About Coherence Use in Natural Language Christophe Fouquer - - PowerPoint PPT Presentation

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

About Coherence Use in Natural Language

Christophe Fouqueré

Laboratoire d’Informatique de Paris-Nord Université Paris 13 - CNRS 7030

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

What does a linguistic concept denote? What does a syntactic or semantic type denote? What kind of coherence is there between elements of such denotations? What are the consequences of using types/formulas?

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Concepts and Types

➥Types as kinds of tags used in linguistic formal theories: Noun, Phrase, Verb, ... e and t (for individuals and truth-values) ➥Types used to analyze, to control inferences. ➥Two terms with same type should be in some sense interchangeable: their ‘duals’ are mutually acceptable contexts. And duals of such a set of contexts should define a type.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Concepts and Types

➥Concepts in Linguistics: ... Grammar: tense, aspect, mood, modality, ... Syntax: phrase, clause, grammatical function, grammatical voice Semantics, Pragmatics ➥Concepts in Natural Language: Being, ..., Table, ... ➥“A conceptualization is an abstract view of the world.”

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Types and Linear Logic

Categorial Grammar is widely used, as such or in variants, as it relates Natural Language as a typed functional language, hence to λ-calculus: linguistic information is encoded in the lexicon via the assignment of syntactic types to lexical items, expressions are either functions or arguments.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Curry-Howard correspondence allows to view linguistic theories as formulas in a suitable Logics. Linear Logic extends the intuitionnistic approach: Full Linear Logic may be viewed as a strongly typed programming language Non-intuitionnism may be interpreted for example as exception handling Formulas may be interpreted as usable resources

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Questions remain: What does a type denote? Is there any relation between elements of (denotations of) concepts and types? The Geometry of Interaction program initiated by JY Girard tries to fully integrate syntax and semantics: ➥logical objects give the denotation of their use. So let us look at ontologies and concepts ...

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Quoting Guarino ("Handbook on Ontologies"): “A body of formally represented knowledge is based on a conceptualization: the objects, concepts, and other entities that are assumed to exist in some area of interest and the relationships that hold among them. A conceptualization is an abstract, simplified view of the world that we wish to represent for some purpose. Every knowledge base, knowledge-based system, or knowledge-level agent is committed to some conceptualization, explicitly or implicitly.”

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Quoting Quine’s slogan ("On what there is"): “To be is to be the value of a bounded variable” ➥The logic to be adopted, according to Quine, is First Order Logic relying on set theory. Hence: concepts and relations are denoted by sets of objects, data that are recorded in the system as instantiating those concepts and relations.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

However: such a choice implies that any change in the extensional picture produces also a change of conceptualization It means that even the turn-over, over the time, of the instances of a concept causes an unending change of the reference conceptualization.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

(works done with Abrusci and Romano) The focus is on the extensional level, i.e. on “real” objects: relations among resources are encoded in a logical framework, hence the logical interpretation should rely on structures richer than sets: Coherence Spaces The interpretation of a concept produces: graph theoretical objects the determination of the extensional counterpart within the collection of resources.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

What is a resource? In concrete / web ontologies: data stored in some base, tags put by a user In Natural Language: words, sentences produced or heard, dialogues, ...

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Coherence Spaces

Coherence Spaces are defined as a denotational semantics for Linear Logic. Definition A coherence space A is a countable graph with vertices |A| and a coherence relation ¨A reflexive and symmetric. A propositional letter is denoted by a coherence space. Connectives are denoted by operations on coherence spaces. What results? A proof is denoted by a clique. A (multiplicative) proof structure (formulas, axioms, cuts) is a proof iff its experiments are coherent wrt the par of the conclusions.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Coherence Spaces: operations

Definition

A⊥ is defined such that |A⊥| = |A| and x ¨A⊥ y iff x = y or x ¨A y A ⊗ B is defined such that |A ⊗ B| = |A| × |B| and (x, x′) ¨A⊗B (y, y′) iff x ¨A y and x′ ¨B y′ A ⊸ B is defined such that |A ⊸ B| = |A| × |B| and (x, x′) ¨A⊸B (y, y′) iff (x ¨A y then x′ ¨B y′ and x = y then x′ = y′) A ⊕ B is defined such that |A ⊕ B| = {0} × |A| ∪ {1} × |B| and (0, x) ¨A⊕B (0, x′) iff x ¨A x′, (1, y) ¨A⊕B (1, y′) iff y ¨B y′, (0, x) ¨A⊕B (1, y).

