A Formalization of Generalized Parameters

in Situated Information

Roussanka Loukanova

Department of Mathematics, Stockholm University, Stockholm, Sweden

Keywords:

Formal Language, Situation Theory, Parametric Information, Generalized Parameters, Partiality, Restrictions,

Memory Variables, Recursion, Situations, Space-time Locations.

Abstract:

The paper introduces a higher-order, type-theoretical formal language L

ST

GP

of information content that is par-

tial, parametric, underspeciﬁed, dependent on situations, and recursive. The terms of the formal language

represent situation-theoretic objects. The language has specialized terms for constrained computations by

mutual recursion. It introduces terms representing nets of parameters that are simultaneously constrained to

satisfy restrictions.

1 INTRODUCTION

The formal language L

ST

GP

is a generalization of the

formal language introduced in (Loukanova, 2015) and

serves as a language for algorithmic semantics by us-

ing semantic structures that are models of informa-

tion by Situation Theory introduced in (Seligman and

Moss, 2011) and (Loukanova, 2014).

The semantic type system of Situation Theory, as

model-theory of information, is important in the in-

formational structures of the theory, because, on the

one hand, its types classify the distribution of objects

in the situation-theoretic domains, and, on the other

hand, the types participate in the informational com-

ponents of the situation-theoretic objects themselves.

We re-introduce that type system, formally, into the

components of the formal language L

ST

GP

, presented in

this paper. By this, the formal types serve the inter-

face between the formal syntax of L

ST

GP

and the situa-

tional, semantic frames. We are designing the syn-

tax of L

ST

GP

to be as close as possible to the multi-

dimensional geometry of abstract, semantic objects.

The types play important role of the algorithmic

semantics in the syntax-semantics interfaces of L

ST

GP

,

where L

ST

GP

-terms of specialized forms determine the

algorithms for computing their denotations. We con-

sider that such features of L

ST

GP

contribute to adequate-

ness of computational semantics of human language,

targeting advanced applications to computational pro-

cessing of human language. This is a central motiva-

tion for our work on extending and developing both

Situation Theory, as a ﬁne-grained model-theory of

information, as well as a specialized formal language

for it, e.g., L

ST

GP

in this paper.

We consider that developing L

ST

GP

, introduced here,

facilitates important aspects of syntax-semantics in-

terfaces of human language. Developing computa-

tional semantics of human language by a direct map-

pings between its syntax and situation-theoretic, i.e.,

semantic set-theoretic, domains, would be difﬁcult,

hard work, even for simple applications. Instead, such

tasks would be facilitated, by having a suitable formal

language, e.g., as the one introduced in (Loukanova,

2015), and extending it by L

ST

GP

, introduced in this pa-

per. It would be advantageous to use such a formal

language that represents the ﬁne-granularity of infor-

mation content in situation-theoretical objects.

The formal language L

ST

GP

has expressions for func-

tions, properties, relations, and types, with explicitly

expressed argument roles, not by linearly ordered ar-

gument slots typically used in traditional formal lan-

guages and semantic model-theories. The argument

roles of functions, relations, and types are associated

with appropriateness constraints. Only objects sat-

isfying the corresponding constraints can ﬁll up the

roles.

We deﬁne terms of generalized recursion over

situation-theoretic objects that can have functional

and relational components with constraints. These

terms have separate, but connected components, re-

spectively, for recursive computations with assign-

ments and propositional restrictions. The recursion

component of a term represents instantiations of para-

metric information in recursion variables, i.e., in

Loukanova, R.

A Formalization of Generalized Parameters in Situated Information.

DOI: 10.5220/0005850303430353

In Proceedings of the 8th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2016) - Volume 1, pages 343-353

ISBN: 978-989-758-172-4

Copyright

c

2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

343

memory cells. The cells, and other parameters that

occur in the objects saved in the cells, can be re-

stricted to satisfy constraints, which also can be linked

by mutual recursion.

The recursion terms represent algorithms for com-

putation of values that are saved in memory slots, i.e.,

in memory variables. The memory variables that oc-

cur in L

ST

GP

terms, represent the structures in memory

sections of a computational entity, which are engaged

in algorithmic computations by using informational

content saved in the memory slots. From the perspec-

tive of neuroscience, L

ST

GP

terms represent neural nets

of recursively linked memory cells for processing and

saving information. Our inspiration for this is from

the work (Kandel et al., 2000) and (Squire and Kan-

del, 2009). Information is saved in the memory cells

of a neural network via mutual recursion. In addition,

the memory cells are restricted to satisfy higher-order,

typed constraints. The components of recursive as-

signments and of recursive constraints represent neu-

ral cells for memory and exchange of partial, para-

metric, information that depends on situations.

We introduce specialized terms that generalize the

situation theoretical notion of a complex, restricted

parameter to a generalized, parametric net. The para-

metric net consists of memory components, which are

restricted to be simultaneously of a given complex

type, and can involve recursive computations.

