Acyclic Recursion with Polymorphic Types and Underpecification
Roussanka Loukanova
Department of Mathematics, Stockholm University, Stockholm, Sweden
Keywords:
Semantics, Algorithms, Intension, Denotation, Recursion, Types, Underspecification, Subtypes, Polymor-
phism.
Abstract:
The paper extends Moschovakis higher-order type theory of acyclic recursion by adding type polymorphism.
We extend the type system of the theory to model parametric information that pertains to underspecified types.
Different kinds of type polymorphism are presented via type variables and recursion constructs for alternative,
disjunctive type assignments. Based on the new type system, we extend the reduction calculus of the theory
of acyclic recursion. We motivate the type polymorphism with examples from English language.
1 INTRODUCTION
A sequence of papers (Moschovakis, 1989;
Moschovakis, 1994; Moschovakis, 1997) intro-
duced a new approach to the notion of algorithm, by
the mathematical concept of recursion. The approach
uses a formal language of recursion and fine semantic
distinctions between denotations and algorithms
computing the denotations. That work was on
untyped theory of algorithms and was the basis of
(Moschovakis, 2006), which introduced typed theory
L
λ
ar
of acyclic recursion, as formalization of the
concepts of algorithmic meaning in typed models.
L
λ
ar
was introduced for primary application to algo-
rithmic semantics of human language, by considering
semantics of programming languages, where a given
denotation can be computed by different algorithms.
Work on extending the expressive power of L
λ
ar
was initiated in (Loukanova, 2016; Loukanova,
2013a; Loukanova, 2011b). The class of formal lan-
guages and theories of typed acyclic recursion, col-
lectively, is a formal system, which we call the typed
theory of acyclic recursion (TTofAR).
TTofAR has potentials for applications to algo-
rithmic semantics of formal and natural languages.
What makes the work highly demanding is one of the
distinctive features of human languages semantic
ambiguities and underspecification. Ambiguous lan-
guage expressions are interpreted depending on con-
text information, and often, even in context, specific
interpretations can be parametric. Thus, intelligent
language processing requires computational integra-
tion of syntax with semantics that allows underspec-
ified representations. Availability of computational
models of context are essential for specific interpreta-
tions and disambiguation. However, it is important to
provide representations in underspecified forms, be-
fore context information can select specific interpre-
tations. Thus, it is important to develop computa-
tional theory that represents parametric information
as such. Furthermore, parametric components can be
about unknown types of objects, not only underspec-
ified objects of fully known types. This paper is on
development of a theory for computational semantics
that maintains parametric types.
Among the formal languages, we consider appli-
cations of TTofAR to semantics of programming lan-
guages, languages used in database systems and vari-
ous areas of Artificial Intelligence (AI). There is real-
ization in contemporary developments and technolo-
gies, related to language processing, that they are
more successful by using language constructs with
parametric or otherwise underspecified types. In pro-
gramming languages and formal languages of type
theory, this is provided by techniques classified as
polymorphism. See, e.g., the classic work (Cardelli
and Wegner, 1985). In large-scale computational
grammars of human language, underspecification has
become important subject, see e.g., (Copestake et al.,
2005), work which, while not fully formalized, has
been used in implementations. For providing such
applications, our paper extends L
λ
ar
by adding poly-
morphism to its type system.
Semantic ambiguities are among the core, open
problems in language processing. While a broad
spectrum of semantic theories have been seeking so-
392
Loukanova, R.
Acyclic Recursion with Polymorphic Types and Underpecification.
DOI: 10.5220/0005749003920399
In Proceedings of the 8th International Conference on Agents and Artificial Intelligence (ICAART 2016) - Volume 2, pages 392-399
ISBN: 978-989-758-172-4
Copyright
c
2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
lutions, here we only mention theories that, accord-
ing to us, have underlain research and applications in
the areas related to processing of human language that
includes computational semantics. Montague’s Inten-
sional Logic (IL), see (Thomason, 1974), introduced
the first significant approach to computational seman-
tics, which also included representation of semantical
ambiguities. IL was extended by (Gallin, 1975) to
handle hidden quantification over possible wolds and
times, which was the basis of extended, Montague
style grammars, e.g., (Muskens, 1995), and more re-
cently (Villadsen, 2010) . On the side of comprehen-
sive, logical approaches to information and semantics,
(Barwise, 1981) was the first to introduce a computa-
tional approach to the concepts of partiality, seman-
tic parameters, and restricted parameters in logic ap-
proaches to information. While TTofAR, including
L
ta
ar
introduced here, extend (Gallin, 1975), and by that
Montague IL, TTofAR delimit Situation Theory to
functional systems. This is done by representing set-
theoretic relations with their characteristic functions,
via standard Curry encodings (when possible and rea-
sonable), and to acyclic recursion, i.e., to computa-
tions that close-off, via mutual recursion that ends af-
ter certain number of steps. TTofAR essentially ex-
tend Gallin logic, and logic systems based on it, by
adding expressiveness for algorithmic semantics and
semantic denotations, i.e., denotational and algorith-
mic semantics. By L
ta
ar
, we extend the expressiveness
of L
λ
ar
for handling underspecified information by type
polymorphism, which is also acyclic with respect to
computations. The system L
ta
ar
offers unique formal-
ization of underspecification in information and se-
mantics, via integration of polymorphism with com-
putations that close-off. This makes it valuable the-
ory, as a mathematical model of algorithms that can
be underspecified, and for applications.
