KE
Uploaded byKilian Evang
929 views

Logical Inference in RTE

This document discusses logical inference in recognizing textual entailment (RTE). It covers topics such as logics, formal languages, semantics, proof theories, propositional and first-order resolution, unification, and paramodulation. The document is authored by Kilian Evang and outlines these topics for applying logical inference to RTE.

Embed presentation

Downloaded 18 times
Logical Inference in RTE Kilian Evang Introduction Logics Formal Languages Semantics Proof Theories Logical 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
Logical Inference Outline in RTE Kilian Evang Introduction Introduction Logics Logics Formal Languages Formal Languages Semantics Semantics Proof Theories Theorem Proving Proof Theories Propositional Theorem Proving Resolution Set Conjunctive Normal Form Propositional Resolution The Resolution Rule Example Set Conjunctive Normal Form First-Order The Resolution Rule Resolution Unification Example Skolemisation Example First-Order Resolution Paramodulation Back Matter Unification Skolemisation Example Paramodulation Back Matter
Logical Inference RTE 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!)
Logical Inference Problems 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
Logical Inference Logics 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?
Logical Inference Essential ingredient 1: a class of formal languages in RTE Kilian Evang A 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
Logical Inference Optional 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
Logical Inference First-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
Logical Inference Essential 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
Logical Inference Theorem proving in RTE Kilian Evang Given 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
Logical Inference Tableau 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¬ .
Logical Inference Resolution 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
Logical Inference Propositional 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
Logical Inference Set Conjunctive Normal Form (set CNF) in RTE Kilian Evang Introduction Every formula can be written as a conjunction of Logics disjunctions of possibly negated atomic formulas. Formal Languages A formula that is not in set CNF: Semantics Proof Theories Theorem Proving Propositional (¬p → q) → (¬r → s) Resolution Set Conjunctive Normal Form The same formula in set CNF: The Resolution Rule Example First-Order Resolution ((¬p ∨ r ∨ s) ∧ (¬q ∨ r ∨ s)) Unification Skolemisation Example In list notation: Paramodulation Back Matter [[¬p, r , s], [¬q, r , s]] The inner lists (conjuncts, disjunctions) are called clauses.
Logical Inference Converting 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
Logical Inference Step 1: Converting into NNF in RTE Kilian Evang Rules 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-Order Example 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)
Logical Inference Step 2: From NNF to CNF in RTE Kilian Evang Rules 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 Example Example 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.
Logical Inference Step 3: From CNF to set CNF in RTE Kilian Evang Introduction Logics Formal Languages Remove 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 Example Remove 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
Logical Inference The Resolution Rule in RTE Kilian Evang Introduction Logics Formal Languages Semantics Proof Theories Theorem Proving The key insight Propositional Resolution Set Conjunctive Normal Form (p ∨ r ) ∧ (q ∨ ¬r ) ⇒ (p ∨ q) The Resolution Rule Example First-Order r and ¬r are called a complementary pair, (p ∨ r ) and Resolution Unification (q ∨ ¬r ) are called complementary clauses. Skolemisation Example Paramodulation Back Matter
Logical Inference The Resolution Rule in RTE Kilian Evang From 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 Conjunctive The 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)
Logical Inference Example in RTE Kilian Evang Suppose we want to prove the following formula: Introduction (p ∨ (q ∧ r )) → ((p ∨ q) ∧ (p ∨ r )) Logics Formal Languages Semantics Proof Theories The 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 ]]
Logical Inference Example in RTE Kilian Evang Introduction Logics Formal Languages Then we apply the resolution rule until we derive the empty Semantics Proof Theories clause 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 Matter Success!
Logical Inference First-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
Logical Inference Unification 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)
Logical Inference Resolution 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
Logical Inference Non-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))]
Logical Inference Skolemisation 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.
Logical Inference in RTE Formula to prove: ∀y¬∃xlove(x, y) → ¬∃x∀ylove(x, y) Kilian Evang Negate: ¬(∀y¬∃xlove(x, y) → ¬∃x∀ylove(x, y)) Introduction Convert 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 Propositional Skolemize away existential quantifiers (no arguments Resolution Set Conjunctive necessary in Skolem term since the existentially quantified Normal Form The Resolution Rule formula is not in the scope of a universally quantified one): Example ∀y∀x¬love(x, y) ∧ ∀ylove(s1 , y) First-Order Resolution Drop universal quantifiers and rename variables: Unification Skolemisation ¬love(x, y) ∧ love(s1 , z) Example Paramodulation Already 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!
Logical Inference Paramodulation 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]
Logical Inference The paramodulation rule in RTE Kilian Evang Introduction Logics Formal Languages Semantics Proof Theories Theorem Proving Propositional Resolution From two clauses [s = t, φ] and [ψ, θ] where some r in ψ is Set Conjunctive Normal Form unifiable 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
Logical Inference References 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
Logical Inference References 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
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

