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Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Logical Inference in RTE
Kilian Evang
2009-06-29
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
OutlineIntroduction
LogicsFormal LanguagesSemanticsProof TheoriesTheorem Proving
Propositional ResolutionSet Conjunctive Normal FormThe Resolution RuleExample
First-Order ResolutionUnificationSkolemisationExampleParamodulation
Back Matter
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
RTE in an ideal world
HTBK
τ χκ →∧
KB
Choose:
Translate:
Prove:
(Also make sure that κ ∧ τ is satisfiable!)
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Problems
◮ capture the (relevant) subleties of natural language in alogical language
◮ encoding a sufficient amount of background knowledge(offline)
◮ choosing the right background knowledge (online)◮ too little: entailment is missed very easily
◮ remedy 1: turn a blind eye on non-entailment when(minimal) model sizes for T and T+H are very similar[Bos & Markert, 2005]
◮ remedy 2: use a shallow approach in parallel (ibid.)
◮ too much: proving becomes computationally expensive◮ remedy: very sophisticated reasoning techniques
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Logics
◮ provide formal languages into which natural languageexpressions (and other knowledge) can be translated:representation
◮ key advantage over natural language: entailment iswell-defined: inference
◮ tradeoff between expressivity for representation andtractability for inference
◮ many different logics exist
◮ but what is a logic?
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Essential ingredient 1: a class of formal languages
A logical language is a formal language defined by
1. a vocabulary, i.e. a set of non-logical symbolsspecific to the concrete application
◮ example: {(love, 2), (customer, 1), (robber, 1),(mia, 0), (vincent, 0), (honey-bunny, 0), (give, 3),(yolanda, 0)}
◮ also depends on the kind of logic used – e.g. standarddescription logics do not allow ternary relations
2. elements only specific to the kind of logic used◮ “logical” symbols
◮ example (first-order logic with equality):variables, ⊤, ⊥, ¬, ∧, ∨, →, ∀, ∃, (, ), ,, =
◮ syntactic rules to build formulas, the elements of thelanguage, from logical and non-logical symbols, e.g.
◮ robber(yolanda)◮ ∀x(robber(x) → love(mia,x))◮ mia = vincent
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Optional ingredient: a semantics
◮ in order to say whether a formula is “true” or “false”,you need a semantics
◮ semantics for logical languages are often defined interms of models
◮ intuitively, a model is a situation
◮ intuitively, a formula is satisfied in a model (“true”) iffit makes a correct statement about the situation
◮ exact satisfaction definition given in terms of set theory
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
First-order models
◮ a first-order model consists of a domain and anassignment function
◮ example domain D = {d1, d2, d3, d4}◮ example assignment function F : F (mia) = d2,
F (honey-bunny) = d1, F (yolanda) = d1,F (vincent) = d4, F (customer) = {d1, d2, d4},F (robber) = {d3, d5}, F (love) = {(d3, d4)},F (give) = {(d2, d1, d4)}}
◮ ∃x(love(x,vincent)) is satisfied in M
◮ love(vincent,mia) is not satisfied in M
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Essential ingredient 2: a proof theory
◮ singles out some formulas and calls them theorems
◮ consists of◮ axioms: formulas considered theorems without proof◮ inference rules: allow to derive new theorems from
known ones
◮ for the same logic, there often exist many different,equivalent proof theories
◮ if the logic has a semantics, a proof theory must bespecified in such a way that it is sound and complete
wrt. the semantics, i.e.: a formula is a theorem iff it istrue in all models
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Theorem proving
Given a formula φ, check whether φ is a theorem.
◮ Why?◮ to detect entailment: to check whether κ ∧ τ entails χ,
check whether (κ ∧ τ) → χ is a theorem◮ to detect contradiction: to check whether κ ∧ τ
contradicts χ, check whether ¬(κ ∧ τ ∧ χ) is a theorem
◮ How?◮ brute force: use a proof theory directly, i.e. generate all
axioms (many!) and apply inference rules until theformula is deduced.