In this paper, we focus on introducing the for-

mal syntax of L

ST

GP

and its motivation. We support

this work with examples. Full presentation of deno-

tational and algorithmic semantics of the formal lan-

guage L

ST

GP

, interpreted in the situation-theoretic mod-

els, requires rather technical means, formal reduction

calculus and inference system, which are outside the

scope of this paper.

We hope that this sufﬁces for using L

ST

GP

for for-

mal streamlining developments of various applica-

tions. In particular, L

ST

GP

can contribute to applications,

where semantic information is important, while it car-

ries partiality, ambiguity, underspeciﬁcation, context-

dependency. Fine-grained, detailed and structured se-

mantic information can be included in such applica-

tions by using terms of L

ST

GP

. Among potential appli-

cations, we would like to point, at ﬁrst place, the po-

tentials for using L

ST

GP

and its specialized variants, for

computational syntax-semantics interfaces in large-

scale grammars of human language, such as HPSG,

LFG, GF, and grammars using Logic Programming.

Other important applications include database, in re-

lational, object-oriented, and hybrid approaches; for-

mal representation and storing of semantic informa-

tion in ontology systems; semantic information in text

processing; information retrieval; etc.

By the examples, which we include in this pa-

per, we address how space-time information can be

included formally, along all components of seman-

tic information. Specialized variables, representing

semantic parameters for space-time locations, which

can be linked to speciﬁc, abstract, real, or virtual sit-

uations that carry partial information.

2 BACKGROUNDS

Originally, Situation Theory was introduced by (Bar-

wise, 1981) as a general theory of information, by

mathematical structures of information. The ideas

of Situation Theory ensued a broad program with

wide spectrum of theoretical research and applica-

tions. The central concepts developed around repre-

senting relational and partial information, and its de-

pendence on situations. The ideas were presented in

great details by (Barwise and Perry, 1983) and (Bar-

wise, 1989). For an informal introduction, see (De-

vlin, 2008). A substantial mathematical presentation

of Situation Theory is given by (Seligman and Moss,

2011). For Situation Theory that we take as a pri-

mary semantic structure of the formal language in

this paper, see (Loukanova, 2014). The non-well-

founded Aczel set theory (Aczel, 1988), with anti-

foundation axiom, is the set-theoretic foundation of

Situation Theory in its full strength that supports cir-

cular information.

Our paper is on a largely open topic of formaliza-

tion of Situation Theory, with computational syntax

and semantic models of ﬁnely-grained information,

initiated in (Loukanova, 2014; Loukanova, 2015).

Higher-order, typed Situation Theory of information

and type-theoretic formal languages for its versions

are opening new theoretical investigations and practi-

cal applications. Computational semantics and com-

putational neuroscience of language are among the

primary applications of Situation Theory by using for-

mal languages for it. This paper is based on our work

on several new directions, in particular: (1) functional

type-theory of recursion (a functional approach); (2)

relational type-theory of situated, partial, and para-

metric information (a relational approach); (3) ap-

plications of these theories to computational syntax-

semantics interfaces in natural and formal languages.

Typical syntax of formal and natural languages

is formulated by rules that express syntactic depen-

dencies between sub-expressions and syntactic cate-

gories. Such syntactic dependencies can be visualized

as appropriately labeled graphs, often as parse trees,

in 2-dimensional plane. In contrast, by the formal lan-

guage L

ST

GP

, we express ﬁne-grained, semantic infor-

PUaNLP 2016 - Special Session on Partiality, Underspeciﬁcation, and Natural Language Processing

344

mation about semantic objects, properties of objects,

relations between them, operations, processes, activi-

ties, which usually take place in 3-dimensional space,

and in time. Situation Theory is abstract model-

theory of information concerning real or virtual situ-

ations, represented by abstract set-theoretical objects.

The formal constructs of L

ST

GP

express, i.e., designate

situation-theoretic objects. (In a similar manner, a

specialized formal language of ﬁrst-order logic is em-

ployed in axiomatic set theories.)

In this paper, we introduce a higher-order, typed

formal language of Situation Theory with memory

variables. Memory variables represent memory slots

where underspeciﬁed objects can be saved. Such un-

derspeciﬁed objects can be simple semantic parame-

ters, or more complex objects with parametric com-

ponents. Informally, while the memory “slots” serve

saving, i.e., memorizing information, they are dy-

namic so that information stored in them can be up-

dated. The memory variables can be thought of be-

ing labels, or addresses of memory slots. The val-

ues saved in the memory variables are subject to two

kinds of “specializations”:

• Memory variables, which correspond to seman-

tic parameters, are constrained to satisfy situation-

theoretic restrictions.

• Values can be assigned to memory variables, re-

spectively to parameters, by recursive computa-

tions.

