Logical Inference                                 in RTE                              Kilian Evang                        ...
Logical InferenceOutline                                in RTE                                    Kilian EvangIntroduction...
Logical InferenceRTE in an ideal world                                  in RTE                                            ...
Logical InferenceProblems                                                                       in RTE                    ...
Logical InferenceLogics                                                             in RTE                                ...
Logical InferenceEssential ingredient 1: a class of formal languages                       in RTE                         ...
Logical InferenceOptional ingredient: a semantics                                        in RTE                           ...
Logical InferenceFirst-order models                                               in RTE                                  ...
Logical InferenceEssential ingredient 2: a proof theory                                  in RTE                           ...
Logical InferenceTheorem proving                                                                in RTE                    ...
Logical InferenceTableau                                                             in RTE                               ...
Logical InferenceResolution                                                              in RTE                           ...
Logical InferencePropositional Resolution                                                 in RTE                          ...
Logical InferenceSet Conjunctive Normal Form (set CNF)                                 in RTE                             ...
Logical InferenceConverting into set CNF                              in RTE                                              ...
Logical InferenceStep 1: Converting into NNF                        in RTE                                                ...
Logical InferenceStep 2: From NNF to CNF                                   in RTE                                         ...
Logical InferenceStep 3: From CNF to set CNF                                  in RTE                                      ...
Logical InferenceThe Resolution Rule                                            in RTE                                    ...
Logical InferenceThe Resolution Rule                                                                  in RTE              ...
Logical InferenceExample                                                                  in RTE                          ...
Logical InferenceExample                                                                                  in RTE          ...
Logical InferenceFirst-order resolution                                                      in RTE                       ...
Logical InferenceUnification in a nutshell                                             in RTE                              ...
Logical InferenceResolution with unification                                           in RTE                              ...
Logical InferenceNon-redundant factors                                                in RTE                              ...
Logical InferenceSkolemisation                                                                in RTE                      ...
Logical Inference                                                                     in RTEFormula to prove: ∀y¬∃xlove(x,...
Logical InferenceParamodulation                                                         in RTE                            ...
Logical InferenceThe paramodulation rule                                             in RTE                               ...
Logical InferenceReferences                                                        in RTE                                 ...
Logical InferenceReferences                                                          in RTE                               ...
Logical Inference                                                    in RTE                                               ...
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Logical Inference in RTE

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Logical Inference in RTE

  1. 1. Logical Inference in RTE Kilian Evang Introduction Logics Formal Languages Semantics Proof TheoriesLogical Inference in RTE Theorem Proving Propositional Resolution Set Conjunctive Normal Form Kilian Evang The Resolution Rule Example First-Order Resolution Unification Skolemisation 2009-06-29 Example Paramodulation Back Matter
  2. 2. Logical InferenceOutline in RTE Kilian EvangIntroduction IntroductionLogics Logics Formal Languages Formal Languages Semantics Semantics Proof Theories Theorem Proving Proof Theories Propositional Theorem Proving Resolution Set Conjunctive Normal FormPropositional Resolution The Resolution Rule Example Set Conjunctive Normal Form First-Order The Resolution Rule Resolution Unification Example Skolemisation ExampleFirst-Order Resolution Paramodulation Back Matter Unification Skolemisation Example ParamodulationBack Matter
  3. 3. Logical InferenceRTE in an ideal world in RTE Kilian Evang Introduction KB Logics Formal Languages Semantics Proof Theories Theorem Proving Propositional Resolution Choose: BK T H Set Conjunctive Normal Form The Resolution Rule Example Translate: First-Order Resolution Unification Prove: κ ∧ τ → χ Skolemisation Example Paramodulation Back Matter(Also make sure that κ ∧ τ is satisfiable!)