The untyped theory of recursion (Moschovakis,
1997; Moschovakis, 1989) is applied in (Hurkens
et al., 1998) to model reasoning. The potentials of
L
λ
ar
, with typed recursion, have been used in vari-
ous applications. Application of L
λ
ar
to logic pro-
gramming in linguistics and cognitive science is given
in (Van Lambalgen and Hamm, 2004). A sequence
of papers (Loukanova, 2016; Loukanova, 2013c;
Loukanova, 2013b; Loukanova, 2013a; Loukanova,
2012a; Loukanova, 2012b; Loukanova, 2011b;
Loukanova, 2011c; Loukanova, 2011d; Loukanova,
2011e; Loukanova, 2011a) and (Loukanova and
Jim
´
enez-L
´
opez, 2012) along extending the original
L
λ
ar
, applied TTofAR to computational semantics and
computational syntax-semantics interface of human
language. By adding polymorphism, L
ta
ar
offers po-
tentials for varieties of applications.
The following sections are concise introduction to
the theory of L
ta
ar
. We give motivation from compu-
tational semantics of human language, because L
ta
ar
has direct potentials for applications to computer-
ized processing of human language, including large-
scale, computational grammars of human language
and NLP for AI.
2 MOTIVATION FOR
UNDERSPECIFIED TYPES
In this section, we present some motivation for intro-
ducing parametric types in a type system, i.e., in this
paper, by extending L
λ
ar
. The general idea is that para-
metric types can be left underspecified when there is
not enough information to designate fully specified
types. Further specification (which still can be para-
metric) of a parametric type can be refined depending
on environment. In order to reach readers from vary-
ing research areas, including AI, we present the use-
fulness of parametric types by examples from human
language, with the special case of English. The mo-
tivation holds for other natural and formal languages,
and applications, e.g., in databases and other informa-
tion sciences, when there is underspecified data de-
pendent on possibilities for further instantiations.
Human language is abundant of different kinds of
ambiguities, e.g., see (Loukanova, 2010) for a clas-
sification along with representative examples. Some
semantic ambiguities can be associated with cor-
responding syntactic categories and parsings, while
there are purely semantic ambiguities that can not
be distinguished syntactically. In any case, seman-
tic interpretations need formal representations by ex-
pressions, i.e., terms of a formal theory that provides
mathematical medium for semantic interpretations.
In formal and computational grammars of hu-
man language, the grammatical function of a modi-
fier in syntax is associated with varieties of syntac-
tic categories. Notoriously, a modifier can contribute
to semantic ambiguities, some of which can corre-
spond to syntactic ambiguities, e.g., as for the sen-
tence (1a) in (1b)–(1d). The semantic difference be-
tween the two parsings (1b) (1c) is explicit. To rep-
resent the semantics of (1b), one needs a term, in
which [in Pittsburgh]
PP
is rendered to a subterm of
type (
e
e
e
e), which is appropriate for both nomi-
nal expressions (i.e., common nouns and more com-
plex NOMs) and verb phrases. Depending on seman-
tic theory and available information, one may assume
that the semantics of (1c) and (1d) are approximately
the same or, alternatively, that (1d) has no reasonable
meaning. So, with this limitation, there is no need
Acyclic Recursion with Polymorphic Types and Underpecification
393
of parametric type to represent the semantics of the
sentence (1a), which can be rendered by using under-
specified terms of L
λ
ar
, but without involving paramet-
ric types.
John bought the car. (1a)
[John [[bought]
V
[the [car [in Pittsburgh]
PP
]
NOM
]
NP
]
VP
]
S
(1b)
[John [[[bought]
V
[the [car]
NOM
]
NP
]
VP
[in Pittsburgh]
PP
]
VP
]
S
(1c)
[[John [[bought]
V
[the [car]
NOM
]
NP
]
VP
]
S
[in Pittsburgh]
PP
]
S
(1d)
Similarly to prepositional phrases (PPs), adverbs syn-
tactically can be either verbal or sentence modifiers.
E.g., the adverb “amusedly in a sentence like “The
comedian was presenting his new joke amusedly. can
contribute to different parsings with corresponding,
different semantic interpretations. The entire class
of grammatical modifiers, across syntactic categories,
would benefit from polymorphic type assignment to
modifiers. In this paper, we present motivation for
type polymorphism with the shorter sentence (2a),
which does not involve tense, aspects, and transitive
verbs in the VP, while shares a common pattern of
grammatical modification. The adverb “disturbingly”
in (2a) can be either a verbal modifier, e.g., as in the
parsing (2b), or a sentence modifier, as in (2c).
