Minimal Recursion Semantics and the Language
of Acyclic Recursion
Roussanka Loukanova
Uppsala, Sweden
Abstract. Moschovakis (2003-2006) developed a logical calculus of the formal
language L
λ
ar
of acyclic recursion, which is a type-theoretical work with many
potential applications. On the implementation side, large-scale grammars for hu-
man languages, e.g. versions of HPSG, have been using semantic representations
casted in the feature-value language Minimal Recursion Semantics (MRS). While
lacking strict formalization, MRS represents successfully ambiguous quantifier
scoping. In this paper, we introduce the basic definitions of MRS by reflecting on
possibilities for formalization of MRS with a version of the language L
λ
ar
.
1 Introduction: Why MRS Representations?
Research presented in this paper targets formalization and development of analysis with
syntax-semantics interface of spoken and written human language (incl. texts larger
than sentences), which continues to be a largely open area, in need of theoretical foun-
dations for reliable coverage.
Versions of constraint-based lexicalist grammar (CBLG), in particular of Head-
Driven Phrase Structure Grammar (HPSG), have achieved significant developments and
accumulated large resources for English and other languages, incl. for Norwegian, Dan-
ish, Arabic, etc. An international consortium, which originates by work in Stanford
1
,
developed a grammar tool, LKB, for writing grammars of human languages. In the
last years, LKB comes with possibility for semantic representation, by using Minimal
Recursion Semantics (MRS), see [3]. By its nature, MRS in HPSG is a notational,
specialized version of Situation Semantics, with feature-value structures representing
information terms.
Among the existing approaches to theory of meaning of natural language (NL),
model-theoretic alternatives provide viability of computational semantics for applica-
tions to the study of language faculty, knowledge representation in general, and in par-
ticular, for representation of linguistic knowledge, and development of intelligent com-
puterized systems. Typically, computational semantics of NL involves rendering of NL
expressions into some formal logic language. First-order languages and logic, while
well-studied and understood, have repeatedly exposed their unsuitability for semantics
of NL, from the perspectives of computability and linguistic adequacy. On the other
hand, higher order languages and typed λ-calculi have pleasant computational proper-
ties, but are still problematic from theoretic and application points as theories of mean-
ing, representation of knowledge and information flow, for which they are under active
1
see <http://www.delph-in.net/lkb/>
Loukanova R..
Minimal Recursion Semantics and the Language of Acyclic Recursion.
DOI: 10.5220/0003309800880097
In Proceedings of the 1st International Workshop on AI Methods for Interdisciplinary Research in Language and Biology (BILC-2011), pages 88-97
ISBN: 978-989-8425-42-3
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

developments, for applications to semantics of artificial languages, e.g., for semantics
of programming languages, and for NL.
Semantic representation, including rendering of NL expressions into formal logic
languages, such as first or higher order languages, have been problematic in systems
for NLP. The variety of such applications is large and growing: semantic transfer ap-
proaches to machine translation (MT), obtaining semantic representations by parsing
NL sentences, generation of NL sentences for given semantic representations, and other
more advancedapplications to automatic understanding of NL, for example,in question-
answer systems, information transfer, information extraction, knowledge representation
systems including knowledge inference, update, etc.
For example, a simplified semantic transfer schemata for MT typically consists of
the following stages, some of which, at least the first two, may be carried on in a com-
positional way:
Parsing an expression of a source NL, which produces:
Semantic representation of the input NL expression in some formal language. The
semantic representation is called source LF.
A transfer component, which converts the source LF into a semantic representation,
called the target LF.
A generator converts the target LFs into expression(s) of the target NL.
In such systems, ideally, a semantic analyzer of the source NL sentences produces se-
mantic representations, called logic forms (LFs) in some formal language, to be used
for generating logically equivalent sentences in a target NL. The basic problems that
emerge are related to mismatch between LFs and NL expressions. The LF produced
by a parser typically carries on the syntactic structure of the input NL expression. For
example, the order of the atomic formulas in a LF, e.g., such as a conjunction, may
correspond to the syntactic structure of the NL expression, while it is irrelevant for the
semantic interpretation. In a simplified approach, a generator can be build to try all
logically equivalent LFs until finds the appropriate ones. Such approaches meet serious
problems, for example, involving spurious ambiguity or unacceptability; e.g., analyses
may produce various logically equivalent LFs some of which correspond to unaccept-
able NL sentences (see Copestake et al. for examples and discussion). Depending on
the formal language chosen, such approaches may inherit some serious drawbacks with
respect to computability: computational inefficiency and/or undecidability of the prob-
lem of logical form equivalence. Some of these problems get pleasantly resolved for a
semantic core of NL, which has a syntactic expression in NL, by a recent development
of a grammatical framework (GF), see [12] and [13].
Some of the classic semantic theories used in NLP may carry on more fundamental
problems, among which, a serious one is the quantifier scope ambiguity. This is demon-
strated by any of the notorious examples, with at least two quantifier NPs, like (1a),
for which there is only one classic context-free parse tree, while having more than one
possible logic forms, representing alternative scoping:
(1) a. [[Every man]
NP
loves [a woman]
NP
]
S
.
b. de dicto reading:
89