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Coherence Spaces and Ontologies

A knowledge base is a triple (O, M, Φ) such that: O is a set of predicate and relation symbols M is a set of individuals (or constants) Φ is defined on O such that Φ(P) ⊂ M and φ(R) ⊂ (M ×M) Definition An ontological compatibility space (OCS) O defined on a KB (O, M, Φ) is a coherence space such that: |O| =

P∈O Φ(P) ∪ R∈O Φ(R) (i.e. elements and pairs)

x ¨O y iff ∃P ∈ O, {x, y} ⊂ Φ(P) x, y ¨O x′, y′ iff ∃R ∈ O, {x, y, x′, y′} ⊂ Φ(R) x ¨O x′, y′ and x, y ¨O x′.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Coherence Spaces and Ontologies

Any concept of a knowledge base, i.e. the extension of a predicate or a relation, is a clique of its OCS. The ⊕ operation on OCS corresponds to the union of two

• ntologies.

The ⊸ operation on OCS corresponds to a mapping of

• ntologies.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Coherence Spaces and Ontologies

Such a framework allows to relate also folksonomies to

• ntologies.

2 resources are in relation if they have some quality in common (maybe subjective) a tag, a concept is represented as a clique Note that the viewpoint may be changed: 2 tags are in relation if there exists a common resource, . . . What is a point? What is a coherence structure?

Mainly logical structures, i.e. proofs, that may be questioned, i.e. reduced by cuts. Hence Ludics or Game Semantics

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Ludics

Ludics is a pre-logical framework upon which logic is built Ludics: from designs to behaviours w.r.t. interaction Proof theory: to proofs from formulas w.r.t. rules Automata: from words to languages w.r.t. acceptance

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

From desseins-proofs to designs

c ` (a⊥ ⊗ b⊥) {{c, a⊥ ⊗ b⊥}} (a ` b) ⊗ ⊥ {a ` b, ⊥} ∅ ⊥ a ` b {{a, b}} c a⊥ ⊗ b⊥ . . .

✎ ✍ ☞ ✌ ✎ ✍ ☞ ✌ ✎ ✍ ☞ ✌ ❍ ❍ ❍ ✟✟ ✟

Γ ⊢ P σ

{{0,1}}

τ

{0,1}

τ.0

∅

τ.1

{{0,1}}

σ.0

{0,... }

σ.1

...

σ.0.0

...

†

✻ ✻ ❍ ❍ ❍ ❨ ✟✟ ✟ ✯ ✻ ✻ ❍ ❍ ❍ ❨ ✻

DESIGN base branch chronicle daemon

✞ ✝ ☎ ✆ ☛ ✡ ✟ ✠ ☛ ✡ ✟ ✠ ✟✟✟ ✻

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Interaction: Normalization of design nets

α β β.1 β.2 α.0 γ

✻ ✻ ❅ ❅ ■

• ✒

✻ ✻ ☛ ✡ ✟ ✠ ☛ ✡ ✟ ✠ ✞ ✝ ☎ ✆

β σ σ.1 σ.2 β.2 τ

✻ ✻ ❅ ❅ ■

• ✒

✻ ✻ ✞ ✝ ☎ ✆ ☛ ✡ ✟ ✠ ✞ ✝ ☎ ✆ ✲

α σ σ.1 σ.2 γ τ

✻ ✻ ❅ ❅ ■

• ✒

✻ ✻ ✞ ✝ ☎ ✆ ☛ ✡ ✟ ✠ ✞ ✝ ☎ ✆ ✻ ✲ ✻ ❅ ❅ ■

• ✒

✻ ✁ ✁ ✁ ✁ ✁ ✁ ☛ ✻ ✻

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Ludics: Objects

Definition A chronicle is a sequence of (alternate) actions (a branch in a proof) A design on a basis ∆ ⊢ Γ is a set of (coherent) chronicles (a (para)proof) An ethics is a set of designs (ways to (para)prove formulas) A behaviour is an ethics closed by biorthogonality (a type, a formula with its set of (para)proofs)

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Designs vs Game Semantics, and Coherence

Plays are ways to travel through (coherent) chronicles as it could be done with an interaction. What difference between Game Semantics and Designs?