◮ better: find a clever, sound, and complete technique tofind the answer by inspecting the formula
◮ still, theorem proving is purely syntactic: we may worryabout models in defining the technique, but not inapplying it
◮ tableau and resolution are such techniques
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Tableau
◮ a refutation method: to prove that φ is a theorem,derive a contradiction from ¬φ
◮ very intuitive: using a variety of specialized rules,decompose the formula step by step until twocontradictory atomic formulas have been derived
◮ a small example for a propositional tableau:
1 F (p ∧ ¬p) √
2 Fp 1, F∧3 F¬p 1, F∧,
√
4 Tp 3, F¬.
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Resolution
◮ a technique at the heart of state-of-the-art theoremprovers such as Prover9 or Vampire
◮ invented by J. Alan Robinson in 1965
◮ originally formulated for first-order logic, adapted toother logics
◮ a refutation method: to prove that φ is a theorem,derive a contradiction from ¬φ
◮ ¬φ must first be transformed to a normal form
◮ resolution then consists of the repeated application of asingle rule
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Propositional Resolution
◮ resolution for propositional logic (the quantifier-freefragment of first-order logic)
◮ atomic formulas like boxer(butch) orlove(vincent,mia) treated as atoms like p or q
◮ always terminates (propositional logic is decidable)
◮ the normal form for propositional resolution is called set
conjunctive normal form (set CNF)
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Set Conjunctive Normal Form (set CNF)
Every formula can be written as a conjunction ofdisjunctions of possibly negated atomic formulas.A formula that is not in set CNF:
(¬p → q) → (¬r → s)
The same formula in set CNF:
((¬p ∨ r ∨ s) ∧ (¬q ∨ r ∨ s))
In list notation:
[[¬p, r , s], [¬q, r , s]]
The inner lists (conjuncts, disjunctions) are called clauses.
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Converting into set CNF
1. convert into negation normal form (NNF)
2. convert from NNF to CNF
3. remove duplicates (from CNF to set CNF)
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Step 1: Converting into NNF
Rules
1. Rewrite ¬(φ ∧ ψ) as ¬φ ∨ ¬ψ2. Rewrite ¬(φ ∨ ψ) as ¬φ ∧ ¬ψ3. Rewrite ¬(φ→ ψ) as φ ∧ ¬ψ4. Rewrite φ→ ψ as ¬φ ∨ ψ5. Rewrite ¬¬ψ as ψ
Example
(¬p → q) → (¬r → s)
4 ⇔ ¬(¬p → q) ∨ (¬r → s)
3 ⇔ (¬p ∧ ¬q) ∨ (¬r → s)
4 ⇔ (¬p ∧ ¬q) ∨ (¬¬r ∨ s)
5 ⇔ (¬p ∧ ¬q) ∨ (r ∨ s)
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Step 2: From NNF to CNF
Rules
1. Rewrite θ ∨ (φ ∧ ψ) as (θ ∨ φ) ∧ (θ ∨ ψ)2. Rewrite (φ ∧ ψ) ∨ θ as (φ ∨ θ) ∧ (ψ ∨ θ)3. Rewrite (φ ∧ ψ) ∧ θ as θ ∧ (φ ∧ ψ)4. Rewrite (φ ∨ ψ) ∨ θ as θ ∨ (φ ∨ ψ)
Example
(¬p ∧ ¬q) ∨ (r ∨ s)
2 ⇔ (¬p ∨ (r ∨ s)) ∧ (¬q ∨ (r ∨ s))
Set notation: [[¬p, r , s], [¬q, r , s]]No duplicates, already in set CNF.
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Step 3: From CNF to set CNF
Remove duplicate literals from each clause, e.g.:
[[p, q, r ,¬s], [p,¬q, p,¬r ]]⇔ [[p, q, r ,¬s], [p,¬q,¬r ]]
Remove duplicate clauses from the list, e.g.
[[t,¬r ], [p, q,¬r ], [t,¬r ]]⇔ [[t,¬r ], [p, q,¬r ]]
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
The Resolution Rule
The key insight
(p ∨ r) ∧ (q ∨ ¬r) ⇒ (p ∨ q)
r and ¬r are called a complementary pair, (p ∨ r) and(q ∨ ¬r) are called complementary clauses.
