Maxime Amblard

Formal modelling of dialogue: how words interact (not only in the dictionary!) EMLex lecture series/S éminaire de l'ATILF

Montague perspective

• intentional logic

• generalized quantifiers (most, few, three, ...)

• first model of the scope ambiguity of quantifiers

• definition of a rigorous syntax semantics interface 15/61

Montague perspective

• intentional logic

• generalized quantifiers (most, few, three, ...)

• first model of the scope ambiguity of quantifiers

• definition of a rigorous syntax semantics interface

• task1 definition of a fragment of English [with categorical grammars]

• task2 specification of the meaning of lexical items [with λ-calcul]

• task3 exhibit how to build semantics representations [with functional application and β-r éduction] 15/61

[Task1] Categorial Grammars

• categories explaining how the word can be compose to build complex structures 16/61

[Task1] Categorial Grammars

• categories explaining how the word can be compose to build complex structures

• differentiate what is left from what is right 16/61
[Task1] Categorial Grammars [Task2] λ-calcul

Functional view of the computation:

• variables are linked by the λ λx.man(x)

• they are markers in formulas

• two terms are composed by the functional application

• β-conversion, α-conversion and η-expansion perform the calculus ((λx.man(x))@(vincent)) man(Vincent)

[Task3] Curry-Howard Isomorphism

Vincent aime Marie (S) S love(vincent, marie) Vincent (NP) NP λP.P@vincent aime Marie (VP) NP \ S λz.love(z, marie) aime (TV) NP \ S / NP λX .λz.X @λx.love(z, x) Marie (NP) NP λP.(P@marie)

18/61

But there is still much to do But there is still much to do But there is still much to do But there is still much to do • inter-sentencial anaphora A man walks in the park. He whistle.

∃x.(man x ∧ walk in the park x) ∧ (whistle x)

• Context Change Potential (CCP) [Heim1983]
The interpretation is done in context and the context is modified by the interpretation.

• Discourse Representation Theory (DRT) [Kamp1981]/File Change Semantics (FCS) [Heim1982] intermediate levels between representation and truth values

• Context Change Potential (CCP) [Heim1983]
The interpretation is done in context and the context is modified by the interpretation.

• • types :

MG TTDL s , d : o Ω γ → (γ → o) → o 24/61
• types :

MG TTDL s , d : o Ω γ → (γ → o) → o n : ι → o ι → Ω np : (ι → o) → o (ι → Ω) → Ω 24/61
Type and sentences combinaison

• types : MG TTDL s , d : o Ω γ → (γ → o) → o n : ι → o ι → Ω np : (ι → o) → o (ι → Ω) → Ω
• composition to build a discourse: update TTDL λDSeφ.De(λe .Se φ)

24/61

Type and sentences combinaison

• types : MG TTDL s , d : o Ω γ → (γ → o) → o n : ι → o ι → Ω np : (ι → o) → o (ι → Ω) → Ω
• composition to build a discourse: update TTDL λDSeφ.De(λe .Se φ)

24/61

Type and sentences combinaison

• types : MG TTDL s , d : o Ω γ → (γ → o) → o n : ι → o ι → Ω np : (ι → o) → o (ι → Ω) → Ω
• composition to build a discourse: update TTDL λDSeφ.De(λe .Se φ)

• context manipulation (variable list): A man i walks in the park. He i whistles . A man i walks in the park. He i whistles. Anaphore discursive en TTDL cont.

• "::" context update ι → γ → γ • "sel" pick
A man i walks in the park. He i whistles. Anaphore discursive en TTDL cont.

A man i walks in the park. He i whistles. • Linguistic studies of mental diseases [START_REF] Chaika | A linguist looks at "schizophrenic" language[END_REF]) and [START_REF] Fromkin | A linguist looks at "a linguist looks at 'schizophrenic language[END_REF] • Pragmatic discontinuities in performing verbal interaction (Trognon and Musiol 1996)

• Discontinuities definitive [START_REF] Musiol | Incoherence et formes psychopathologique dans l'interaction verbale schizophrenique[END_REF]): pathological use of discourse planning for patients with schizophrenia (paranoid) under interdisciplinary approaches

• Building of a linguistic resource on mental pathology

• semi-supervised interviews • neuro-cognitive tests • double eye-trackers
• Epistemological and philosophical studies (norm, madness, rationality)

• Identify these purposes with:

• formal models • A lot of administrative steps:

• CPP of the area of the medical institution (including a finalise description of the all protocol) • CNIL

• Data should not be use for/against the patient • Patient involvement (significant loss of participation >55%)

• Heavy protocol 39/61

Semi-Supervised Interview Schizophrenic / Psychologist B 2 How long have you been living there?

A 3 For five years.

58/61

Example 1/2 A 1 I live in Paris.

B 2 How long have you been living there?

A 3 For five years.

A 1 . q B 2 . a A 3 ce = λc. A 3 B 2 ( A 1 c) ce Thanks!

→ β A 3 B 2 ( A 1 ce) 58/61

••••••

  Discourse Representation Theory (DRT) [Kamp1981]/File Change Semantics (FCS) [Heim1982] intermediate levels between representation and truth values • Dynamic Predicate Logic (DPL) [Groenendijk1991] • Type Theoretic Dynamic Logic (TTDL) (de Groote 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus • Primitive types • ι: individual / entity • o: proposition / truth value • γ: left context 23/61 Type Theoretic Dynamic Logic • Type Theoretic Dynamic Logic (TTDL) (de Groote 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus • Primitive types • ι: individual / entity • o: proposition / truth value • γ: left context Type Theoretic Dynamic Logic (TTDL) (de Groote 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus • Primitive types • ι: individual / entity • o: proposition / truth value • γ: left context s = o ? Type Theoretic Dynamic Logic (TTDL) (de Groote 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus • Primitive types • ι: individual / entity • o: proposition / truth value • γ: left context Type Theoretic Dynamic Logic (TTDL) (de Groote 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus • Primitive types • ι: individual / entity • o: proposition / truth value • γ: left context s = o ? Type Theoretic Dynamic Logic (TTDL) (de Groote 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus • Primitive types • ι: individual / entity • o: proposition / truth value • γ: left context s = o ? Type Theoretic Dynamic Logic (TTDL) (de Groote 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus • Primitive types • ι: individual / entity • o: proposition / truth value • γ: left context s = o ? Type Theoretic Dynamic Logic (TTDL) (de Groote 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus • Primitive types • ι: individual / entity • o: proposition / truth value • γ: left context → (γ → o) → o λeφ.∃x. candidate(x) ∧ φ(x :: e)

∧

  update TTDL = λABeφ.Ae(λe .Be φ) ∃ λPeφ.∃x.Px(x :: i walks in the park. He i whistles. NP λQeφ.∃x.(man x ∧ Qx(x :: e)φ) λQ.∃x.(man x ∧ Qx) N→NP a λPQeφ.∃x.Pxe(λe .Qx(x :: e )φ) λPQ.∃x.(Px ∧ Qx) N man λxeφ.man x ∧ φe λx.man x NP λQeφ.∃x.(man x ∧ Qx(x :: e)φ) λQ.∃x.(man x ∧ Qx) N→NP a λPQeφ.∃x.Pxe(λe .Qx(x :: e )φ) λPQ.∃x.(Px ∧ Qx) N man λxeφ.man x ∧ φe λx.man x S λeφ.∃x.(man x ∧ walk in the park x ∧ φ(x :: e)) ∃x.(man x ∧ walk in the park x) NP a man λQeφ.∃x.(man x ∧ Qx(x :: e)φ) λQ.∃x.(man x ∧ Qx) NP→S walk in the park λS.S(λxeφ.walk in the park x ∧ φe) λS.S(λx.walk in the park x) S λeφ.∃x.(man x ∧ walk in the park x ∧ φ(x :: e)) ∃x.(man x ∧ walk in the park x) NP a man λQeφ.∃x.(man x ∧ Qx(x :: e)φ) λQ.∃x.(man x ∧ Qx) NP→S walk in the park λS.S(λxeφ.walk in the park x ∧ φe) λS.S(λx.walk in the park x)

S

  λeφ.(whistle(sel e) ∧ φe) ∃x.(whistle x ∧ x =?) NP he λPeφ.P(sel e)eφ λP∃x.(Px ∧ x =?) NP→S whistle λS.S(λxeφ.whistle x ∧ φe) λS.S(λx.whistle x)

S

  λeφ.(whistle(sel e) ∧ φe) ∃x.(whistle x ∧ x =?) NP he λPeφ.P(sel e)eφ λP∃x.(Px ∧ x =?) NP→S whistle λS.S(λxeφ.whistle x ∧ φe) λS.S(λx.whistle x) 28/61

  λeφ.∃x.(man x ∧ walk x ∧ whistle(sel(x :: e)) ∧ φ(x :: e)) ∃x.(man x ∧ walk x) ∧ ∃x.(whistle x ∧ x =?) -1 ∧ -2 -1 λeφ.∃x.(man x ∧ walk x ∧ φ(x :: e)) ∃x.(man x ∧ walk x) -2 λeφ.(whistle(sel e) ∧ φe) ∃x.(whistle x ∧ x =?) 29/61

  λeφ.∃x.(man x ∧ walk x ∧ whistle(sel(x :: e)) ∧ φ(x :: e)) ∃x.(man x ∧ walk x) ∧ ∃x.(whistle x ∧ x =?) -1 ∧ -2 -1 λeφ.∃x.(man x ∧ walk x ∧ φ(x :: e)) ∃x.(man x ∧ walk x) -2 λeφ.(whistle(sel e) ∧ φe) ∃x.(whistle x ∧ x =?)