Jerry barks disturbingly. (2a)
[[Jerry]
NP
[[barks]
VP
disturbingly]
VP
]
S
(2b)
[[Jerry [barks]
VP
]
S
disturbingly]
S
(2c)
Semantically, with the parsing (2b), the sentence can
be used to describe a situation where the action of
barking itself is explicitly disturbing as a physical ac-
tion, e.g., Jerry himself may be disturbed and sound
that way. In such cases, we can render the adverb
“disturbingly” into a constant of type appropriate for
modifiers of VP, i.e., ((
e
e
e
t) (
e
e
e
t)), as in (10),
and the sentence into a formal term of L
λ
ar
, as in (11b)–
(11f).
In other contexts, a parsing corresponding to (2c),
with the adverb “disturbingly” as a sentence modifier,
requires a different type (
e
t
e
t), as in (12), which is
appropriate for semantics of sentences, represented by
(13a)–(13f). E.g., this is needed when the sentence is
used to describe a situation that is disturbing, not nec-
essarily because the barking action is disturbing per
se, but in other aspects. E.g., Jerry himself may not be
disturbed, and his action of barking may not be physi-
cally disturbing by itself at all, while the situation de-
scribed by the component sentence [Jerry [barks]
VP
]
S
is disturbing for other participants, i.e., agents, in the
described environment.
For these examples and their representation in L
λ
ar
,
see (Loukanova, 2012a), which initiated representa-
tion om grammatical modifiers by underspecified re-
cursion. Here we need to stress that the technique
for using underspecified L
λ
ar
-terms, where the under-
specification covers only the presence of free, i.e., un-
derspecified, recursion variables of L
λ
ar
, as presented
in (Loukanova, 2012a), requires extending the theory
L
λ
ar
by adding parametric types and extending the en-
tire reduction system of L
λ
ar
. This is the topic of this
paper.
We introduce the extended formal system in the
following sections. Then, in Section 7, we demon-
strate the superiority of the extended theory L
λ
ar
. At
first, we give alternative formal representations of the
semantics of (2b)–(2b) by using the original L
λ
ar
, with-
out parametric types. Then, we present the semantic
ambiguity of (2a) as semantic underspecification, by
terms of the extended L
λ
ar
.
In this paper, we show that the use of the technique
of underspecification, extended by adding parametric,
underspecified types, is representative for handling
semantic ambiguity. In particular, we apply it for rep-
resenting underspecified ambiguities of the grammat-
ical function of modification in human language. In-
stead of rendering a sentence into alternative formal
terms, for all possible interpretations, when it is not
known which one may be appropriate in different con-
texts, and with respect to different agents, it is better
to render all of them into a single underspecified term
with underspecified recursion variables and paramet-
ric types. Then, the parameters can be instantiated
suitably depending on contexts and agents having ad-
ditional information.
We shall provide renderings of this sentence in
Section 7, after we present the necessary definitions
of the extended, polymorphic, formal system L
λ
ar
.
3 TYPES
We extend the type system of L
λ
ar
, by adding a formal
sub-language Types
L
ta
ar
(abbreviated as Types).
Basic Type Constants: BTypes = { e, t, s },
where e is for entities (individuals); t for truth values;
s for states, i.e., context information
Type Variables: TV = {ζ
i
| i = 0, 1, . . . }
Types Types
L
ta
ar
: defined recursively (in BNF)
θ
:
e | t | s | σ | (τ σ) | (3a)
τ
0
where {ϑ
1
:
type
= T
1
, . . . , ϑ
m
:
type
= T
m
} (3b)
where τ, σ, τ
0
Types; ϑ
j
TV; T
j
are nonempty,
finite sets of types ( j = 0, . . . , m), such that { ϑ
1
:
type
=
ICAART 2016 - 8th International Conference on Agents and Artificial Intelligence
394
T
1
, . . . , ϑ
m
:
type
= T
m
} satisfies the Acyclicity Constraint
AC 1.
Acyclicity Constraint AC 1. For any pairwise dif-
ferent ϑ
j
TV and nonempty, finite sets T
j
of types
( j = 0, . . . , m), {ϑ
1
:
type
= T
1
, . . . , ϑ
m
:
type
= T
m
} is acyclic
iff there is rank: {ϑ
1
, . . . , ϑ
m
} N, such that
if ϑ
j
occurs freely in T
i
then rank(ϑ
j
) < rank(ϑ
i
).
We call the type terms that have free occurrences
of type variables underspecified types. Some useful
notations:
e t, the type of characteristic functions (4a)
of sets of individuals
e
e (s e), the type of state dependent (4b)
entities, i.e., entity “concepts”
e
t (s t), the type of state dependent truth,
i.e., of state dependent propositions
e
τ (s τ), for τ TV (4c)
The type (5) is for curryed functions from objects of
type τ
1
, . . . , τ
n
to objects of type σ; (5) has type vari-
ables as components, i.e. it is underspecified.
(τ
1
·· · (τ
n
σ)) σ, τ
i
TV, n 0 (5)
The type variables, e.g., τ TV, which we use as a
specialized kind of recursion variables, introduced in
the following section, represent “unknown” or “partly
known” types, for which no, or only some, informa-
tion is available. E.g.:
ϒ
1
(τ
1
τ
2
) where {τ
1
:
type
= e, τ
2
:
type
= t} (6a)
ϒ
2
τ where {τ
:
type
= (τ
1
τ
2
),
τ
1
:
type
= e, τ
2
:
type
= t}
(6b)
4 SYNTAX
L
ta
ar
has typed vocabulary, i.e., for each τ Types:
Constants K: K
τ
= {c
0
τ
, . . . , c
τ
k
, . . . },
Pure variables PV: PV
τ
= {v
0
, v
1
, . . .},
Recursion variables RV: RV
τ
= {r
0
, r
1
, . . .},
Pure and recursion variables can be of underspec-
ified types, by type variables and alternative options
via sets of types assigned to type variables, in (3b).
