Computational tools derived from Ludics Use in web ... · Ludics C-Designs Generators and Automata...

Ludics C-Designs Generators and Automata Web and c-Ludics (web) Ontologies and c-Ludics

Computational tools derived from LudicsUse in web technologies

Christophe Fouqueré

Laboratoire d’Informatique de Paris-NordUniversité de Paris 13 - CNRS 7030


Web is (also) a dialog !

The web is a place of (digital) dialogues between human beingsand/or programs.

We need abstractions of processes and data such that we caninfer and compute:

why? to better understand what is in play,what? properties of computational objects: equivalences ofprograms, discovery of structures in web data, . . .how? by means of (finite) descriptions of interactions:retrieval of information, web service usage.


1 Ludics

2 C-Designs

3 Generators and Automata

4 Web and c-Ludics

5 (web) Ontologies and c-Ludics


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


Ludics: Objects

DefinitionA 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)


Designs and Normalization

A design D is a set of coherent chronicles:

åinteraction with a design E succeeds and reduces to adaimon if there exists in D a path dual to some path in theinteracting design E (and this design is in the dual).

åinteraction succeeds if there exists a path dual to somepath in the interacting design (and the result is whatremains).

A design may model a (simple) process (as a web service)or datum (as a concept in a concrete ontology).


Ludics: Interaction as Normalization

b

b2

b20

b1

z

b

s

s1

b2

b20

b1

s2

z

s

s1

z

s2

z

(boldface for positive actions)


Ludics: Interaction as dual plays

b

b2

b20

b1

z

b

s

s1

b2

b20

b1

s2

z

s

s1

z

s2

z

(visited paths in each design)


Designs vs Game Semantics

Plays are ways to travel through (coherent) chronicles as itcould 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


Consequences on Ethics

An ethics is a set of designs:

åinteraction succeeds and reduces to a daimon if there exists,for each design in the ethics, a path dual to some path in theinteracting design.

An ethics models an additive/disjunctive situation, e.g. a webservice that may have different executions w.r.t. what exists inits current context.

In terms of incremental/learning process, adding a designmay delete plays:ånon-monotonicity in terms of plays of designsåmonotonicity in terms of ethics

How?


Consequences on Ethics

Update of an over-generalization:

åSuppose you consider that ‘eat’ may be a verb or a noun (as‘drink’ is), hence one design for the two possibilities, then youmay reconsider your position by adding a new design:

The original design has two slices: one for verb and one fornounYou add a design that considers only one slice.

Update of an unknown information:

åAdverbs may be used with ‘eat’. ‘up’ is one of them.The original design contains the whole set of suchadverbs. But without ‘content’: just a daimonYou add a design replacing such a daimon by a design thatintegrates some content.


Computational Ludics (K. Terui, 2008)

Goal: Logical reconstruction of computability andcomplexity theory

Why based on Ludics: monism, interaction, existentialismMonism: One kind of object versus automata/word,program/data, grammar/sentence, . . .Interaction:

Primitive of computation (cf Curry-Howard), acceptanceviewed as orthogonalitySymmetry of the process (versus functions)

Existentialism: not necessarily closed objects


Computational Ludics: Preliminary Results

Correspondence with automata theory

Computational presentation of (generalized) Ludics (andgeneralization of completeness properties)

Introduction of design generators (hence finite presentationof infinite designs)


Ludics vs Automata

Correspondence:

set of Actions vs AlphabetDesign vs Word

Behaviour vs LanguageFormula/Type vs (eg reg.) Expression

Orthogonality vs Acceptance

D ⊥ R iff [[D,R]] = Dai iff D∗ is accepted by R∗


Generalizing Ludics (1)

First of all, Terui defines:

A term language for design presentation.

A language for generating designs, hence giving thepossibility to speak of, say, regular expressions.

åThere is a possibility to deal with infinite reductions.


Generalizing Ludics (2)

Second:Loci addresses are absolute and reflect a chronicle orderå In an action, a focus and its ramification are replaced bya name together with a set of (distinct) variables. Hence,variables may be replaced/reduced by designs during anormalization step.