More Related Content

PDF
ENTROPY STABLE ENO SCHEMES
byHamed Zakerzadeh
 
PPTX
Modeling Improved Syllabification Algorithm for Amharic
byGuy De Pauw
 
PDF
Glue Semantics for Proof Theorists
byValeria de Paiva
 
PDF
Jarrar.lecture notes.aai.2012s.descriptionlogic
bySinaInstitute
 
PDF
A Cross-Lingual Annotation Projection Approach for Relation Detection
bySeokhwan Kim
 
PDF
Description Logics in RTE
byKilian Evang
 
PDF
Ecml2010 Slides
byfanizzi
 
PDF
Zif bolker_w2
byBen Bolker
 
ENTROPY STABLE ENO SCHEMES
byHamed Zakerzadeh
 
Modeling Improved Syllabification Algorithm for Amharic
byGuy De Pauw
 
Glue Semantics for Proof Theorists
byValeria de Paiva
 
Jarrar.lecture notes.aai.2012s.descriptionlogic
bySinaInstitute
 
A Cross-Lingual Annotation Projection Approach for Relation Detection
bySeokhwan Kim
 
Description Logics in RTE
byKilian Evang
 
Ecml2010 Slides
byfanizzi
 
Zif bolker_w2
byBen Bolker
 

What's hot

KEY
Reading the mind’s eye: Decoding object information during mental imagery fr...
byThomas Serre
 
PDF
Knowledge Organizers of Cell Biology
byMeena Kharatmal
 
PDF
ICPW2007.deWaard
bypragmaticweb
 
KEY
Study proposal: Dohorap
byDominic Portain
 
PDF
Learning Morphological Rules for Amharic Verbs Using Inductive Logic Programming
byGuy De Pauw
 
PDF
A Graph-based Cross-lingual Projection Approach for Weakly Supervised Relatio...
bySeokhwan Kim
 
PDF
Multivariate analyses & decoding
bykhbrodersen
 
PDF
Section1-1
by春男 洪
 
PPTX
Neural Correlates of Nouns and Verbs: fMRI Study Design
byLaura Gwilliams
 
DOC
12-Multistrategy-learning.doc
bybutest
 
PDF
Csr2011 june14 12_00_hansen
byCSR2011
 
KEY
Logic02 logicalconnectives
bythisisnotatextbook
 
PDF
Mlearn2009 [kompatibilitási mód]
byGábor Kismihók
 
PDF
Ontology Based Content Management in a Mobilized LearningEnvironment
byCorvinno Technology Transfer Center Nonprofit Public Ltd.
 
PDF
Iccs 2010
byMeena Kharatmal
 
Reading the mind’s eye: Decoding object information during mental imagery fr...
byThomas Serre
 
Knowledge Organizers of Cell Biology
byMeena Kharatmal
 
ICPW2007.deWaard
bypragmaticweb
 
Study proposal: Dohorap
byDominic Portain
 
Learning Morphological Rules for Amharic Verbs Using Inductive Logic Programming
byGuy De Pauw
 
A Graph-based Cross-lingual Projection Approach for Weakly Supervised Relatio...
bySeokhwan Kim
 
Multivariate analyses & decoding
bykhbrodersen
 
Section1-1
by春男 洪
 
Neural Correlates of Nouns and Verbs: fMRI Study Design
byLaura Gwilliams
 
12-Multistrategy-learning.doc
bybutest
 
Csr2011 june14 12_00_hansen
byCSR2011
 
Logic02 logicalconnectives
bythisisnotatextbook
 
Mlearn2009 [kompatibilitási mód]
byGábor Kismihók
 
Ontology Based Content Management in a Mobilized LearningEnvironment
byCorvinno Technology Transfer Center Nonprofit Public Ltd.
 