  4. 4. Logical InferenceProblems in RTE Kilian Evang Introduction Logics ◮ capture the (relevant) subleties of natural language in a Formal Languages Semantics logical language Proof Theories Theorem Proving ◮ encoding a sufficient amount of background knowledge Propositional Resolution (offline) Set Conjunctive Normal Form ◮ choosing the right background knowledge (online) The Resolution Rule Example ◮ too little: entailment is missed very easily First-Order ◮ remedy 1: turn a blind eye on non-entailment when Resolution Unification (minimal) model sizes for T and T+H are very similar Skolemisation [Bos & Markert, 2005] Example Paramodulation ◮ remedy 2: use a shallow approach in parallel (ibid.) Back Matter ◮ too much: proving becomes computationally expensive ◮ remedy: very sophisticated reasoning techniques
  5. 5. Logical InferenceLogics in RTE Kilian Evang Introduction Logics Formal Languages ◮ provide formal languages into which natural language Semantics Proof Theories expressions (and other knowledge) can be translated: Theorem Proving Propositional representation Resolution Set Conjunctive ◮ key advantage over natural language: entailment is Normal Form The Resolution Rule well-defined: inference Example First-Order ◮ tradeoff between expressivity for representation and Resolution Unification tractability for inference Skolemisation Example ◮ many different logics exist Paramodulation Back Matter ◮ but what is a logic?
  6. 6. Logical InferenceEssential ingredient 1: a class of formal languages in RTE Kilian EvangA logical language is a formal language defined by Introduction 1. a vocabulary, i.e. a set of non-logical symbols Logics specific to the concrete application Formal Languages Semantics ◮ example: {(love, 2), (customer, 1), (robber, 1), Proof Theories Theorem Proving (mia, 0), (vincent, 0), (honey-bunny, 0), (give, 3), Propositional (yolanda, 0)} Resolution Set Conjunctive ◮ also depends on the kind of logic used – e.g. standard Normal Form The Resolution Rule description logics do not allow ternary relations Example First-Order 2. elements only specific to the kind of logic used Resolution ◮ “logical” symbols Unification Skolemisation ◮ example (first-order logic with equality): Example Paramodulation variables, ⊤, ⊥, ¬, ∧, ∨, →, ∀, ∃, (, ), ,, = Back Matter ◮ syntactic rules to build formulas, the elements of the language, from logical and non-logical symbols, e.g. ◮ robber(yolanda) ◮ ∀x(robber(x) → love(mia, x)) ◮ mia = vincent
  7. 7. Logical InferenceOptional ingredient: a semantics in RTE Kilian Evang Introduction Logics Formal Languages Semantics ◮ in order to say whether a formula is “true” or “false”, Proof Theories Theorem Proving you need a semantics Propositional Resolution ◮ semantics for logical languages are often defined in Set Conjunctive Normal Form terms of models The Resolution Rule Example ◮ intuitively, a model is a situation First-Order Resolution ◮ intuitively, a formula is satisfied in a model (“true”) iff Unification Skolemisation it makes a correct statement about the situation Example Paramodulation ◮ exact satisfaction definition given in terms of set theory Back Matter
  8. 8. Logical InferenceFirst-order models in RTE Kilian Evang Introduction Logics ◮ a first-order model consists of a domain and an Formal Languages Semantics assignment function Proof Theories Theorem Proving ◮ example domain D = {d1 , d2 , d3 , d4 } Propositional Resolution ◮ example assignment function F : F (mia) = d2 , Set Conjunctive Normal Form The Resolution Rule F (honey-bunny) = d1 , F (yolanda) = d1 , Example F (vincent) = d4 , F (customer) = {d1 , d2 , d4 }, First-Order Resolution F (robber) = {d3 , d5 }, F (love) = {(d3 , d4 )}, Unification Skolemisation F (give) = {(d2 , d1 , d4 )}} Example Paramodulation ◮ ∃x(love(x, vincent)) is satisfied in M Back Matter ◮ love(vincent, mia) is not satisfied in M