The terms Terms
L
ta
ar
(Terms): Recursively, in BNF:
A
:
c
τ
| x
τ
| (7a)
B
(στ)
(C
σ
)
τ
| (7b)
λv
σ
(B
τ
)
(στ)
| (7c)
A
σ
0
0
where {ϑ
1
:
type
= T
1
, . . . , ϑ
m
:
type
= T
m
}
{ p
σ
1
1
:
= A
σ
1
1
, . . . ,
p
σ
n
n
:
= A
σ
n
n
}
σ
0
(7d)
where for n, m 0, c
τ
K
τ
; x
τ
PV
τ
RV
τ
; v
σ
RV
σ
; A, B, A
σ
i
i
Terms (i = 0, . . . , n); p
i
RV
σ
i
(i =
1, . . . , n); ϑ
j
TV; T
j
Types are nonempty, finite
sets of types ( j = 1, . . . , m); such that the sequences:
{ϑ
1
:
type
= T
1
, . . . , ϑ
m
:
type
= T
m
} and
{ p
σ
1
1
:
= A
σ
1
1
, . . . , p
σ
n
n
:
= A
σ
n
n
} satisfy the Acyclicity
Constraint AC 2.
Acyclicity Constraint AC 2. the sequences
{ϑ
1
:
type
= T
1
, . . . , ϑ
m
:
type
= T
m
} and
{ p
σ
1
1
:
= A
σ
1
1
, . . . , p
σ
n
n
:
= A
σ
n
n
} are (jointly) acyclic iff
for a function rank: { ϑ
1
, . . . , ϑ
m
, p
1
, . . . , p
n
} N,
1. if p
j
occurs freely in A
i
, then rank(p
j
) < rank(p
i
)
2. if ϑ
j
occurs freely in T
i
, then rank(ϑ
j
) < rank(ϑ
i
)
3. for p
σ
i
i
:
= A
σ
i
i
, if ϑ
j
occurs freely in σ
i
, then
rank(ϑ
j
) < rank(p
i
)
Informally, { p
σ
1
1
:
= A
σ
1
1
, . . . , p
σ
n
n
:
= A
σ
n
n
} defines
recursive step-wise computations of the values of A
σ
i
i
(i.e., of their denotations), which are saved in p
i
. The
requirement rank(p
j
) < rank(p
i
), allows the value of
A
σ
i
i
saved in p
σ
i
i
to depend on the value of A
σ
j
j
saved
in p
σ
i
j
, which occurs freely in A
σ
i
i
, and on the values of
p
l
, ϑ
k
with rank lower than rank(p
j
). By the acyclic-
ity AC 2, the computations defined by the recur-
sion assignments { p
σ
1
1
:
= A
σ
1
1
, . . . , p
σ
n
n
:
= A
σ
n
n
}, along
with the type constraints {ϑ
1
:
type
= T
1
, . . . , ϑ
m
:
type
= T
m
},
terminate after a finite number of steps.
A type τ can be underspecified in two ways. (1)
τ can have free occurrences of type variables in it.
An underspecified type τ
0
can be specified by adding
specifications of some of its free variables, in the form
of type constraints via type assignments: (2) Each
type assignment ϑ
j
:
type
= T
j
binds the type variable ϑ
j
disjunctively, so it can still carry disjunctively under-
specified information, when T
j
has more than one el-
ement in it.
The terms (7d) of L
ta
ar
are essential for encoding
two-fold semantic information: denotational values
and algorithmic steps for computation of the deno-
tational values. We leave the precise definitions of
denotational and algorithmic semantics outside of the
scope of this paper. Informally, the denotation of a
term of the form (7d), is computed from the parts
A
σ
i
i
, by mutual recursion following the rank, saving
the values on the way in p
σ
i
i
.
Acyclic Recursion with Polymorphic Types and Underpecification
395
5 REDUCTION CALCULUS
This section introduces the reduction system of L
ta
ar
,
which extends L
λ
ar
.
Informally, congruence between terms (types) is
the smallest relation
c
(
t
c
) between terms (types)
that is reflexive, symmetric, transitive, and closed un-
der: term formation, renaming bound variables, re-
ordering of assignments, congruence of types. A type
τ Types is proper iff it is not a type variable, i.e.,
τ 6∈ TV, otherwise it is immediate.