The L

ST

GP

has terms for functions, types, relations,

and recursive computations with restrictions. Typi-

cally, by traditions, formal languages use symbols and

other more complex expressions for functions, prop-

erties, and relations, with linearly ordered argument

slots that represent argument roles, according to con-

ventional agreements. The formal language L

ST

GP

di-

verges from this tradition, by explicit association of

the argument roles to functions, properties, relations,

and types. The argument roles are associated with ap-

propriateness constraints. Only objects satisfying the

corresponding constraints can ﬁll up the roles.

These features of L

ST

GP

are more direct representa-

tions of operations and relations between objects in

nature and especially, in bio-structures. The type sys-

tem of L

ST

GP

models structures of basic and complex

objects, and information in nature.

The terms of the formal language L

ST

GP

are demon-

strated with examples. We give intuitions about the

denotational and algorithmic semantics of L

ST

GP

. In

particular, we provide examples of computational

patterns in The work (Loukanova, 2015) introduces

a similar formal language L

ST

GP

for Situation The-

ory. The language L

ST

GP

introduced in this paper is

a proper extension of L

ST

GP

. The restricted recursion

L

ST

GP

-terms constrain memory locations individually,

by constraints of the form (p : T ). The language L

ST

GP

covers restrictions over sets of objects, e.g., by propo-

sitional constraints ({p

1

, . . . , p

m

} : T ), where T is a

term for a type with m argument roles. By such a gen-

eralized constraint, the memory variables p

1

, . . . , p

m

represent objects that are restricted to be simultane-

ously of the type T . E.g., p

1

, . . . , p

m

can be memory

nets for memorizing such objects.

Thus, the language L

ST

GP

has terms for designating

a new kind of semantic objects — complex units of

informational parameters, that consist of sets of pa-

rameters that are simultaneously constrained to sat-

isfy recursive conditions, instead of singleton param-

eters with recursive restrictions. Such terms, in effect,

extend Situation Theory with complex parametric ob-

jects.

3 TYPES AND VOCABULARY OF

L

ST

GP

As in Situation Theory, the class Typ es , which can be

a proper class, i.e., a non-well-founded set, is deﬁned

at recursive stages, by starting with a set of primitive

(basic) types. The choice of the basic types depends

on technical choices and applications of L

ST

GP

. Note

that some of the expressions of the formal language

L

ST

GP

, deﬁned later, will be its complex types.

Primitive Types. We take a set, BTypes, of primitive

(basic) types:

BTypes = { IND, REL, FUN, ARGR, LOC,

POL, PAR, INFON, SIT,

PROP, SET, TYPE, |=}

(1)

where:

• IND is the type for individuals and expressions

denoting individuals;

• REL, for relations, primitive and complex, and

expressions denoting such;

• FUN, for functions, primitive and complex, and

function expressions;

• ARGR, for abstract argument roles of proper-

ties, relations, functions, and types basic and com-

plex (Typically, in formal languages and theories, ar-

gument roles are represented by argument slots, ac-

cording to some ﬁxed, conventional linear order.);

• LOC, for space-time locations;

• POL, for two polarities denoted by 1 and 0

(which are markers for a property, relation, activity,

action, or process, taking place, not truth values.);

A Formalization of Generalized Parameters in Situated Information

345

• PAR, for basic and complex parameters, as se-

mantic underspeciﬁed objects, and for expressions de-

noting semantic parameters;

• INFON, for situation-theoretic objects that are

basic or complex information units, and for expres-

sions denoting such;

• SIT, for situations;

• PROP, for abstract objects that are propositions,

and for expressions denoting such;

• TYPE, for primitive and complex types and type

expressions;

• SET, for sets and set expressions;

• |= is a designated type called “supports”, which

is used to state and express that informational content

holds in, or is supported by, a certain situation s

(more details later).

Typed Vocabulary. The vocabulary of L

ST

GP

consists

of typed constants and two kinds of variables in all

types, as follows.

Constants: L

ST

GP

has a countable set of constants,

K

τ

= { c

τ

0

, c

τ

1

, . . . }, for all τ ∈ Types, and K is the set

of all constants:

K =

[

τ∈Types

K

τ

(2)

Some of the sets of constants can be ﬁnite, and some

empty (e.g., for complex types). Here we assume

non-empty sets of constants: K

IND

6=

/

0 — for primi-

tive individuals, K

LOC

6=

/

0 — for space-time locations,

K

REL

6=

/

0 — for primitive relations, K

FUN

6=

/

0 — for

primitive functions, K

POL

= {0, 1} – for polarity val-

ues

1

.

Note that in Situation Theory as semantic model-

theory, there is a non-empty domain Rel

n

of primitive

relations with n argument roles, for each natural num-

ber n ∈ N. A given primitive relation r is represented

by some unique set-theoretic object, e.g., an atomic

element (urelement), not by a set of the ordered n-

tuples of objects assumed to stand in that relation.

The reason is that, in Situation Theory, n objects can

stand in the relation r, only in some situation s. How-

ever, we can deﬁne the extension of the relation r in a

given situation s, as the set of the ordered n-tuples of

objects, which stand in the relation n, in the situation

s.