  9. 9. Logical InferenceEssential ingredient 2: a proof theory in RTE Kilian Evang Introduction Logics ◮ singles out some formulas and calls them theorems Formal Languages Semantics ◮ consists of Proof Theories Theorem Proving ◮ axioms: formulas considered theorems without proof Propositional Resolution ◮ inference rules: allow to derive new theorems from Set Conjunctive known ones Normal Form The Resolution Rule Example ◮ for the same logic, there often exist many different, First-Order equivalent proof theories Resolution Unification ◮ if the logic has a semantics, a proof theory must be Skolemisation Example specified in such a way that it is sound and complete Paramodulation Back Matter wrt. the semantics, i.e.: a formula is a theorem iff it is true in all models
  10. 10. Logical InferenceTheorem proving in RTE Kilian EvangGiven a formula φ, check whether φ is a theorem. Introduction ◮ Why? Logics Formal Languages ◮ to detect entailment: to check whether κ ∧ τ entails χ, Semantics Proof Theories check whether (κ ∧ τ ) → χ is a theorem Theorem Proving ◮ to detect contradiction: to check whether κ ∧ τ Propositional Resolution contradicts χ, check whether ¬(κ ∧ τ ∧ χ) is a theorem Set Conjunctive Normal Form ◮ How? The Resolution Rule Example ◮ brute force: use a proof theory directly, i.e. generate all First-Order axioms (many!) and apply inference rules until the Resolution Unification formula is deduced. Skolemisation Example ◮ better: find a clever, sound, and complete technique to Paramodulation find the answer by inspecting the formula Back Matter ◮ still, theorem proving is purely syntactic: we may worry about models in defining the technique, but not in applying it ◮ tableau and resolution are such techniques
  11. 11. Logical InferenceTableau in RTE Kilian Evang Introduction Logics ◮ a refutation method: to prove that φ is a theorem, Formal Languages Semantics derive a contradiction from ¬φ Proof Theories Theorem Proving ◮ very intuitive: using a variety of specialized rules, Propositional Resolution decompose the formula step by step until two Set Conjunctive contradictory atomic formulas have been derived Normal Form The Resolution Rule Example ◮ a small example for a propositional tableau: First-Order Resolution √ Unification 1 F (p ∧ ¬p) Skolemisation Example 2 Fp 1, F∧ Paramodulation √ 3 F ¬p 1, F∧ , Back Matter 4 Tp 3, F¬ .
  12. 12. Logical InferenceResolution in RTE Kilian Evang Introduction Logics ◮ a technique at the heart of state-of-the-art theorem Formal Languages Semantics provers such as Prover9 or Vampire Proof Theories Theorem Proving ◮ invented by J. Alan Robinson in 1965 Propositional Resolution ◮ originally formulated for first-order logic, adapted to Set Conjunctive Normal Form The Resolution Rule other logics Example ◮ a refutation method: to prove that φ is a theorem, First-Order Resolution derive a contradiction from ¬φ Unification Skolemisation Example ◮ ¬φ must first be transformed to a normal form Paramodulation Back Matter ◮ resolution then consists of the repeated application of a single rule
  13. 13. Logical InferencePropositional Resolution in RTE Kilian Evang Introduction Logics Formal Languages Semantics Proof Theories ◮ resolution for propositional logic (the quantifier-free Theorem Proving fragment of first-order logic) Propositional Resolution Set Conjunctive ◮ atomic formulas like boxer(butch) or Normal Form The Resolution Rule love(vincent, mia) treated as atoms like p or q Example First-Order ◮ always terminates (propositional logic is decidable) Resolution Unification ◮ the normal form for propositional resolution is called set Skolemisation Example conjunctive normal form (set CNF) Paramodulation Back Matter
  14. 14. Logical InferenceSet Conjunctive Normal Form (set CNF) in RTE Kilian Evang IntroductionEvery formula can be written as a conjunction of Logicsdisjunctions of possibly negated atomic formulas. Formal LanguagesA formula that is not in set CNF: Semantics Proof Theories Theorem Proving Propositional (¬p → q) → (¬r → s) Resolution Set Conjunctive Normal FormThe same formula in set CNF: The Resolution Rule Example First-Order Resolution ((¬p ∨ r ∨ s) ∧ (¬q ∨ r ∨ s)) Unification Skolemisation ExampleIn list notation: Paramodulation Back Matter [[¬p, r , s], [¬q, r , s]]The inner lists (conjuncts, disjunctions) are called clauses.