Designs are cut-free, however a design may be thenormalized form between two designså Explicit cuts are added to the term language.

Designs are identity-freeå As soon as variables are part of the term language,replacing a variable by another one is nothing else but anidentity.


Term Language

Girard/Curien’s original designs:Normal, η-expanded, linear lambda terms (maybeinfinitary).Actions built with ramifications: I ∈ Pf (N).(in Desseins) a node is addressed by a locus.(in Desseins) children of negative nodes may have a(maybe infinite) set of negative actions with same focus

C-designs:Arbitrary terms.Actions built from a given signature: A = (A,ar) where

A is a set of names,ar : A → N gives an arity to each name.

A term is referenced by a name.A specific constant corresponds to undefined positivesubnodes, hence divergence: Ω


Term Language: Actions

Actions are polarized:

A positive action is one of the following:

z 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 )


Term Language: C-designs

The sets of positive and negative designs are coinductivelydefined by:

P ::= z daemon

| Ω divergence

| N0|u〈−→Ni〉 proper action

N ::= x variable

|∑

u u(−→xi ).Pu (negative) choices


Term Language: Ludics designs as C-designs

In Ludics:

ξ00 ` ξ1` ξ011

z

ξ01 `` ξ0, ξ1

ξ10 ` ξ2` ξ1, ξ2

ξ ` or(−, ξ, 0,1)

(+, ξ0, 0,1)

(−, ξ01, 1)

z

(−, ξ, 1,2)

(+, ξ1, 0)

As a c-design: a(x1, x2).B

B = x1 | b〈∑

u(−→xi ).Ω,C +

∑u 6=c u(

−→xi ).Ω〉

C = c(z1).z

z

+ a′(x ′1, x′2).B′

B′ = x ′1 | b′〈∑

u(−→xi ).Ω〉

or a(x1, x2).(

x1 | b〈∑

u(−→xi ).Ω , c(z1).z +

∑u 6=c u(

−→xi ).Ω〉)

+a′(x ′1, x′2).(x ′1 | b′〈

∑u(−→xi ).Ω〉

)+∑

u 6=a,a′ u(−→xi ).Ω


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|u0〈N1 . . .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 JPK, J·K is extended to allc-designs.Orthogonality:

N ⊥ P if P[N/x ] ⇓ z


Term Normalization: a Linear Example

a(−, ξ, 0,1)

(+, ξ0, 0,1)

(−, ξ01, 1)

z

a′(−, ξ, 1,2)

(+, ξ1, 0)

a(+, ξ, 0,1)

(−, ξ1, 2)

z

(−, ξ0, 0,1)

(+, ξ01, 1)

(a(x1, x2).

(x1 | b〈

∑u(−→xi ).Ω , c(z1).z +

∑u 6=c u(

−→xi ).Ω〉)

+ a′(x ′1, x′2).(x ′1 | b′〈

∑u(−→xi ).Ω〉

)+ . . .

)a〈b(y1, y2).

(y2|c〈

∑u(−→xi ).Ω〉

)+∑

u 6=b u(−→xi ).Ω , b”(y”1).z +

∑u 6=b” u(

−→xi ).Ω〉

→(

b(y1, y2).(y2|c〈

∑u(−→xi ).Ω〉

)+∑

u 6=b u(−→xi ).Ω

)b〈∑

u(−→xi ).Ω , c(z1).z +

∑u 6=c u(

−→xi ).Ω〉

→(

c(z1).z +∑

u 6=c u(−→xi ).Ω

)c〈∑

u(−→xi ).Ω〉

→ z



(−, ξ, 0,1)

(+, ξ0, 0,1)

(−, ξ01, 1)

z

(−, ξ, 1,2)

(+, ξ1, 0)

(+, ξ, 0,1)

(−, ξ1, 2)

z

(−, ξ0, 0,1)

(+, ξ01, 1)

b b

(a(x1, x2).