Reduction Rules for Types:
Congruence (t-cong)
If τ
c
σ, then τ
t
σ
Transitivity (t-trans)
If τ
t
σ and σ
t
θ then τ
t
θ
Compositionality
If τ
1
t
τ
2
and σ
1
t
σ
2
, then (t-rep12)
(τ
1
σ
1
)
t
(τ
2
σ
2
)
If τ
i
t
σ
i
, for i = 0, . . . , n, then (t-rep3)
τ
o
where {ϑ
1
:
type
= τ
1
, . . . , ϑ
n
:
type
= τ
n
}
t
σ
0
where {ϑ
1
:
type
= σ
1
, . . . , ϑ
n
:
type
= σ
n
}
The Head Rule for Types (t-head)
τ
0
where {
ϑ
:
type
=
τ }
where {
θ
:
type
=
σ }
t
τ
0
where {
ϑ
:
type
=
τ ,
θ
:
type
=
σ }
given that no ϑ
i
occurs free in any σ
j
,
for i = 1, . . . , n, j = 1, . . . , m
The Beki
ˇ
c-Scott Rule for Types (t-B-S)
τ
0
where {ϑ
:
type
=
σ
0
where {
θ
:
type
=
σ }
,
ϑ
:
type
=
τ }
t
τ
0
where {ϑ
:
type
= σ
0
,
θ
:
type
=
σ ,
ϑ
:
type
=
τ }
given that no θ
i
occurs freely in any τ
j
,
for i = 1, . . . , n, j = 1, . . . , m
S-subtype Rule (S-subt)
For any type variables t
1
, . . . , t
m
TV and any
proper types τ
1
, . . . , τ
n
,
{t
1
, . . . , t
m
, τ
1
, . . . , τ
n
}
t
{t
1
, . . . , t
m
, ϑ
1
, . . . , ϑ
n
} where {ϑ
1
:
type
= τ,
. . . , ϑ
n
:
type
= τ}
Reduction Rules for Terms: The set of the re-
duction rules for Terms
L
ta
ar
extends the L
λ
ar
-reduction
(Moschovakis, 2006), by adding one more rule (type-
r) defined here, to the L
λ
ar
rules (cong), (trans), (rep1),
(rep3), (rep3), (head), (B-S), (recap), (ap), (λ).
The Type Rule (type-r)
If τ
t
σ, then A
τ
A
σ
6 KEY THEORETICAL
FEATURES OF L
ta
ar
We give some of the major results (with proofs in an
extended paper) that are essential for the algorithmic
semantics of the language L
ta
ar
. We say that a type
τ Types is irreducible iff
for all σ Types, if τ σ, then τ
t
c
σ (8)
We say that a term A Terms is irreducible iff
for all B Terms, if A B, then A
c
B (9)
Theorem 1 (Canonical Form Theorem for Types).
For each type term τ Types, there is a unique, up
to congruence, irreducible type term cf(τ), such that:
1. (a) cf(τ) τ or
(b) cf(τ) τ
0
where {ϑ
1
:
type
= τ
1
, . . . , ϑ
n
:
type
= τ
n
}
2. τ
t
cf(τ)
3. if τ
t
σ and σ is irreducible, then σ
t
c
cf(τ), i.e.,
cf(τ) is the unique up to congruence irreducible
type to which τ can be reduced.
Theorem 2 (Canonical Form Theorem). For each
term A
τ
, there are a unique, up to congruence, irre-
ducible type cf(τ) and a unique, up to congruence,
irreducible term cf(A
τ
), such that:
1. (a) cf(A
τ
) A
cf(τ
) or
(b) cf(A
τ
) A
τ
0
0
where {p
1
:
= A
τ
1
1
, . . . , p
n
:
= A
τ
n
n
},
where cf(τ)
c
τ
0
c
cf(τ
0
) and
τ
i
c
cf(τ
i
), for i = 1, . . . , n.
2. A
τ
cf(A
τ
)
3. if A
τ
B
σ
and B
σ
is irreducible, then B
σ
c
cf(A
τ
), i.e., cf(A
τ
) is the unique up to congruence
irreducible term of irreducible type to which A
τ
can be reduced.
The proofs of the above theorems use induction
by the definitions of Types
L
ta
ar
, Terms
L
ta
ar
, and the re-
duction rules. They are too long to be included in this
paper, and will be provided in an extended work. See
(Moschovakis, 2006) for the proof of the Canonical
Form Theorem in L
λ
ar
(without polymorphism), corre-
sponding to Theorem 2.
The canonical forms have a distinguished feature
that is part of their algorithmic role: they provide
algorithmic patterns for computations of the denota-
tions of all basic components, i.e., algorithms for se-
mantic computations. The more general terms pro-
vide algorithmic patterns represented by sub-terms
for the fundamental computational operators — func-
tional application, λ-abstraction, component values
that are saved in recursion variables upon the recur-
sive computations. The assignments of values to the
ICAART 2016 - 8th International Conference on Agents and Artificial Intelligence
396
recursion variables of lower ranks provide the specific
basic data that feeds-up the successive computational
steps.
The more general terms and sub-terms classify
language expressions with respect to their semantics
and determine the algorithms for computing their de-
notations. The canonical forms represent the decom-
position of the algorithmic steps to the most basic
ones.
Informally, two terms A, B Terms are algorith-
mically equivalent, i.e., algorithmically synonymous,
A B, iff (a) when A and B have algorithmic senses
(i.e., A and B are proper), then their denotations
(which are equal) are computed by the same algo-
rithm; (b) otherwise (i.e., A and B are immediate), A
and B have the same denotations.