Another note is in due that, in Situation theory, the

domain A

IND

of the individuals is a collection that,

depending on speciﬁc applications, can be either a

set or a proper class, i.e., non-well-founded set. See

(Aczel, 1988) about Aczel non-well-founded set the-

ory. Furthermore, we shell assume that A

IND

includes

1

To distinguish the polarities from propositional truth val-

ues, one may take two other distinctive symbols, e.g.,

K

POL

= {−, +}.

numbers, e.g., all natural numbers, N = {0, 1, . . . }, in-

tegers, real numbers, etc. One may choose also that

A

IND

is a proper class that includes the universe of all

sets (which is not a set), or a speciﬁc sub-collection

of sets, e.g., the hereditarily countable sets. A more

ﬁne-grained Situation Theory may have such objects

like sets, in a domain of type SET, which can either a

sub-type of the type IND, or a separate type. We leave

these choices open here.

Pure Variables: The language L

ST

GP

has a set of

typed pure variables, PV

τ

= { v

τ

0

, v

τ

1

, . . . }, for all types

τ ∈ Types, and PV is the set of the pure variables:

PV =

[

τ∈Types

PV

τ

(3)

Restricted Memory Variables: The formal lan-

guage L

ST

GP

has a distinctive set MV

τ

of typed variables

MV

τ

= { p

τ

0

, p

τ

1

, . . . }, for all types τ ∈ Types. The set

of all restricted memory variables is:

MV =

[

τ∈Types

MV

τ

, (4)

The variables of MV are also simply called memory

variables or recursion variables. Thus, the L

ST

GP

vari-

ables are typed, and the sets of all variables are clas-

siﬁed as follows:

Vars

τ

= PV

τ

∪MV

τ

, (5a)

Vars = PV ∪MV (5b)

For each of the basic types, we take a set of basic,

restricted memory variables, which can be used for

assigning, i.e., “saving”, information and objects in

them, by respecting the corresponding types. The

typing of the memory variables, serves among other

computational tasks, also reserving speciﬁc shape or

construction structure of the memory slots, as unde-

termined blue-prints. E.g., PAR

IND

, PAR

LOC

, PAR

REL

,

PAR

FUN

, PAR

INFON

, PAR

SIT

, PAR

PROP

, etc.

We also interprete the memory variables as mem-

ory slots carrying information about partly known ob-

jects, where the information available in a memory

slot is subject to update. Alternatively, memory vari-

ables are for objects under developmental changes.

For semantic reasons, we distinguish between re-

stricted variables, which are syntactic objects, and se-

mantic parameters as semantic objects, while such de-

tails are not in the subject of this paper. Sometimes,

when the context makes it clear, the restricted, mem-

ory variables can be called parameters. Typically, un-

less otherwise speciﬁed either explicitly or by the syn-

tax of the expressions, we shall use letters x, y, z, with

or without subscripts, to vary over pure variables of

any types, and letters p, q, r, with or without sub-

scripts, to vary over memory variables of any type.

PUaNLP 2016 - Special Session on Partiality, Underspeciﬁcation, and Natural Language Processing

346

Deﬁnition 1 (Argument roles). Argument roles.

1. We take a set of expressions (basic or complex,

by relatively simple rules of construction), called ba-

sic argument roles, which can be associated with

some of the expressions, such as the constants for ba-

sic relations, functions, and types:

BRoles = {ρ

1

, . . . , ρ

m

, . . . }. (6)

2. ARoles is the class of basic and complex ar-

gument roles, which includes the set of the basic ar-

gument roles, BRoles ⊂ ARoles, and other argument

roles, denoted by [ξ], for ξ ∈ PV, that are assigned to

complex relations, functions, and types via rules of

term formations (given later)

ARoles = BRoles ∪ {[ξ] | ξ ∈ PV } (7)

Note that the argument roles [ξ] are argument

roles generated by using only pure variables from

the collection PV, which are typed. The argument

roles [ξ] of relations, functions, and types, and terms

denoting such, get generated via the term formation

rules given later. Note that the elements of the

set ARoles, esp. for the primitives, are specialized

expressions, which can vary depending on speciﬁc

applications of L

ST

GP

.

Example 3.1. As usually in relational approaches, we

take relation constants such as smile, read, read-to,

and give. We can associate sets of argument roles with

them, in a simple way, without including constraints

over the expressions ﬁlling the argument roles, and

by using conventional linking and understanding be-

tween the actual roles representing “who-does-what-

to-whom”, which is used in many applications involv-

ing functions and relations:

ARGR(read) = {arg

1

, arg

2

} (8a)

ARGR(read-to) = {arg

1

, arg

2

, arg

3

} (8b)

ARGR(give) = {arg

1

, arg

2

, arg

3

} (8c)

Rules for syntax-semantics interface can provide

the necessary links between {arg

1

, arg

2

, arg

3

}, and

the actual roles “who-does-what-to-whom”.