(x1 | b〈

∑u(−→xi ).Ω , c(z1).z +

∑u 6=c u(

−→xi ).Ω〉)

+ a′(x ′1, x′2).(x ′1 | b′〈

∑u(−→xi ).Ω〉

)+ . . .

)a〈b(y1, y2).

(y2|c〈

∑u(−→xi ).Ω〉

)+∑

u 6=b u(−→xi ).Ω , b”(y”1).z +

∑u 6=b” u(

−→xi ).Ω〉

→(

b(y1, y2).(y2|c〈

∑u(−→xi ).Ω〉

)+∑

u 6=b u(−→xi ).Ω

)b〈∑

u(−→xi ).Ω , c(z1).z +

∑u 6=c u(

−→xi ).Ω〉

→(

c(z1).z +∑

u 6=c u(−→xi ).Ω

)c〈∑

u(−→xi ).Ω〉

→ z



(−, ξ, 0,1)

(+, ξ0, 0,1)

(−, ξ01, 1)

z

(−, ξ, 1,2)

(+, ξ1, 0)

(+, ξ, 0,1)

(−, ξ1, 2)

z

(−, ξ0, 0,1)

(+, ξ01, 1)cc

(a(x1, x2).

(x1 | b〈

∑u(−→xi ).Ω , c(z1).z +

∑u 6=c u(

−→xi ).Ω〉)

+ a′(x ′1, x′2).(x ′1 | b′〈

∑u(−→xi ).Ω〉

)+ . . .

)a〈b(y1, y2).

(y2|c〈

∑u(−→xi ).Ω〉

)+∑

u 6=b u(−→xi ).Ω , b”(y”1).z +

∑u 6=b” u(

−→xi ).Ω〉

→(

b(y1, y2).(y2|c〈

∑u(−→xi ).Ω〉

)+∑

u 6=b u(−→xi ).Ω

)b〈∑

u(−→xi ).Ω , c(z1).z +

∑u 6=c u(

−→xi ).Ω〉

→(

c(z1).z +∑

u 6=c u(−→xi ).Ω

)c〈∑

u(−→xi ).Ω〉

→ z



(−, ξ, 0,1)

(+, ξ0, 0,1)

(−, ξ01, 1)

z

(−, ξ, 1,2)

(+, ξ1, 0)

(+, ξ, 0,1)

(−, ξ1, 2)

z

(−, ξ0, 0,1)

(+, ξ01, 1)

(a(x1, x2).

(x1 | b〈

∑u(−→xi ).Ω , c(z1).z +

∑u 6=c u(

−→xi ).Ω〉)

+ a′(x ′1, x′2).(x ′1 | b′〈

∑u(−→xi ).Ω〉

)+ . . .

)a〈b(y1, y2).

(y2|c〈

∑u(−→xi ).Ω〉

)+∑

u 6=b u(−→xi ).Ω , b”(y”1).z +

∑u 6=b” u(

−→xi ).Ω〉

→(

b(y1, y2).(y2|c〈

∑u(−→xi ).Ω〉

)+∑

u 6=b u(−→xi ).Ω

)b〈∑

u(−→xi ).Ω , c(z1).z +

∑u 6=c u(

−→xi ).Ω〉

→(

c(z1).z +∑

u 6=c u(−→xi ).Ω

)c〈∑

u(−→xi ).Ω〉

→ z


Term Normalization in the Large

In the previous example:Variables are used linearly.They specify an immediate sub-cut.

The framework of c-ludics is more general:Variables may appear several times.Cuts may be put for later usage.

As in the following example...


Term Normalization: a Non-linear Example

(a(x).

(x | b〈c().

(x | b〈c().z +

∑u 6=c u(

−→xi ).Ω〉)〉))

a〈b(y). (y | c〈〉)〉

→ (b(y). (y | c〈〉)) b〈c().(

(b(y). (y | c〈〉)) b〈c().z +∑

u 6=c u(−→xi ).Ω〉

)→ c().

