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. FCLUnaryConnective FCLBinaryConnective Arg Arg Connective: enum{} Connective: enum{, , , } 1..* 1..* Arg 1..2 QuantifierExpression Quantifier: enum{,} Full Classical First-Order Logic (FCFOL): syntax Syntax Functor FCLConnective FCFOLFormula FCFOLAtomicFormula FCFOLTerm Functor * FCFOLNonFunctionalTerm FCFOLFunctionalTerm PredicateSymbol Functor Arg 1..* FunctionSymbol FOLVariable ConstantSymbol X,Y (p(f(X),Y)  q(g(a,b)))  (U,V Z ((X = a)  r(Z))  (U = h(V,Z))))

4. Arg Arg INFCFOLFormula CNFCFOLFormula 1..* 1..* Functor =  Functor =  CNFCFOLClause INFCFOLClause INFCLPLHS NegativeLiteral INFCLPRHS Functor =  Functor =  Functor =  Functor =  Functor =  FCFOL Normal Forms Implicative Normal Form (INF) Premisse * * Conclusion FCFOLAtomicFormula FCFOLTerm Functor FCFOLFunctionalTerm * FCFOLNonFunctionalTerm PredicateSymbol * Functor FunctionSymbol FOLVariable ConstantSymbol * * Literal Conjunctive Normal Form (CNF)

5. CNFCFOLFormula INFCFOLFormula Functor =  Functor =  INFCFOLClause CNFCFOLClause INFCFOLLHS NegativeLiteral Functor =  Functor =  Functor =  Functor =  Horn CFOL (HCFOL) Implicative Normal Form (INF) * Premisse * Conclusion IntegrityConstraint context IntegrityConstraint inv IC: Conclusion.ConstantSymbol = false DefiniteClause context DefiniteClause inv DC: Conclusion.ConstantSymbol  false Fact context Fact inv Fact: Premisse -> size() = 1 and Premisse -> ConstantSymbol = true FCFOLAtomicFormula Conjunctive Normal Form (CNF) * * Literal IntegrityConstraint context IntegrityConstraint inv IC: Literal->forAll(oclIsKindOf(NegativeLiteral)) context DefiniteClause inv DC: Literal.oclIsKindOf(ConstantSymbol)->size() = 1 DefiniteClause context Fact inv Fact: Literal->forAll(oclIsKindOf(ConstantSymbol)) Fact

6. TruthValue Arg Arg Value: enum{true,false} 1..* 1..* Arg 1..2 * EntitySet Entity * RelationName ComplexRelation SimpleRelation EntityName * * * ComplexEntityProperty SimpleEntityProperty EntityPropertyName PredicateMapping ConstantMapping FunctionMapping FCFOL semantics: cognitive interpretations Syntax FCFOLFormula FCFOLAtomicFormula FCFOLTerm FCFOLNonFunctionalTerm FCFOLFunctionalTerm FCFOLNonGroundTerm FOLVariable ConstantSymbol PredicateSymbol FunctionSymbol FCFOLGroundTerm Semantics

7. TruthValue Arg Arg Value: enum{true,false} 1..* 1..* Arg 1..2 FormulaMapping AtomMapping NounGroundTermMapping GroundTermMapping TruthMapping FCFOL semantics: cognitive interpretations Syntax FCFOLFormula FCFOLAtomicFormula FCFOLTerm FCFOLNonFunctionalTerm FCFOLFunctionalTerm FCFOLNonGroundTerm PredicateSymbol FOLVariable ConstantSymbol FunctionSymbol FCFOLGroundTerm * EntitySet Entity * RelationName ComplexRelation SimpleRelation EntityName * * * ComplexEntityProperty SimpleEntityProperty EntityPropertyName 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),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)},...}

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. TruthValue Arg Arg Value: enum{true,false} 1..* 1..* Arg 1..2 Herbrandsemantics 1..* HerbrandBase HerbrandUniverse HerbrandInterpretation AtomValuation HerbrandModel FCFOL semantics: Herbrand interpretations Syntax FCFOLFormula FCFOLAtomicFormula FCFOLTerm FCFOLNonFunctionalTerm FCFOLFunctionalTerm FCFOLNonGroundTerm FOLVariable ConstantSymbol PredicateSymbol FunctionSymbol 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),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)))

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. X/b X/b p p p p p p p p p fail X/a Y/a a b a a a a a a X X X X X X Y X b b X/f(c,Z) X/f(c,d) Y/a Y/a Z/d p p p p X/p(a,X) X X X/p(a,X) p fail Y Y f f a a f f a p Guarantees termination c c Z Z c c Z d c p 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(X+c)) with {X/b}

20. Characteristics of logics and knowledge 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

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

23. Characteristics of logics and knowledge representation languages

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