Records in Ludics Myriam Quatrini & Eugenia Sironi Workshop - - PowerPoint PPT Presentation

Ludics: main objects; interesting properties Records in c-Ludics

Records in Ludics

Myriam Quatrini & Eugenia Sironi Workshop LOCI: Type Theory with records and Ludics Queen Mary University of London June 16-17

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics

Context: Ludics and Dialogue theories

One of the main aims of LOCI is the development of a theory of dialogue and its applications for linguistic phenomenons in the ludics framework. Until now we mainly considered one aspect of Ludics : “it is a theory of interaction”. But, in order to deepen our formalisation we also have to exploit its relevance to represent computationnal objects.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics

This talk:

to introduce some properties of Ludics, the ones which seem relevant to deal with computationnal theories to describe the proposition to deal with the notion of “record” in the ludical framework as it was already set in the seminal article of Girard [Locus Solum, 2001] to expose the transposition of such objects : record, in computationnal Ludics. A framework due to K. Terui in which the objects and concepts of Ludics are set in a relevant way to a computationnal treatment.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Ludics: a theory of logic

Ludics is a theory elaborated by J-Y. Girard to reconstruct logic starting from the notion of interaction. The logical main notions: formulas, proofs are not “ a priori” given but recovered, “rebuilt” from the notion of interaction (the cut-rule). Ludics was developed starting from concepts of Linear Logic:

• polarisation: linear logic rules have polarities, either + or

−, thus making proofs sequences of polarized steps.

• focalisation: results coming from Theoretical Computer

Science [?] lead us to focalized proofs, that is proofs as alternating sequences of steps. Such an alternance is close to the one of plays in a game. Ludics may be seen from a game semantics approach.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

In Ludics the notion of proof is subsumed by the one of design which may be seen as a proof search or as a strategy. As a strategy a design is a set of plays (chronicles) The plays themselves are alternating sequences of moves (actions).

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Actions as moves

An action (move) is given by three datas:

• a polarity (one player’s view being fixed, this player’s moves

are positive, his/her opponent’s are negative),

• a focus, that is the location (locus) of the move, and
• a ramification, which represents the finite set of locations

which can be reached in one step. A special positive move is provided by the so called da¨ ımon, denoted by †. (which will enable to define the winning position

• f the opponent)

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Chronicle: an alternated sequence of actions

+, 0, {0} −, 00, {1, 2} +, 001, {0} −, 0010, {0} +, 002, {0} c1 +, 0, {0} −, 00, {1, 2} +, 3, {0} −, 30, {0} † c2 −, 0, {0} +, 00, {1, 2} −, 001, {0} +, 3, {0} −, 30, {0} c3 In a chronicle the positive actions are:

• either justified ((+, 001, {0}) ; (+, 002, {0}))
• or initial ((+, 0, {0}) or (+, 3, {0})).

the negative actions are justified by the immediate previous action (except the starting one).

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

The base of a chronicle:

+, 0, {0} −, 00, {1, 2} +, 001, {0} −, 0010, {0} +, 002, {0} c1 +, 0, {0} −, 00, {1, 2} +, 3, {0} −, 30, {0} † c2 −, 0, {0} +, 00, {1, 2} −, 001, {0} +, 3, {0} −, 30, {0} c3 The base of the chronicle is a sequent Γ ⊢ ∆ of loci where the set Γ contains all the initial focus of a negative action (so at most one) and ∆ contains all the initial focus of a positive action (so a finite number). c1 is based on ⊢ 0 (which is a positive base); c2 is based on ⊢ 0, 3 (which is also a positive base); c3 is based on 0 ⊢ 3 (which is a negative base).

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Coherence relation on chronicles

−, ǫ, {2} +, 2, {8, 9} −, 28, {0} +, 280, ∅ −, 29, {0} +, 290, ∅ c1 −, ǫ, {2} +, 2, {8, 9} −, 29, {0} +, 290, ∅ −, 28, {0} +, 280, ∅ c2 −, ǫ, {2} +, 2, {8, 9} −, 28, {0} +, 280, ∅ −, 29, {0} † c3 Two chronicles c1 and c2 are coherent, when one may put them together in the same strategy. In the foregoing example, the chronicles are pairwise coherent except that c1 and c3.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Examples of strategies based on ǫ ⊢