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
The Resolution Rule
From two complementary clauses[p1, · · · , pn, r , pn+1, · · · , pm] and[q1, · · · , qj ,¬r , qj+1, · · · , qk ], deduce[p1, · · · , pn, pn+1, · · · , pm, q1, · · · , qj , qj+1, · · · , qk ]
The process of resolution
1. apply the resolution rule to some pair of complementaryclauses
2. remove duplicates from the result
3. add the result to the set of clauses
4. start over, unless◮ the empty clause has been derived (success)◮ no unprocessed complementary pair remains (failure)
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Example
Suppose we want to prove the following formula:
(p ∨ (q ∧ r)) → ((p ∨ q) ∧ (p ∨ r))
The first step is to transform its negation into set CNF:
¬((p ∨ (q ∧ r)) → ((p ∨ q) ∧ (p ∨ r)))
⇔ (p ∨ (q ∧ r)) ∧ ¬((p ∨ q) ∧ (p ∨ r))
⇔ (p ∨ (q ∧ r)) ∧ (¬(p ∨ q) ∨ ¬(p ∨ r))
⇔ (p ∨ (q ∧ r)) ∧ ((¬p ∧ ¬q) ∨ (¬p ∧ ¬r))⇔ ((p ∨ q) ∧ (p ∨ r)) ∧ (((¬p ∧ ¬q) ∨ ¬p) ∧ ((¬p ∧ ¬q) ∨ ¬r))⇔ · · ·⇔ ((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
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Example
Then we apply the resolution rule until we derive the emptyclause or no unprocessed complementary pair remains:
[[p, q], [p, r ], [¬p], [¬p,¬r ], [¬q,¬p], [¬q,¬r ]]⇔ [[p, q], [p, r ], [¬p], [¬p,¬r ], [¬q,¬p], [¬q,¬r ], [q]]⇔ [[p, q], [p, r ], [¬p], [¬p,¬r ], [¬q,¬p], [¬q,¬r ], [q], [r ]]⇔ [[p, q], [p, r ], [¬p], [¬p,¬r ], [¬q,¬p], [¬q,¬r ], [q], [r ], [¬r ]]⇔ [[p, q], [p, r ], [¬p], [¬p,¬r ], [¬q,¬p], [¬q,¬r ], [q], [r ], [¬q], []]
Success!
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
First-order resolution
◮ theoremhood in first-order logic is only semi-decidable:the algorithm will eventually halt if the formula is atheorem, but may never halt if the formula is not atheorem
◮ still useful
◮ new preprocessing phase
1. transform into NNF, with two additional rules:rewrite ¬∀xφ as ∃x¬φ, ¬∃xφ as ∀x¬φ
2. discard existential quantification, replace variables by aunique placeholder (skolemisation)
3. discard universal quantification, treat variables asimplicitly universally quantified (rename if necessary)
4. put the result into set CNF
◮ new resolution phase◮ resolution with unification
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Unification in a nutshell
◮ making two terms identical by replacing variables,using the most general substitution possible
◮ robber(vincent) and customer(x)are not unifiable: different relation symbols
◮ robber(vincent) and robber(mia)are not unifiable: different constant arguments
◮ love(x, y) and love(mia, z) are unifiable. Whichsubstitution?
◮ [x/mia, y/vincent, z/vincent]?Bad idea, too specific.
◮ [x/mia, y/z] is the most general unifier (mgu).Result: love(mia, z)
◮ also: love(father(x),mia) and love(x,mia) are notunifiable: would create a cycle (“occurs check” needed)
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Resolution with unification
◮ example: ∀x(love(x,mia)) ∧ ¬love(vincent,mia)◮ we should be able to refute that
◮ normal form: [[love(x,mia)], [¬love(vincent,mia)]]◮ what tells us there’s a contradicition here – after we
dropped the universal quantifier?