Typically, the basic relations and types are asso-

ciated with argument roles that have to satisfy con-

straints for their appropriate ﬁlling. We stress that

there is no assumption that the argument roles of the

relations, functions, and types are ordered.

Argument Roles with Appropriateness Con-

straints.

1. Every relation and type constant and pure or

memory variable γ, is associated with a set ARGR(γ)

of typed expressions arg

i

, i = 1, . . . , n, for argument

roles:

ARGR(γ) = {T

1

: arg

1

, . . . , T

n

: arg

n

}

for all γ ∈ K

REL

∪ K

TYPE

∪ Vars

REL

∪Vars

TYPE

(9)

where n ≥ 0, arg

i

∈ ARoles, and T

i

are sets of basic

types. The expressions arg

1

, . . . , arg

n

are called the

argument roles (or the argument slots) of γ. Each T

i

is

speciﬁc for the corresponding argument role arg

i

of γ,

i = 1, . . . , n. T

i

is called the appropriateness constraint

of arg

i

, i = 1, . . . , n.

2. Every function constant and function variable

γ, i.e., γ ∈ A

FUN

∪ Vars

FUN

, is associated with two sets

of sub-typed expressions:

ARGR(γ) = {T

1

: arg

1

, . . . , T

n

: arg

n

} (10a)

Value(γ) = {T

n+1

: arg

n+1

} (10b)

where n ≥ 0., arg

i

∈ ARoles, and T

i

are sets of basic

types. The expressions arg

1

, . . . , arg

n

are called the

argument roles (or the argument slots) of γ. The ex-

pression arg

n+1

is called the value role of γ. Each T

i

is speciﬁc for the corresponding argument role arg

i

of γ, i = 1, . . . , n + 1. T

i

is called the appropriateness

constraint of arg

i

, i = 1, . . . , n + 1.

More sensible approaches, in compare to Ex-

ample (3.1), use more explicit, informative naming

of the speciﬁc argument roles for the primitive

relations. E.g., semantic roles of the verbs denoting

actions, processes, etc., have been subject of syntactic

representations in Transformational Grammar (TG)

and Government and Binding Theory (GB, GBT), by

the so called Theta Theory and Θ-roles in (Chomsky,

1993) and (Dowty, 1979). For a more recent work on

the topic, see (Jaworski and Przep

´

orkowski, 2014).

Situation-theoretic approaches to semantics of human

language, e.g., in HPSG grammars, have been using

a traditional convention, by which the semantic

argument roles of a relation denoted by a verb are

named by using English word-formation rules, e.g.,

adding the sufﬁxes “er”, “ed”, etc. (sometimes by

misspelling), respectively for the agent and patient.

See Example (11a)–(11d).

Example 3.2.

ARGR(IND : smile) = { IND : smiler } (11a)

ARGR(read) = { IND : reader,

IND : readed },

(11b)

ARGR(read-to) = { IND : reader

IND : readed,

IND : listener }

ARGR(give) = { IND : giver

IND : recipient,

IND : what }

(11c)

ARGR(γ) = {T

1

: arg

1

, . . . ,

T

n

: arg

n

}

(11d)

(for γ ∈ K

REL

, n ∈ N)

A Formalization of Generalized Parameters in Situated Information

347

Note 1. Note that complex types can be associated

with relation, type, and function terms, including con-

stants and variables, via more complex terms, i.e., via

restriction terms, introduced later.

Theorem 1. (1) BRoles can be countable, given that

the sets of basic relations, functions, and types are

countable, and each have a countable set of argument

roles.

(2) BRoles can be (equinumerously) generated

from a ﬁnite base of symbols, by assuming Situation

Theory with basic relations, functions, and types that

have ﬁnite number of argument roles, and that the sets

of the basic relations, functions and types are count-

able.

(3) BRoles can be restricted to a ﬁnite set given

that the sets of primitive relations, functions and types

are ﬁnite.

Proof. By using Cantor Theorem for countable sets.

4 TERMS OF L

ST

GP

The classes of Terms

τ

, for τ : TYPE, are deﬁned recur-

sively, at stages with respect to the recursive levels of

type constructions.

Constants (as terms). If c ∈ K

τ

, then c ∈ Terms

τ

,

denoted as c : τ. There are no free and no bound vari-

ables in c. Variables (as terms). If x ∈ Vars

τ

, then,

x ∈ Terms

τ

. denoted x : τ. The only occurrence of

the variable x in the term x is free; there are no bound

occurrences of variables in x.