ǫ 2 28 29 280 290 {2} {8, 9} {0} {0} ∅ ∅ ǫ 2 28 29 280 † {2} {8, 9} {0} {0} ∅ The strategy D1 The strategy K1 contains the chronicles c1 and c2 contains the chronicles c3 and c2 They are both based on ǫ ⊢.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

(Designs) A design D, based on Γ ⊢ ∆, is a set of chronicles based on Γ ⊢ ∆, such that the following conditions are satisfied: The chronicles are pairwise coherent. The set is prefix closed. A chronicle without extension in D ends with a positive action. D is non-empty when the base is positive (in that case all the chronicles begin with a unique positive action).

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Examples: some designs based on ǫ ⊢

D1 = {c1,c2,...} ǫ 2 28 29 290 280 D2 = {d,...} ǫ 4 45 450 ∅ D3 = {e,...} ǫ 6 66 660 ∅

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Strategy-like versus proof-like presentation

The design D = D1 ∪ D2 ∪ D3 is also a design based on ǫ ⊢. Strategy-like design

D = ǫ 2 28 29 290 280 ǫ 4 45 450 ǫ 6 66 660

Proof-like design

D =

∅

⊢ 280 28 ⊢

∅

⊢ 290 29 ⊢ ⊢ 2

∅

⊢ 450 45 ⊢ ⊢ 4

∅

⊢ 660 66 ⊢ ⊢ 6 ǫ ⊢ Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Proof-like presentation : design as dessin

A design (as dessin) based on Γ ⊢ ∆ is a tree of sequents built by means of the three following rules:

• DA¨

IMON

†

⊢ ∆

• POSITIVE RULE

...ξ.i ⊢ ∆i ...

(ξ,I)

⊢ ∆,ξ

• NEGATIVE RULE

⋅⋅⋅ ⊢ ξ.I,∆I ...

(ξ,N)

ξ ⊢ ∆

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

(Nets, cut-net, closed cut-net) A net is a finite set of designs on disjoint bases. A cut-net is a net with cuts: an addresse present only

• nce in a negative position of a base and once in a positive
• ne of another base.

Example : D1 ξ ⊢ σ D2 σ ⊢ ρ,τ D3 ρ ⊢ α In a closed cut-net all addresses in bases are parts of some cut. Example : F0 ⊢ ξ F1 ξ ⊢

Myriam Quatrini & Eugenia Sironi Records in Ludics

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Interaction in closed case

Informel definition: It simply corresponds to a travel which starts from the first positive node and, applies, at each proper positive action, the following: moves to the corresponding negative one, (if there is one, if not, interaction fails), moves upward to the unique action which follows, and so on, and if the travel meets †, the interaction successfully terminates

• n it.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

The interaction may terminate like the normalization between D2 and F:

ǫ {4} 4 {5} 45 {0} 450 ∅ † ǫ 4 {5} 45 450 ∅ It simply corresponds to a travel which starts from the first positive node, moves to the corresponding negative one, moves upward to the unique action which follows, and so on, and successfully terminates on †.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

The interaction may fail like the normalization between D1 and F:

ǫ 4 45 450 † ǫ 2 28 29 280 290 ? the travel starts from the first positive node at each proper positive action, moves to the corresponding negative one, if there is one. Otherwise the normalization fails !

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

The interaction may not terminate

ǫ {0} {0} 00 000 00 . . . ⋮

• 00. . .

{0} ǫ {0} 00 000 00 . . . 00 . . . ⋮ ⋯ ⋯ ⋯

At each step an action (+,0...,{0}) meets the action (−,0...,{0}) and the interaction continues. . .

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

(Orthogonality) Two designs D and E respectively based on ⊢ ξ and ξ ⊢, are said othogonal when [[D,E]] = {†}. D denotes the set of all designs E such that D and E are

• rthogonal.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

From the notion of orthogonality: an order relation

It is then possible to compare two designs according to their counter-designs: D ⪯ S iff D⊥ ⊂ S⊥ Example:

ǫ 2 28 29 280 290 {2} {8, 9} {0} {0} ∅ ∅ ǫ 2 28 29 280 † {2} {8, 9} {0} {0} ∅ D1 K1

D1 ⪯ K1

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Another reason for being more defined

D1 =

∅

⊢ 280 28 ⊢

∅

⊢ 290 29 ⊢ ⊢ 2 ǫ ⊢ D =

∅

⊢ 280 28 ⊢

∅

⊢ 290 29 ⊢ ⊢ 2

∅

⊢ 450 45 ⊢ ⊢ 4

∅

⊢ 660 66 ⊢ ⊢ 6 ǫ ⊢

D1 ⪯ D

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Formulas/Types

(Behaviour) Let E be a set of designs on same base, E is a behaviour if E = E⊥⊥. A behaviour is positive (resp. negative) if the base of its designs is positive (resp. negative).