A strategy is a set of (half)realizations of interaction A design is a set of (half)potentialities of interaction

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Designs vs Game Semantics, and Coherence

A design D is a set of coherent chronicles: ➥interaction with a design E succeeds and reduces to a daimon if there exists in D a path dual to some path in the interacting design E (and this design is in the dual). ➥interaction succeeds if there exists a path dual to some path in the interacting design (and the result is what remains). If we suppose ProperNoun is modelled as a design, it may be used with a design typable as ProperNoun ⊸ NounPhrase.

Note that the reduction to be computable should be a finite process. This constraint is overcome when considering C-designs.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Ethics and Coherence

An ethics is a set of designs: ➥interaction succeeds and reduces to a daimon if there exists, for each design in the ethics, a path dual to some path in the interacting design. An ethics models an additive/disjunctive situation. In terms of incremental/learning process, adding a design may delete plays: ➥non-monotonicity in terms of plays of designs ➥monotonicity in terms of ethics How?

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Ethics and Coherence

Update of an over-generalization: ➥Suppose you consider that ‘eat’ may be a verb or a noun (as ‘drink’ is), hence one design for the two possibilities, then you may reconsider your position by adding a new design: The original design has two slices: one for verb and one for noun You add a design that considers only one slice. Update of an unknown information: ➥Adverbs may be used with ‘eat’. ‘up’ is one of them. The original design contains the whole set of such

• adverbs. But without ‘content’: just a daimon

You add a design replacing such a daimon by a design that integrates some content.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Ethics and Coherence

Which ‘coherence’ can we associate to an ethics? In general, an ethics is a not-yet completely defined behaviour, i.e. type, i.e. concept. Let us consider two designs beginning with two negative actions with disjoint ramifications: ⇒ The dual must be the positive daimon. ⇒ Its closure is ‘top’. That is to say, such a set of designs is in some way incoherent, say irrational concept. Otherwise the ‘concept’ is rationally defined, i.e. a characterization of a part of the universe.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Computational Ludics (K. Terui, 2008): Goal

Correspondence with automata theory Computational presentation of (generalized) Ludics (and generalization of completeness properties) Introduction of design generators (hence finite presentation

• f infinite designs)

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Generalizing Ludics (1)

First of all, Terui defines: A term language for design presentation. A language for generating designs, hence giving the possibility to speak of, say, regular expressions. ➥There is a possibility to deal with infinite reductions.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Generalizing Ludics (2)

Second: Loci addresses are absolute and reflect a chronicle order ➥ In an action, a focus and its ramification are replaced by a name together with a set of (distinct) variables. Hence, variables may be replaced/reduced by designs during a normalization step. Designs are cut-free, however a design may be the normalized form between two designs ➥ Explicit cuts are added to the term language. Designs are identity-free ➥ As soon as variables are part of the term language, replacing a variable by another one is nothing else but an identity.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Language: Actions

Actions are polarized: A positive action is one of the following:

• daemon

Ω

divergence

u

proper positive action, u ∈ A

A negative action is one of the following: x

variable

u(x1, . . . , xar(u))

proper negative action

(x1, . . . , xn noted − → xi )

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Language: C-designs

The sets of positive and negative designs are coinductively defined by: P ::=

• daemon

| Ω

divergence

| N0|u− → Ni

proper action

N ::= x

variable

|

• u u(−

→ xi ).Pu

(negative) choices

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Language: Ludics designs as C-designs

In Ludics:

ξ00 ⊢ ξ1 ⊢ ξ011

• ξ01 ⊢

⊢ ξ0, ξ1 ξ10 ⊢ ξ2 ⊢ ξ1, ξ2 ξ ⊢

• r

(−, ξ, {0, 1}) (+, ξ0, {0, 1}) (−, ξ01, {1})

• (−, ξ, {1, 2})

(+, ξ1, {0})

As a c-design:

a(x1, x2).B B = x1 | b u(− → xi ).Ω, C +

u=c u(−

→ xi ).Ω C = c(z1).

• +a′(x′

1, x′ 2).B′

B′ = x′

1 | b′ u(−

→ xi ).Ω

• r

a(x1, x2).

• x1 | b u(−

→ xi ).Ω , c(z1). +

u=c u(−

→ xi ).Ω

• +a′(x′

1, x′ 2).