⇒••

  (deprecation of cognitive flexibility and inhibition).• Oculomotor behavior (double Eye-Trackers) The representation need more than logical semanticsRepresentationUse of SDRT + thematic boxes (grey ones) an dernier euh (→) j'savais pas comment faire j' étais perdue et pourtant j'avais pris mes m édicaments j'suis dans un état vous voyez m ême ma bouche elle est s èche j'suis dans un triste état I didn't know what to do. I was lost. V83 Vous êtes quand m ême bien (↑) G84 J'pense que ma t ête est bien mais on croirait à moiti é (↓) la moiti é qui va et la moiti é qui va pas j'ai l'impression de c ¸a vous voyez (↑) V85 D'accord G86 alors c'est la conscience peut être la conscience est ce que c'est c ¸a (↑) V87 Vous savez c ¸a arrive à tout le monde d'avoir des moments biens et des moments o ù on est perdu Everybody is lost at times. G88 Oui j'ai peur de perdre tout le monde Yes I am afraid I lose everybody. V89 Mais ils vont plut ôt bien vos enfants (↑) G90 Ils ont l'air ils ont l'air mais ils ont des allergies ils ont (→) mon petit fils il s'est cass é le bras à l' école tout c ¸a type of frames: γ findv : γ → v × (v → γ) type of frames: γfindv : γ → v × (v → γ) → v × (v → γ) answer av = v × (v → γ) → γ

•

  Increase the phenomena analyzed in SLAMtk Especially work on syntax and lexical statistics • Try DDN approaches on the SLAM corpus Need more ressources in French • Deeply study the human annotations of the corpus • Increase the coverage of the corpus in volume and number of pathologies studied Collection of data at the Montperrin Hospital of Aix-En-Provence • Define remedial help process • Refine the analysis of dysfunction, opening towards a cognitive interpretation and give more complex context for the interpretation • Defining robust semantics grammars for TTDL • Definition of a TTDL for dialogue framework Ongoing work on questions and answers with Maria Boritchev • (French translation of Fracas) 61/61

But there is still much to do

	• Proper name • Proper name
	Vincent vs λP.P@Vincent Vincent vs λP.P@Vincent
	• Pronoms • Pronoms
	x x
	• scope quantifier ambiguity • scope quantifier ambiguity
	Everyboxer loves a woman Everyboxer loves a woman
	• verbal ellipse • verbal ellipse
	Mary went to the party and Vincent to Mary went to the party and Vincent to
	• gapping
	Mary loves Vincent, and Honey-Bunny Pumpkin
	19/61 19/61

But there is still much to do

	• Proper name
	Vincent vs λP.P@Vincent
	• Pronoms
	x
	• scope quantifier ambiguity
	Everyboxer loves a woman
	• verbal ellipse
	Mary went to the party and Vincent to
	• gapping
	Mary loves Vincent, and Honey-Bunny Pumpkin
	• presupposition
	19/61

But there is still much to do

	• Proper name
	Vincent vs λP.P@Vincent
	• Pronoms
	x
	• scope quantifier ambiguity
	Everyboxer loves a woman
	• verbal ellipse
	Mary went to the party and Vincent to
	• gapping
	Mary loves Vincent, and Honey-Bunny Pumpkin
	• presupposition
	• inter-sentencial phenomanas
	19/61

But there is still much to do

	• Proper name
	Vincent vs λP.P@Vincent
	• Pronoms
	x
	• scope quantifier ambiguity
	Everyboxer loves a woman
	• verbal ellipse
	Mary went to the party and Vincent to
	• gapping
	Mary loves Vincent, and Honey-Bunny Pumpkin
	• presupposition
	• inter-sentencial phenomanas
	• ...
	19/61

And on the other side of the diagram: LANGUAGE-WORLD

	Limits of montageovian approaches
	• Donkey sentences
	We add information:	Every farmer who owns a donkey beats it
	• time, tense and aspect: Allen logic, Reichenbach, van Benthem
	• event, Davidson
	• plurial
	• modalities
	• ...
		20/61

(∃x∃y.(farmer x ∧ donkey y ∧ own x y )) → beat x y

corpus hard to constitute [Amb. et al journ ée ATALA 2014]

	SLAM SLAM A relatively large corpus A
			La Rochelle La Rochelle		Lyon Lyon			Total Total
	• Corpus • Corpus	Schizophrenics Schizophrenics	♂ 15 ♂ 15	♀ 3 ♀ 3	tot 18 tot 18	♂ 22 ♂ 22	♀ 9 ♀ 9	tot 31 tot 31	49 49
		Controls Controls	15 15	8 8	23 23	4 4	4 4	8 8	31 31
		Total Total	30 30	11 11	41 41	26 26	13 13	39 39	80 80
	• Formalization • Formalization 31 575 speeches / 375 000 words
	• question the cognitive reality of semantico-pragmatic models,
	• automatically identify unusual uses of the language La Rochelle	Lyon
	• Epistemology # speeches • Epistemology • question the normative concepts of rationality and logicity # words # speeches S 3 863 11 145 46 859 119 762 4 062 4 433 66 725 79 081 # words T 7 282 72 903 371 12 356
	• study interpretation under linguistic interaction, and the status of implicit norms P + S 3 819 11 517 30 293 138 571 4 098 4 480 33 686 37 842 P + T 7 698 108 278 382 4 156
	Total	22 662	258 333			8 913		116 923
									34/61 38/61 38/61

• Type Theoretic Dynamic Logic (TTDL) (de Groote 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus

From Montague to Dynamic Semantics Dynamic Semantics

• Context Change Potential (CCP) [Heim1983] The interpretation is done in context and the context is modified by the interpretation. • How to use these representations?

Summary Summary

• Usefulness of these representations?

30/61

Summary

• Semantics

• Compositionality (Frege)

• Logical approaches (Montague)

(1) John loves Mary love(John, Mary)

• How to use these representations?

• Usefulness of these representations?

(2) If a farmer owns a donkey, he beats it

Summary

• Semantics

• Compositionality (Frege)

• Logical approaches (Montague)

(1) John loves Mary love(John, Mary)

• How to use these representations?

• Usefulness of these representations?

(2) If a farmer owns a donkey, he beats it ∃x(∃y.(farmer x ∧ donkey y ∧ own x y ) → beat x y )

Summary

• Semantics

• Compositionality (Frege)

• Logical approaches (Montague)

(1) John loves Mary love(John, Mary)

• How to use these representations?

• Usefulness of these representations?

(2) If a farmer owns a donkey, he beats it ∃x(∃y.(farmer x ∧ donkey y ∧ own x y ) → beat x y )

• Cognitive reality, conceptual reality? ...

30/61

Can we understand madness?

The SLAM project -Schizophrenia and Language: Analyse and Modelling A131 Yes you think it's because they were in politics (↑) B132 Yes, so well yeah there was C too who committed murder, uh huh (→) he was there too, the one in B but well (→) it, that, it's because of politics again

Discontinuity example

The schizophrenic switch twice from a theme to another one:

Discontinuity example

The schizophrenic switch twice from a theme to another one:

• politic death (symbolic)

• death (literal)

Discontinuity example

The schizophrenic switch twice from a theme to another one:

• politic death (symbolic)

• death (literal)

The two themes are relied but they express two different realities.

Two conjectures

• Schizophrenics are logically consistent. Hence discontinuities appear in the process which produce the representation, thus at the pragmatic level.

• Underspecification (ambiguity) plays a central role

Two conjectures

• Schizophrenics are logically consistent. Hence discontinuities appear in the process which produce the representation, thus at the pragmatic level.

• Underspecification (ambiguity) plays a central role Slogan: "A choice is never a definitive one!"

Two conjectures

• Schizophrenics are logically consistent. Hence discontinuities appear in the process which produce the representation, thus at the pragmatic level.

• Underspecification (ambiguity) plays a central role Slogan: "A choice is never a definitive one!" Phonological, morphological, lexical, discourse, ...

SDRT [Asher


Être Faut

Parce Gagnant

Quand Qu'autrement

Est On

Fini Perdant C'est

Quoi

↓) ET C'EST VRAIMENT TR ÈS TR ÈS COMPLIQU É (→) LA POLITIQUE C'EST QUELQUE CHOSE QUAND ON S'EN OCCUPE

Oh yeah (↑) and complicated (↑) and it's really very very complicated (→) politics, it's really something when you get into it, have to win or else when you lose, well, you're finished (↓

Conversation example (english only)

B124 Oh yeah (↑) and complicated (↑) and it's really very very complicated (→) politics, it's really something when you get into it, have to win or else when you lose, well, you're finished (↓)

A125 Yes B126 JCD is dead, L is dead, P is dead uh (...)

A127 So you think they're dead because they lost (↑)

B128 No they won but if they're dead, it's their disease well it's it's (→)

A129 Yeah it's because they had a disease, it's not because they were in politics (↑) B130 Yes I mean (→)

A131 Yes you think it's because they were in politics (↑) B132 Yes, so well yeah there was C too who committed murder, uh huh (→) he was there too, the one in B but well (→) it, that, it's because of politics again • A lot of administrative steps :

• CPP of the area of the medical institution (including a finalise description of the all protocol) • CNIL

• Data should not be use for/against the patient • Patient involvement (significant loss of participation >55%)

• Heavy protocol

Semi-Supervised Interview Schizophrenic / Psychologist They are thematic islands

• Schizophrenics are logically consistent. Hence discontinuities appear in the process which produce the representation, thus at the pragmatic level.

• Underspecification (ambiguity) plays a central role Rise without attachement 1/2 G82 l'an dernier euh (→) j'savais pas comment faire j' étais perdue et pourtant j'avais pris mes m édicaments j'suis dans un état vous voyez m ême ma bouche elle est s èche j'suis dans un triste état I didn't know what to do. I was lost. V83 Vous êtes quand m ême bien (↑) G84 J'pense que ma t ête est bien mais on croirait à moiti é (↓) la moiti é qui va et la moiti é qui va pas j'ai l'impression de c ¸a vous voyez (↑) V85 D'accord G86 Ou alors c'est la conscience peut être la conscience est ce que c'est c ¸a (↑) V87 Vous savez c ¸a arrive à tout le monde d'avoir des moments biens et des moments o ù on est perdu Everybody is lost at times. G88 Oui j'ai peur de perdre tout le monde Yes I am afraid I lose everybody. V89 Mais ils vont plut ôt bien vos enfants (↑) G90 Ils ont l'air ils ont l'air mais ils ont des allergies ils ont (→) mon petit fils il s'est cass é le bras à l' école tout c ¸a Why ?