Myriam Quatrini & Eugenia Sironi Records in Ludics

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Some connectives

Let M and N be two negative behaviours on the same base ξ ⊢. Let P and Q be some positive behaviour on the same base ⊢ ξ. M&N ∶= M ∩ N. P ⊕ Q ∶= (P ∪ Q)⊥⊥. P ⊗ Q = {(+,ξ,I ∪ J)w such that: (+,ξ;I)w ∈ D ∈ P (+,ξ,J)w ∈ E ∈ Q and I ∩ J = ∅}⊥⊥. The neutral element of the multiplicative conjonction 1, on the base ⊢ ξ, is the behaviour: {{(+,ξ,∅)}}⊥⊥.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Example

Design Proof

∅

⊢ 280 28 ⊢

∅

⊢ 290 29 ⊢ ⊢ 2 ǫ ⊢

∅

⊢ 1 ↓ 1⊥ ⊢

∅

⊢ 1 ↓ 1⊥ ⊢ ⊢↑ 1⊗ ↑ 1 ↓ (↑ 1⊗ ↑ 1)⊥ ⊢ The design D1 belongs to the negative behaviour M1 =↑ (↑ 1⊗ ↑ 1).

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Another types

Let M =↑ 1⊗ ↑ 1 & ↓↑ 1 & ↓↑ 1. Design Proof

∅

⊢ 280 28 ⊢

∅

⊢ 290 29 ⊢ ⊢ 2

∅

⊢ 450 45 ⊢ ⊢ 4

∅

⊢ 660 66 ⊢ ⊢ 6 ǫ ⊢

∅

⊢ 1 ↓ 1⊥ ⊢

∅

⊢ 1 ↓ 1⊥ ⊢ ⊢↑ 1⊗ ↑ 1

∅

⊢ 1 ↓ 1⊥ ⊢ ⊢↓↑ 1

∅

⊢ 1 ↓ 1⊥ ⊢ ⊢↓↑ 1 M⊥ ⊢ The design D belongs to the behaviour M. And it belongs also to the behaviour M1.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Incarnation and connectives

(Incarnation) Let B be a behaviour, D be a design in B. ∣D∣B = ⋂{R ⊂ D and R ∈ B} is the incarnation of D with respect to the behaviour B. A design D is material in a behaviour B if D = ∣D]B. The incarnation of a behaviour B, denoted ∣B∣, is the set of its material designs. Let M and N be two negative behaviours on the same base: M&N = M ∩ N ∣M&N∣ = ∣M∣ × ∣N∣ Let P and Q be two positive behaviours on the same base: P ⊕ Q = (P ∪ Q)⊥⊥ P ⊕ Q = P ∪ Q ∪ {Dai}

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Examples

Design Proof

∅

⊢ 280 28 ⊢

∅

⊢ 290 29 ⊢ ⊢ 2

∅

⊢ 450 45 ⊢ ⊢ 4

∅

⊢ 660 66 ⊢ ⊢ 6 ǫ ⊢

∅

⊢ 1 ↓ 1⊥ ⊢

∅

⊢ 1 ↓ 1⊥ ⊢ ⊢↑ 1⊗ ↑ 1

∅

⊢ 1 ↓ 1⊥ ⊢ ⊢↓↑ 1

∅

⊢ 1 ↓ 1⊥ ⊢ ⊢↓↑ 1 M⊥ ⊢

∅

⊢ 280 28 ⊢

∅

⊢ 290 29 ⊢ ⊢ 2 ǫ ⊢

∅

⊢ 1 ↓ 1⊥ ⊢

∅

⊢ 1 ↓ 1⊥ ⊢ ⊢↑ 1⊗ ↑ 1 M⊥

1 ⊢

The design D belongs to the behaviour M and to the behaviour M1. The design D is material w.r.t M but is not material w.r.t M1. The design D1 is material w.r.t. the behaviour M1.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