7 FORMAL PRESENTATION OF
MOTIVATION
Now, we shall supplement the above formal defini-
tions with examples and further motivation. The for-
mal system allows flexible choices depending on ap-
plication needs. For example, we can render the ad-
verb “disturbingly” into a constant disturbingly
1
as in
(10), where the type is specific without any parame-
ters.
disturbingly
render
disturbingly
1
: ((
e
e
e
t) (
e
e
e
t))
(10)
Then, with the rendering (10), we can render the sen-
tence parsing (11a) into the terms in (11b)–(11f):
[[Jerry]
NP
[barks disturbingly]
VP
]
S
render
(11a)
[disturbingly
1
(bark)](jerry) (11b)
[disturbingly
1
(b) where
{b
:
= bark}](jerry) (ap, rep1)
(11c)
disturbingly
1
(b)(jerry) where
{b
:
= bark} (recap)
(11d)
[disturbingly
1
(b)( j) where { j
:
= jerry}]
where {b
:
= bark} (ap, rep3)
(11e)
cf
disturbingly
1
(b)( j)
where { j
:
= jerry, b
:
= bark} (head)
(11f)
Typically, in the existing systems of computational
semantics and computational grammars that incorpo-
rate semantic representations, in order to represent
alternative syntactic categories, type assignments, or
meanings of a lexeme (word), that lexeme is repre-
sented in the system by a set of different lexical items,
e.g., by marking a common lexeme sequence with
some label or index, correspondingly. Similarly, we
can render the adverb “disturbingly” into a second
L
λ
ar
-constant disturbingly
2
as in (12), which is needed
for the same sentence with the alternative parsing in
(13a) and its corresponding interpretation represented
by (13b)–(13f). As in (10) and (11b)–(11f), the types
of the constant disturbingly
2
in (12) and the terms in
(13b)–(13f) are specific, without any parameters.
disturbingly
render
disturbingly
2
: (
e
t
e
t) (12)
The type of the rendering (12) is suitable for rendering
of the sentence as in (13a)–(13f):
[[Jerry [barks]
VP
]
S
disturbingly]
S
render
(13a)
disturbingly
2
bark(jerry)
(ap) (13b)
disturbingly
2
(b) where
{b
:
= bark(jerry)} (ap, rep3)
(13c)
disturbingly
2
(b) where
{b
:
= [bark( j) where
{ j
:
= jerry}]} (B-S)
(13d)
cf
disturbingly
2
(b) where
{b
:
= bark( j), j
:
= jerry}
(13e)
disturbingly
2
(b) where
{ j
:
= jerry, b
:
= bark( j)}
(13f)
While, the above alternative renderings can be appro-
priate in specific environments, with respect to spe-
cific agents, such specific information may not be
available. In such cases it is more adequate to ren-
der the sentence into a formal term that is underspec-
ified and open to be instantiated when suitable infor-
mation is available. We can not render the verb “dis-
turbingly” into a parameter that could be instantiated
either as disturbingly
1
or disturbingly
2
in the standard
L
λ
ar
since every parameter and every constant have
to be of specific types as in (10) and (12). Our ex-
tended formal system TTofAR, with parametric types,
formalizes such possibilities, e.g., to render the verb
“disturbingly” to a constant disturbingly of paramet-
ric type, as in (14). The variable ζ is a type variable,
which can be instantiated when there is suitable infor-
mation available.
disturbingly
render
disturbingly : (
e
ζ
e
ζ) (14)
Note that, instead of assigning a simple type variable
to the constant disturbingly, we have used the under-
specified (i.e., parametric) type term (
e
ζ
e
ζ), since
we “know” some information about the parametric
type, which we can represent by parametric type ex-
pression. The type (
e
ζ
e
ζ) is functional and requires
Acyclic Recursion with Polymorphic Types and Underpecification
397
that the term to be modified and the resulted, modi-
fied, term to be of the same “unknown” type
e
ζ. This
is characteristic for any modifier expressions in artifi-
cial and natural languages. In addition,
e
ζ (s ζ) is
state dependent. One could just use simple, paramet-
ric modifier type ζ ζ).
Then, the sentence can be rendered to an under-
specified term as in (15b), where the type of the con-
stant disturbingly is parametric and underspecified.
[Jerry barks disturbingly]
S
render
(15a)
A where { j
:
= jerry, B
:
= bark,
D
:
= disturbingly}
(15b)
When sufficient information is available, the type
variable ζ can be instantiate suitable, either as (
e
e
e
t),
as in (16), or as
e
t. Thus, the already specified term
(16) corresponds to the denotation of the terms in
(11a)–(11f).
A where {ζ
:
type
= (
e
e
e
t)}{ j
:
= jerry,
B
:
= bark,
D
:
= disturbingly}
(16)
We have presented an example from human lan-
guage, as a pattern of handling language ambiguities.
Semantic ambiguities are abundant in human lan-
guages, and make language processing, understand-
ing, and communications, via computerized system,
one of the difficult areas in theories and applications.