Infon Terms. For every relation term (basic or com-

plex) ρ ∈ Terms

REL

, associated with argument roles

ARGR(ρ) = {T

1

: arg

1

, . . . , T

n

: arg

n

} (12)

and every sequence of terms ξ

1

, . . . , ξ

n

such that

ξ

1

∈ Terms

T

1

, . . . , ξ

n

∈ Terms

T

n

(13)

(i.e., terms ξ

1

, . . . , ξ

n

that satisfy the correspond-

ing appropriateness constraints of the argument roles

of ρ, denoted also as T

1

: ξ

1

, . . . , T

n

: ξ

n

), and ev-

ery space-time term τ ∈ Terms

LOC

, i.e., LOC : τ, ev-

ery polarity term t ∈ Terms

POL

, i.e., POL : t, i.e.,

t ∈ {0, 1} ∪ PAR

POL

, the expression (14a) is an infon

term, i.e., an element of Terms

INFON

, alternatively de-

noted with type assignment in (14b).

ρ, T

1

: arg

1

: ξ

1

, . . . ,

T

n

: arg

n

: ξ

n

,

LOC : Loc : τ,

POL : Pol : t ∈ Terms

INFON

(14a)

ρ, T

1

: arg

1

: ξ

1

, . . . , T

n

: arg

n

: ξ

n

,

LOC : Loc : τ, POL : Pol : t : INFON

(14b)

All free (bound) occurrences of variables in ρ, ξ

1

, . . . ,

ξ

n

, τ are also free (bound) in the infon term (14a).

The notation (14b) is used when the type label-

ing is relevant. We use the notation ρ, arg

1

:

ξ

1

, . . . , arg

n

: ξ

n

, Loc : τ;t when the type constraints

are understood; or ρ, ξ

1

, . . . , ξ

n

, τ;t when also

there is an understood order of the arguments.

Proposition Terms. For every type term (basic or

complex) γ ∈ Terms

TYPE

, associated with argument

roles ARGR(γ) = {T

1

: arg

1

, . . . , T

n

: arg

n

}, and ξ

1

∈

Terms

T

1

, . . . , ξ

n

∈ Terms

T

n

, i.e., terms ξ

1

, . . . , ξ

n

that

satisfy the corresponding appropriateness constraints

of the argument roles of γ, denoted T

1

: ξ

1

, . . . , T

n

: ξ

n

,

the expression in (15a) is a proposition term, in a post-

ﬁx notation, i.e., an element of Terms

PROP

, alterna-

tively denoted in (15b) with the type assignment that

it is a term of type PROP.

({T

1

: arg

1

: ξ

1

, . . . ,

T

n

: arg

n

: ξ

n

} : γ) ∈ Terms

PROP

(15a)

({T

1

: arg

1

: ξ

1

, . . . ,

T

n

: arg

n

: ξ

n

} : γ) : PROP

(15b)

All free (bound) occurrences of variables in γ, ξ

1

,

. . . , ξ

n

, are also free (bound) in the proposition term

in (15a) and (15b).

Often, we shall skip the braces around the argu-

ment role assignments in (15a) and (15b). As custom-

ary in formal languages, depending on circumstances,

postﬁx notation of the proposition terms (as above in

(15a) and (15b)) can be alternated with preﬁx, or inﬁx

notations.

Notation 1. In some cases, the full preﬁx notations

(16a)–(16b) are more convenient.

(γ, { T

1

: arg

1

: ξ

1

, . . . , T

n

: arg

n

: ξ

n

}) : PROP (16a)

(γ, T

1

: arg

1

: ξ

1

, . . . , T

n

: arg

n

: ξ

n

) : PROP (16b)

Complex Proposition Terms. Terms for proposi-

tion are formed by using the usual logic connectives,

¬, ∧, ∨, etc.

Notation 2. Negated propositions as in (17a) are

sometimes denoted by (17b).

¬({T

1

: arg

1

: ξ

1

, . . . , T

n

: arg

n

: ξ

n

}γ) (17a)

({T

1

: arg

1

: ξ

1

, . . . , T

n

: arg

n

: ξ

n

} 6: γ) (17b)

Notation 3. The prefex notation (18a) is used when

the type constraints over the argument roles are un-

derstood. The notation (18b) is used when also there

is an understood order of the argument roles.

PUaNLP 2016 - Special Session on Partiality, Underspeciﬁcation, and Natural Language Processing

348

(γ, arg

1

: ξ

1

, . . . , arg

n

: ξ

n

) (18a)

(γ, ξ

1

, . . . , ξ

n

) (18b)

Notation 4. A postﬁx notation (19a) can be used

when the type constraints over the argument roles

are understood; and (19b) when the proposition is

negated. The notations (19c) and (19d) are used when

also there is an understood order of the arguments,

since they conform with a traditional notation of type

association to terms in the case of n = 1. A precaution

is due with the later notation since it is an abbreviated

notation of a proposition term of the formal language

L

ST

GP

not a type assignment to a L

ST

GP

term.