• x′

1 | b′ u(−

→ xi ).Ω

• +

u=a,a′ u(−

→ xi ).Ω

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Normalisation: Extension of λ-calculus

In λ-calculus: (λx1 . . . xn.P)N1 . . . Nn − → P[N1/x1, . . . , Nn/xn] In C-Ludics: Reduction rule of c-designs:

• u u(x1 . . . xar(u)).Pu|u0N1 . . . Nar(u0)

− → Pu0[N1/x1, . . . , Nar(u0)/xar(u0)] Normalization: P ⇓ Q if P − →∗ Q and Q is neither a cut nor Ω If such a Q exists, it is noted P, · is extended to all c-designs. Orthogonality: N ⊥ P if P[N/x] ⇓

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Normalization: a Non-linear Example

• a(x).
• x | bc().
• x | bc(). +

u=c u(−

→ xi ).Ω

• ab(y). (y | c)

→ (b(y). (y | c)) bc().

• (b(y). (y | c))

bc(). +

u=c u(−

→ xi ).Ω

• →

c().

• (b(y). (y | c))

bc(). +

u=c u(−

→ xi ).Ω

• c

→ b(y). (y | c) bc(). +

u=c u(−

→ xi ).Ω →

• c(). +

u=c u(−

→ xi ).Ω

• c

→

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Normalization: a Non-linear Example

• a(x).
• x | bc().
• x | bc(). +

u=c u(−

→ xi ).Ω

• ab(y). (y | c)

→ (b(y). (y | c)) bc().

• (b(y). (y | c))

bc(). +

u=c u(−

→ xi ).Ω

• →

c().

• (b(y). (y | c))

bc(). +

u=c u(−

→ xi ).Ω

• c

→ b(y). (y | c) bc(). +

u=c u(−

→ xi ).Ω →

• c(). +

u=c u(−

→ xi ).Ω

• c

→

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Normalization: a Non-linear Example

• a(x).
• x | bc().
• x | bc(). +

u=c u(−

→ xi ).Ω

• ab(y). (y | c)

→ (b(y). (y | c)) bc().

• (b(y). (y | c))

bc(). +

u=c u(−

→ xi ).Ω

• →

c().

• (b(y). (y | c))

bc(). +

u=c u(−

→ xi ).Ω

• c

→ b(y). (y | c) bc(). +

u=c u(−

→ xi ).Ω →

• c(). +

u=c u(−

→ xi ).Ω

• c

→

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Normalization: a Non-linear Example

• a(x).
• x | bc().
• x | bc(). +

u=c u(−

→ xi ).Ω

• ab(y). (y | c)

→ (b(y). (y | c)) bc().

• (b(y). (y | c))

bc(). +

u=c u(−

→ xi ).Ω

• →

c().

• (b(y). (y | c))

bc(). +

u=c u(−

→ xi ).Ω

• c

→ b(y). (y | c) bc(). +

u=c u(−

→ xi ).Ω →

• c(). +

u=c u(−

→ xi ).Ω

• c

→

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Normalization: a Non-linear Example

• a(x).
• x | bc().
• x | bc(). +

u=c u(−

→ xi ).Ω

• ab(y). (y | c)

→ (b(y). (y | c)) bc().

• (b(y). (y | c))

bc(). +

u=c u(−

→ xi ).Ω

• →

c().

• (b(y). (y | c))

bc(). +

u=c u(−

→ xi ).Ω

• c

→ b(y). (y | c) bc(). +

u=c u(−

→ xi ).Ω →

• c(). +

u=c u(−

→ xi ).Ω

• c

→

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Normalization: a Non-linear Example

• a(x).
• x | bc().
• x | bc(). +

u=c u(−

→ xi ).Ω

• ab(y). (y | c)

→ (b(y). (y | c)) bc().

• (b(y). (y | c))

bc(). +

u=c u(−

→ xi ).Ω

• →

c().