• Study conventionnal vs pathological uses

• rebuild more consistent speeches (syntactically)

Results

Distribution of disfluencies in an interview 27/45

Results (% of disfluencies) S C S+C P+S P+C P Corpus Lyon by speeches 0, 5417 0, 5589 0, 545 0, 1400 0, 1513 0, by words 0, 032 0, 0168 0, 0288 0, 0144 0, 0138 0, Corpus La Rochelle by speeches 0, 7117 0, 484 0, 5842 0, 3338 0, 7369 0, by words 0, 0595 0, 0468 0, 0524 0, 0421 0, 0496 0,

Results (% of disfluencies)

S C S+C P+S P+C P Corpus Lyon by speeches 0, 5417 0, 5589 0, 545 0, 1400 0, 1513 0, by words 0, 032 0, 0168 0, 0288 0, 0144 0, 0138 0, Corpus La Rochelle by speeches 0, 7117 0, 484 0, 5842 0, 3338 0, 7369 0, by words 0, 0595 0, 0468 0, 0524 0, 0421 0, 0496 0, 0,04 0,68 0,11 0,73 0,15 0,76 0,14 0,74 S 0,05 0,69 0,06 0,70 0,07 0,72 0,08 0,71 0,06 0,71 0,10 0,72 P 0,02 0,64 0,06 0,68 FR : ratio of the number of lemmas to the total number of forms (T)TR : ratio of the number of lemmas to the total number of different forms (types)

Organization of 3 human annotation campaigns

• Identification of decisive discontinuities

• SDRT representation

Results

• Huge difficulties for discontinuities

(

  B124) Oh ouais (↑) et pis compliqu é (↓) et c'est vraiment tr ès tr ès compliqu é (→) 'est quelque chose quand on s'en occupe faut être gagnant parce qu'autrement quand on est perdant c'est fini quoi (↓) J. C. D. est mort, L. est mort, P. est mort euh (...) Ils sont morts parce qu'ils ont perdu à votre avis (↑) Non ils gagnaient mais si ils sont morts, c'est la maladie quoi c'est c'est (→) Non ils gagnaient mais si ils sont morts, c'est la maladie quoi c'est c'est (→) Ouais c'est parce qu'ils étaient malades, c'est pas parce qu'ils faisaient de la politique (↑) Si vous pensez que c'est parce qu'ils faisaient de la politique (↑) Oui tiens oui il y a aussi C. qui a accompli un meurtre l à (→) il était pr ésent lui aussi qui est à B. mais enfin (→) c'est encore à cause de la politique c ¸a Oui tiens oui il y a aussi C. qui a accompli un meurtre l à (→) il était pr ésent lui aussi qui est à B. mais enfin (→) c'est encore à cause de la politique c ¸a Oui tiens oui il y a aussi C. qui a accompli un meurtre l à (→) il était pr ésent lui aussi qui est à B. mais enfin (→) c'est encore à cause de la politique c ¸a

••

  Limit human actions : → Disfluencies, Distagger (CONSTANT et DISTER 2010) → POS and lemmas, MElt (DENIS et SAGOT 2009) Limit human actions : → Disfluencies, Distagger (CONSTANT et DISTER 2010) → POS and lemmas, MElt (DENIS et SAGOT 2009)

•••••••B

  ι : individual / entity • o : proposition / truth value • γ : left context s = o Type Theoretic Dynamic Logic• Type Theoretic Dynamic Logic (TTDL) (DE GROOTE 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus • Primitive types • ι : individual / entity • o : proposition / truth value • γ : left context Type Theoretic Dynamic Logic (TTDL) (DE GROOTE 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus • Primitive types • ι : individual / entity • o : proposition / truth value • γ : left context Type Theoretic Dynamic Logic (TTDL) (DE GROOTE 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus • Primitive types • ι : individual / entity • o : proposition / truth value • γ : left context Type Theoretic Dynamic Logic (TTDL) (DE GROOTE 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus • Primitive types • ι : individual / entity • o : proposition / truth value • γ : left context Type Theoretic Dynamic Logic (TTDL) (DE GROOTE 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus • Primitive types • ι : individual / entity • o : proposition / truth value • γ : left context Type Theoretic Dynamic Logic (TTDL) (DE GROOTE 2006) : montagovian framework, with dynamicity which add continuation in λ-calculus • Primitive types • ι : individual / entity • o : proposition / truth value • γ : left context → (γ → o) → o λeφ.∃x. candidate(x) ∧ φ(x :: e)• a feature v • type of frames : γ findv : γ → v × (v → γ) type of frames : γ findv : γ → v × (v → γ) How long have you been living there ? A For five years.A 1 . q B 2 . a A 3 ce = λc. A 3 B 2 ( A 1 c) ce→ β A 3 B 2 ( A 1 ce)

corpus hard to constitute [Amb. et al journ ée ATALA 2014]

	A relatively large corpus A
			La Rochelle La Rochelle		Lyon Lyon			Total Total
		Schizophrenics Schizophrenics	♂ 15 ♂ 15	♀ 3 ♀ 3	tot 18 tot 18	♂ 22 ♂ 22	♀ 9 ♀ 9	tot 31 tot 31	49 49
		Controls Controls	15 15	8 8	23 23	4 4	4 4	8 8	31 31
		Total Total	30 30	11 11	41 41	26 26	13 13	39 39	80 80
	31 575 speeches / 375 000 words
		La Rochelle						Lyon
		# speeches	# words		# speeches	# words
	S T	3 863 11 145 7 282	46 859 119 762 72 903	4 062 4 433 371	66 725 081 12 356
	P + S 3 819 11 517 P + T 7 698	30 293 138 571 108 278	4 098 4 480 382	33 686 842 4 156
	Total	22 662	258 333			8 913		116
									12/45

-Discontinuities annotation guide -SDRT Talking with patient with schizophrenia [AMR TALN 2011] [AMR Evol. Psychiatrique 2012] [AMR congr ès de linguistique romane 2013] [AMR Dialogue, Rationality and Formalism Springer 2014] [AMR Philosophie et langage 31 2014]

	SLAM Modeling Modeling
	Use of SDRT + thematic boxes (grey ones) Use of SDRT + thematic boxes (grey ones)
	Two interlocutors, thus two (spontaneous) views on the exchange.
	Records	A 1 A 1	narr narr	A 3 A 3
	Discourse interpretation by
	transcription guide normal subject el el		el el	Schizophrenic
	? Corpus (3 rd person) hypothesis : pragmatic correctness -Disfluencies B 2 B 2	-POS (1 st person) -Stemming pragmatic incorrectness B 4 B 4
	⇓				-Syntactic parsing ⇑
	semantics incorrectness	hypothesis : semantic correctness question question
	• Brain activity (EEG) contradictory contents : look like a contradiction	A 5 A 5	coherent content : possibility of interpretation B 6 rep B 6 rep
	⇒ The representation need more than logical semantics
					14/45 16/45

• Oculomotor behavior (double Eye-Trackers)

Processing dialogue : access to subparts of the interaction for update.

• Defining robust semantics grammars for TTDL • Definition of a TTDL for dialogue framework Ongoing work on questions and answers with Maria Boritchev • (French translation of Fracas) 45/45

Thanks !

OUAIS (↑) ET PIS COMPLIQU É (

Discontinuity example

The schizophrenic switches twice from a theme to another one : 11/45

Discontinuity example

The schizophrenic switches twice from a theme to another one :

• politic death (symbolic)

• death (literal)

Discontinuity example

The schizophrenic switches twice from a theme to another one :

• politic death (symbolic)

• death (literal)

The two themes are relied but they express two different realities.

Two conjectures

• Schizophrenics are logically consistent.

Hence discontinuities appear in the process which produce the representation, thus at the pragmatic level.

• Underspecification (ambiguity) plays a central role Slogan : "A choice is never a definitive one !"

Two conjectures

• Schizophrenics are logically consistent. Hence discontinuities appear in the process which produce the representation, thus at the pragmatic level.

• Underspecification (ambiguity) plays a central role Slogan : "A choice is never a definitive one !" Phonological, morphological, lexical, discourse, ... Why ?

• Study conventionnal vs pathological uses

• rebuild more consistent speeches (syntactically)

Results :

• Patients with schizophrenia produce slightly more disfluencies

• But they have no specific behavior for POS / lemmas Distagger f-score : 95,5 %, precision : 95,3 %, recall : 95,8 % (CONSTANT et DISTER 2010)

1. 'euh'

(1) moi c ¸a m'est presque plus euh difficile et euh anti-naturel de parler

Repeat

(2) j' arrive à être à être concentr ée quand il faut faire quelque chose

self-corrections

(3) enfin je sais pas trop le les termes 4. starters • Impossibility of disidentification

• Task with a small context : randomise speeches • Inability to anonymize the history and the geography

• Patient reality

• Formal analysis of language = define a standard • Deviate = dysfunction • But, every speaker is confronted daily with language disorders from healthy people • The diagnosis can not suffer from approximations A current Extension : developing complex context (REBUSCHI 2015) :

• Discursive context that depends on interaction and dynamics of interaction ;

• Doxatic context which takes up all the presuppositions, beliefs about the world and the projection of the beliefs of the speakers ;

• Pragmatic context that is interpreted by the situation of the interaction (the speaker who says "I" in playing a role does not say "me" to designate himself, but to designate the individual he plays) ;

• Material and social context in which the idea is to consider both the framework of interactions and all the influences which build it.

Toward a formal treatment

Type Theoretic Dynamic Logic

Dialogue Composer

Use of :

• TTDL for compositionality

• Frame Semantics for representation of the content 39/45

Dialogue Composer

Use of :

• TTDL for compositionality

• Frame Semantics for representation of the content

• Ongoing work : defining such a framework and apply it to the SLAM corpus

Utterances type

Example 1/2 A I live in Paris.

B How long have you been living there ?

A For five years.