Records: an example

Suppose you want represent the record: ⎛ ⎜ ⎝ coord = (8,9) colour = green shape = cercle ⎞ ⎟ ⎠

The atomic date types are :

• the type : Colour = ⊕8

i=1 ↓↑ 1i based on ⊢ 4

for the colours, where the data “green” is encoded by i = 5;

• the type : Shape = ⊕3

i=1 ↓↑ 1i based on ⊢ 6

for the shapes, where the data “cercle” is encoded by i = 6;

• the type : Coord = ⊕∞

i=1 ↑ 1i ⊗ ⊕∞ k=1 ↑ 1k based on ⊢ 2

for the coordinates, where the data “(n,m)” is encoded by i = n and k = m.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

We then use the type R =↑ Coord & ↑ Shape & ↑ Colour, to type the datas : ⎛ ⎜ ⎝ coord = (8,9) colour = green shape = cercle ⎞ ⎟ ⎠ And the record (coord = (8,9);colour = green;shape = cercle) is the design D:

∅

⊢ 280 28 ⊢

∅

⊢ 290 29 ⊢ ⊢ 2

∅

⊢ 450 45 ⊢ ⊢ 4

∅

⊢ 660 66 ⊢ ⊢ 6 ǫ ⊢ which is material in R.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics Ludics as a Game Theory Interaction Connectives and incarnation

We also have that: ∣D∣↓Coord = D1; ∣D∣↓Colour = D2; ∣D∣↓Shape = D3. Where,

∅

⊢ 280 28 ⊢

∅

⊢ 290 29 ⊢ ⊢ 2 ǫ ⊢ D1

∅

⊢ 450 45 ⊢ ⊢ 4 ǫ ⊢ D2

∅

⊢ 660 66 ⊢ ⊢ 6 ǫ ⊢ D3

If we are interested only in one field, for example the colour of the object, we just need to consider its incarnation w.r.t. ↓ Colour.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

C-Ludics

Terui’s c-designs extend Girard’s designs as λ−terms generalised. At the place of (λxM)N → M[N/x] we write λ(x).M∣λ < N >→ M[N/x] We replace the couple λ/λ of λ-calculus with the couple a/a for any name a ∈ A A is a signature i.e. the pair (A,ar) where A is a set of names and ar ∶ A → N is a function which gives an arity to each name

• f A.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

C-designs

The c-designs are co-inductively defined as Negative c-designs are defined as x ∣ ∑a∈A a( ⃗ xa).Pa Positive c-designs are defined as

✠ ∣ Ω ∣ No∣a < N1,...,Nn >

where for each name a ∈ A Pa is a positive c-design, and for i = 0,...,n Ni is a negative c-design. ∑a∈A a( ⃗ xa).Pa corresponds to λa ⃗ xa.Pa No∣a < N1,...,Nn > corresponds to (λa ⃗ xa.Pa)N1 ...Nn

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

A cut is a c-design of the form ∑a∈A a(⃗ x).Pa∣a < N1,...,Nn >. A variable x occurring as No∣a < N1,...,x,...,Nn > in a c-design T is called an identity in T. If T = x T is itself an identity.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

The reduction relation

To eliminate cuts in c-designs we define a reduction relation. The reduction relation → is defined on positive c-designs as follows (∑a( ⃗ xa).Pa)a < ⃗ N >→ Pa[⃗ N/ ⃗ xa] We write P ⇓ Q if P reduces to Q in a finite numer of steps. (i.e. P →⋆ Q) and Q is neither a cut nor Ω (P converges to Q). Otherwise we write P ⇑ (P diverges).

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

Orthogonality

A c-design is closed if it has no free variables. A positive c-design P is atomic if it has got at most one free variable (called x0). A negative c-design N is atomic if it doesn’t have free variables. An atomic, positive c-design P and an atomic,negative one N are orthogonal and written P ⊥ N if substituting N to x0 in P we obtain a c-design which converges to ✠ i.e. P[N/x0] ⇓✠.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

Order relations

The observational ordering ≼ is the largest binary relation R

• ver c-designs such that

1

if ✠ R T then T =✠

2

if Ω R T then T is positive

3

if No∣a < N1,...,Nn > R T then T =✠ or T = Mo∣a < M1,...,Mn > and Ni R Mi for i = 0,...,n

4

If x R T then T = x.