Iccs 2010
byMeena Kharatmal
 

Viewers also liked

PPTX
Lecture5
byÁlvaro Cárdenas
 
PPTX
Inference rulesproofmethods
byRajendran
 
PPT
Artificial intelligence and knowledge representation
bySajan Sahu
 
PPTX
Proofs by contraposition
byAbdur Rehman
 
PDF
CPSC 125 Ch 2 Sec 1
byDavid Wood
 
PPTX
Expert Systems
byAmir NikKhah
 
PPTX
Expert system (unit 1 & 2)
byLakshya Gupta
 
PPTX
Bca ii dfs u-3 tree and graph
byRai University
 
PPT
Backtracking
bycristian torres
 
PPT
Expert System
byuniversity of education,Lahore
 
PPT
Нечеткие знания в экспертных системах
byСпецсеминар "Искусственный Интеллект" кафедры АЯ ВМК МГУ
 
PDF
NUMA-optimized Parallel Breadth-first Search on Multicore Single-node System
byYuichiro Yasui
 
PPTX
Ai 2
bybayaws
 
PPTX
Экспертные системы
byОтшельник
 
PPT
Prpositional2
bygueste5006ef
 
PDF
Mayo aug1, jsm slides (3)
byjemille6
 
PDF
Logical Abduction and an Application on Business Rules Management
byTobias Trapp
 
PPTX
Анализ тональности с помощью ДСМ метода
byDima Kostyaev
 
PPTX
Ідентифікація багатофакторних залежностей на основі нечіткої бази знань з рі...
byРоман Тилець
 
PDF
The Science of UX Design
byZack Naylor
 
Lecture5
byÁlvaro Cárdenas
 
Inference rulesproofmethods
byRajendran
 
Artificial intelligence and knowledge representation
bySajan Sahu
 
Proofs by contraposition
byAbdur Rehman
 
CPSC 125 Ch 2 Sec 1
byDavid Wood
 
Expert Systems
byAmir NikKhah
 
Expert system (unit 1 & 2)
byLakshya Gupta
 
Bca ii dfs u-3 tree and graph
byRai University
 
Backtracking
bycristian torres
 
Expert System
byuniversity of education,Lahore
 
Нечеткие знания в экспертных системах
byСпецсеминар "Искусственный Интеллект" кафедры АЯ ВМК МГУ
 
NUMA-optimized Parallel Breadth-first Search on Multicore Single-node System
byYuichiro Yasui
 
Ai 2
bybayaws
 
Экспертные системы
byОтшельник
 
Prpositional2
bygueste5006ef
 
Mayo aug1, jsm slides (3)
byjemille6
 
Logical Abduction and an Application on Business Rules Management
byTobias Trapp
 
Анализ тональности с помощью ДСМ метода
byDima Kostyaev
 
Ідентифікація багатофакторних залежностей на основі нечіткої бази знань з рі...
byРоман Тилець
 