Perspectives

Perspectives 1/2 • Define remedial help process

• Refine the analysis of dysfunction, opening towards a cognitive interpretation and give more complex context for the interpretation


Julien Doublet
email: julien.doublet@orange-ftgroup.com

Olivier Lepetit
email: olivier.lepetit@orange-ftgroup.com

CONTACTLESS PALMPRINT AUTHENTICATION USING CIRCULAR GABOR FILTER AND APPROXIMATED STRING MATCHING

Keywords: Biometrics, Image Segmentation, Pattern Recognition, Color and Texture

More and more research have been developed very recently for automatic hand recognition. This paper proposes a new method for contactless hand authentication in complex images. Our system uses skin color and hand shape information for hand detection process. Next, the palm is extracted and characterized by circular Gabor filter. Finally, the palm features are matched by a new normalized approximated string matching. The experimental results present an error rate lower than 1.2% with a population of 49 people. We show that our method improves the performances of two palmprint methods 2D-Gabor and PPOC on our database.

Introduction

Recently, biometrics recognition have attracted a lot of research and many works are reported in the literature. Biometrics technology is used to identify people from their physical or behavioral characteristics. These characteristics have three major specificities, they must be distinctive to differentiate each people, to be stable in the course of time and to be defined without a lot of constraints for users. So, various technologies were developed based on fingerprints, iris, face, voice, signature, gait or hand. Hand biometrics present many advantages compared to other biometrics technologies. Its characteristics are relatively stable and present a lot of discriminating features such as principal lines, wrinkles, ridges or hand shape. In addition, they present an high user acceptability and the hand can be obtained from low-resolution images with cheaper devices. Many algorithms have been defined for hand authentication based on palmprint [START_REF] Duta | Matching of palmprint[END_REF], [START_REF] Kong | Palmprint feature extraction using 2-d Gabor filters[END_REF], [START_REF] Wu | Wavelet Based Palmprint Recognition[END_REF] or hand-shape recognition [START_REF] Yoruk | Shape-Based Hand Recognition[END_REF], [START_REF] Kumar | Personal Verification using palmprint and geometry biometric[END_REF]. All these algorithms can be decomposed in four stages: the acquisition, the hand segmentation (and palm definition if necessary), the features extraction and the comparison with the reference. Classically, the acquisition is a hand scan. Then, the hand is segmented by a fixed threshold and the palm is generally defined from hand shape curvature [START_REF] Kong | Palmprint feature extraction using 2-d Gabor filters[END_REF]. Next, various methods have been proposed for recognition. They are Sobel and morphological operations, Fourier spectrum, 2D Gabor phase [START_REF] Kong | Palmprint feature extraction using 2-d Gabor filters[END_REF], wavelet signatures [START_REF] Wu | Wavelet Based Palmprint Recognition[END_REF], eigenpalm and eigenfingers features [START_REF] Ribaric | Fratric: A Biometric Identification System Based on Eigenpalm and Eigenfinger Features[END_REF], phase and orientation code [START_REF] Wu | Fusion of Phase And Orientation Information for Palmprint Authentication[END_REF], etc. The last papers present really good results in recognition rate using generally a comparison method based on Hamming distance adjusted following the features. In all these systems, the hand acquisition is in fact the principal limit. It is not hygienic, all the users touch the same glass, and some artifacts can be created during the acquisition according to the pressure of users on the plate glass. In this paper, we propose a contactless acquisition with a webcam to reduce these limitations and to get a quick and easy image acquisition. Hand detection process from these acquisitions is a complex task. Many methods are defined for hand detection in man-machine interactions like skin blob tracking, active contours, mean shift or condensation algorithm but provide an approximated detection of each finger. To increase the segmentation quality, we propose a method combining information of skin color and hand shape. This segmentation phase is described in section 2. In section 3, we defined a new feature extraction by circular Gabor filter and a novel palmprint matching using normalized approximated string matching. Section 4 contains some experimental results and section 5 provides the conclusions.

Preprocessing

Preprocessing is used to define the central part of the hand (the palm). The most part of preprocessing algorithm is decomposed in three steps. Firstly the hand is binarized by thresholding. Next, the contour of the hand is extracted and some key points are defined. Finally, from these points a coordinate system is defined and the palm is extracted. In our system, the architecture is kept but the methods are changed for a contactless system.

Accurate hand detection

In biometrics recognition systems, the first step consists on a definition of an accurate and fast segmentation process. In this section, we describe our hand detection method without tracking. This detection uses a fusion of skin color modeling and a robust active shape model.

Skin color segmentation

Classically the skin color is modeled by Bayes classifiers or Gaussian Mixture Models [START_REF] Vezhnevets | A survey on pixel-based skin color detection techniques[END_REF]. Contrary to these methods, our skin color is modeled by machine learning and for a good compromise between the execution time and the precision of detection, we use a neural network (NN). The NN entries are composed by three neurons, one for each color component of pixels in RGB domain. The NN output is the probability that a pixel is a skin pixel. The learning phase permits to model the skin tone in RGB domain. In parallel, a Principal Components Analysis on a skin pixels database defines a specific color space named skin space. After the learning, the NN can detect the pixels looking like skin. For each pixel of an image, the NN computes the probability that each pixel is a skin pixel. This process builds the probabilities map. To obtain a segmentation close to real time, a multiresolution process builds the probabilities map. This process computes the probabilities map at a lower resolution. Next the precision is increased in computing the probabilities only for the pixels near skin and non-skin transition in better resolution. Many methods are defined using the probabilities map to clearly identify the detected skin object (hand or face) and to get an accurate segmentation. The principal method consists in segmentation by histogram thresholding on the probabilities map [START_REF] Sural | Segmentation and histogram generation using the HSV color space for image retrieval[END_REF][START_REF] Singh | A robust skin color based face detection algorithm[END_REF]. In our approach, a morphological dilatation with a circular element of radius one pixel is carried out on the probabilities map. Next, the probabilities map is segmented by histogram thresholding. This threshold is fixed to 0.5: each pixel is considered as skin color if the neural network output value is more that 0.5 else it is classified as background tone. Finally, as our hand detection process will be used for biometrics recognition, we suppose that only one hand is presented to the system. So the region with the largest surface is preserved and considered as the hand. The regions with an important size and elongation are so conserved. They can be some fingers disconnected from the hand for the users wearing some rings. To reconnect these regions to the hand, each preserved region is approximated by an ellipse and we search in the direction of the major axis if the hand is close. If the region is near the hand, an adjusted patch is added to reconnect the finger to the hand. This patch has a width equals to the width of the minor axis and a length depending on the distance between the finger and the hand. It is oriented with the major axis direction and centered between the hand and the finger. This process is illustrated in Figure 1.

Color active shape model

The segmentation by skin color cannot carry out the task of hand detection in a robust way. So, a specific active shape model is defined to cancel this problem and to solve the two major difficulties of active shape model [START_REF] Cootes | Statistical models of appareance for computer vision[END_REF]. These problems are the contour initialization which must be close to the real form and the model convergence in the detection phase.

Classically, the form to detect is defined by a set of points: the landmarks. In training phase, the average form and the contour variations are computed by principal components analysis on an hand database annotated by these landmarks. In segmentation phase, the contour is initialized by the characteristic points of the hand: the five fingertip points and the four points located in the valley between two adjacent fingers. These points are calculated from the skin segmentation by distance analysis on the contour. Next, two other points are automatically added close to wrist from these points. The other landmarks defining with more precision the hand shape are disposed between those. Thus, the model X is obtained by the 11 initial points and M intermediate points between those. X is defined by X[0], X[1], . . . , X[(M + 1) × 10] where X[i] is the ith landmark. After the initialization phase, the model is deformed. To control the problem of model divergence implying that the model does not follow the real hand contours, a weight is applied to the deformations to limit the shape constraints. In addition, the gradient is computed in the skin color space by Di Zenzo algorithm. Next, this gradient is balanced by the coefficient of the probabilities map pixels. So the background objects do not perturb the contour evolution. The experiments show that a good compromise between the execution time and the detection precision is obtained by fixing M at 12. This complete detection process is illustrated on Figure 2(a-d).

Palm definition

After hand detection, it is necessary to define the palm independently of the distance between the hand and the capture device. Our extraction is based on hand dimensions and the palm extraction method described in [START_REF] Kong | Palmprint feature extraction using 2-d Gabor filters[END_REF]. In [START_REF] Kong | Palmprint feature extraction using 2-d Gabor filters[END_REF], two values are fixed: the distance between the points O1 and O2 and the palm size ||A1A2|| (Figure 3). These con-stant values in traditional recognition systems are defined here according to hand sizes. They are determined from the hand width, computed by the Euclidean distance between the points X[L1] and X[L2] where L1 and L2 are the indexes fixed after experiments at 30 and 125. Thus, ||O1O2|| and ||A1A2|| are defined by:

||O1O2|| = α||X[L1]X[L2]|| ||A1A2|| = β||X[L1]X[L2]|| (1)
Where α and β are the dimension coefficients chosen at 1/10 and 2/3, respectively. Next, the palm is resized with a fixed size N × N . The extracted palm contains some principal lines which can be determined by a specific palm line extraction. These lines are not unique to each individual, thus it is necessary to use the ridges and the secondary lines of the palm. This information cannot be extracted from the palm with images in low resolution, so a global palm characterization is suitable.

Robust circular Gabor filter

The Gabor filter is a well known tool in texture analysis.