({arg

1

: ξ

1

, . . . , arg

n

: ξ

n

} : γ) (19a)

({arg

1

: ξ

1

, . . . , arg

n

: ξ

n

} 6: γ) (19b)

(ξ

1

, . . . , ξ

n

: γ) (19c)

(ξ

1

, . . . , ξ

n

6: γ) (19d)

A precaution is due for proposition terms with n >

1. (19c), and respectively, (19d), does not mean that

each one of the objects ξ

i

is (is not) of type γ, since the

type γ has n arguments. The proposition terms (19c)

and (19d), i.e., (ξ

1

, . . . , ξ

n

: γ) and (ξ

1

, . . . , ξ

n

6: γ) de-

note statements that the objects {ξ

1

, . . . , ξ

n

}, together

as a set (or a group), respectively are not, of type γ,

by ﬁlling the corresponding argument roles. I.e., in

these notations, each ξ

i

ﬁlls the corresponding argu-

ment role arg

i

, and linearity is only visual notation.

Application Terms. For every function term (ba-

sic or complex) γ ∈ Terms

FUN

, associated with ar-

gument roles ARGR(γ) = { T

1

: arg

1

, . . . , T

n

: arg

n

}

and a value role Value(γ) = {T

n+1

: val }, and every

ξ

1

∈ Terms

T

1

, . . . , ξ

n

, ξ

n+1

∈ Terms

T

n+1

, the expres-

sion in (20a), respectively (20b), is an application

term:

γ{T

1

: arg

1

: ξ

1

, . . . ,

T

n

: arg

n

: ξ

n

} ∈ Terms

T

n+1

(20a)

γ{T

1

: arg

1

: ξ

1

, . . . , T

n

: arg

n

: ξ

n

} : T

n+1

(20b)

The notation (20b), which includes the type associ-

ation, is used when the type labeling is relevant. In

addition, the term (21a), respectively in (21b), is a

proposition term.

γ{T

1

: arg

1

: ξ

1

, . . . ,

T

n

: arg

n

: ξ

n

} = ξ

n+1

∈ Terms

PROP

(21a)

γ{T

1

: arg

1

: ξ

1

, . . . ,

T

n

: arg

n

: ξ

n

} = ξ

n+1

: PROP

(21b)

All free (bound) occurrences of variables in γ, ξ

1

, . . . ,

ξ

n+1

are also free (bound) in the application terms.

Note that the application expression in (20a) does

not by default designate the value, i.e., the result of a

process of valuation of the function application. Fur-

thermore, we allow partial functions, i.e., the value

of a function application term γ{T

1

: arg

1

: ξ

1

, . . . , T

n

:

arg

n

: ξ

n

} may not exist. In such a case, the denotation

of the corresponding proposition term (21a) might be

undeﬁned, or a special object that designates error.

When there is an understood order of the function ar-

guments, we may use the traditional terms of func-

tional application: γ(ξ

1

, . . . , ξ

n

) : T

n+1

.

λ-Terms. For every term (basic or complex) Φ ∈

Terms and any pure variables ξ

1

, . . . , ξ

n

∈ PV, the ex-

pression λ{ξ

1

, . . . , ξ

n

}Φ is a λ-abstraction term, i.e.:

λ{ξ

1

, . . . , ξ

n

}Φ ∈ Terms (22a)

[ξ

1

], . . . , [ξ

n

] are expressions for the argument roles of

λ{ξ

1

, . . . , ξ

n

}Φ, and are associated with correspond-

ing appropriateness constraints as follows:

ARGR(λ{ξ

1

, . . . , ξ

n

}Φ) =

{T

1

: [ξ

1

], . . . , T

n

: [ξ

n

]}

(23)

where, for each i ∈ { 1, . . . , n}, T

i

is the union of all

sets (of types) that are the appropriateness constraints

of all the argument roles that occur in Φ, and such

that ξ

i

ﬁlls up them, without being bound. (Note that

ξ

i

may ﬁll more than one argument role in Φ.)

All free occurrences of ξ

1

, . . . , ξ

n

in Φ are bound

in the term λ{ ξ

1

, . . . , ξ

n

}Φ. All other free (bound)

occurrences of variables in Φ are free (bound) in the

term λ{ξ

1

, . . . , ξ

n

}Φ.

Case 1: Complex Relations with Complex Ar-

guments. In the case when Φ ∈ Terms

INFON

the ex-

pression λ{ξ

1

, . . . , ξ

n

}Φ is a complex-relation term,

i.e.:

λ{ξ

1

, . . . , ξ

n

}Φ ∈ Terms

REL

(24a)

λ{ξ

1

, . . . , ξ

n

}Φ : REL (24b)

Case 2: Complex Types with Complex Argu-

ments. In the case when Φ ∈ Terms

PROP

, the expres-

sion λ{ξ

1

, . . . , ξ

n

}Φ is a complex-type term, i.e.:

λ{ξ

1

, . . . , ξ

n

}Φ ∈ Terms

TYPE

(25a)

λ{ξ

1

, . . . , ξ

n

}Φ : TYPE (25b)

The expressions for types, as result of abstraction for-

mation over pure variables in proposition terms, are

signiﬁcantly distinct from other λ-terms, and we use

the following expressions for abstraction types:

[T

1

: ξ

1

, . . . , T

n

: ξ

n

| Φ] ∈ Terms

TYPE

(26a)

[T

1

: ξ

1

, . . . , T

n

: ξ

n

| Φ] : TYPE (26b)

When the type assignment is understood, we use the

notation (27).