The Science of UX Design
byZack Naylor
 

Similar to Logical Inference in RTE

PPTX
Knowledge Representation, Inference and Reasoning
bySagacious IT Solution
 
PPT
Propositional And First-Order Logic
byankush_kumar
 
PDF
Logic
byShiwani Gupta
 
PPT
C2.0 propositional logic
byMelaku Bayih Demessie
 
PPT
PROPOSISTION LOGIC with reference to AI .
byRaghavendraPrasad179187
 
PPTX
Jarrar: Propositional Logic Inference Methods
byMustafa Jarrar
 
PDF
16_FirstOrderLogic.p_4_moduleModuleNotespdf
byShylaBg1
 
PPTX
Resolution(decision)
bySlideshare
 
PPTX
1004_theorem_proving_2018.pptx on the to
byfariyaPatel
 
PPTX
Knowledge Representation and Reasoning.pptx
byMohanKumarP34
 
PDF
Logic for everyone
byble nature
 
PPT
Logic
byFlora Mae Angtud-Galenzoga
 
PPTX
Geometry 02 Reasoning and Proofs an.pptx
byRushikeshNimgire
 
PDF
6.c-CMPS 403-F19-Session 6-Resolution.pdf
byAmirMohamedNabilSale
 
PPT
3 fol examples v2
byDigvijay Singh
 
PDF
Linear logic (and Linear Lisp)
bySosuke MORIGUCHI
 
PPT
wddd
byNguyen Tuan
 
PDF
Main
byseanmcl
 
PPT
PropositionalLogic.ppt
byMArunyNandinikkutty
 
PDF
A Superposition Oriented Theorem Prover
byTony Lisko
 
Knowledge Representation, Inference and Reasoning
bySagacious IT Solution
 
Propositional And First-Order Logic
byankush_kumar
 
Logic
byShiwani Gupta
 
C2.0 propositional logic
byMelaku Bayih Demessie
 
PROPOSISTION LOGIC with reference to AI .
byRaghavendraPrasad179187
 
Jarrar: Propositional Logic Inference Methods
byMustafa Jarrar
 
16_FirstOrderLogic.p_4_moduleModuleNotespdf
byShylaBg1
 
Resolution(decision)
bySlideshare
 
1004_theorem_proving_2018.pptx on the to
byfariyaPatel
 
Knowledge Representation and Reasoning.pptx
byMohanKumarP34
 
Logic for everyone
byble nature
 
Logic
byFlora Mae Angtud-Galenzoga
 
Geometry 02 Reasoning and Proofs an.pptx
byRushikeshNimgire
 
6.c-CMPS 403-F19-Session 6-Resolution.pdf
byAmirMohamedNabilSale
 
3 fol examples v2
byDigvijay Singh
 
Linear logic (and Linear Lisp)
bySosuke MORIGUCHI
 
wddd
byNguyen Tuan
 
Main
byseanmcl
 
PropositionalLogic.ppt
byMArunyNandinikkutty
 
A Superposition Oriented Theorem Prover
byTony Lisko
 

Recently uploaded

PPTX
Combining Induction and Transduction for Abstract Reasoning.pptx
byallen578468
 
PDF
Making Sense of Raster: From Bit Depth to Better Workflows
bySafe Software
 
PDF
Knowing and Doing: Knowledge graphs, AI, and work
bymarainglezakis1
 
PDF
Security Technologys: Access Control, Firewall, VPN
byShielaLasala
 
PPTX
Ethics in AI - Artificial Intelligence Fundamentals.pptx
byemenyiblessing
 
PDF
DIGITAL FORENSICS - Notes for Everything.pdf
bypankajkumavatbeit
 
PDF
Innovative AI Solutions for Business Growth.pdf
byVox Works
 
PDF
Session 1 - Solving Semi-Structured Documents with Document Understanding
byDianaGray10
 
PDF
The year in review - MarvelClient in 2025
bypanagenda
 
PPTX
The Future of IT Service Management AI Automation & Beyond.pptx
byChetu
 
PDF
TPPmark2025 Kenta Inoue's answer 12/04/2025
byKenta Inoue
 
PDF
Accelerating Responsible AI Adoption in Public Sector and Private Organizations.
byYasir Naveed Riaz
 
PDF
Safeguarding AI-Based Financial Infrastructure
byYasir Naveed Riaz
 
PDF
Ransomware_Resilience_Strategic_Playbook.pdf
byAfonso Henrique Rodrigues Alves
 
PDF
The major tech developments for 2026 by Pluralsight, a research and training ...
byChris Skinner
 
PDF
December Patch Tuesday
byIvanti
 
PDF
Igniting the Future: Copilot trends, agentic transformation and product roadm...
byUni Systems S.M.S.A.
 
PDF
eResource Scheduler Enterprise Resource Management and Scheduling Software.pdf
byeResource Scheduler
 
PPTX
RISE with SAP for Automotive - S4 HANA Value.pptx
byAmira Jemi
 
PDF
AI Technology important knowledge ......
byutkarshj2007
 
Combining Induction and Transduction for Abstract Reasoning.pptx
byallen578468
 
Making Sense of Raster: From Bit Depth to Better Workflows
bySafe Software
 
Knowing and Doing: Knowledge graphs, AI, and work
bymarainglezakis1
 
Security Technologys: Access Control, Firewall, VPN
byShielaLasala
 
Ethics in AI - Artificial Intelligence Fundamentals.pptx
byemenyiblessing
 
DIGITAL FORENSICS - Notes for Everything.pdf
bypankajkumavatbeit
 
Innovative AI Solutions for Business Growth.pdf
byVox Works
 
Session 1 - Solving Semi-Structured Documents with Document Understanding
byDianaGray10
 
The year in review - MarvelClient in 2025
bypanagenda
 
The Future of IT Service Management AI Automation & Beyond.pptx
byChetu
 
TPPmark2025 Kenta Inoue's answer 12/04/2025
byKenta Inoue
 
Accelerating Responsible AI Adoption in Public Sector and Private Organizations.
byYasir Naveed Riaz
 
Safeguarding AI-Based Financial Infrastructure
byYasir Naveed Riaz
 
Ransomware_Resilience_Strategic_Playbook.pdf
byAfonso Henrique Rodrigues Alves
 
The major tech developments for 2026 by Pluralsight, a research and training ...
byChris Skinner
 
December Patch Tuesday
byIvanti
 
Igniting the Future: Copilot trends, agentic transformation and product roadm...
byUni Systems S.M.S.A.
 
eResource Scheduler Enterprise Resource Management and Scheduling Software.pdf
byeResource Scheduler
 