The Gabor filters are a gaussian modulated by a wave. In oriented Gabor filters, this wave is an oriented complex sinusoidal signal. A typical formula expression is:

G(x, y) = g(x, y) × exp 2πif x cos θ + y sin θ (2)
where g(x, y) = 1 2πσ 2 exp

-(x 2 + y 2 ) 2σ 2 (3)
The parameters f and θ represent the frequency and the orientation of the sinusoidal signal respectively. g(x, y) is a Gaussian with the scale parameter σ. The Gabor filters have an important specific quality: they provide an accurate time and frequency location. In addition, they are robust to variation of brightness and contrast. So, these filters have been used with very high recognition rate in multiple biometrics applications with fingerprints, iris [START_REF] Daugman | High Confidence Visual Recognition of Persons by a Test of Statistical Independence[END_REF] or palmprint [START_REF] Kong | Palmprint feature extraction using 2-d Gabor filters[END_REF]. Recently, the performances of palmprint recognition using Gabor filters have been improved using a combination of different orientation filters [START_REF] Wu | Fusion of Phase And Orientation Information for Palmprint Authentication[END_REF]. In this paper, the palm features are extracted by a circular Gabor filter. This filter is a Gaussian modulated by a circular wave. So, this filter can be defined as [START_REF] Zhang | Invariant Texture Segmentation Via Circular Gabor Filters[END_REF]:

G c (x, y) = g(x, y) × exp 2πif x 2 + y 2 (4)
where f is the frequency of the circular Gabor filter and g(x, y) is defined by (3). The real part of oriented Gabor filter and circular Gabor filter with f = 0.3666, σ = 1.4045 and θ = 0 • (for oriented filter) are shown in Fig. 4. For more luminosity robustness, the filter is turned to zero DC (direct current). So, a robust circular filter of size (2n + 1) 2 can be explained by:

Ω c (x, y) = G c (x, y) - i=n i=-n j=n j=-n G c (i, j) (2n + 1) 2 (5)
This robust circular Gabor filter is convoluted with the palm image to product the palmprint features. Contrary to the methods of Zhang and al. [START_REF] Kong | Palmprint feature extraction using 2-d Gabor filters[END_REF][7], the features are not binarized so all information is kept. With all information, the variation between two data is preserved. So for example, two pixel values -0.001 and +0.001 that are really close are not considered very different contrary to a binary matching based on negative or positive values.

Normalized approximated string matching

From the features, many methods are employed like energies computation, Euclidean distance or some pixel based comparison (as Hamming distance) if the features are binarized. In our method, an approximated string matching [START_REF] Hoad | Video representations for effective retrieval from large collections[END_REF] in another context is used to match our features.

Approximated string matching

AMSA is based on Levenshtein distance which is a generalization of Hamming distance. Initially, the AMSA permits to compare two vectors in computing a scoring matrix with a scoring function. This function determines the similarity between two elements. The scoring matrix is filled following a match or a mismatch between the elements of the vectors. The matching score is the high value in scoring matrix. The AMSA is robust to translation, deletion, substitution or addition of elements in the vectors. The algorithm 1 presents the matrix filling where T is the threshold defining match or mismatch, f 1 (d) is the match scoring function and f 2 (d) is the mismatch scoring function. The algorithm have two vectors V 1 and V 2 of size s 1 and s 2 respectively in input. In Table 1, an example of matrix filling for two vectors V 1 = {8, 3, 11, 32, 21, 7, 6, 15, 20} and V 2 = {11, 31, 20, 7, 1, 15} with the parameters T = 1, f 1 (d) = 4 and f 2 (d) = 6 is shown. In this example, the scoring matching between V 1 and V 2 is equal to 16. The functions f 1 and f 2 can be complex and depend of the difference d in Algorithm 1.

for i ← 0 to s 1 do M (0, i) = 0 end for j ← 0 to s 2 do M (j, 0) = 0 end Matrix computation; for i ← 1 to s 1 do for j ← 1 to s 2 do d = ||V 1 (i) -V 2 (j)|| if d ≤ T then M (i, j) = M (i -1, j -1) + f 1 (d) else M (i, j) = M ax 0, M (i, j - 1) -f 2 (d), M (i -1, j) -f 2 (d) end end end

Palmprint matching

The matching procedure is applied on vectors. So, a palmprint feature P which is the real and imaginary part of the convolution of the robust circular Gabor filter with the palm is represented as a vector of size S = N × N × 2. We propose to modify the definition of the matrix computation to increase the weight of long matching sequence between two vectors. So, the definition of the matrix computation is changed by a function using the last calculated scores. To compare two palmprint features P 1 and P 2 , we used the Algorithm 2 where f 1 (x, d) = x × 2 + 1 is the match scoring function and f 2 (x, d) = x/2 is the mismatch scoring function. The coefficients functions are fixed for a very quick execution. Sure without constraints, the coefficients of these functions can be fixed following the difference d.

The maximum value of scoring match is 2 S+1 -1. So, a normalized distance between two palmprint features P 1 and P 2 , where the perfect matching is 0 and the poorest matching is 1, is defined by:

D(P 1 , P 2 ) = 1 -max 0<i≤S 0<j≤S M (i, j) /(2 S+1 -1) (6)
(i) -P 2 (j)|| if d ≤ T then M (i, j) = f 1 M (i -1, j -1), d else M (i, j) = f 2 M ax(M (i, j -1), M (i -1, j)), d

end end end 5 Experimentations

A specific database is made to validate our approach. All images of the database were acquired by a webcam Philips ToUcam Pro 740K with a size of 640 × 480. The database contains 490 images of hands, some of these presenting hands with rings, coming from 49 people. 10 images are acquired for each individual of the database. The capture restrictions are that the users must present their hands in front of the camera with theirs fingers separated. The users are not accustomed to the system, so in practice, a bad acquisition is automatically rejected by the system. On this database, the proposed method is compared to two systems of Zhang and al.: 2D Gabor filters [START_REF] Kong | Palmprint feature extraction using 2-d Gabor filters[END_REF] and PPOC [START_REF] Wu | Fusion of Phase And Orientation Information for Palmprint Authentication[END_REF]. For these two methods, the palms size is 128 × 128 to form DB1 and for our method the palms size is 32 × 32 named DB2 to limit the computation and the reference size. Figure 5 shows two examples of hand images with corresponding palm in DB1. Each palm in the database is compared with all the other palm images. To increase the robustness of the system and limit the impact of inaccurate hand detection, five palms are used as enrollment. So, to compare two palmprint features P 1 and P 2 , four features of the same user that the second feature are randomly added to the second one to form a test set {P 1 2 , • • • , P 5 2 } with P 1 2 = P 2 . The matching distance is a defined by the minimum match- ing score between the palmprint features:

D f (P 1 , P 2 ) = min 1≤i≤5 D(P 1 , P i 2 ) (7)
To limit the random impact, the total comparison process is done 10 times. So, the total number of comparison is 1202950. From the comparison process, the distance matching distribution curves are defined. These curves represent the frequency of matching score between the references coming from the same user named the genuine curve and coming from two different users named the impostor curve. These curves are showed in Figure 6 (a).

From a matching D f (P 1 , P 2 ), a threshold δ must be fixed to defined if the two compared features P 1 and P 2 come from the same users or not. So, if the matching score is lower than the threshold δ then the two features come from the same user else they are declared as coming from two different users. A comparison is incorrect if a user is accepted or rejected wrongly. Two rates characterize the accuracy of the system: the false acceptance rate (FAR) defined by the ratio of people authenticated wrongly and the false rejection rate (FRR) defined by the ratio of people rejected wrongly. The proposed method based on circular Gabor filter and string matching is evaluated with a set of parameters. So, the optimal parameters obtained are f = 0.0916, σ = 5.6179 and T = 0.25. The corresponding optimal comparison rate on DB2 is presented in Figure 6 (b) and shows an equal error rate (when FAR=FRR) equals to 1.2%. Finally, our system is compared to the methods of Zhang and al. 2D-Gabor filters [START_REF] Kong | Palmprint feature extraction using 2-d Gabor filters[END_REF] and Palmprint Phase and Orientation Code (PPOC) [START_REF] Wu | Fusion of Phase And Orientation Information for Palmprint Authentication[END_REF] with optimal parameters and five palms as enrollment. The optimal results presented in table 2 show that our system outperformed the methods of Zhang and al. in our database. The complete process of recognition (segmentation, characteristics extraction and matching) is carried out in less than 1 second on Pentium M with 1.60GHz.

Conclusions

In this paper, we present a new method of biometrics hand recognition for a contactless system. First of all, the hand segmentation is explained. It is carried out thanks to an integration of the skin color components and a shape model. Then, the authentication process based on circular Gabor filter and robust string matching is exposed. The features extraction permits to get the information in all directions in a limited time. Next, the modified string matching computes a distance robust to a limited bad palm extraction and noisy images. The complete process is validated after experiments on a database of 49 people. It shows an error rate of 1.2% with an execution time lower than 1 second. In order to manage space rotation of the hand, an invariant method with the perspective and the shear will have to be required. Moreover, the test database will have to be diversified and supplemented to confirm our approaches. These two tracks are currently the subject of our work.

Figure 1 .

 1 Figure 1. Fingers re-connection: (a) binarized map (b) patch adjustment (c) binarized map with patch

Figure 2 .Figure 3 .

 23 Figure 2. Hand detection (a) original image (b) skin segmentation (c) initial hand shape (d) final hand shape

Figure 4 .

 4 Figure 4. Oriented (a) and circular (b) Gabor function

Algorithm 2 :

 2 Palmprint matching algorithm Input: Two palmprint features P 1 [1, . . . , S] and P 2 [1, . . . , S], a threshold T , a match scoring function f 1 and a missmatch scoring function f 2 . Output: Scoring matrix M between the two sets. begin Initialization; Scoring matrix M [0 . . . S, 0 . . . S] for i ← 0 to S do M (0, i) = 0 end for j ← 0 to S do M (j, 0) = 0 end Matrix computation; for i ← 1 to S do for j ← 1 to S do d = ||P 1

Figure 5 .

 5 Figure 5. Two examples in our database and corresponding palm from DB1

Figure 6 .

 6 Figure 6. Authentication test results: distance matching distribution (a) and ROC (b)

Table 1

 1 Two Sets V 1 [1, . . . , s 1 ] and V 2 [1, . . . , s 2 ], a threshold T , a match scoring function f 1 and a mismatch scoring function f 2 . Output: Scoring matrix M between the two sets.

. Scoring matrix example Algorithm 1: Approximated string matching algorithm Input: begin Initialization; Scoring matrix M [0 . . . s 1 , 0 . . . s 2 ]

Table 2 .

