A Translation from Logic to English with
Dynamic Semantics?
Elizabeth Coppock and David Baxter
Cycorp, Inc., Austin TX 78731, USA
Abstract. We present a procedure for translating standard predicate
logic into English. The procedure generates both referring expressions
and non-referring expressions, including both referential and bound vari-
able anaphora. Non-referring expressions correspond to short-term dis-
course referents [1], which present a special set of challenges for a natural
language generation system: (i) they have limited `lifespans' and (ii) the
determiner with which they are introduced (every, some, any, no) is sen-
sitive to the logical context. Our system addresses these challenges using
dynamically updated information states.
Keywords. natural language generation, dynamic semantics, predicate
logic, quantication, anaphora
1 Introduction
Things change as discourse progresses. One type of change that occurs is that
entities become part of the discourse context when they are rst mentioned. This
aects the form that referring expressions take; the cat in (1) is introduced with
an indenite, and subsequent references to it take the form of pronouns [2{9].
(1) I saw a cat and kept it. It ran away.
Another type of change that occurs during discourse is that quantiers may
begin or cease to take scope. This occurs in the following example [10, 11]:
(2) Everybody found a cat and kept it. It ran away.
On the reading of (2) in which dierent people found (potentially) dierent cats,
the it in the second sentence cannot corefer with the it in the rst sentence.
This is because, on that reading, the discourse referent introduced by a cat is
associated with a quantier whose scope does not extend past the rst sentence.
Dynamic semantics [10, 11, 5, 12{14], as opposed to truth-conditional seman-
tics, e.g. [15], takes the dynamic nature of texts very seriously: \Very generally,
the dynamic view on meaning comes to this: the meaning of a sentence is the
? Thanks to Lucas Champollion, Elias Ponvert, David Beaver, and two anonymous
reviewers for thoughtful suggestions. This work was partially supported under
DARPA's Rapid Knowledge Formation program.

2 Elizabeth Coppock and David Baxter
change an utterance of it brings about, and the meanings of non-sentential ex-
pressions consist in their contributions to this change" [14], p. 182. But as this
quotation suggests, dynamic semantics has so far taken the perspective of inter-
pretation rather than generation of natural language.
Dynamically updated information states are useful for speakers as well as
hearers. Like hearers, speakers must keep track of what discourse referents have
already been mentioned; a speaker who does not do so cannot determine when
to use a pronoun. Speakers must also keep track of what quantiers and logical
operators take scope over the current expression, so that they know when it is
no longer possible to use a pronoun, as in (2), and in order to decide between
determiners like every, some, any, and no, as we will discuss further below.
In this paper, we combine natural language generation with dynamic seman-
tics. Specically, the problem we aim to solve is to dene a translation from
standard predicate logic to English. The input may be any formula of predicate
logic, and the output is a string of English text. We are not concerned with
generating every possible English translation (cf. logical completeness), but the
translations that are generated should be faithful and natural representations of
the input formula (cf. logical soundness). \Faithfulness" can be evaluated as fol-
lows: if a formula  entails another formula , and G() is the natural language
translation of , then native speakers should have the intuition that G() im-
plies G( ) (cf. [16]). At the same time, we aim to keep track of the changes that
occur as the discourse progresses. Our solution is a logic-to-English translation
procedure that uses dynamically updated information states.
2 Problems with non-referential `discourse referents'
Because the possible inputs to the logic-to-English translation problem includes
quanticational formulas, it is necessary to manage both short-term and long-
term discourse referents in Karttunen's sense [1]: \the appearance of an indenite
noun phrase [or any noun phrase, for our purposes] establishes a `discourse refer-
ent' just in case it justies the occurrence of a coreferential pronoun or a denite
noun phrase later in the text." This denition allows the study of coreference
to proceed \independently of any general theory of extralinguistic reference" (p.
367). On this denition, discourse referents can be established by expressions
that genuinely refer as well as non-referential expressions such as nobody. Dis-
course referents that do not correspond to proper referents come with a special
set of challenges, motivating a dynamic approach to natural language genera-
tion: they have limited `lifespans' (x2.1) and they are sensitive to the presence
of operators with limited scope (x2.2).
2.1 Lifespan limitations
Non-referential discourse referents have a limited `lifespan' [1]. This is exempli-
ed in (2), repeated here:
(3) Everybody found a cat and kept it. It ran away. [11]