3 AUSTIN: a conversational abstract machine
3.2 Sentences and utterances .1 Sentences
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
When (
S
, ≤, triggerFor, answerFor) is a set of sentences, and E a subset ofS
:- ReductionsOf (E) = {s ∈
S
| (∃ s’ ∈E) s ≤ s’}- TriggersFor (E) = {s ∈
S
| (∃ s’∈E) s triggerFors’}- AnswersForReductionsOf (E) = {s ∈
S
| (∃ s1∈E) (∃ s2∈S
) ((sanswerFors2) ∧ (s2≤ s1))}In the informal examples given before, the relation reductionOf corresponds to what we intuitively describe as “more informative”:
- // Linda lives in Paris // reductionOf // a woman lives in a city //
- // where does Linda live? // reductionOf // where does some woman live? //
The relation answerFor corresponds to the usual intuition, also it must be noticed that a long sentence may answer a short question:
- // Linda lives in Paris // answerFor // where does Linda live? //
- // Linda lives in Paris in a beautiful apartment owned by her family since the beginning of the past century // answerFor // where does Linda live? //
The less intuitive relation triggerFor must be understood as: “is a specialized invocation context for”:
- // Linda lives in Paris // triggerFor // a woman lives in a city //
answerFor and triggerFor will only be used in tests, while reductionOf will be a constraint on emissions of utterances by the participants in the conversation; these three relations playing three distinct roles in the operational semantics of the calculus are illustrated by comparison algorithms in the generative grammar given in [3.9].
3.2.2 Utterances
Among the infinite possible sentences, a few are effectively uttered by “emitters” during the conversation; to express this, we need the notion of utterance which embeds a sentence together with additional information. An utterance may or not be removed from the conversation: only in the last case will it be considered as a valuable trigger or answer for other utterances! A definitive utterance will stay in the conversation until its end, while a non-definitive utterance will eventually be removed.
It is possible that two different emitters utter the same sentence, the first as definitive, the second as non-definitive; this gives birth to two different utterances: an utterance is in fact a pair (sentence, status)! In case the same pair (sentence, status) has been uttered concurrently by different emitters, we will consider only one utterance with several
emitters and store their list inside the utterance; this will facilitate the analysis of the set of all sentences present at a given state.
Definition 2: An utterance is a pair (sentence, status), where:
- “sentence” is taken in
S
- “status” is taken in {0, 1}; status = 0 means definitive; status = 1 means non-definitive An utterance will be denoted (s, bool).
An utterance has a list L of emitters which will be further formalized as pairs (conversational pattern, context).
U
(ω) denotes the set of all utterances when conversation has reached a given state “ω”.Let “s” be a sentence, Q be a subset of
S
, the following definitions aim at a concise presentation of the calculus:- PotentialTriggersFor (s, ω) = {u = (s’, 1) ∈
U
(ω) | (∃ s”∈S
) ((s”≤ s’) ∧ (s”triggerFors))} isthe set of all non-definitive utterances which can accept as a reduction a trigger for “s” - DefinitiveTriggersFor (s, ω) = {u = (s’, 0) ∈
U
(ω) | s’triggerFors} is the set of all definitive utterances which embed a trigger for “s”- PotentialAnswersFor (Q, ω) = {u = (s’, 1) ∈
U
(ω) | (∃ s∈Q) (∃ s”∈S
) ((s”≤ s’) ∧ (s”answerFor s))} is the set of all non-definitive utterances which accept as a reduction an answer for a sentence of Q
- DefinitiveAnswersFor (Q, ω) = {u = (s’, 0) ∈
U
(ω) | (∃ s∈Q) (s’answerFors)} is the set of all definitive utterances which embed an answer for a sentence of QInformal examples:
- (// Linda lives in Paris //, 0) ∈ DefinitiveTriggersFor (// a human lives in a place // ) - (// Linda lives in a city //, 1) ∈ PotentialTriggersFor (// a human lives in a place //)
because this non-definitive utterance may be removed and replaced by (// Linda lives in Paris //, 0)
- (// Linda lives in Paris //, 0) ∈ DefinitiveAnswersFor (// does Linda lives in Paris? // ) - (// Linda lives in a city //, 1) ∈ PotentialAnswersFor (// does Linda lives in Paris? //)
because this non-definitive utterance may be removed and replaced for instance by (// Linda lives in Paris //, 0) or (// Linda lives in Tokyo //, 0)
- (// a woman lives in Paris //, 1) ∈ PotentialAnswersFor (// does Linda lives in Paris? //) because this non-definitive utterance may be removed and replaced for instance by (// Linda lives in Paris //, 0) or (// Mathilda lives in Paris //, 0)
3.2.3 Changes of state in the set of utterances
Let us come back to the short conversation described in [3.1.2]
Its initial state is composed by the following utterances, a directive emitted by a “user”
and a set of commissives associated to the patterns:
- (// what is the number of primary colours? // , 0, {(
☺
, ∗)}) is a definitive question emitted by (☺
, ∗)- (// what are the instances of x? // , 1, {(χA, ∗)}) is a non-definitive question emitted by (χA, ∗)
- (// the number of instances of x is n // , 1, {(χA, ∗)}) is a non-definitive assertion emitted by (χA, ∗)
- (// y is an instance of x // , 1, {(χB, ∗), (χC, ∗)}) is a non-definitive assertion emitted by (χB, ∗) and (χC, ∗)
Its final state consists of definitive assertions answering definitive questions55:
- (// what is the number of primary colours? // , 0, {(
☺
, ∗)}) emitted by (☺
, ∗) - (// what are the primary colours? // , 0, {(χA, a1)}) emitted by (χA, a1)- (//red is an instance of primary colours // , 0, {(χB, b1)})
- (// yellow is an instance of primary colours // , 0, {(χB, b1), (χC, b1)}) emitted by (χB, b1) and (χC, b1)
- (// blue is an instance of primary colours // , 0, {(χC, b1)})
- (// the number of primary colours is 3 // , 0, {(χA, a1)}) definitive assertion Two operations correspond to the basic rewriting actions in the set of utterances:
“addition” and “removal, this last one being allowed only for non-definitive utterances;
definitive utterances are kept in the conversation!
Definition 3: Let “e” be an emitter in a conversation, either (χI, a) or (χI, ∗) or (
☺
, ∗) , the two operations on utterances are the following:- add ((s, bool), e): “e” is added to the list L associated to (s, bool); in case no such utterance already existed, the utterance is “created”;
- remove ((s, 1), e): “e” is removed from the list L associated to (s, bool); in case L becomes empty, the utterance is “erased”.
55 The non-definitive assertions and questions “disappear” as soon as their list of emitters becomes empty: according to Definition 2 they are no longer considered as utterances.