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 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. 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 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

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 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. 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 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.

15. 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

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 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

20. 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)

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 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. 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

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 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

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