Representing semantic information is particularly dif-
ficult, because resolving multiple interpretations is
highly context dependent, and not only inefficient,
but also unpredictable without sufficient context in-
formation. In this section, we have presented these
problems with a pattern example, and demonstrated
formalization of ambiguous, multiple interpretations,
computationally, with the type polymorphism of L
ta
ar
.
Specific interpretations can be obtained from the for-
mal terms, via context and agents providing instanti-
ations.
8 CONCLUSIONS AND FUTURE
WORK
The notion of referencial intension, which we also
call algorithmic intension, in the languages of recur-
sion, including L
ta
ar
covers the computational aspect of
the concept of meaning the algorithmic steps by
which denotations are computed. The algorithmic in-
tension, Int(A), of a meaningful term A is the tuple of
functions (a recursor) that is determined by its canoni-
cal form cf(A) A
0
where {p
1
:
= A
1
, . . . , p
n
:
= A
n
}.
Thus, the languages of recursion offer a formalisa-
tion of semantic layers: denotations and algorithmic
steps for computation of denotations. When spe-
cific denotations depend on context information that
is unknown, the algorithms can be represented by L
ta
ar
terms in canonical forms with free type and recursion
variable.
L
ta
ar
= Type Specification = Intensions (Algorithms) = Denotations
| {z }
Algorithmic Semantics
By L
ta
ar
, we have extended, the concept of algo-
rithm to a generalized notion of underspecified al-
gorithm, with respect to polymorphism, and in addi-
tion, for acyclic computations, i.e., polymorphic algo-
rithms that potentially close-off, which we consider
unique among theoretic formalizations of semantic
concepts.
We plan to extend the formal system presented in
this paper in two directions. On the theoretical side,
we work on extending the system with other reduction
rules and calculi. Such work broadens the range of
applications of L
λ
ar
to computational semantics of hu-
man languages, as well as to various areas of AI. E.g.,
we consider that it is important to add constraints over
underspecified types, along other kinds of constraints
over possible instantiations of free recursion variable,
e.g., in the line of (Loukanova, 2013a).
On the application side, we target applications
to computational syntax-semantics interface in nat-
ural language processing, e.g., along the lines of
(Loukanova, 2011d; Loukanova, 2011e; Loukanova,
2011a; Loukanova, 2012b) and (Loukanova and
Jim
´
enez-L
´
opez, 2012). Ambiguities and under-
specification are characteristic for human languages.
(Loukanova, 2012a) presented ambiguity among
some human language modifiers. A vast range of am-
biguous modifiers require to be rendered with terms
of varying semantic types. The type system in this
paper provides formalization of such underspecifica-
tion. We work on extending furthermore the expres-
sive power of the formal system L
ta
ar
introduced in this
paper, for covering other classes of ambiguities.
REFERENCES
Barwise, J. (1981). Scenes and other situations. The Journal
of Philosophy, 78:369–397.
Cardelli, L. and Wegner, P. (1985). On understanding types,
data abstraction, and polymorphism. ACM Computing
Surveys (CSUR), 17(4):471–523.
Copestake, A., Flickinger, D., Pollard, C., and Sag, I.
(2005). Minimal recursion semantics: an introduction.
Research on Language and Computation, 3:281–332.
ICAART 2016 - 8th International Conference on Agents and Artificial Intelligence
398
Gallin, D. (1975). Intensional and Higher-Order Modal
Logic. North-Holland.
Hurkens, A. J. C., McArthur, M., Moschovakis, Y. N.,
Moss, L. S., and Whitney, G. T. (1998). The logic of
recursive equations. The Journal of Symbolic Logic,
63(2):451–478.
Loukanova, R. (2010). Computational Syntax-Semantics
Interface. In Bel-Enguix, G. and Jim
´
enez-L
´
opez,
M. D., editors, Language as a Complex System: Inter-
disciplinary Approaches, pages 111–150. Cambridge
Scholars Publishing.
Loukanova, R. (2011a). From Montague’s Rules of Quan-
tification to Minimal Recursion Semantics and the
Language of Acyclic Recursion. In Bel-Enguix, G.,
Dahl, V., and Jim
´
enez-L
´
opez, M. D., editors, Biology,
Computation and Linguistics New Interdisciplinary
Paradigms, volume 228 of Frontiers in Artificial Intel-
ligence and Applications, pages 200–214. IOS Press,
Amsterdam; Berlin; Tokyo; Washington, DC.
Loukanova, R. (2011b). Modeling Context Information
for Computational Semantics with the Language of
Acyclic Recursion. In P
´
erez, J. B., Corchado, J. M.,
Moreno, M., Juli
´
an, V., Mathieu, P., Canada-Bago,
J., Ortega, A., and Fern
´
andez-Caballero, A., editors,
Highlights in Practical Applications of Agents and
Multiagent Systems, volume 89 of Advances in Intel-
ligent and Soft Computing. Springer, pages 265–274.
Springer.
Loukanova, R. (2011c). Reference, Co-reference and
Antecedent-anaphora in the Type Theory of Acyclic
Recursion. In Bel-Enguix, G. and Jim
´
enez-L
´
opez,
M. D., editors, Bio-Inspired Models for Natural and
Formal Languages, pages 81–102. Cambridge Schol-
ars Publishing.