((b(y). (y | c〈〉)) b〈c().z +

∑u 6=c u(

−→xi ).Ω〉)

c〈〉

→ b(y). (y | c〈〉) b〈c().z +∑

u 6=c u(−→xi ).Ω〉

→(

c().z +∑

u 6=c u(−→xi ).Ω

)c〈〉

→ z


Term Normalization: Ludics style

(a(x).

(x | b〈c().

(x | b〈c().z +

∑u 6=c u(

−→xi ).Ω〉)〉))

a〈b(y). (y | c〈〉)〉

(−, ξ, 0)

(+, ξ0, 0)

(−, ξ01, )

(+, ξ0, 0)

(−, ξ01, )

z

(+, ξ, 0)

(−, ξ0, 0)

(+, ξ01, )

a a



→ (b(y). (y | c〈〉)) b〈c().(

(b(y). (y | c〈〉)) b〈c().z +∑

u 6=c u(−→xi ).Ω〉

)

(−, ξ, 0)

(−, ξ0, 0)

(+, ξ01, )

(+, ξ0, 0)

(−, ξ01, )

(−, ξ0, 0)

(+, ξ01, )

(+, ξ0, 0)

(−, ξ01, )

z

(+, ξ, 0)

b b

b b



→ c().(

(b(y). (y | c〈〉)) b〈c().z +∑

u 6=c u(−→xi ).Ω〉

)c〈〉

(−, ξ, 0)

(−, ξ0, 0)

(+, ξ01, )

(+, ξ0, 0)

(−, ξ01, )

(−, ξ0, 0)

(+, ξ01, )

(+, ξ0, 0)

(−, ξ01, )

z

(+, ξ, 0)

cc

b b



→ b(y). (y | c〈〉) b〈c().z +∑

u 6=c u(−→xi ).Ω〉

(−, ξ, 0)

(−, ξ0, 0)

(+, ξ01, )

(+, ξ0, 0)

(−, ξ01, )

(−, ξ0, 0)

(+, ξ01, )

(+, ξ0, 0)

(−, ξ01, )

z

(+, ξ, 0)

b b



→(

c().z +∑

u 6=c u(−→xi ).Ω

)c〈〉

(−, ξ, 0)

(−, ξ0, 0)

(+, ξ01, )

(+, ξ0, 0)

(−, ξ01, )

(−, ξ0, 0)

(+, ξ01, )

(+, ξ0, 0)

(−, ξ01, )

z

(+, ξ, 0)

cc



→ z

(−, ξ, 0)

(−, ξ0, 0)

(+, ξ01, )

(+, ξ0, 0)

(−, ξ01, )

(−, ξ0, 0)

(+, ξ01, )

(+, ξ0, 0)

(−, ξ01, )

z

(+, ξ, 0)


Generators

A design generator is given as a set of recursive definitions ofpositive or negative c-designs:

DefinitionA design generator G is a triple (P,N ,g) such that:

P and N are disjoint sets∀P ∈ P, g(P) is

either z

or Ωor N0 | u〈N1, . . . ,Nn〉 with Ni ∈ N

∀N ∈ N , g(N) iseither a variable xor∑

u(−→xu).Pu with Pu ∈ P


Generators: Reduction

Reduction rule and normalization may be defined ongenerators in the same way as they are on c-designs.

The reduction rule∑

u u(x1 . . . xar(u)).Pu|u0〈N1 . . .Nar(u0)〉 isextended by corecursion to c-designs defined by generators.

The procedure is not effective as replacement may be infinite(as a design may contain an infinite number of the samevariable).

Instead of the previous reduction, one may define reduction bymeans of stack of environments: substitutions are notexplicitly done but kept in the environment.


Generators: an Example

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,

JP[−−→η(xi)/

−→xi ]K = P

Jη(N[−−→η(xi)/

−→xi ])K = N


Generators: Basic Data Types

Integers, lists, ... may be encodedfrom a set of names: eg zero, succ, consand a ‘functional’ name ↑

so that data c-designs d =↑ (x). (x | a〈d1, . . . ,dn〉) encodebasic data:

0∗ =↑ (x). (x | zero〈〉)(n + 1)∗ =↑ (x). (x | succ〈n∗〉)...

