First-Order Classical Deduction

1
First-Order Classical Deduction

• Jacques Robin

2
Outline

• 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

3
Full Classical First-Order Logic (FCFOL) syntax
Syntax
Functor
FCLConnective
FCFOLFormula
FCFOLAtomicFormula
FCFOLTerm
Functor

PredicateSymbol
FCFOLFunctionalTerm
FCFOLNonFunctionalTerm
Functor
Arg
1..
FunctionSymbol
ConstantSymbol
FOLVariable
?X,Y (p(f(X),Y) ? ?q(g(a,b))) ? (?U,V ?Z ((X a)
? r(Z)) ? (U h(V,Z))))
4
FCFOL Normal Forms
Implicative Normal Form (INF)
Premisse


Conclusion
FCFOLAtomicFormula
FCFOLTerm
Functor
FCFOLFunctionalTerm

PredicateSymbol
FCFOLNonFunctionalTerm

Functor
FunctionSymbol
ConstantSymbol
FOLVariable


Literal
Conjunctive Normal Form (CNF)
5
Horn CFOL (HCFOL)
Implicative Normal Form (INF)
Premisse


Conclusion
IntegrityConstraint
context IntegrityConstraint inv IC
Conclusion.ConstantSymbol false
DefiniteClause
context DefiniteClause inv DC Conclusion.Constant
Symbol ? false
Fact
context Fact inv Fact Premisse -gt size() 1 and
Premisse -gt ConstantSymbol true
FCFOLAtomicFormula
Conjunctive Normal Form (CNF)


Literal
IntegrityConstraint
context IntegrityConstraint inv IC
Literal-gtforAll(oclIsKindOf(NegativeLiteral))
DefiniteClause
context DefiniteClause inv DC Literal.oclIsKindOf
(ConstantSymbol)-gtsize() 1
Fact
context Fact inv Fact Literal-gtforAll(oclIsKindOf
(ConstantSymbol))
6
FCFOL semantics cognitive interpretations
Syntax
FCFOLFormula
FCFOLAtomicFormula
FCFOLTerm
FCFOLNonGroundTerm
FCFOLFunctionalTerm
FCFOLNonFunctionalTerm
ConstantSymbol
FOLVariable
PredicateSymbol
FunctionSymbol
FCFOLGroundTerm
Semantics
7
FCFOL semantics cognitive interpretations
Syntax
FCFOLFormula
FCFOLAtomicFormula
FCFOLTerm
FCFOLNonGroundTerm
FCFOLFunctionalTerm
FCFOLNonFunctionalTerm
PredicateSymbol
FunctionSymbol
ConstantSymbol
FOLVariable
FCFOLGroundTerm

EntitySet
Entity

RelationName
SimpleRelation
ComplexRelation
EntityName



EntityPropertyName
SimpleEntityProperty
ComplexEntityProperty
Semantics
8
FCFOL semantics cognitive interpretations
FCFOLFormula
semantics
FCFOLCognitiveInterpretation
TruthMapping
FormulaMapping
AtomMapping
NounGroundTermMapping
GroundTermMapping
PredicateMapping
FunctionMapping
ConstantMapping
9
FCFOL semantics Herbrand 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),b
  roOf(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),...

10
FCFOL semantics Herbrand interpretations

• 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 model 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

11
FCFOL semantics Herbrand interpretations
Syntax
FCFOLFormula
FCFOLAtomicFormula
FCFOLTerm
FCFOLNonGroundTerm
FCFOLFunctionalTerm
FCFOLNonFunctionalTerm
PredicateSymbol
FunctionSymbol
ConstantSymbol
FOLVariable
FCFOLGroundTerm
Semantics
12
Reasoning 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

13
Propositionalization

• 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)

14
Propositionalization

• 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),jo
  e))) ? (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)))

15
Propositionalization

• 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)

16
First-Order Term Unification
Failure by Occur-Check
17
Lifted 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

18
FCFOL theorem proving by repeated lifted
resolution and factoring example
19
Deduction 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(Xc)) with X/b

20
Characteristics of logics and knowledge
representation languages

• Commitments
• ontological meta-conceptual elements to model
  agents environment
• epistemological meta-conceptual elements to
  model agents 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

21
Characteristics of logics and knowledge
representation languages

• 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?)

22
Characteristics of logics and knowledge
representation languages
Logic / Knowledge Representation Language Ontological commitment Epistemic commitment Name Hypothesis 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 /-
23
Characteristics of logics and knowledge
representation languages
Logic / Knowledge Representation Language Ontological commitment Epistemic commitment Name Hypothesis 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
24
Characteristics 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

25
Some theoretical results about logics and
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