPL Ground terms do not extend expressive power of CPL

Alone, they are merely syntactic sugar i.e, clearer for the knowledge engineer but equivalent to constant

symbols for an inference engine ex, anc(joe,broOf(dan)) ancJoeBroOfDan,

loc(agent,step(3),coord(2,2)) locAgent3_2_2

How to reason in CFOL? Reuse CPL reasoning approaches, principles and engines!

Fully (formulas propositionalization) transforms CFOL formulas into CPL formulas as preprocessing step

Partially (inference method generalization) lift CPL reasoning engines with new, variable handling component

(unification) all CPL approaches free of exhaustive truth value enumeration can be

lifted to CFOL

PropositionalizationPropositionalization

Variable substitution function Subst(,k): Given a set of pairs variable/constant, Subst(,k) = formula obtained from k by substituting its variables

with their associated constants in Subst({X/a,Y/b}, X,Y,Z p(X,Y) q(Y,Z)) (Z p(a,b) q(b,Z))

Substitutes CFOL formula k by conjunction of ground formulas ground(k) generated as follows: For each universally quantified variable X in k and each v Uh(k)

Add Subst({X/v},k) to the conjunction For each existentially quantified variable Y in k

Add Subst({Y/s},k) to the conjunction where s is a new Skolem ground term, i.e. s Uh(k)

Skolem term to eliminate existentially quantified variable Y in scope of outer universal quantifier Q must be function of the variables quantified by Q

ex, Y X,Z p(X,Y,Z) becomes X,Z p(X,a,Z))but X,Z Y p(X,Y,Z) becomes X,Z p(X,f(X,Z),Z)


Get prop(k) from ground(k) by turning each ground atomic formula into an equivalent constant symbol through concatenation of its predicate, function and constant symbol

Example: k = parent(joe,broOf(dan)) parent(broOf(dan),pat)

(A,D anc(A,D) (parent(A,D) (P anc(A,P) parent(P,D)))) ground(k) = parent(joe,broOf(dan)) parent(broOf(dan),pat)

(anc(joe,joe) (parent(joe,joe) (anc(joe,s1(joe,joe) parent(s1(joe,joe),joe)))

(anc(joe,broOf(dan)) (parent(joe,broOf(dan)) (anc(joe,s2(joe,broOf(dan))) parent(s2(joe,broOf(dan)),joe))) ... ... (anc(pat,pat) (parent(pat,pat) (anc(pat,sn(pat,pat)) parent(sn(pat,pat),pat))))

prop(k) = parentJoeBroOfDan parentBroOfDanPat (ancJoeJoe (parentJoeJoe (ancJoeS1JoeJoe parentS1JoeJoeJoe))) (ancJoeBroOfDan (parentJoeBroOfDan (ancJoeS2JoeBroOfDan parentS2JoeBroOfDanJoe ... ... (ancPatPat (parentPatPat (ancPatSnPatPat parentSnPatPatPat)))


k |=CFOL k’ iff prop(k) |=CPL prop(k’)

Fixed-depth Herbrand base: Uh(k,d) = {f Uh(k) | depth(f) = d}

Fixed-depth propositionalization: prop(k,d) = {c1 ... cn | ci built only from elements in Uh(k,d)}

Thm de Herbrand: prop(k) |=CPL prop(k’) d, prop(k,d) |=CPL prop(k’,d)

For infinite prop(k) prove prop(k) |=CPL prop(k’) iteratively: try proving prop(k,0) |=CPL prop(k’,0),

then prop(k,1) |=CPL prop(k’,1),

... until prop(k,d) |=CPL prop(k’,d)

First-Order Term UnificationFirst-Order Term Unification

p

a X

p

Y b

p

a X

p

Y f

c Z

X/f(c,Z)

Y/a

p

a f

c Z

p

a b

X/b

Y/a

p

a X

p

X b

fail

X/b

X/a

p

a X

p

Y f

c Z

p

a f

c d

X/f(c,d)

Y/a

Z/d

p

a X

Xfail

X/p(a,X)

Failure by Occur-Check

p

a X

X X/p(a,X) p

a p

c pGuarantees termination

Lifted inference rulesLifted inference rules

Bi-direction CPL rules trivially lifted as valid CFOL rules by substituting CPL formulas inside them by CFOL formulas

Lifted modus ponens: Subst(,p1), ..., Subst(,pn), (p1 ... pn c) |= Subst(,c)

Lifted resolution: l1 ... li ... lk, m1 ... mj ... mk, Subst(,li) = Subst(,mj)

|= Subst(, l1 ... li-1 li-1... lk m1 ... mj-1 mj-1... mk) CFFOL inference methods (theorem provers):

Multiple lifted inference rule application Repeated application of lifted resolution and factoring

CHFOL inference methods (logic programming): First-order forward chaining through lifted modus ponens First-order backward chaining through lifted linear unit resolution

guided by negated query as set of support Common edge over propositionalization: focus on relevant

substitutions

FCFOL theorem proving by repeated FCFOL theorem proving by repeated lifted resolution and factoring: lifted resolution and factoring:

exampleexample

Deduction with equalityDeduction with equality

Axiomatization: Include domain independent sub-formulas defining equality in the KB (X X = X) (X,Y X = Y Y = X) (X,Y,Z (X = Y Y = Z) X = Z)

(X,Y X = Y (f1(X) = f1(Y) ... fn(X) = fn(Y)) (X,Y,U,V (X= Y U = V) f1(X,U) = f1(Y,V) ... fn(X,U) = fn(Y,V)) ...