  15. 15. Logical InferenceConverting into set CNF in RTE Kilian Evang Introduction Logics Formal Languages Semantics Proof Theories Theorem Proving Propositional Resolution 1. convert into negation normal form (NNF) Set Conjunctive Normal Form 2. convert from NNF to CNF The Resolution Rule Example 3. remove duplicates (from CNF to set CNF) First-Order Resolution Unification Skolemisation Example Paramodulation Back Matter
  16. 16. Logical InferenceStep 1: Converting into NNF in RTE Kilian EvangRules Introduction 1. Rewrite ¬(φ ∧ ψ) as ¬φ ∨ ¬ψ Logics 2. Rewrite ¬(φ ∨ ψ) as ¬φ ∧ ¬ψ Formal Languages Semantics Proof Theories 3. Rewrite ¬(φ → ψ) as φ ∧ ¬ψ Theorem Proving Propositional 4. Rewrite φ → ψ as ¬φ ∨ ψ Resolution Set Conjunctive 5. Rewrite ¬¬ψ as ψ Normal Form The Resolution Rule Example First-OrderExample Resolution Unification Skolemisation Example Paramodulation (¬p → q) → (¬r → s) Back Matter 4 ⇔ ¬(¬p → q) ∨ (¬r → s) 3 ⇔ (¬p ∧ ¬q) ∨ (¬r → s) 4 ⇔ (¬p ∧ ¬q) ∨ (¬¬r ∨ s) 5 ⇔ (¬p ∧ ¬q) ∨ (r ∨ s)
  17. 17. Logical InferenceStep 2: From NNF to CNF in RTE Kilian EvangRules Introduction Logics 1. Rewrite θ ∨ (φ ∧ ψ) as (θ ∨ φ) ∧ (θ ∨ ψ) Formal Languages Semantics 2. Rewrite (φ ∧ ψ) ∨ θ as (φ ∨ θ) ∧ (ψ ∨ θ) Proof Theories Theorem Proving 3. Rewrite (φ ∧ ψ) ∧ θ as θ ∧ (φ ∧ ψ) Propositional Resolution 4. Rewrite (φ ∨ ψ) ∨ θ as θ ∨ (φ ∨ ψ) Set Conjunctive Normal Form The Resolution Rule ExampleExample First-Order Resolution Unification Skolemisation Example Paramodulation (¬p ∧ ¬q) ∨ (r ∨ s) Back Matter 2 ⇔ (¬p ∨ (r ∨ s)) ∧ (¬q ∨ (r ∨ s))Set notation: [[¬p, r , s], [¬q, r , s]]No duplicates, already in set CNF.
  18. 18. Logical InferenceStep 3: From CNF to set CNF in RTE Kilian Evang Introduction Logics Formal LanguagesRemove duplicate literals from each clause, e.g.: Semantics Proof Theories Theorem Proving [[p, q, r , ¬s], [p, ¬q, p, ¬r ]] Propositional Resolution ⇔ [[p, q, r , ¬s], [p, ¬q, ¬r ]] Set Conjunctive Normal Form The Resolution Rule ExampleRemove duplicate clauses from the list, e.g. First-Order Resolution Unification [[t, ¬r ], [p, q, ¬r ], [t, ¬r ]] Skolemisation Example Paramodulation ⇔ [[t, ¬r ], [p, q, ¬r ]] Back Matter
  19. 19. Logical InferenceThe Resolution Rule in RTE Kilian Evang Introduction Logics Formal Languages Semantics Proof Theories Theorem ProvingThe key insight Propositional Resolution Set Conjunctive Normal Form (p ∨ r ) ∧ (q ∨ ¬r ) ⇒ (p ∨ q) The Resolution Rule Example First-Orderr and ¬r are called a complementary pair, (p ∨ r ) and Resolution Unification(q ∨ ¬r ) are called complementary clauses. Skolemisation Example Paramodulation Back Matter
