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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