• (b(y). (y | c))

bc(). +

u=c u(−

→ xi ).Ω

• c

→ b(y). (y | c) bc(). +

u=c u(−

→ xi ).Ω →

• c(). +

u=c u(−

→ xi ).Ω

• c

→

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Normalization: Ludics style

• a(x).
• x | bc().
• x | bc(). +

u=c u(−

→ xi ).Ω

• ab(y). (y | c)

(−, ξ, {0}) (+, ξ0, {0}) (−, ξ01, {}) (+, ξ0, {0}) (−, ξ01, {})

• (+, ξ, {0})

(−, ξ0, {0}) (+, ξ01, {}) a a

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Normalization: Ludics style

• a(x).
• x | bc().
• x | bc(). +

u=c u(−

→ xi ).Ω

• ab(y). (y | c)

(−, ξ, {0}) (+, ξ0, {0}) (−, ξ01, {}) (+, ξ0, {0}) (−, ξ01, {})

• (+, ξ, {0})

(−, ξ0, {0}) (+, ξ01, {}) a a

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Normalization: Ludics style

→ (b(y). (y | c)) bc().

• (b(y). (y | c))

bc(). +

u=c u(−

→ xi ).Ω

• (−, ξ, {0})

(−, ξ0, {0}) (+, ξ01, {}) (+, ξ0, {0}) (−, ξ01, {}) (−, ξ0, {0}) (+, ξ01, {}) (+, ξ0, {0}) (−, ξ01, {})

• (+, ξ, {0})

b b b b

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Normalization: Ludics style

→ c().

• (b(y). (y | c))

bc(). +

u=c u(−

→ xi ).Ω

• c

(−, ξ, {0}) (−, ξ0, {0}) (+, ξ01, {}) (+, ξ0, {0}) (−, ξ01, {}) (−, ξ0, {0}) (+, ξ01, {}) (+, ξ0, {0}) (−, ξ01, {})

• (+, ξ, {0})

c c b b

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Normalization: Ludics style

→ b(y). (y | c) bc(). +

u=c u(−

→ xi ).Ω

(−, ξ, {0}) (−, ξ0, {0}) (+, ξ01, {}) (+, ξ0, {0}) (−, ξ01, {}) (−, ξ0, {0}) (+, ξ01, {}) (+, ξ0, {0}) (−, ξ01, {})

• (+, ξ, {0})

b b

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Normalization: Ludics style

→

• c(). +

u=c u(−

→ xi ).Ω

• c

(−, ξ, {0}) (−, ξ0, {0}) (+, ξ01, {}) (+, ξ0, {0}) (−, ξ01, {}) (−, ξ0, {0}) (+, ξ01, {}) (+, ξ0, {0}) (−, ξ01, {})

• (+, ξ, {0})

c c

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Term Normalization: Ludics style

→

• (−, ξ, {0})

(−, ξ0, {0}) (+, ξ01, {}) (+, ξ0, {0}) (−, ξ01, {}) (−, ξ0, {0}) (+, ξ01, {}) (+, ξ0, {0}) (−, ξ01, {})

• (+, ξ, {0})

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

C-design Generators

A design generator is given as a set of recursive definitions of positive or negative c-designs: Reduction rule and normalization may be defined on generators in the same way as they are on C-designs.

Example The fax may be generated by ({}, {N}, η) where η(N) =

• u(−

→ xu).

• N | u−

− − → η(xu)

• Let P and N without cuts and identities,

P[− − → η(xi)/− → xi ] = P η(N[− − → η(xi)/− → xi ]) = N

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Generators: Basic Data Types

Standard functional programs may be interpreted by (finite) C-designs, maybe with recursive definitions. This may be extended by non-intuitionnistic operations. However computation remains deterministic (multiple results may obviously be modelled).

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

NL and C-Ludics

Natural Language, as described in Ludics style (e.g. Lecomte-Quatrini), may be represented in C-Ludics: a positive action a is an utterance of a speaker, a negative action a(− → xa) is a waited/received utterance, the specific c-design a(x).Ω means that the receiver refuses the dialogue as soon as the utterance a is received, the specific c-design a(x). means that the dialogue is properly closed: the receiver acknowledges. As C-Designs include Ludics and Lambda-Calculus, semantic

• r syntactic analysis may also be modelled.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

NL and C-Ludics

We change slightly the syntax: Definition α, le, γ, . . . are tag names, P ::=

• daimon

| Ω

divergence

| γ(− → Nγ)@N0

proper case, cut if N0 is not a variable

N ::= x

variable

|

• γ(γ(−

→ xγ) ⇒ Pγ)

sum of functionalities

The function ar gives the arity of each tag, i.e. the number of values that has to be sent/received for a given tag. N0 should contain the (address of) the code for executing γ(− → xγ).