 2 Comparison of systems proposed by Zhang and al. and our method


Thomas Gheno

Brian Gleeson

Kinetics of Al 2 O 3 -scale growth by oxidation and dissolution in molten silicate

Keywords: Oxide growth, Selective oxidation, Silicate deposit

During the high temperature oxidation of an alloy, the growth of a protective oxide scale selectively consumes one of the alloying elements. This can be accelerated if a molten deposit is present and dissolves the scale. The kinetics of Al 2 O 3 -scale growth and dissolution in Na silicate were studied by considering the Al mass balance at the alloy/scale and scale/silicate interfaces. The analysis was applied to a single-phase γ-NiCoCrAlY alloy exposed to a SiO 2 -Na 2 SO 4 deposit at 1100 • C, using an oxidation rate constant and effective Al diffusivity measured after deposit-free oxidation of the same alloy, and independent measurements of Al solubility and diffusivity in Na silicate. The simulated Al 2 O 3 -scale thickness and Al depletion profile matched well with the experimental data. The model was then used to assess Al 2 O 3 -scale failure on two alloys with particularly large Cr contents (27 and 35 at. %), which underwent internal oxidation when exposed to the SiO 2 -Na 2 SO 4 deposit.

Introduction

Alloys and coatings used at high temperatures withstand corrosion through the formation of slow-growing, compact and adherent oxide scales, e.g., Cr 2 O 3 or Al 2 O 3 . Such a protective scaling behavior may be challenged if a deposit condenses on the component surface and reacts with the thermally grown oxide. Indeed, at steadystate, protective scale growth is typically limited by solid-state diffusion in the oxide, and proceeds at a rate inversely proportional to the scale thickness, i.e.,

dX dt = k p X ( 1
)
where X is the thickness, t the time and k p the parabolic constant. Any process consuming the scale would result in an increased oxidation rate. In recent studies [START_REF] Gheno | High temperature reaction of MCrAlY coating compositions with CaO deposits[END_REF][START_REF] Gheno | Modes of deposit-induced accelerated attack of MCrAlY systems at 1100 °C[END_REF][START_REF] Gheno | On the reaction mechanism of MCrAlY alloys with oxide-sulfate deposits at 1100 °C[END_REF], we showed that Al 2 O 3 -scale forming NiCoCrAlY compositions in air were susceptible to accelerated corrosion in the presence of oxide-sulfate deposits. Two modes of attack were identified, which involved Al

Modeling

The consequences of Al 2 O 3 dissolution in molten Na silicate (NS) are evaluated by modeling oxide growth kinetics in a manner similar to that adopted in the study of solid-state reaction with CaO [START_REF] Gheno | High temperature reaction of MCrAlY coating compositions with CaO deposits[END_REF]. A thickening rate law is first determined by adding the contribution of oxide dissolution to that of oxide formation, in a regime where dissolution is limited by diffusion in the silicate. The Al mass balance is then derived by considering the Al flux in the alloy, the oxide scale and the silicate. The situation is illustrated in Fig. 1.

The dissolution rate of Al 2 O 3 is related to the Al flux entering the silicate,

J NS Al x=0 , via (dX) d dt = - V Al 2 O 3 2 J NS Al x=0 (2)
where V Al 2 O 3 is the molar volume of Al 2 O 3 (the factor of 1/2 on the right-hand side accounts for the fact that there are two moles of Al per mole of Al 2 O 3 ). This flux is calculated by solving the diffusion equation in the silicate. Assuming the molar volume of the silicate, V NS , is independent of its composition, Fick's law may be written as

J NS Al = - D NS Al V NS ∂N NS Al ∂x (3)
N NS Al (x, t) = N o/s Al erfc   x 2 D NS Al t   (5)
It follows that at the Al 2 O 3 /NS interface,

∂N NS Al ∂x x=0 = - N o/s Al πD NS Al t (6)
which is combined with Eqs. ( 2) and (3) to give

(dX) d dt = - 1 2 V Al 2 O 3 V NS N o/s Al D NS Al πt (7)
The net Al 2 O 3 growth is obtained by adding the production rate, Eq. ( 1), and the dissolution rate, Eq. ( 7):

dX dt = k p X - k d √ t (8)
where the dissolution constant k d is

k d = 1 2 V Al 2 O 3 V NS N o/s Al D NS Al π (9)
Equation ( 8) is analogous to that obtained for Al 2 O 3 reaction with CaO [START_REF] Gheno | High temperature reaction of MCrAlY coating compositions with CaO deposits[END_REF]. Considering that X(t = 0) = 0, it is solved to yield the new rate law

X 2 = 2k t (10)
k = k 2 d + k p -k d k 2 d + 2k p (11)
The simple form of Eq. ( 10) derives from the choice of parabolic kinetics to describe intrinsic scale growth: Eq. ( 1) was used with the underlying assumptions that scale growth was limited by diffusion in the scale from the start, and that the oxide diffusivity was constant with time. Strictly speaking, the requirement of a finite reaction rate at t = 0 involves the existence of a preexisting scale; this and other transient effects would generate constant and linear terms in the rate law [START_REF] Monceau | Determination of parabolic rate constants from a local analysis of mass-gain curves[END_REF]. Furthermore, Al 2 O 3 -scale growth is typically limited by grain boundary diffusion, and oxide grain growth would lead to sub-parabolic kinetics [START_REF] Rhines | Role of grain growth in the oxidation of nickel[END_REF][START_REF] Naumenko | Correlation between the microstructure, growth mechanism, and growth kinetics of alumina scales on a FeCrAlY alloy[END_REF]. Nonetheless, results inferred from the assumption of parabolic kinetics from t = 0 are valuable owing to their simplicity, and often prove useful in practice. Within this assumption, Al 2 O 3 growth also follows parabolic kinetics in the oxidation-dissolution regime, with a rate constant given by Eq. [START_REF] Bockris | The structure of the liquid silicates: partial molar volumes and expansivities[END_REF]. This is now used to calculate the Al concentration at the metal/oxide interface, N m/o Al , according to Wagner's analysis [START_REF] Wagner | Theoretical analysis of the diffusion processes determining the oxidation rate of alloys[END_REF]. The latter was developed for binary alloys and used a constant interdiffusion coefficient in the alloy. The case of n-element alloys, n > 2, is more complex, as interdiffusion is described with a set of (n -1) 2 independent coefficients. For the sake of simplicity, the present analysis considers instead a unique, effective Al diffusion coefficient in any given (single-phase) NiCoCrAlY alloy, denoted as D a Al . As shown later, the approach is useful if such an effective coefficient can be measured in the same alloy after oxidation with no deposit. N m/o Al is then given by [START_REF] Wagner | Theoretical analysis of the diffusion processes determining the oxidation rate of alloys[END_REF] N

0 Al -N m/o Al 1 -N m/o Al = F 1 2 k c D a Al ( 12
)
where F is the auxiliary function defined by F (u) = √ πu(1 -erf u) exp u 2 and k c the parabolic constant for metal recession. The latter is defined as

d 2 = 2k c t ( 13
)
where d is the depth of the metal/oxide interface relative to the original alloy surface.

In the absence of dissolution, the mass balance for the Al contained in the oxide scale is given as

2X V Al 2 O 3 = d V a (14)
with V a the molar volume of the alloy. This yields

k c = 2V a V Al 2 O 3 2 k p ( 15
)
When the Al 2 O 3 dissolves in NS, the mass balance in Eq. ( 14) becomes

2X V Al 2 O 3 + n NS Al = d V a ( 16
)
where n NS Al is the amount of Al in the silicate per unit surface area, i.e., the accumulated flux of Al into the silicate over time. Combining Eqs. ( 3) and ( 6), the Al flux at the Al 2 O 3 /NS interface can be written as

J NS Al x=0 = N o/s Al V NS D NS Al πt (17)
which yields [START_REF] Philibert | Atom movements -Diffusion and mass transport in solids[END_REF] n

NS Al = t 0 J NS Al (t) dt = 2 N o/s Al V NS D NS Al t π (18)
Using the definitions of k d and k , n NS Al can be written as

n NS Al = 2Xk d V Al 2 O 3 2 k (19)
This is introduced into Eq. ( 16) to give

2X V Al 2 O 3 1 + k d 2 k = d V a (20)
Noting that, by definition, X = √ 2k t is a solution of Eq. ( 8), k verifies

k √ 2k = k p √ 2k -k d (21)
The latter can be rearranged to give

1 + k d 2 k = k p k . Substituting into Eq. ( 20
) yields 2X V Al 2 O 3 k p k = d V a (22)
Equation ( 22) together with the definition of k and k c , Eqs. ( 11) and ( 13), respectively, provide the following expression for k c

k c = 2V a V Al 2 O 3 2 k 2 p k (23)
Thus changing from intrinsic oxidation, Eq. ( 15), to oxidation-dissolution, Eq. ( 23), amounts to replacing k p by k 2 p /k when evaluating Al consumption. The same expression was arrived at in the case of solid-state reaction with CaO [START_REF] Gheno | High temperature reaction of MCrAlY coating compositions with CaO deposits[END_REF] -the analogy reflects the fact that both Ca aluminate growth and Al diffusion in NS are diffusion-controlled processes.

Materials and experiments

Experimental data were obtained from NiCoCrAlY alloys, based on the fcc-A1 solid solution γ-(Ni,Co) and the B2 compound β-(Ni,Co)Al, reacted with and without SiO 2 -10 wt. % Na 2 SO 4 deposits in CO 2 -20H 2 O-1.6O 2 at 1100 • C. The experimental procedure is detailed in Ref. [START_REF] Gheno | Modes of deposit-induced accelerated attack of MCrAlY systems at 1100 °C[END_REF]. Briefly, 50 h isothermal corrosion experiments were conducted in a tube furnace under flowing gas, using cast alloys. The gas flow rate was 100 ml/min, with a linear velocity of 2 mm/s at 1100 • C, and a total pressure slightly over 1 atm. The alloys were cast and vacuum-annealed at the Materials Preparation Center of the Ames Laboratory [9]. Prior to exposure, 1 mm thick specimens were ground using SiC paper to a P1200-grit finish. Laboratory-grade powders were ground and applied as an ethanol-based slurry, producing 30 ± 5 mg/cm 2 deposits after evaporation of the ethanol. Sodium was applied as a sulfate, but it reacted with SiO 2 to form a molten silicate in the S-free gas at 1100 • C.

At the end of the 50 h experiments, metallographic cross-sections were prepared and analyzed by electron probe microanalysis (EPMA) using a JEOL JXA 8530F. Measured intensities were processed via a built-in ZAF calibration procedure, using pure metals as standards. Two alloy composition profiles were recorded below the Al 2 O 3 scale in each specimen.