DefinitionA function is a negative c-design f [x1, . . . , xn] such that, for alldata c-designs d1, . . . ,dn, Jf [d1, . . .dn]K is either a data designor the undef design, ie ↑ (x).Ω.


Generators

Remark that(↑ (x).f [x ]) | ↑d −→ f [d ]

Standard functional operations are represented by c-designs,maybe with recursive definitions:

case x of...

let d = e in...

duplication of data

Cuts are necessary to obtain finite specifications !


Standard C-designs and Finite Automata (1)

A standard c-design (ie a Girard’s design) is a c-design that islinear, cut-free, identity-free,6= Ω, with a finite set of free variables.

A deterministic finite automaton M is a tuple (Σ,Q,δ,q0,QF ):Σ is an alphabetQ is a finite setδ : Q × Σ→ Q is the transition function,

q a−→ q′ iff δ(q,a) = q′

q0 ∈ Q is the initial state and QF ⊂ Q the set of final statesM accepts a word a1 . . . an

if ∃qi ∈ Q s.t. q0a1−→ q1

a2−→ . . .an−→ qf ∈ QF


Standard C-designs and Finite Automata (2)

Theorem (Terui)Let M be a DFA, there exists a finitely generated positivestandard c-design P such that M accepts w iffJP[w∗/x ]K = z

and vice-versa.

The proof is based on the following:A name nil for the empty string

For each state q, two c-designs q+ and q−

g(q+i ) = x | ↑〈q−i 〉

g(q−i ) =∑

a∈A a(x).q+ja + nil().z, if ∀a ∈ A,qi

a−→ qja and qi ∈ QF

g(q−i ) =∑

a∈A a(x).q+ja + nil().Ω, if ∀a ∈ A,qi

a−→ qja and qi 6∈ QF


C-designs and Turing Machines

Theorem (Terui)Let P be a finitely generated positive c-design, there existsa Turing Machine M such that M accepts w iffJP[w∗/x ]K = z

Let M be a Turing Machine, there exists a finitelygenerated positive identity-free linear c-design M suchthat M accepts w iff JP[w∗/x ]K = z


C-designs and Turing Machines

In other words:

Girard’s finite designs recognize exactly regularlanguages

Cuts are necessary to recover the full expressivepower of Turing machines


C-designs and Grammars: an Example (1)

Example (Language anbn, n > 0)A string w is encoded as a negative c-design w∗:

ε∗ =↑ (x).x |nil〈〉(a.w)∗ =↑ (x).x |a〈w∗〉

Use of a stack to postpone what remains to be done:G[t ] = M[nil().z, t ]M[z, t ] = t | ↑〈a(x).M ′[z, x ]〉M ′[z, t ] = t | ↑ 〈

b(x).(x |↑〈z〉)+a(x).M[b(x).(x |↑〈z〉), ↑ (y).y |a〈x〉] 〉

We write Nil as a shorthand for nil().z.


Web and c-Ludics

To stick to web style, we change slightly the syntax:

Definitionα, β, γ, . . . are tag names,

P ::= z 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 ofvalues that has to be sent/received for a given tag.N0 should contain the (address of) the code for executing γ(

−→xγ).


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)):

DefinitionThe 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 acut nor Ω.


Web Interpretation: User Side

A daimon corresponds to a user stop: she properly exitsthe web dialogue.

A divergence takes place whenever the action proposedby the server is not recognized by her browser.

Otherwise, a proper positive action denotes the linkclicked by the user at some stage of the dialogue (this onecharacterized by an address) together with addresses forthe continuation and other data, e.g. form data. Negativeactions have ϑ as tag if update is the only action performedby the browser.


Web Interpretation: Server Side

A daimon should not occur as it corresponds to a serverfailure: it has no available answer even if it should.

Divergence corresponds to a request that is not takeninto account, a standard ‘Page not available’ should besent by the overall system.

