A first colourful example

2. each conversational pattern consists in a pre-defined graph of “nodes” which process initial context and ask questions, wait until further information is delivered (by other patterns) according to the “answerFor” relation, then process this additional information, ask other questions and wait again …

3. the calculus is globally monotonous because of the “reductionOf” relation (this relation is a partial order denoted “<=”) imposed on the rewriting.

- (χB,∗)emits: c // y is an instance of x //

- (χC,∗)emits: c // y is an instance of x //

We immediately notice that ‘c’ is emitted both by (χB,∗) and (χC,∗), the reader may in first approximation consider that two instances of ‘c’are present at the beginning of the conversation. However, this does not mean (χB,∗) and (χC,∗) will keep on emitting the same sentences during the conversation (as will be shown).

Most of these utterances will be replaced during the conversation by more informative ones; this is explicitly mentioned by the property non-definitive (boolean = 1) associated to them.

Table 6 also indicates that some sentences play the role of triggers for participants, which means that whenever they are perceived by a participant, a conversational process starts.

Table 6

0 = definitive 1 = non definitive emitted by

a1 // what is the number of primary colours? // (☺^{, *)} a1 triggerFor a

a // what is the number of instances of x? // χ^A

b // what are the instances of x? // (χ^{A, *)} χ^B; χ^C

c // y is an instance of x // (χ^{B, *); (}χ^{C, *)}

d // the number of instances of x is n // (χ^{A, *)}

b1 <= b b1 triggerFor b c1 <= c c1 answerFor b1 c2 <= c c2 answerFor b1 c3 <= c c3 answerFor b1 0

0

0

(χ^{A, a1)}

0 (χ^{C, b1)}

(χ^{B, b1); (}χ^{C, b1)}

c2 // yellow is an instance of primary colours //

c3 // blue is an instance of primary colours //

is a starter for relation to other sentences

(χ^{B, b1)}

1 1 1 1 sentence

0

b1 // what are the instances of primary colours? //

c1 // red is an instance of primary colours //

utterance

Our calculus relies on the fact that each participant is invoked only when its starter (either a question e.g. a// what is the number of instances of x? // or an assertion e.g. a’ //

here is a set of instances of x //) is triggered by one of the sentences already emitted in the conversation:

- (χA,∗) starts a conversational process whenever a sentence recognized as a trigger for a // what is the number of instances of x? // appears in the conversation; Table 6 indicates that the initial question a1 // is in such a relation⁵³ with ‘a’.

- (χB,∗) and (χC,∗) start a conversational process whenever a sentence recognized as trigger for b // what are the instances of x? // appears in the conversation; Table 6 indicates that the question b1 //emitted by(χA, a1) is in such a relation with ‘b’.

Then, new definitive (boolean = 0) utterances are emitted during conversation:

- (χA, a1), representing (χA,∗) working in the context “a1”, emits: b // what are the instances of primary colours? //

- (χB, b1) emits: c1 // red is an instance of primary colours //

- (χB, b1) emits: c2 // yellow is an instance of primary colours //

- (χC, b1) emits: c2 // yellow is an instance of primary colours //

- (χC, b1) emits: c3 // blue is an instance of primary colours //

- (χA, a1) emits: d1 // the number of instances of primary colours is 3 //

Table 6 indicates how some sentences play the role of answers for other sentences: ‘c1’,

‘c2’ and ‘c3’ play the role of answers⁵⁴ for the question ‘b1’.

We can now take a look at Table 7 to understand how the conversational processes occur:

- in a first step, (χA, ∗) is invoked by ‘a1’ recognized as “triggerFor” its starter

‘a’, this generates (χA, a1) which emits ‘b1’ and ‘d1’ respectively more informative than ‘b’ and ‘d’ ; ‘b’ and ‘d’ are removed. We write “b1 ≤ b” and say

“b1 reductionOf b”; one major constraint in our calculus being that participants always “reduce” sentences.

53 We want this information to be computable instead of needing explicit declaration; therefore we define a

“triggerFor” relation between sentences and take hypotheses for the associated test to be decidable.

54 This point is another major aspect of our calculus, and we shall ask for this information to be computable language instead of needing explicit declaration; therefore we define an “answerFor” relation between sentences and take hypotheses for the associated test to be decidable.

- when (χA, a1) emits ‘b1’ it is recognized as “triggerFor” ‘b’ which plays the role of starter for both (χB, ∗) and (χC, ∗); therefore these patterns are both invoked by ‘b1’.

- when (χB,∗) is invoked by ‘b1’, it emits ‘c1’ and ‘c2’ more informative than ‘c’

which is removed;

Table 7

- ‘c1’, ‘c2’ and ‘c3’ are recognized as the totality of possible answers for ‘b1’; altogether they activate the pendant step of (χA, a1);

- when (χA, a1) is given ‘b1’ as an answer to its intermediary question, it replaces ‘d1’ by the less ambiguous ‘d2’ and since no other participant can be triggered, the conversation ends.

Let us recall the major points learnt from this first example:

- each conversational pattern owns one starter;

- each conversational pattern emits at the beginning of the conversation abstract versions of further utterances under the shape of commissives;

- each rewriting consists in reductions (more informative versions) of these abstractions.

rewriting action

( b1 // what are the instances of primary colours? // , 0) (χ^{A, a1)}

( b // what are the instances of x? // , 1) (χ^{A, *)}

( d1 // the number of instances of primary colours is n // , 0) (χ^{A, a1)}

( d // the number of instances of x is n // , 1) (χA, *) ( c1 // red is an instance of primary colours // , 0) (χ^{B, b1)}

( c2 // yellow is an instance of primary colours // , 0) (χ^{B, b1)}

( c // y is an instance of x // , 1) (χ^{B, *)}

( c2 // yellow is an instance of primary colours // , 0) (χC, b1) ( c3 // blue is an instance of primary colours // , 0) (χ^{C, b1)}

( c // y is an instance of x // , 1) (χ^{C, *)}

( d2 // the number of instances of primary colours is 3 // , 0) (χ^{A, a1)}

( d1 // the number of instances of primary colours is n // , 1) (χA, a1) invocation of χA by a1

invocation of χB by b1

invocation of χC by b1

answer {c1, c2, c3} is given to question b1 of χA

transition in conversation

added by removed by

( sentence, boolean )

So far, we took no interest neither in how (χA,∗) calculates the number of instances, nor in how (χB,∗) and (χC,∗) produce their different answers to the same question … in [3.3], we shall explain how “conversational pattern” are composed of interdependent nodes, the internal functional structure of each node and how they interact.

But we must first formalize sentences and ensure that the key relations “reductionOf”,

“answerFor” and “triggerFor” are computable!

3.2 Sentences and utterances