Results and discussion

Alloy composition effects were studied by varying the β phase fraction and the Cr content. In general, β-rich alloys were immune to the Na silicate (NS), and formed external Al 2 O 3 scales, while some γ-rich alloys suffered internal oxidation [START_REF] Gheno | Modes of deposit-induced accelerated attack of MCrAlY systems at 1100 °C[END_REF]. The situation of exclusive external scaling will be considered to evaluate the validity of the oxidation-dissolution analysis. The effects of alloy composition on scale failure will be discussed subsequently.

First, the dissolution constant k d is determined from the solubility and diffusivity of Al in the silicate, as well as its molar volume. Analysis by SEM-EDS of NS deposits left after corrosion experiments gave a Na 2 O fraction of 15 mol. %, close to the liquid composition expected at 1100 • C [START_REF] Wu | Optimization of the thermodynamic properties and phase diagrams of the Na 2 O-SiO 2 and K 2 O-SiO 2 systems[END_REF]. This was used with density and expansivity measurements of Na 2 O -SiO 2 solutions [START_REF] Bockris | The structure of the liquid silicates: partial molar volumes and expansivities[END_REF] to provide V NS = 27.5 cm 3 /mol at 1100 • C. The influence of dissolved Al on the molar volume, although possibly significant, is neglected due to lack of adequate data. In corrosion experiments, the NS thickness atop the specimens was found to be variable, and sometimes 5) to Al concentration profiles measured after Al 2 O 3 dissolution in Na silicate (Fig. 2).

V NS (cm 3 /mol) [START_REF] Bockris | The structure of the liquid silicates: partial molar volumes and expansivities[END_REF] 27.5 V Al2O3 (cm 3 /mol) [START_REF]Physical constants of inorganic compounds[END_REF] 25. 5) to the profile using a least-squares method are given in Table 1, along with the resulting k d (Eq. ( 9)). A slight variance is observed in Fig. 2 between the shape of the NS fitted profile and the experimental data, likely due to variations of V NS or D NS Al with the Al fraction. The analysis of oxidation-dissolution kinetics is now applied to Ni-28Co-19Cr-11Al-0.1Y (at. %), a single-phase γ alloy at 1100 • C, which maintained exclusive Al 2 O 3scale formation (Fig. 3). The oxidation constant k p was determined from the average scale thickness measured after a 50 h oxidation experiment conducted in the same conditions as the SiO 2 -Na 2 SO 4 exposure, but with no deposit. It is recognized that thickness measurements after a single reaction time provide no indication as to the time-dependence of the oxidation process. This aspect was not studied experimentally. Rather, as indicated in Section 2, intrinsic scaling kinetics were assumed to be parabolic throughout the analysis. The k p value was thus used with Eq. ( 11) to obtain k , which in turn was used to calculate the expected Al 2 O 3 thickness in the oxidation-dissolution regime via Eq. ( 10). These parameters are summarized in Table 2. The average thickness measured in Ni-28Co-19Cr-11Al-0.1Y after exposure to SiO 2 -Na 2 SO 4 , 0.8 ± 0.1 µm, is in good agreement with the calculated value of 1.0 µm.

The analysis is further evaluated by considering the effect of Al is used with the effective coefficient D NS Al :

N a Al (y, t) -N m/o Al N 0 Al -N m/o Al = erf y 2 D a Al t -erf k c 2D a Al 1 -erf k c 2D a Al (24)
In Eq. ( 24), N a Al (y, t), N m/o Al and N 0 Al are the Al atom fractions at a distance y from the original metal surface, at the metal/oxide interface and in the bulk alloy, respectively (see Fig. 1). In the absence of a deposit, N m/o Al and k c are given by Eqs. ( 12) and [START_REF] Nijdam | Modelling of composition and phase changes in multiphase alloys due to growth of an oxide layer[END_REF], respectively. It follows that for a given N 0 Al , and using k p from Al 2 O 3 thickness measurements (Table 2), N a Al (y, t) only depends on D a Al . Equation (24) was thus fitted to the Al depletion profiles measured by EPMA after oxidation of Ni-28Co-19Cr-11Al-0.1Y, by adjusting D a Al , with a least-squares method. The fitted profile is shown in Fig. 4 along with the experimental data. The procedure yielded D a Al = 1.2 × 10 -10 cm 2 /s. This value was then used together with the k p and k from oxidation and dissolution data (Table 2) to calculate k c , N m/o Al and finally N a Al (y, t) in the oxidation-dissolution regime, through Eqs. ( 12), ( 23) and (24). The resulting profile is plotted in red in Fig. 4 together with the EPMA data obtained after the SiO 2 -Na 2 SO 4 exposure. The agreement is excellent, showing that the mass balance underlying the analysis of oxidation-dissolution kinetics is correct, and that using a constant, effective diffusion coefficient in the alloy is an acceptable approximation in this case.

Application of the model: Al

2 O 3 -scale failure
The analysis is now used to study the role of alloy composition in the dissolutioninduced Al 2 O 3 failure. Upon exposure to SiO 2 -Na 2 SO 4 , two of the alloys studied underwent internal oxidation: Ni-30Co-27Cr-12Al-0.1Y (γ-13 vol. % β at 1100 • C), where internal oxidation was localized, and Ni-33Co-35Cr-7Al-0.1Y (single-phase 12), ( 23) and ( 24) to calculate the profiles in the oxidation-dissolution regime, plotted as red lines.

γ at 1100 • C), where it was nearly uniform (Fig. 3). This reflects the importance of the initial Al content in maintaining a sufficient Al flux to the metal/oxide interface. Yet the Ni-28Co-19Cr-11Al-0.1Y alloy studied above had a slightly lower Al content than that of Ni-30Co-27Cr-12Al-0.1Y, and still maintained an external Al 2 O 3 scale. This suggests that Al 2 O 3 -scale failure may have been favored by the high Cr levels in Ni-30Co-27Cr-12Al-0.1Y and Ni-33Co-35Cr-7Al-0.1Y. Chromium would potentially affect the Al flux balance either via a reduction of Al diffusivity in the alloy, or via modifications of the molten silicate properties, such as its viscosity.

Chromia, like Al 2 O 3 , is amphoteric: depending on the silicate composition, it may behave as a basic or acidic oxide, respectively acting either as a network modifier (releasing O 2-) or as a network former (capturing free O 2-), which would decrease or increase the melt viscosity. At low p O 2 , the basic form CrO, i.e., Cr(+II), is expected to predominate. Potential effects of Cr oxide were studied by adding Cr 2 O 3 or Cr to Na 2 Si 3 O 7 powder (past saturation) in Al 2 O 3 crucible dissolution experiments. Chromium metal was used to buffer a low p O 2 , as would be the case if the Cr in an alloy reacts rapidly with a deposit. Unoxidized Cr particles were still present at the end of the 4 h exposure. An Na 2 Si 3 O 7 -5 wt. % CaO mixture was included in the run in order to test for any effect of CaO, which is commonly present in silicate deposits. Profiles measured by SEM-EDS after 4 h at 1100 • C are included in Fig. 2. No significant effect of the silicate composition was observed on either N 0 Al or D NS Al -Al did not diffuse any faster in the presence of Cr. This should be reflected in the Al mass balance, which is verified by calculating k c for alloys Ni-30Co-27Cr-12Al-0.1Y and Ni-33Co-35Cr-7Al-0.1Y.

As for Ni-28Co-19Cr-11Al-0.1Y, k p values were determined from average thicknesses measured after deposit-free oxidation of each alloy. Those and the resulting k values are included in Table 2. After exposure of Ni-30Co-27Cr-12Al-0.1Y and Ni-33Co-35Cr-7Al-0.1Y to SiO 2 -Na 2 SO 4 , wherever an external Al 2 O 3 was still present, its thickness had significant local variations, from ∼ 0.1 µm to values as large as those measured with no deposit (i.e., ∼ 2 µm); no meaningful average could be determined. These variations indicate that the alloys did not uniformly follow the reaction path considered in the present model, a point which will be returned to subsequently.

For the sake of simplicity, the analysis of Al depletion profiles is only carried out for the single-phase γ alloy Ni-33Co-35Cr-7Al-0.1Y. Fitting Eq. ( 24) to the Al concentration profiles measured by EPMA after oxidation with no deposit yielded D a Al = 5.9 × 10 -11 cm 2 /s. This value was then used in Eqs. ( 12), ( 23) and (24) together with the alloy k p and k values (Table 2) to simulate the Al profile in the dissolution-oxidation regime. The result is shown together with the corresponding experimental data in Fig. 4. The agreement is reasonable, although the shape of the simulated profile does not quite match the experimental data. This is possibly a shortcoming of using a constant, effective diffusion coefficient across such a wide composition range.

The value of D a Al obtained for Ni-33Co-35Cr-7Al-0.1Y is about half that found for Ni-28Co-19Cr-11Al-0.1Y (Table 2). The difference is significant and, in addition to the lower N 0 Al , the lower D a Al contributed to reduce

N m/o
Al in Ni-33Co-35Cr-7Al-0.1Y, to the extent that it was close to zero in zones were Al 2 O 3 was still present (Fig. 4). In such a situation, local fluctuations in the oxidation process, which are unavoidable in practice, would account for Al

Future Considerations

The present analysis successfully modeled the kinetics of Al 2 O 3 -scale growth during concurrent oxidation and dissolution in molten Na silicate, despite being based on a very simple treatment of scale growth and diffusion in the alloy and the silicate. Regarding diffusion in the alloy, the rationale was that in a given alloy, the diffusivity is the same whether a deposit is present or not; this prompted the use of an effective D a Al fitted to deposit-free profiles to calculate profiles in the oxidation-dissolution regime. One limitation of this method is that although the bulk composition is indeed the same, the composition range spanned in the depletion region is greater in the presence of a deposit, because of the enhanced Al consumption. As a consequence, the average D a Al must be different, as noted above in the case of alloy Ni-33Co-35Cr-7Al-0.1Y.