◮ it’s the fact that the terms can be unified – we areallowed to treat this as a complementary pair
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Non-redundant factors
◮ whenever adding a new clause in propositionalresolution, we need to remove duplicates inside it
◮ in first-order resolution, we also need to take care ofterms that could become duplicates by unification
◮ example:[A(m),A(y),B(n, x),B(y, z),¬C (w),¬C (f (z))]
◮ two possible most general variable substitutions thatmake the clause non-redundant:
◮ [y/m,w/f (z)]◮ [y/n, z/x,w/f (z)]
◮ both must be used, resulting non-redundant factorsare 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
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Skolemisation
◮ recall: before transforming a formula to CNF, existentialquantifiers are dropped; bound variables are replaced byplaceholders
◮ rationale: ∃x(φ(x)) iff there is some “witness” s withφ(s)
◮ crucial: s must be a name we didn’t use before, newlyintroduced to vocabulary
◮ also: assumption that we can do with a single witnessmay be too bold
◮ example: ∀x∃y(love(x, y) ∧ ¬love(y, x))◮ individual not loving back depends on the unlucky lover◮ solution: choose s1(x) as placeholder (containing all
variables that are universally bound at the position ofthe existential quantifier as arguments). s1 then denotesa function mapping every combination of individuals toan appropriate witness. Such placeholders are known asSkolem terms.
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Formula to prove: ∀y¬∃xlove(x, y) → ¬∃x∀ylove(x, y)Negate: ¬(∀y¬∃xlove(x, y) → ¬∃x∀ylove(x, y))Convert to negation normal form:∀y¬∃xlove(x, y) ∧ ¬¬∃x∀ylove(x, y)∀y∀x¬love(x, y) ∧ ¬¬∃x∀ylove(x, y)∀y∀x¬love(x, y) ∧ ∃x∀ylove(x, y)Skolemize away existential quantifiers (no argumentsnecessary in Skolem term since the existentially quantifiedformula is not in the scope of a universally quantified one):∀y∀x¬love(x, y) ∧ ∀ylove(s1, y)Drop universal quantifiers and rename variables:¬love(x, y) ∧ love(s1, z)Already in set clause normal form – write in list notation:[[¬love(x, y)], [love(s1, z)]]Apply resolution with unification (mgu: [x/s1, y/z]):[[¬love(x, y)], [love(s1, z)], []]Success!
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
Paramodulation
◮ technique as described cannot deal with equality
◮ example:(yolanda = honey-bunny∧robber(yolanda)) →robber(honey-bunny) is a theorem, but will not beproved if = is treated as just another binary predicate
◮ state-of-the-art theorem provers use an additional rule,paramodulation
◮ given A = B , permits to substitute B for terms unifiablewith A in formulas
◮ intelligent restrictions needed to counter explosion ofsearch space, see [Nieuwenhuis & Rubio, 2001]
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
The paramodulation rule
From two clauses [s = t, φ] and [ψ, θ] where some r in ψ isunifiable with s with the most general unifier σ, deduce[φ,ψ[r/s], θ]σ.
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
References
Blackburn, P. & J. Bos (2005)Representation and Inference for Natural Language. A
First Course in Computational Semantics
CSLI
Bos, J. & K. Markert (2005)Recognising Textual Entailment with Logical InferenceIn Proceedings of EMNLP 2005
http://aclweb.org/anthology-new/H05-1079
Gallier, Jean (2003)Resolution in First-Order LogicIn Logic for Computer Science. Foundations of
Automatic Theorem Proving
http://www.cis.upenn.edu/ cis510/tcl/chap8.pdf
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
References
Jones, R.B. (1998)What is Logic?http://www.rbjones.com/rbjpub/logic/log001.htm
Nieuwenhuis, R. & A. Rubio (2001)Paramodulation-based theorem provingIn Handbook of Automated Reasoning
MIT Press
Sakharov, A. & E.W. WeissteinPropositional CalculusFrom MathWorld
http://mathworld.wolfram.com/PropositionalCalculus.html
Logical Inference
in RTE
Kilian Evang
Introduction
Logics
Formal Languages
Semantics
Proof Theories
Theorem Proving
Propositional
Resolution
Set ConjunctiveNormal Form
The Resolution Rule
Example
First-Order
Resolution
Unification
Skolemisation
Example
Paramodulation
Back Matter
∀x(member(x,rte-class) → thank(kilian, x))