1. [every man]
i
[[a woman]
j
[e
i
loves e
j
]
S
]
S
]
S
2. ∀x(man(x) → ∃y(woman(y) ∧ love(x, y)))
c. de re reading:
1. [[a woman]
j
[[every man]
i
[e
i
loves e
j
]
S
]
S
]
S
.
2. ∃y(woman(y) ∧ ∀x(man(x) → love(x, y)))
Among the quantifiers and quantifier scope ambiguity problems for NLP the fol-
lowing ones are ongoing:
1. A typical (context-free) parser gives only one parse tree structure, which corre-
sponds to multiple LFs, without any direct compositional way to derive them.
2. A classic style treatment of quantifiers (as in the lines of classic Montague’s PTQ,
see [8]) results in computational inefficiency, in particular, if all readings of a NL
expression with several quantifiers have to be derived at each level of processing
the sentence and its components.
Linguistic studies of underspecificationin human languages (NL) have been broadly
reviewed in [2]. Some approaches, closely related to the topic of this paper, have been
tried in logic type theories to represent multiple scoping by techniques for underspeci-
fied representation: for an example in the line of a Montagovian approach, see [10]; for
representation of quantifier ambiguities and, in general, of partiality of information in
Situation Semantics, by using Situation Theory, see [4]. A simplified version of a quan-
tifier storage technique was implemented in HPSG, e.g., see [11], which in recent years
evolved in elaborated MRS representation, see [3]. More recently, a new approach has
been initiated in [5], [6], and [7].
A demonstrative example for underspecified scoping is depicted by the following
unconnected graph (“underspecified tree”), which carries information about the “bare”-
predicate structure of the sentence, where the “disconnected” quantifiers carry indexing
information about the corresponding subject–complement argument roles they fill up.
(2) Underspecified tree (actually a graph) structure:
[every man]
NP
, i
every man
[a woman]
NP
, j
a woman
x
i
loves x
j
, S
x
i
, NP loves x
j
, VP
loves x
j
The set of the two indexed NP sub-graphs represents syntactically the ‘logic’ storage
needed for computing the resolved logic form of the sentence, e.g., see [4]. The S sub-
tree represents the underspecified semantic basis of the sentence. By this partially con-
nected graph we have a syntactical representation of the underspecified logic form:
(3) a. Quantifier storage:
[every man]
NP
, i
every man
[a woman]
NP
, j
a woman
90

b. Underspecified semantic basis:
x
i
loves x
j
, S
x
i
, NP loves x
j
, VP
loves x
j
2 MRS Representations and Notations
MRS uses a language with n-ary conjunction. Since binary conjunction ∧ causes spu-
rious ambiguity in parsing NL expressions, for efficiency of processing, ∧ is taken as
an n-ary operator, which is represented implicitly: any list of atomic formulas is inter-
preted as a conjunction.
Elementary predication (EP) is any atomic formula or a conjunction of atomic for-
mulas. EPs are tagged with labels. Thus the MRS formula (4a) is represented by the
tagged tree (4b):
(4) a. every(x)

∧

big(x), white(x), horse(x)