Otherwise, each sublocus of the update tag ϑ is a supportfor a negative action: each such sublocus corresponds to aframe, the ramification associated to a negative actionbeing a set of links, whereas the set of subloci is a layout.Frames remain available for the dialogue, except the oneswhere user clicks occur.


Web 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 sidea,b,c, . . . on the user side

User Servera := (ϑ(x)⇒ b) u := ϑ(v1)@xb := β(c)@x v1 := (β(x)⇒ w)c := (ϑ(x1, . . . , xn)⇒ d) w := ϑ(x1, . . . ,xn)@xd := δ0(e)@x1 x1 := (δ0(x)⇒ y) + (δ1(x)⇒ . . . ) + (ε(x)⇒ Dais)

. . .xn := (δ′(x)⇒ y’) + (ε′(x)⇒ Undefs)

e := (ϑ(x1, x2)⇒ Daic) y := ϑ(z1,z2)@xDaic := z Dais := z y’ := . . . Undefs := Ω


Web in c-Ludics: Execution Example

a

b

c

d

e

Daicz

user stop

ϑ

δ0

userchoice(restrictedby website)

ϑ

β

ϑu

v1

w

x1

y

z1

ϑ

z2

δ0

. . .

δ1

Daisz

ε

ϑ

. . . xn

y’

δ′

Undefs

Ω

ε′

β

ϑ

......

...

layouts ordata

bad serverspecifica-tion

‘Page notavailable’

a link

β(c)@v1

ϑ(x1, . . . ,xn)@c

page update

δ0(e)@x1

user request

ϑ(z1,z2)@e

User Server


Web in c-Ludics: more than Ludics

Auxiliary arguments may be added for sending or receiving(addresses of) values.Eg sending and displaying an HTML datum of address h on theclient browser:

u’ := ϑ(v1,h)@x on the server side to send datum h,a’ := (ϑ(x , y)⇒ b’) on the client side to receive it,b’ := (let _=(display y) in (β(c)@x)) for displaying thedatum.

The interaction may look like the following:

u’[a’/x ] = ϑ(v1,h)@(ϑ(x , y)⇒ b’)−→ b’[v1/x ,h/y ] = (let _=(display h) in (β(c)@v1))

−→ β(c)@v1 (with h displayed on the client side)

−→ . . .


Ontologies and c-Ludics

Ontologies, as used in the web, may well fit in such a logicalframework:

Quoting Guarino ("Handbook on Ontologies"):

“A body of formally represented knowledge is based on aconceptualization: the objects, concepts, and other entities thatare assumed to exist in some area of interest and therelationships that hold among them.

A conceptualization is an abstract, simplified view of the worldthat we wish to represent for some purpose.

Every knowledge base, knowledge-based system, orknowledge-level agent is committed to some conceptualization,explicitly or implicitly.”


Quoting Quine’s slogan ("On what there is"):“To be is to be the value of a bounded variable”

åThe logic to be adopted, according to Quine, is First OrderLogic relying on set theory.

Hence:concepts and relations are denoted by sets of objects,data that are recorded in the system as instantiating thoseconcepts and relations.


However:

such a choice implies that any change in the extensionalpicture produces also a change of conceptualization

It means that even the turn-over, over the time, of theinstances of a concept causes an unending change of thereference conceptualization.


(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 logicalframework,hence the logical interpretation should rely on structuresricher than sets: Coherence Spaces

The interpretation of a concept produces:graph theoretical objectsthe determination of the extensional counterpart within thecollection of resources.

What is a resource? In concrete / web ontologies: data storedin some base, tags put by a user


Ontologies may then be represented in c-Ludics:

requests and concepts should be considered syntax andsemantics, modelled in a unique kind of objectsusing interaction objects as elementary objects allows fortaking into account particularities of requesting informationover the webgenerators are rational elementary denotationsrequests and concepts are double orthogonal of suchassembled blocks

This may provide means for a complete approach fromdialogue to ontologies, integrating nicely a possibility formodelling learning.


Thanks for your attention

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