5

if (∑a( ⃗ xa).Pa) R T then T = ∑a( ⃗ xa).Qa with Pa R Qa for every a ∈ A

Myriam Quatrini & Eugenia Sironi Records in Ludics

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Observational ordering corresponds to the preorder relation

• ver designs i.e.

Given two designs D,E that correspond to the c-designs d,e if D ≼ E then d ≼ e. Now let’s talk about a class of c-designs that corresponds to Girard’s original designs.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

Standard c-designs

A c-design T is standard if it is total (T ≠ Ω) The set of free variables of T, fv(T) is finite linear i.e. for any subdesign of T of the form N0∣a < N1,...Nn > the sets fv(N0),...fv(Nn) are pairwise disjoint. cut-free identity-free Standard c-designs correspond to original designs of Girard

• ver the signature RAM = (Pf(N),∣

∣) where ∣ ∣ gives the cardinality of I ∈ Pf(N)) and viceversa.

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Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

Strong separation

A standard c-design T admits strong separation if there is an anti-design [T c] such that T ⊥ [T c] and U ⊥ [T c] for any standard, ✠-free c-design U such that T ⋠ U. This means that there is an anti-design which characterizes T by orthogonality .

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

data-designs

In order to represent data (as for instance natural numbers) Terui introduces a set of negative c-designs called data-designs. We consider a signature A that contains a fixed unary name ↑. The set of data designs consists of negative c-designs, (with the only negative action ↑), coinductively defined as d =↑ (x).x∣a < d1,...,dn > where x ∉ fv(di) for i = 1,...,n. Fixed a 0-ary name zero and an unary name suc we have 0⋆ =↑ (x).x∣zero (n + 1)⋆ =↑ (x).x∣suc < n⋆ >

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

Properties of data-designs

Data designs satisfy some properties. data-designs are stardard every finite data design can be duplicated any pair of distinct data designs are incomparable with respect to ≼ hence d1 ≼ d2 implies d1 = d2 for any data designs d1,d2. all finite data-designs enjoy strong separation functions over data-designs are defined as follows. An n-any function design is a negative c-design F[x1,...,xn] such that fv(F) ⊆ {x1,...,xn} and [[F[d1,...,dn]]] is either a data-design or ↑ (x).Ω for any data design d1,...,dn

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

The result of strong separation over finite data designs can be expressed as follows Let d be a finite data-design. For any negative standard c-design N which is ✠-free (d)c

x0 ⊥ N ⇔ d ≼ N.

where given a finite data-design d, the positive atomic c-desing (d)c

x0 is defined adding a ✠ when the visit of d is terminated.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

Representation of records

We want to represent records using data designs, but this definition of data designs isn’t enough because the only negative action involved is ↑ (x), and records can contain more then one field. We need more negative actions. So we fix a signature A which contains a name a for every field of a record that we want to represent. We define the set of data-record designs that is an extension of the set of data-designs, adding a negative action a(x) for every field of a record that we want to represent.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

Data-record designs

The set of data-record designs consists of negative designs d,d1,...,dn is coinductively defined as follows d = ∑

c1,...,ck

ci(xi).xi∣a < d1,...,dn > The record coord : (3,4), colour : green, shape : circle is represented by the following data-record design coord(x).x∣(3,4)

2 <↑ (x).x∣3 0,↑ (x).x∣4 0 > +colour(y).y

∣green1 <↑ (x).x∣green0 > +shape(t).t∣circle

1 <↑ (x).x∣circle 0 >

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

What happened to properties?