(X,Y X = Y (p1(X) p1(Y) ... pm(X) pm(Y))

(X,Y,U,V (X= Y U = V) p1(X,U) p1(Y,V) ... pm(X,U) pm(Y,V)) ...

New inference rule (parademodulation): l1 ... lk t = u, m1 ... mn(...,v,...)

|= Subst(unif(t,v), l1 ... lk m1 ... mn(...,y,...))

ex, Extend unification to check for equality

ex, if a = b + c, then p(X,f(a)) unifies with p(b,f(X+c)) with {X/b}

Characteristics of logics and Characteristics of logics and knowledge representation languagesknowledge representation languages

Commitments: ontological: meta-conceptual elements to model agent’s environment epistemological: meta-conceptual elements to model agent’s beliefs

Hypothesis and assumptions: Unique name or equality theory open-world or closed-world

Monotonicity: if KB |= f, then KB g |= f Semantic compositionality:

semantics(a1 c1 a2 c2 ... cn-1 an) = f(semantics(a1), ... ,semantics(an) ex, propositional logic truth tables define functions to compute

semantics of a formula from semantics of its parts Modularity

semantics(ai) independent from its context in larger formulas

ex, semantics(a1) independent of semantics(a2), ... , semantics(an) in contrast to natural language


Expressive power: theoretical (in terms of language and grammar theory) practical: concisely, directly, intuitively, flexibly, etc.

Inference efficiency: theoretical limits practical limits due to availability of implemented inference

engines Acquisition efficiency:

easy to formulate and maintain by human experts possible to learn automatically from data (are machine learning

engines available?)


Logic / Knowledge Representation Language

Ontological commitment Epistemic commitment

NameHypothesis

World Hypothesis

Decidable Modular

Classical Propositional Logic facts true, false unique name open yes yes

Classical First-Order Logic entities, relations and functions among entities

true, false entity name = open semi yes

Classical High-Order Logic entities, relations and functions among entities, relations and functions

true, false entity, relation, function name =

open no yes

Propositional Temporal Logic properties, time points, time intervals

true, false unique name open ? yes

Propositional Modal Logic facts true, false, possible,

necessary

unique name open yes yes

Bayesian Networks facts [0,1] unique name closed yes no

Definite Logic Programs entities, relations and functions among entities

true, false unique name closed semi yes

Event Calculus entities, relations and functions among entities, time points, time intervals,

events

true, false, unknown

unique name closed semi +/-


Logic / Knowledge Representation Language

Ontological commitment Epistemic commitment

NameHypothesis

World Hypothesis

Decidable Modular

UML classes, objects, attributes, methods, associations, generalizations, compositions, aggregations

? ? ? ? yes

OCL classes, objects, ..., invariant constraints, pre and post condition constraints

? ? ? ? yes

Bayesian Logic Programs entities, functions and relations of entities

[0,1] unique closed semi no

CHRD entities, functions and relations of entities

true, false unique closed semi yes

Transaction Frame Logic classes, objects, attributes, methods, generalizations, entities, functions and relations among entities, functions, relations, classes, objects, attributes, methods, generalizations

true, false, unknown

object = closed semi yes

Characteristics of inference enginesCharacteristics of inference engines

Engine inference: f |-E g, if engine E infers g from f

Engine E sound for logic L: f |-E g only if f |=L g

Engine E fully complete for logic L: if f |=L g, then f |-E g

if f |L g, then (f g) |-E false

Engine E refutation-complete for logic L: if f |=L g, then f |-E g

but if f |L g, then either (f g) |-E false or inference never terminates (equivalent to halting problem)

Engine inference complexity: exponential, polynomial, linear, logarithmic in KB size

Some theoretical results about logics Some theoretical results about logics and inference methodsand inference methods

Results about logic: Satisfiability of full classical propositional logic formula is decidable

but exponential Entailment between two full classical first-order logic formulas is

semi-decidable Entailment between two full classical high-order logic formulas is

undecidable Results about inference methods:

Truth-table model checking, multiple inference rule application resolution-factoring application and DPLL are sound and fully complete for full classical propositional logic

WalkSAT sound but fully incomplete for full classical propositional logic

Forward-chaining and backward chaining sound, fully complete and worst-case linear for Horn classical propositional logic

Lifted resolution-factoring sound, refutation complete and worst case exponential for full classical first-order logic

Ontologies Reasoning Components Agents Simulations First-Order Classical Deduction Jacques Robin.

Documents

Transcript of Ontologies Reasoning Components Agents Simulations First-Order Classical Deduction Jacques Robin.