A thorough analysis would involve numerical modeling of multi-component diffusion using composition-dependent diffusion coefficients. Since the Al concentration at the alloy/scale interface depends on the Al flux in the alloy, it includes contributions of the Cr and Co concentration gradients, via off-diagonal terms of the diffusiv-ity matrix. Therefore, in lieu of Wagner's equations, the numerical model would have to solve the diffusion and oxidation-dissolution problems concurrently, with a finitedifference scheme such as those used by Nesbitt [START_REF] Nesbitt | Predicting minimum aluminum concentrations for protective scale formation on Ni-base alloys II. Cyclic oxidation[END_REF] or Nijdam et al. [START_REF] Nijdam | Modelling the thermal oxidation of ternary alloys-compositional changes in the alloy and the development of oxide phases[END_REF] for example. This can be extended to multi-phase alloys using a coupled thermodynamic-kinetic model [START_REF] Nijdam | Modelling of composition and phase changes in multiphase alloys due to growth of an oxide layer[END_REF]. Commercial programs such as Dictra [START_REF] Borgenstam | DICTRA, a tool for simulation of diffusional transformations in alloys[END_REF] provide the tools for such computation, provided that the oxidation problem is implemented correctly, and that appropriate thermodynamic and mobility data are available.

As a test of the latter aspect, concentration profiles in the single-phase γ alloys Ni-28Co-19Cr-11Al-0.1Y and Ni-33Co-35Cr-7Al-0.1Y oxidized with and without deposit were simulated in Dictra, using the experimentally measured interfacial and bulk concentrations as boundary conditions. The diffusivities were computed from the CRALDAD thermodynamic database, which was optimized for high Co and Cr levels in the NiCoCrAl system [START_REF] Gheno | Experimental study and thermodynamic modeling of the Al-Co-Cr-Ni system[END_REF][START_REF] Liu | Thermodynamic modeling of Al-Co-Cr, Al-Co-Ni, Co-Cr-Ni ternary systems towards a description for Al-Co-Cr-Ni[END_REF], and the MobNi1 [19] and NIST Ni Superalloy [START_REF] Campbell | Development of a diffusion mobility database for Ni-base superalloys[END_REF] mobility databases (both gave similar results). In all cases, the simulated profiles matched the experimental profiles less well than when using an effective D a Al (the computed DNi

Al,Al values were higher than our D a Al values by a factor of 10). The simulations improved when using the interdiffusion data by Nesbitt and Heckel [START_REF] Nesbitt | Interdiffusion in Ni-rich, Ni-Cr-Al alloys at 1100 • C and 1200 • C II. Diffusion-coefficients and predicted concentration profiles[END_REF] for NiCrAl alloys, despite the absence of Co in these data (it was replaced by Ni in our calculations). We conclude that efforts are needed to better account for the effects of high Co contents in mobility databases.

2 O 3

 23 reaction with CaO to form Ca aluminates, and Al 2 O 3 dissolution in molten Na silicate. Both processes cause an increased Al consumption and can lead to internal oxidation if the Al flux required for the scale to grow exceeds that available from the alloy. Thus Al 2 O 3 -scale failure is expected to occur in alloys with an Al concentration below a certain threshold. The conditions for this are analyzed in the following.

Figure 1 :

 1 Figure 1: Schematic Al concentration profiles developed in a system where an Al 2 O 3 scale is formed by oxidation of an alloy and consumed by dissolution in a molten silicate.

  with D NS Al and N NS Al the diffusion coefficient and mole fraction of Al in NS, respectively. Further assuming that D NS Al is constant, the time-dependent diffusion equastate, N NS Al at the Al 2 O 3 /NS interface, N o/s Al , is constant and equal to the Al solubility in NS. Equation (3) is then solved to yield[START_REF] Philibert | Atom movements -Diffusion and mass transport in solids[END_REF]

4. 1

 1 Evaluation of the model: external Al 2 O 3 -scale growth

  -19Cr-11Al-0.1Y Ni-30Co-27Cr-12Al-0.1Y Ni-33Co-35Cr-7Al-0.1Y

Figure 3 :

 3 Figure 3: Cross-section micrographs of the corrosion products formed after 50 h exposure to SiO 2 -10Na 2 SO 4 in CO 2 -20H 2 O-1.6O 2 at 1100 • C. The Ni-28Co-19Cr-11Al-0.1Y (γ) alloy maintained an external Al 2 O 3 scale, while Ni-30Co-27Cr-12Al-0.1Y (γ-13 vol. % β) and Ni-33Co-35Cr-7Al-0.1Y (γ) locally underwent internal oxidation (alloy compositions in at. %).

2 O 3

 23 dissolution on Al consumption in the alloy: the interfacial Al concentration is calculated from the constants k p and k , via Eqs. (12) and (23), and compared with Al concentration profiles measured by EPMA below the Al 2 O 3 scale after the SiO 2 -Na 2 SO 4 exposure. Equation (12) requires an effective diffusion coefficient for Al in the alloy. In the absence of adequate literature data, D NSAl is obtained from the depletion profiles measured after the deposit-free oxidation of the same alloy. The expression derived by Wagner for the depletion profile developed upon a oxidation of a binary alloy[START_REF] Wagner | Theoretical analysis of the diffusion processes determining the oxidation rate of alloys[END_REF]

Table 1 :

 1 Aluminum concentration profiles measured by SEM-EDS from Al 2 O 3 crucibles dissolving in silicate mixtures, after 4 h annealing at 1100 • C. The best fit of Eq. (5) to the experimental data is showed by a solid line. Parameters used in the calculation of the Al 2 O 3 dissolution constant, k d , via Eq. (9). The Al solubility and diffusivity in Na silicate, N

		wt. %
		LSS2 SiO 2 -26Na 2 O
	𝑵 𝐀𝐥 𝐍𝐒 , 𝐚𝐭. %	LSS3 LSS4 LSS5 simu SiO 2 -24Na 2 O-5CaO SiO 2 -24Na 2 O-5Cr 2 O 3 best fit SiO 2 -24Na 2 O-5Cr
	Figure 2: o/s Al and D NS Al , were adjusted by
	fitting Eq. (

  no greater than the diffusion length of Al, such that the Al concentration reached a plateau (larger than 0). It is noted that such an Al accumulation in the silicate reduces the Al flux, and thus locally affects the Al 2 O 3 dissolution and thickening rates. In order to obtain reliable values of solubility and diffusivity in the general case of a thick NS deposit, some Na 2 Si 3 O 7 powder was annealed in an Al 2 O 3 crucible, with no metal specimen, for 4 h at 1100 • C. The Al concentration profile resulting from the dissolution of the Al 2 O 3 crucible was measured by SEM-EDS (Fig.2). Values of N

			6
	N	o/s Al (at. %)	6.8
	D NS Al (cm 2 /s)	4.8 × 10 -10
	k d (cm 2 /s)	3.9 × 10 -7
	o/s Al and D NS Al obtained by fitting Eq. (

Table 2 :

 2 Al 2 O 3 -scale thicknesses and thickening constants in the oxidation and oxidationdissolution regimes, corresponding to 50 h deposit-free and SiO 2 -Na 2 SO 4 deposit exposures at 1100 • C, respectively.

	Alloy composition, at. % Ni-28Co-19Cr Ni-33Co-35Cr	Ni-30Co-27Cr
	(phases at 1100 °C)	-11Al-0.1Y (γ) -7Al-0.1Y (γ) -12Al-0.1Y (γ-β)
	No deposit
	X (µm), measured	2.0 ± 0.4	1.8 ± 0.3	1.9 ± 0.4
	k p (cm 2 /s)	1.2 × 10 -13	9.0 × 10 -14	1.0 × 10 -13
	D a Al (cm 2 /s)	1.2 × 10 -10	5.9 × 10 -11
	SiO 2 -Na 2 SO 4 deposits
	k (cm 2 /s), Eq. (11)	2.7 × 10 -14	1.8 × 10 -14	2.1 × 10 -14
	X (µm), calculated	1.0	0.8	0.9
	X (µm), measured	0.8 ± 0.1	n/a a	n/a a

a Local variations were too important for a meaningful average to be determined.

  Aluminum concentration profiles measured by EPMA in single-phase γ alloys oxidized with and without SiO 2 -Na 2 SO 4 deposit, 50 h in CO 2 -20H 2 O-1.6O 2 at 1100 • C (symbols). The black lines are best fits of (24) to the deposit-free data (see text for details). The adjusted D a Al were used with Eqs. (

		Ni-28Co-19Cr-11Al-0.1Y	Ni-33Co-35Cr-7Al-0.1Y
			EPMA, no deposit
			EPMA, SiO 2 -Na 2 SO 4 deposit
	N Al 𝑵 𝐀𝐥 𝐚 , at. % , 𝐚𝐭. %	EPMA, no deposit	𝐚 𝑵 𝐀𝐥 , 𝐚𝐭. %
		EPMA, SiO 2 -Na 2 SO 4 deposit
		fitted to the EPMA data	fitted to the EPMA data
		calculated	calculated
		y, µm
	Figure 4:

  2 O 3 breakdown on the rest of the alloy surface. Yet, even if D a Al tends to be lower in high Cr alloys, the Ni-30Co-27Cr-12Al-0.1Y alloy had a significantly larger Al reservoir to start with, such that a low D a Al would not, by itself, cause Al 2 O 3 failure. Indeed, values of N Al 2 O 3 was present were about 6-8 at. % -in these zones, Al 2 O 3 was far from being unstable. This in itself indicates that local Al 2 O 3 -scale failure on this alloy cannot be understood solely based on steady-state diffusion in the metal and the silicate. Instead, the high Cr concentration must have affected the transient oxidation behavior in a way that altered the initial conditions of the oxidation-dissolution process.

	m/o Al	measured in
	zones where external

Acknowledgements

This work was supported by the Department of Energy through the University Turbine Systems Research (UTSR) Program run by the National Energy Technology Laboratory, Award Number DE-FE0007271. The authors thank Greta Lindwall (NIST, USA) and Bonnie Lindahl (CEA, France) for stimulating discussions about diffusion modeling, and for providing the mobility data and running the Dictra simulations.