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Web and C-Ludics

Interaction between a negative C-design

γ(γ(−

→ xγ) ⇒ Pγ) and a positive C-design γ(− → Nγ)@x is equivalent to computing the reduction of the positive C-design γ(− → Nγ)@(

γ(γ(−

→ xγ) ⇒ Pγ)): Definition The reduction rule is given by: γ(− → Nγ)@(

γ(γ(−

→ xγ) ⇒ Pγ)) − → Pγ[− → Nγ/− → xγ] Interaction succeeds if there is a normal form that is neither a cut nor Ω.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

NL Interpretation: Speaker Side

A daimon corresponds to the end of an utterance. A divergence takes place whenever an utterance of the hearer is not recognized. Otherwise, a proper positive action denotes the utterance

• f a word at some stage of the dialogue (this one

characterized by an address). Negative actions have ϑ as tag if the hearer is not supposed to speak!

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

NL Interpretation: Hearer Side

A daimon should not occur as it corresponds to interrupting the listening. Divergence corresponds to a word that is not taken into account. Otherwise, each sublocus of the update tag ϑ is a support for a negative action: each such sublocus corresponds to an utterance of some kind, the ramification associated to a negative action being a set of possible words, whereas the set of subloci denotes various media. These remain available: interpretation of a multimedia situation may be done by interleaving the computations.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

NL in C-Ludics: an Example

C-Ludics loci denote memory addresses, either code or data (boldface for positive C-designs): u, v, w, . . . on the server side a, b, c, . . . on the user side

User Server a := (ϑ(x) ⇒ b) u := ϑ(v1)@x b := Det(c)@x v1 := (Det(x) ⇒ w) c := (ϑ(x1) ⇒ d) w := ϑ(x1)@x d := N(e)@x1 x1 := (N(x) ⇒ y) + (A(x) ⇒ . . . ) + (ǫ(x) ⇒ Undefs) e := (ϑ(x1) ⇒ Daic) y := ϑ(z1)@x Daic := Dais := y’ := . . . Undefs := Ω

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

NL in C-Ludics: Execution Example

a b c d e

Daic

• user stop

ϑ

N

ϑ

Det

ϑ u v1 w x1 y z1 ϑ

N

. . .

A Undefs

Ω ǫ′ ϑ

Det

ϑ . . .

Unknown word Det(c)@v1 ϑ(x1)@c Acknowledge N(e)@x1 Spoken word ϑ(z1)@e

Speaker Hearer

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

NL in C-Ludics: more than Ludics

Auxiliary arguments may be added for sending or receiving (addresses of) values: ➥The ‘hearer’ may also emit words, hence ϑ is not the only possibility. Values may be non-linearly used: ➥Part of the ‘hearer’ design may have space for building interpretations (syntactic, semantic, ...) Multimedia analysis may take place: ➥see next slide

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

NL in C-Ludics: Execution Example Ctd

a b c d e

Daic

• ϑ

N

ϑ

Det

ϑ u v1 w x1 y z1 ϑ z2

N

. . .

A Undefs

Ω ǫ′ ϑ . . . xn y’ δ′

Det

ϑ . . . . . . . . .

sensors Det(c)@v1 ϑ(x1, . . . , xn)@c N(e)@x1 ϑ(z1, z2)@e

Multimedia Person

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Back to Coherence

What is a type, a concept? An explicit, rational explanation one may give in a similar fashion for a whole set of objects (real or not)! In other words, a type/concept may be represented as a generator that is used in the definition of a whole set of

• bjects.

As such, this generator is some kind of formula one may reason about. Thanks to the logical foundation where contexts and duals are

• f the same kind, a generator denotes a double orthogonal wrt

the contexts where it may appear to form a design.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Back to Coherence

What is the relation with coherence spaces? There is a coherence relation between two points if they share the same generator. This may provide means for a complete approach from dialogue to ontologies, integrating nicely a possibility for modelling learning.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

In summary: types and concepts should be considered syntax and semantics, modelled in a unique kind of objects using interaction objects as elementary objects allows for taking into account particularities of Natural Language, e.g. specific phrases generators are building blocks of elementary objects types and concepts are double orthogonal of such building blocks C-Ludics may provide sufficient place for dealing with such characteristics. However, underdeterminate analyses still remains a challenge.

Concepts/Types in NL Coherence Spaces Ludics C-Designs NL and C-Ludics Summary

Thanks for your attention