, sleep(x)

b. h
0
: every(x)(h
1
, h
2
)
h
1
: big(x), white(x), horse(x) h
2
: sleep(x)
The above MRS representations (4a) and (4b) can be written in the following nota-
tion, by using assignments to resemble terms in Moschovakis language L
λ
ar
:
(5) h
0
where { h
0
:= every (x, h
1
, h
2
),
h
1
:= ∧(big(x), white(x), horse(x)), h
2
:= sleep(x) }
Note that, in Moschovakis’ calculus of L
λ
ar
, coordination terms containing conjunction
(the value of h
1
above) undergo further reduction to canonical forms. In a resolved
term, the head subterm does not need to be assigned to any location like h
0
above. In a
realistic NL grammar, top locations (labels) simplify the compositional derivations, but
add “computational” steps.
MRS uses three kinds of variables:
1. Variables, called also parameters, for quantification over and reference to indi-
viduals denoted by NPs and for filling up argument slots of relations, when the
arguments designate individuals;
2. Labels, for tagging EPs and filling up argument slots of relations, when the argu-
ments are for EPs;
3. Free labels for labels to which no EPs are assigned.
Respectively, the language L
λ
ar
has two sorts of variables:
91

1. Pure variables, which are to be quantified, i.e. corresponding to the MRS variables
for individuals;
2. Recursion (location) variables to which EPs are assigned, i.e. these correspond to
the MRS labels. For example: h := sleep(x), where h is a location variable, i.e. a
label, and x is a pure variable.
Different MRS Representations of the NL Sentence.
(6) Every dog chases some white cat.
(7) a. De Re reading, in predicate language:
some(y)[(white(y)∧ cat(y)) ∧
every(x) (dog (x) → chase(x, y))]
b. De Re reading, by a MRS tree:
h
5
: some(y)
h
7
: white(y), cat(y) h
1
: every (x)
h
3
: dog(x) h
4
: chase(x, y)
c. De Re reading, in MRS term:
h
5
: some(y, h
7
, h
1
), h
7
: white(y), cat(y),
h
1
: every(x, h
3
, h
4
), h
3
: dog(x), h
4
: chase(x, y)
d. De Re reading, in the language L
λ
ar
:
h
5
where{ h
5
:= some(h
7
, h
1
)
h
7
:= λy (p(y)&q(y)),
p := λy white(y), q := λy cat(y),
h
1
:= λy every(h
3
(y), h
4
(y)),
h
3
:= λy λx dog(x), h
4
:= λy λx chase(x, y)}
(8) a. De Dicto reading, in predicate language:
every(x)[dog(x), some(y)(∧(white(y), cat(y)), chase(x, y))]
b. De Dicto reading, in MRS:
h
1
: every(x)
h
3
: dog(x) h
5
: some(y)
h
7
: white(y), cat(y) h
4
: chase(x, y)
c. De Dicto reading, by a MRS term:
h
1
: every(x, h
3
, h
5
), h
3
: dog(x),
h
5
: some(y, h
7
, h
4
), h
7
: white(y), cat(y), h
4
: chase(x, y)
d. De Dicto reading, in the language L
λ
ar
:
h
1
where { h
1
:= every (h
3
, h
5
), h
3
:= λx dog (x),
h
5
:= λx some(h
7
, h
4
(x)), h
7
:= λy (p(y)&q(y)),
p := λy white(y), q := λy cat(y),
h
4
:= λxλy chase(x, y)}
92

Underspecified Representation. The underspecified quantification can be depicted by
the labeled graph with nodes that are unconnected:
(9) a. Underspecified MRS graph:
•h
1
: every(x)
•h
3
: dog(x) •h
A
•h
5
: some(y)
•h
7
: white(y), cat(y) •h
B
•h
4
: chase(x, y)
b. Underspecified MRS term:
h
1
: every(x, h
3
, h
A
), h
3
: dog(x),
h
5
: some(y, h
7
, h
B
), h
7
: white(y), cat(y),
h
4
: chase(x, y)
There are exactly two ways of assigning EPs to the variables h
A
and h
B
to form
well formed MRS terms and corresponding tree representations I.e., the system
of equations has exactly two solutions corresponding to:
1. h
A
: h
5
, h
B
: h
4
;
2. h
A
: h
4
, h
B
: h
1
.
Note that, in MRS, the variables h
A
and h
B
are called handles.
c. Underspecified L
λ
ar
-term: (for example, by using the rules of the reduction
calculus)
h
0
(u) where { h
1
:= λy every(h
3
, λx h
A
(x)(y)),
h
5
:= λx some(h
7
, λy h
B
(x)(y)), h
3
:= dog,
h
7
:= λy (p(y)&q(y)), p := λy white(y), q := λy cat(y),
h
4
:= λxλy chase(x, y)}
This system of equations has exactly two solutions, in case it is extended, cor-
respondingly, by adding the following assignments inside the scope of the re-
cursion operator where:
1. h
B
:= h
4
, h
A
:= λxλz h
5
(x), h
0
:= h
1
;
2. h
A
:= h
4
, h
B
:= λz h
1
, h
0
:= h
5
.
3 Basic Definitions of MRS
I will introduce MRS as currently given in Copestake et al., but in Moschovakis terms.
A language of MRS includes, along other symbols and types, a set Rel of relation
symbols. MRS, by using Moschovakis terminology, has two sorts of variables:
Definition 1 (Variables).
– a set of pure variables:
V
pure
= x, y, z, . . .
Pure variables are to be quantified and for reference to individuals.
93