Loukanova, R. (2011d). Semantics with the Language of
Acyclic Recursion in Constraint-Based Grammar. In
Bel-Enguix, G. and Jim
´
enez-L
´
opez, M. D., editors,
Bio-Inspired Models for Natural and Formal Lan-
guages, pages 103–134. Cambridge Scholars Publish-
ing.
Loukanova, R. (2011e). Syntax-Semantics Interface for
Lexical Inflection with the Language of Acyclic Re-
cursion. In Bel-Enguix, G., Dahl, V., and Jim
´
enez-
L
´
opez, M. D., editors, Biology, Computation and Lin-
guistics New Interdisciplinary Paradigms, volume
228 of Frontiers in Artificial Intelligence and Applica-
tions, pages 215–236. IOS Press, Amsterdam; Berlin;
Tokyo; Washington, DC.
Loukanova, R. (2012a). Algorithmic Semantics of Am-
biguous Modifiers by the Type Theory of Acyclic Re-
cursion. IEEE/WIC/ACM International Conference
on Web Intelligence and Intelligent Agent Technology,
3:117–121.
Loukanova, R. (2012b). Semantic Information with Type
Theory of Acyclic Recursion. In Huang, R., Ghorbani,
A. A., Pasi, G., Yamaguchi, T., Yen, N. Y., and Jin, B.,
editors, Active Media Technology - 8th International
Conference, AMT 2012, Macau, China, December 4-
7, 2012. Proceedings, volume 7669 of Lecture Notes
in Computer Science, pages 387–398. Springer.
Loukanova, R. (2013a). Algorithmic Granularity with
Constraints. In Imamura, K., Usui, S., Shirao, T.,
Kasamatsu, T., Schwabe, L., and Zhong, N., edi-
tors, Brain and Health Informatics, volume 8211 of
Lecture Notes in Computer Science, pages 399–408.
Springer International Publishing.
Loukanova, R. (2013b). Algorithmic Semantics for Pro-
cessing Pronominal Verbal Phrases. In Larsen, H. L.,
Martin-Bautista, M. J., Vila, M. A., Andreasen, T., and
Christiansen, H., editors, Flexible Query Answering
Systems, volume 8132 of Lecture Notes in Computer
Science, pages 164–175. Springer Berlin Heidelberg.
Loukanova, R. (2013c). A Predicative Operator and Un-
derspecification by the Type Theory of Acyclic Re-
cursion. In Duchier, D. and Parmentier, Y., editors,
Constraint Solving and Language Processing, volume
8114 of Lecture Notes in Computer Science, pages
108–132. Springer Berlin Heidelberg.
Loukanova, R. (2016). Specification of underspecified
quantifiers via question-answering by the theory of
acyclic recursion. In Andreasen, T., Christiansen, H.,
Kacprzyk, J., Larsen, H., Pasi, G., Pivert, O., De Tr
´
e,
G., Vila, M. A., Yazici, A., and Zadro
˙
zny, S., editors,
Flexible Query Answering Systems 2015, volume 400
of Advances in Intelligent Systems and Computing,
pages 57–69. Springer International Publishing.
Loukanova, R. and Jim
´
enez-L
´
opez, M. D. (2012). On
the Syntax-Semantics Interface of Argument Mark-
ing Prepositional Phrases. In P
´
erez, J. B., S
´
anchez,
M. A., Mathieu, P., Rodr
´
ıguez, J. M. C., Adam, E.,
Ortega, A., Moreno, M. N., Navarro, E., Hirsch, B.,
Lopes-Cardoso, H., and Juli
´
an, V., editors, Highlights
on Practical Applications of Agents and Multi-Agent
Systems, volume 156 of Advances in Intelligent and
Soft Computing, pages 53–60. Springer Berlin / Hei-
delberg.
Moschovakis, Y. N. (1989). The formal language of recur-
sion. The Journal of Symbolic Logic, 54(04):1216–
1252.
Moschovakis, Y. N. (1994). Sense and denotation as algo-
rithm and value. In Oikkonen, J. and Vaananen, J.,
editors, Lecture Notes in Logic, number 2 in Lecture
Notes in Logic, pages 210–249. Springer.
Moschovakis, Y. N. (1997). The logic of functional recur-
sion. In Logic and Scientific Methods, pages 179–207.
Kluwer Academic Publishers. Springer.
Moschovakis, Y. N. (2006). A logical calculus of meaning
and synonymy. Linguistics and Philosophy, 29:27–89.
Muskens, R. (1995). Meaning and Partiality. Studies in
Logic, Language and Information. CSLI Publications,
Stanford, California.
Thomason, R. H., editor (1974). Formal Philosophy: Se-
lected Papers of Richard Montague. Yale University
Press, New Haven, Connecticut.
Van Lambalgen, M. and Hamm, F. (2004). The Proper
Treatment Of Events. Explorations in Semantics.
Wiley-Blackwell, Oxford.
Villadsen, J. (2010). Nabla: A Linguistic System based
on Type Theory, volume 3 of Foundations of Com-
munication and Cognition (New Series). LIT Verlag
M
¨
unster-Hamburg-Berlin-Wien-London.
Acyclic Recursion with Polymorphic Types and Underpecification
399