Differently from data designs is possible to compare two data-record designs i.e. d ≼ e ⇏ d = e for any d,e data-record designs. For instance let e be a data-record designs which represents the record colour ∶ blue, shape ∶ circle, and d the data-record design which represents the record colour ∶ blue i.e.

d = colour(x).x∣blue1 <↑ (y).y∣blue0 > +shape(t).Ω and e = colour(x).x∣blue1 <↑ (y).y∣blue0 > +shape(t).t∣circle1 <↑ (z).z∣circle0 >

d ≼ e and d ≠ e.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

Functions over data-record designs

We extend the definition of function over data-designs to data-record designs as follows. An n-any function-record is a negative c-design F[x1,...,xn] such that fv(F) ⊆ {x1,...,xn} and [[F[d1,...,dn]]] is either a data-record design or ∑a( ⃗ xa).Ω for any data-record design d1,...,dn

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

Example 1 : Fshape[x]

Let’s define a function Fshape[x] which taken a data-record design d of the type coord & colour & shape isolates the part

• f d which represents the field shape.

d = coord(x).x∣(m,n)2 <↑ (z).z∣m0,↑ (z).z∣n0 > +colour(y).y∣c1 <↑ (z).z∣c0 > +shape(t).t∣s1 <↑ (z).z∣s0 > We set Fshape[x] = shape(t).(x∣shape < ∑a(x1,...,xn).t∣a < x1,...,xn >>) Fshape[x] is not identity-free satisfies the definition of n-ary function-record.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

Example 2 : G[x]

G[x] is such that taken a data-record design d as above, controls the part of d which represent the field colour and if the colour is blue, it isolates the part of d which represents the field shape; otherwise it diverges i.e. If c1 represents the colour blue then G[d] ⇓ shape(t).t∣s1 <↑ (z).z∣s0 > otherwise G[d] ⇓ shape(t).Ω. G[x] = shape(t).(x∣colour < blue(y).(x∣shape < Faxt >) G[x] is not linear, then is not standard. But this doesn’t create any problem with the definition of n-ary function-record. Indeed fv(G[x]) ⊆ {x} and for any data-record design d [[G[d]]] is a data-record design or the divergence (shape(t).Ω).

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

Strong separation for data-record designs?

Differently from data-record ones, finite data-record designs don’t always admit strong separation, i.e. there is a finite data-record design d such that there is no anti-design [dc] such that d ⊥ [dc] and N ⊥ [dc] for any standard, ✠-free c-design N such that d ⋠ N. For instance let d,N′,N′′ be data-record designs which represents respectively the records colour ∶ blue, shape ∶ circle colour ∶ blue shape ∶ circle then d ⋠ N′ and d ⋠ N′′ but there is no anti-design d′ such that d′ ⊥ d, d′ ⊥ N′ and d′ ⊥ N′′.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

strong separation and incarnation

We go back to data-designs, consider the behavior d⊥ and its incarnation ∣d⊥∣, and observe that (d)c

x0 ∈ (d)⊥ and for every

negative, standard, ✠-free c-design d′ if d ≼ d′ ((d)⊥ ⊆ (d′)⊥) then d′ ⊥ (d)c

• x0. So we wonder if d⊥ can be generated by the

c-design (d)c

x0 i.e. ∣d⊥∣ = {(d)c x0}✠. This would mean : if a

c-design T ∈ d⊥ ⇒ (d)c

x0 ≼ T.

This is not true. Indeed if we consider the data-design which represents the record coord ∶ (8,9) d =↑ (x).x∣8,9 <↑ (y).y∣80,↑ (y).y∣90 > According to the definition (d)c

x0 = x0∣ ↓< {8,9}(x1,x2).x1∣ ↓< {80}.x2∣ ↓< {90}.✠>>>.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

The c-design e = x0∣ ↓< {8,9}(x1,x2).x2∣ ↓< {90}.x1∣ ↓< {80}.✠>>> is

• rthogonal to d but (d)c

x0 ⋠ e.

(d)c

x0 and e correspond to two designs which only differs for the

• rder of some rules.

Myriam Quatrini & Eugenia Sironi Records in Ludics

Ludics: main objects; interesting properties Records in c-Ludics C-Ludics an extension of Ludics and λ-calcul Records with data-designs

It looks possible for d⊥ to be generated by a certain (finite) number of c-designs (the number depends on the structure of d1,...,dn if d =↑ (x).x∣a < d1,...,dn >). We are looking for a weaker version of strong separability for data-record designs. Maybe the following? Given a finite data-record design d there are m positive c-designs (d)1

x0,...,(d)m x0 such that (d)i x0 ⊥ d for

i = 1,...,m and for every standard ✠-free c-design d′′ such that d ⋠ d′′ there is i ∈ {1,...,m} such that (d)i

xo ⊥ d′′

Myriam Quatrini & Eugenia Sironi Records in Ludics