– a set of recursion or variables:
V
rec
= h
0
, h
1
, . . .
Recursion variables are called also location in L
λ
ar
, or labels and handles in MRS.
In the above informal introduction of MRS representations, it is clear that the vari-
ables called handles and labels are of the same formal sort, similar to location variables
in the language L
λ
ar
of recursion. Notationally, in the MRS syntactic constructs, the
distinction between handles and labels is with respect to the positions taken by these
variables. Note that in MRS, ‘location’ variables are used for labeling EPs and for fill-
ing up scopal argument slots of relation symbols. In L
λ
ar
, location variables are used in
the construction of recursive terms.
Definition 2 (Elementary predication (EP)). This definition corresponds closely to the
one given in Copestake et al. (p.12):
label : relation(arg
1
, . . . , arg
n
, sc arg
1
, . . . , sc arg
m
), where
1. label ∈ V
rec
and is called the label of the EP;
2. relation ∈ Rel is an (n + m)-argument relation symbol;
3. arg
1
, . . . , arg
n
∈ V
pure
are the ordinary variable (i.e. non-scopal) arguments of
relation;
4. sc arg
1
, . . . , sc arg
m
∈ V
rec
are the scopal arguments of relation.
Examples:
(10) a. h : every(y, h
1
, h
2
)
b. h : sleep(x)
c. h : probably(h)
Now, by considering the examples given in the paper, the above definition implies a
wrong interpretation of the quantifier symbols like some and every as 3-place argument
relations.
MRS has no λ-abstraction terms and no types corresponding to those of typed λ-
calculus. Versions of CBLG similar to HPSG, use a SEM feature INDEX which, up to
some extend, corresponds to λ-abstraction.
Revised
2
Definition 2 (Elementary predication)
h : relation(a
1
, . . . , a
n
, h
1
, . . . , h
m
), where
1. h ∈ V
rec
and is called the label of the EP;
2. relation ∈ Rel is a relation symbol;
3. a
1
, . . . , a
n
∈ V
pure
are variables which either fill up argument slots of the relation
symbol relation or, in case of a quantifier relation, are the variables bound by it;
4. h
1
, . . . , h
m
∈ V
rec
are variables filling up the scopal arguments slots of relation.
Some Abbreviations and Notations: In MRS, If a variable h ∈ V
rec
labels an EP, it
is called a label; if h ∈ V
rec
fills up an argument position of a relation, it is called a
handle. A bag of EPs that have the same label is interpreted as a conjunction. A bag of
co-labeled EPs h : E1, . . . , h : E
n
is denoted by h : E1, . . . , E
n
.
2
I am giving a minimal revision in order to keep this introduction to MRS close to Copestake
et al.
94