  20. 20. Logical InferenceThe Resolution Rule in RTE Kilian EvangFrom two complementary clauses Introduction[p1 , · · · , pn , r , pn+1 , · · · , pm ] and Logics Formal Languages[q1 , · · · , qj , ¬r , qj+1 , · · · , qk ], deduce Semantics Proof Theories[p1 , · · · , pn , pn+1 , · · · , pm , q1 , · · · , qj , qj+1 , · · · , qk ] Theorem Proving Propositional Resolution Set ConjunctiveThe process of resolution Normal Form The Resolution Rule Example 1. apply the resolution rule to some pair of complementary First-Order Resolution clauses Unification Skolemisation 2. remove duplicates from the result Example Paramodulation 3. add the result to the set of clauses Back Matter 4. start over, unless ◮ the empty clause has been derived (success) ◮ no unprocessed complementary pair remains (failure)
  21. 21. Logical InferenceExample in RTE Kilian EvangSuppose we want to prove the following formula: Introduction (p ∨ (q ∧ r )) → ((p ∨ q) ∧ (p ∨ r )) Logics Formal Languages Semantics Proof TheoriesThe first step is to transform its negation into set CNF: Theorem Proving Propositional Resolution ¬((p ∨ (q ∧ r )) → ((p ∨ q) ∧ (p ∨ r ))) Set Conjunctive Normal Form⇔ (p ∨ (q ∧ r )) ∧ ¬((p ∨ q) ∧ (p ∨ r )) The Resolution Rule Example⇔ (p ∨ (q ∧ r )) ∧ (¬(p ∨ q) ∨ ¬(p ∨ r )) First-Order Resolution⇔ (p ∨ (q ∧ r )) ∧ ((¬p ∧ ¬q) ∨ (¬p ∧ ¬r )) Unification Skolemisation Example⇔ ((p ∨ q) ∧ (p ∨ r )) ∧ (((¬p ∧ ¬q) ∨ ¬p) ∧ ((¬p ∧ ¬q) ∨ ¬r )) Paramodulation⇔ ··· Back Matter⇔ ((p ∨ q) ∧ (p ∨ r ) ∧ (¬p ∨ ¬p) ∧ (¬p ∨ ¬r ) ∧ (¬q ∨ ¬p) ∧ (¬q ∨ ¬r ))CNF: [[p, q], [p, r ], [¬p, ¬p], [¬p, ¬r ], [¬q, ¬p], [¬q, ¬r ]]Set CNF: [[p, q], [p, r ], [¬p], [¬p, ¬r ], [¬q, ¬p], [¬q, ¬r ]]
  22. 22. Logical InferenceExample in RTE Kilian Evang Introduction Logics Formal LanguagesThen we apply the resolution rule until we derive the empty Semantics Proof Theoriesclause or no unprocessed complementary pair remains: Theorem Proving Propositional Resolution [[p, q], [p, r ], [¬p], [¬p, ¬r ], [¬q, ¬p], [¬q, ¬r ]] Set Conjunctive Normal Form⇔ [[p, q], [p, r ], [¬p], [¬p, ¬r ], [¬q, ¬p], [¬q, ¬r ], [q]] The Resolution Rule Example⇔ [[p, q], [p, r ], [¬p], [¬p, ¬r ], [¬q, ¬p], [¬q, ¬r ], [q], [r ]] First-Order Resolution⇔ [[p, q], [p, r ], [¬p], [¬p, ¬r ], [¬q, ¬p], [¬q, ¬r ], [q], [r ], [¬r ]] Unification Skolemisation Example⇔ [[p, q], [p, r ], [¬p], [¬p, ¬r ], [¬q, ¬p], [¬q, ¬r ], [q], [r ], [¬q], []] Paramodulation Back MatterSuccess!