RISE with SAP for Automotive - S4 HANA Value.pptx
byAmira Jemi
 
AI Technology important knowledge ......
byutkarshj2007
 

Logical Inference in RTE

  • 1.
    Logical Inference in RTE Kilian Evang Introduction Logics Formal Languages Semantics Proof Theories Logical 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.
    Logical Inference Outline in RTE Kilian Evang Introduction Introduction Logics Logics Formal Languages Formal Languages Semantics Semantics Proof Theories Theorem Proving Proof Theories Propositional Theorem Proving Resolution Set Conjunctive Normal Form Propositional Resolution The Resolution Rule Example Set Conjunctive Normal Form First-Order The Resolution Rule Resolution Unification Example Skolemisation Example First-Order Resolution Paramodulation Back Matter Unification Skolemisation Example Paramodulation Back Matter
  • 3.
    Logical Inference RTE inan 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.
    Logical Inference Problems 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.
    Logical Inference Logics 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.
    Logical Inference Essential ingredient1: a class of formal languages in RTE Kilian Evang A 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.
    Logical Inference Optional 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.
    Logical Inference First-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.
    Logical Inference Essential ingredient2: 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.
    Logical Inference Theorem proving in RTE Kilian Evang Given 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.
    Logical Inference Tableau 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.
    Logical Inference Resolution 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.
    Logical Inference Propositional 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.
    Logical Inference Set ConjunctiveNormal Form (set CNF) in RTE Kilian Evang Introduction Every formula can be written as a conjunction of Logics disjunctions of possibly negated atomic formulas. Formal Languages A formula that is not in set CNF: Semantics Proof Theories Theorem Proving Propositional (¬p → q) → (¬r → s) Resolution Set Conjunctive Normal Form The same formula in set CNF: The Resolution Rule Example First-Order Resolution ((¬p ∨ r ∨ s) ∧ (¬q ∨ r ∨ s)) Unification Skolemisation Example In list notation: Paramodulation Back Matter [[¬p, r , s], [¬q, r , s]] The inner lists (conjuncts, disjunctions) are called clauses.
  • 15.
    Logical Inference Converting intoset 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.
    Logical Inference Step 1:Converting into NNF in RTE Kilian Evang Rules 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-Order Example 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.
    Logical Inference Step 2:From NNF to CNF in RTE Kilian Evang Rules 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 Example Example 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.
    Logical Inference Step 3:From CNF to set CNF in RTE Kilian Evang Introduction Logics Formal Languages Remove 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 Example Remove 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.
    Logical Inference The ResolutionRule in RTE Kilian Evang Introduction Logics Formal Languages Semantics Proof Theories Theorem Proving The key insight Propositional Resolution Set Conjunctive Normal Form (p ∨ r ) ∧ (q ∨ ¬r ) ⇒ (p ∨ q) The Resolution Rule Example First-Order r and ¬r are called a complementary pair, (p ∨ r ) and Resolution Unification (q ∨ ¬r ) are called complementary clauses. Skolemisation Example Paramodulation Back Matter
  • 20.
    Logical Inference The ResolutionRule in RTE Kilian Evang From 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 Conjunctive The 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.
    Logical Inference Example in RTE Kilian Evang Suppose we want to prove the following formula: Introduction (p ∨ (q ∧ r )) → ((p ∨ q) ∧ (p ∨ r )) Logics Formal Languages Semantics Proof Theories The 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.
    Logical Inference Example in RTE Kilian Evang Introduction Logics Formal Languages Then we apply the resolution rule until we derive the empty Semantics Proof Theories clause 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 Matter Success!
  • 23.
    Logical Inference First-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.
    Logical Inference Unification ina 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.
    Logical Inference Resolution withunification 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.
    Logical Inference Non-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.
    Logical Inference Skolemisation 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.
    Logical Inference in RTE Formula to prove: ∀y¬∃xlove(x, y) → ¬∃x∀ylove(x, y) Kilian Evang Negate: ¬(∀y¬∃xlove(x, y) → ¬∃x∀ylove(x, y)) Introduction Convert 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 Propositional Skolemize away existential quantifiers (no arguments Resolution Set Conjunctive necessary in Skolem term since the existentially quantified Normal Form The Resolution Rule formula is not in the scope of a universally quantified one): Example ∀y∀x¬love(x, y) ∧ ∀ylove(s1 , y) First-Order Resolution Drop universal quantifiers and rename variables: Unification Skolemisation ¬love(x, y) ∧ love(s1 , z) Example Paramodulation Already 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.
    Logical Inference Paramodulation 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.
    Logical Inference The paramodulationrule in RTE Kilian Evang Introduction Logics Formal Languages Semantics Proof Theories Theorem Proving Propositional Resolution From two clauses [s = t, φ] and [ψ, θ] where some r in ψ is Set Conjunctive Normal Form unifiable 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.
    Logical Inference References 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.
    Logical Inference References 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.
    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