Definition 3 (Immediate Outscoping Relation between EPs). Given a bag of EPs M ,
and two EPs E, E
′
∈ M , E immediately outscopes E
′
iff one of the scopal arguments
of E is identical to the label of E
′
. I.e., for some p, p
′
∈ Rel and l, h ∈ V
rec
, E = l :
p(. . . , h , . . .) immediately outscopes E
′
= h : p
′
(. . .) and it is said that l immediately
outscopes h and E
′
.
Definition 4. Given two conjunctions of EPs M and M
′
, M immediately outscopes
M
′
iff there are E ∈ M and E
′
∈ M
′
such that E immediately outscopes E
′
.
Definition 5 (Outscoping Relation). Outscoping relation over a set of EPs is the tran-
sitive closure of the immediate outscoping relation between EPs in that set.
Definition 6 (MRS Structure). A MRS structure is a tuple hGT , LT , R, C i, where
– R is a bag of EPs.
– GT ∈ V
rec
is a label (recursion variable) such that there is no label h ∈ V
rec
which
occurs in R and outscopes GT . GT is called the global top of M.
– LT ∈ V
rec
is the topmost label in R, with respect to the outscoping relation over
the labels in R, and which is not the label of a floating (see later) EP. LT is called
the local top of M .
– C is a set of constraints (introduced later on) satisfied by the outscoping order in
R.
Definition 7 (Scope-resolved MRS Structure). A scope-resolved MRS structure is an
MRS structure such that:
1. The MRS structure forms a tree of EP conjunctions, where dominance is deter-
mined by the outscope ordering on EP conjunctions.
2. The global and local top labels and all handle arguments (i.e. recursion arguments)
are identified with an EP label (in Moschovakis terminology: there are no free re-
cursion variables).
4 Conclusions: Advancing New Developments
Arguments for “Flat” Semantic Representation that Pends Further Development.
The original arguments for introducing MRS representation have been efficiency with-
out loss of information. Further theoretic and development work is needed for the fol-
lowing initiations:
Underspecification. MRS permits underspecification in representations of quantifier
scopes so that a single MRS construct represents multiple scopes without loss of gram-
matical information available in the structure of NL expressions.
Flat Semantic Representation. “Flat” MRS representations, which consist of the most
basic facts, without loss of information, improve the efficiency of NLP. For example,
flat representations have been favored in systems such as Generation from semantic
representations and machine translation with semantic transfer.
Currently, MRS is in development stage for use in HPSG. For example, MRS has
been extensively implemented in grammars for English, Norwegian and Danish by us-
ing LKB.
95

Further Development of MRS and L
λ
ar
Representation.
Syntax-Semantics Interface. MRS representations can be written in a feature-value
language for formal and computational grammar of NL, such asCBLG (e.g., HPSG).
CBLG incorporates several linguistic components: vocabulary, lexicon, syntax, and se-
mantic representations, in a unified way, which is a basis for compositional syntax-
semantics interface.
Logic Foundation. MRS offers semantic representations which are close to the canon-
ical forms of terms in the formal language of Acyclic Recursion. This gives opportu-
nities to formalize MRS and develop new implementations by re-using existing CBLG
resources.
Rendering. Render relation is the translation from NL into the language L
λ
ar
of acyclic
recursion. Feature-value lexicalist approach to grammar theory, such as CBLG (HPSG,
LFG, etc.), is a good computational approach to syntax, which provides procedures for
rendering into logic forms and is linguistically and semantically adequate.
Indexing. Indexing procedure from NL into L
λ
ar
can be provided, e.g., by appropriately
development of Binding Theory in CBLG (HPSG)
With appropriate adjustments of MRS, Moschovakis Acyclic Recursion can provide
appropriate formalization of MRS. Using a version of Acyclic Recursion can contribute
to developing MRS itself, for example, to:
1. formal representation of quantifiers;
2. representation of abstraction in MRS (resembling λ-abstraction);
3. finding a better incorporation of utterance and described situations into the MRS
expressions (and MRS feature structures used in CBLG (HPSG)).
4. representing higher order relations for modifiers that are not conjunctively inter-
preted: alleged, former, ...;
5. representing higher order relations denoted by lexemes creating oblique contexts,
such as know, believe, ... .
Relation to other Type-theoretic Developments for NLP. In recent years, a pow-
erful type-theoretical grammar formalism for natural, i.e. human, language processing
(NLP), Grammatical Framework (GF), see [12] and [13], has been under active devel-
opment. Future work, which is tightly related to the subject of this paper, is research
on the placement of GF in the family of CBLG, with respect to syntax, semantics, and
syntax-semantics inter-dependencies, its theoretic foundations, and applications.
The work on computational semantics presented in this paper is in a direction of
theoretical developments, for providing foundations for, and extending, current appli-
cations in the lines of CBLG (e.g., HPSG and GF), and for new ones.
96

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