  23. 23. Logical InferenceFirst-order resolution in RTE Kilian Evang Introduction ◮ theoremhood in first-order logic is only semi-decidable: Logics the algorithm will eventually halt if the formula is a Formal Languages theorem, but may never halt if the formula is not a Semantics Proof Theories theorem Theorem Proving Propositional ◮ still useful Resolution Set Conjunctive ◮ new preprocessing phase Normal Form The Resolution Rule Example 1. transform into NNF, with two additional rules: First-Order rewrite ¬∀xφ as ∃x¬φ, ¬∃xφ as ∀x¬φ Resolution 2. discard existential quantification, replace variables by a Unification Skolemisation unique placeholder (skolemisation) Example Paramodulation 3. discard universal quantification, treat variables as Back Matter implicitly universally quantified (rename if necessary) 4. put the result into set CNF ◮ new resolution phase ◮ resolution with unification
  24. 24. Logical InferenceUnification in a nutshell in RTE Kilian Evang Introduction ◮ making two terms identical by replacing variables, Logics using the most general substitution possible Formal Languages Semantics ◮ robber(vincent) and customer(x) Proof Theories Theorem Proving are not unifiable: different relation symbols Propositional Resolution ◮ robber(vincent) and robber(mia) Set Conjunctive Normal Form are not unifiable: different constant arguments The Resolution Rule Example ◮ love(x, y) and love(mia, z) are unifiable. Which First-Order substitution? Resolution Unification ◮ [x/mia, y/vincent, z/vincent]? Skolemisation Example Bad idea, too specific. Paramodulation ◮ [x/mia, y/z] is the most general unifier (mgu). Back Matter Result: love(mia, z) ◮ also: love(father(x), mia) and love(x, mia) are not unifiable: would create a cycle (“occurs check” needed)
  25. 25. Logical InferenceResolution with unification in RTE Kilian Evang Introduction Logics Formal Languages Semantics Proof Theories ◮ example: ∀x(love(x, mia)) ∧ ¬love(vincent, mia) Theorem Proving Propositional ◮ we should be able to refute that Resolution Set Conjunctive ◮ normal form: [[love(x, mia)], [¬love(vincent, mia)]] Normal Form The Resolution Rule Example ◮ what tells us there’s a contradicition here – after we First-Order dropped the universal quantifier? Resolution Unification ◮ it’s the fact that the terms can be unified – we are Skolemisation Example allowed to treat this as a complementary pair Paramodulation Back Matter
  26. 26. Logical InferenceNon-redundant factors in RTE Kilian Evang Introduction ◮ whenever adding a new clause in propositional Logics resolution, we need to remove duplicates inside it Formal Languages Semantics ◮ in first-order resolution, we also need to take care of Proof Theories Theorem Proving terms that could become duplicates by unification Propositional Resolution ◮ example: Set Conjunctive Normal Form [A(m), A(y), B(n, x), B(y, z), ¬C (w), ¬C (f (z))] The Resolution Rule Example ◮ two possible most general variable substitutions that First-Order make the clause non-redundant: Resolution Unification ◮ [y/m, w/f (z)] Skolemisation Example ◮ [y/n, z/x, w/f (z)] Paramodulation Back Matter ◮ both must be used, resulting non-redundant factors are added to the list of clauses: ◮ [A(m), B(n, x), B(m, z), ¬C (f (z))] ◮ [A(m), A(n), B(n, x), ¬C (f (x))]
  27. 27. Logical InferenceSkolemisation in RTE Kilian Evang ◮ recall: before transforming a formula to CNF, existential Introduction quantifiers are dropped; bound variables are replaced by Logics placeholders Formal Languages Semantics ◮ rationale: ∃x(φ(x)) iff there is some “witness” s with Proof Theories Theorem Proving φ(s) Propositional Resolution ◮ crucial: s must be a name we didn’t use before, newly Set Conjunctive Normal Form introduced to vocabulary The Resolution Rule Example ◮ also: assumption that we can do with a single witness First-Order Resolution may be too bold Unification Skolemisation ◮ example: ∀x∃y (love(x, y) ∧ ¬love(y, x)) Example ◮ individual not loving back depends on the unlucky lover Paramodulation Back Matter ◮ solution: choose s1 (x) as placeholder (containing all variables that are universally bound at the position of the existential quantifier as arguments). s1 then denotes a function mapping every combination of individuals to an appropriate witness. Such placeholders are known as Skolem terms.
  28. 28. Logical Inference in RTEFormula to prove: ∀y¬∃xlove(x, y) → ¬∃x∀ylove(x, y) Kilian EvangNegate: ¬(∀y¬∃xlove(x, y) → ¬∃x∀ylove(x, y)) IntroductionConvert to negation normal form: Logics∀y¬∃xlove(x, y) ∧ ¬¬∃x∀ylove(x, y) Formal Languages∀y∀x¬love(x, y) ∧ ¬¬∃x∀ylove(x, y) Semantics Proof Theories∀y∀x¬love(x, y) ∧ ∃x∀ylove(x, y) Theorem Proving PropositionalSkolemize away existential quantifiers (no arguments Resolution Set Conjunctivenecessary in Skolem term since the existentially quantified Normal Form The Resolution Ruleformula is not in the scope of a universally quantified one): Example∀y∀x¬love(x, y) ∧ ∀ylove(s1 , y) First-Order ResolutionDrop universal quantifiers and rename variables: Unification Skolemisation¬love(x, y) ∧ love(s1 , z) Example ParamodulationAlready in set clause normal form – write in list notation: Back Matter[[¬love(x, y)], [love(s1 , z)]]Apply resolution with unification (mgu: [x/s1 , y/z]):[[¬love(x, y)], [love(s1 , z)], []]Success!
  29. 29. Logical InferenceParamodulation in RTE Kilian Evang Introduction Logics ◮ technique as described cannot deal with equality Formal Languages Semantics ◮ example: Proof Theories Theorem Proving (yolanda = honey-bunny ∧ robber(yolanda)) → Propositional robber(honey-bunny) is a theorem, but will not be Resolution Set Conjunctive proved if = is treated as just another binary predicate Normal Form The Resolution Rule Example ◮ state-of-the-art theorem provers use an additional rule, First-Order paramodulation Resolution Unification Skolemisation ◮ given A = B, permits to substitute B for terms unifiable Example Paramodulation with A in formulas Back Matter ◮ intelligent restrictions needed to counter explosion of search space, see [Nieuwenhuis & Rubio, 2001]
  30. 30. Logical InferenceThe paramodulation rule in RTE Kilian Evang Introduction Logics Formal Languages Semantics Proof Theories Theorem Proving Propositional ResolutionFrom two clauses [s = t, φ] and [ψ, θ] where some r in ψ is Set Conjunctive Normal Formunifiable with s with the most general unifier σ, deduce The Resolution Rule Example[φ, ψ[r/s], θ]σ. First-Order Resolution Unification Skolemisation Example Paramodulation Back Matter
  31. 31. Logical InferenceReferences in RTE Kilian Evang Introduction Blackburn, P. & J. Bos (2005) Logics Representation and Inference for Natural Language. A Formal Languages Semantics First Course in Computational Semantics Proof Theories Theorem Proving CSLI Propositional Resolution Bos, J. & K. Markert (2005) Set Conjunctive Normal Form Recognising Textual Entailment with Logical Inference The Resolution Rule Example In Proceedings of EMNLP 2005 First-Order Resolution http://aclweb.org/anthology-new/H05-1079 Unification Skolemisation Gallier, Jean (2003) Example Paramodulation Resolution in First-Order Logic Back Matter In Logic for Computer Science. Foundations of Automatic Theorem Proving http://www.cis.upenn.edu/ cis510/tcl/chap8.pdf
  32. 32. Logical InferenceReferences in RTE Kilian Evang Introduction Logics Jones, R.B. (1998) Formal Languages Semantics What is Logic? Proof Theories Theorem Proving http://www.rbjones.com/rbjpub/logic/log001.htm Propositional Resolution Nieuwenhuis, R. & A. Rubio (2001) Set Conjunctive Normal Form Paramodulation-based theorem proving The Resolution Rule Example In Handbook of Automated Reasoning First-Order Resolution MIT Press Unification Skolemisation Sakharov, A. & E.W. Weisstein Example Paramodulation Propositional Calculus Back Matter From MathWorld http://mathworld.wolfram.com/PropositionalCalculus.html
  33. 33. Logical Inference in RTE Kilian Evang Introduction Logics Formal Languages Semantics Proof Theories Theorem Proving Propositional Resolution∀x(member(x, rte-class) → thank(kilian, x)) Set Conjunctive Normal Form The Resolution Rule Example First-Order Resolution Unification Skolemisation Example Paramodulation Back Matter

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