A Formalization of Generalized Parameters in Situated Information

349

Notation 5.

[ξ

1

, . . . , ξ

n

| Φ] (27)

Case 3: Complex Function Terms (Also Oper-

ation Terms) with Complex Arguments. For every

ϕ ∈ Terms

τ

where τ ∈ Types, τ 6≡ INFON, τ 6≡ PROP,

and for any pure variables ξ

1

, . . . , ξ

n

∈ PV, the expres-

sion λ{ξ

1

, . . . , ξ

n

}ϕ is a complex-function term:

λ{ξ

1

, . . . , ξ

n

}ϕ ∈ Terms

FUN

(28a)

λ{ξ

1

, . . . , ξ

n

}ϕ : FUN (28b)

Every function term λ{ ξ

1

, . . . , ξ

n

}ϕ has a role for the

function value, Value(λ{ξ

1

, . . . , ξ

n

}ϕ) = {τ : val },

and argument roles [ξ

1

], . . . , [ξ

n

], which are associ-

ated with corresponding appropriateness constraints

as follows:

ARGR(λ{ξ

1

, . . . , ξ

n

}ϕ) =

{T

1

: [ξ

1

], . . . , T

n

: [ξ

n

]}

(29a)

Value(λ{ξ

1

, . . . , ξ

n

}ϕ) = { τ : val} (29b)

where, for each i ∈ {1, . . . , n }, T

i

is the union of all

sets (of types) that are the appropriateness constraints

of all the argument roles that occur in ϕ, and such

that ξ

i

ﬁlls up them, without being bound. (Note that

any of the pure variables ξ

i

may ﬁll more than one

argument role in ϕ.)

Constrained Recursion Terms. For any type terms

C

k

: TYPE, k = 1, . . . , m (m ≥ 0), with argument roles

ARGR(C

k

):

ARGR(C

k

) = { T

k,1

: arg

k,l

1

, . . . , T

k,l

k

: arg

k,l

k

}

(l

k

≥ 1)

(30)

memory (recursion) variables q

k, j

∈ MV

T

k, j

(i.e., q

k, j

:

T

k, j

), j = 1, . . . , l

k

, terms A

i

: σ

i

, i = 0, . . . , n (n ≥ 0),

and pairwise different memory variables p

i

∈ MV

σ

i

(i.e., p

i

: σ

i

), i = 1, . . . , n, such that the two sequences

(31a) and (31b)

{(q

1,1

, . . . , q

1,l

1

: C

1

), . . . ,

(q

m,1

, . . . , q

m,l

m

: C

m

)}

(31a)

{p

1

:

= A

1

, . . . , p

n

:

= A

n

} (31b)

(jointly) satisfy the Acyclicity Constraint 1, the ex-

pression in (32a), respectively (32b), is a restricted

recursion term of type σ

0

:

A

0

suchthat { (q

1,1

, . . . , q

1,l

1

: C

1

), . . . ,

(q

m,1

, . . . , q

m,l

m

: C

m

)}

where {p

1

:

= A

1

, . . . ,

p

n

:

= A

n

} ∈ Terms

σ

0

(32a)

A

0

suchthat { (q

1,1

, . . . , q

1,l

1

: C

1

), . . . ,

(q

m,1

, . . . , q

m,l

m

: C

m

)}

where {p

1

:

= A

1

, . . . , p

n

:

= A

n

}

: σ

0

(32b)

Acyclicity Constraint 1. The sequences of type con-

straints (31a) and assignments (31b) are (jointly)

acyclic iff there is a ranking function

rank: {

m

[

k=1

{q

k, j

}

l

k

j=1

[

{p

i

}

n

i=1

} −→ N (33)

such that:

1. for all q

k, j

, q

k

0

, j

0

∈

S

m

k=1

{q

k, j

}

l

k

j=1

, if q

k

0

, j

0

oc-

curs freely in C

k

, then rank(q

k

0

, j

0

) < rank(q

k, j

)

2. for all p

i

, p

j

∈ {p

i

}

n

i=1

, if p

j

occurs freely in

A

i

, then rank(p

j

) < rank(p

i

).

Note that the acyclicity constraint is a proper part of

the recursive deﬁnition of the L

ST

GP

-terms.

All free occurrences of p

1

, . . . , p

n

in A

0

, . . . , A

n

,

C

1

, . . . , C

m

are bound in the term (32b). All other

free (bound) occurrences of variables in A

0

, . . . , A

n

,

C

1

, . . . , C

m

are free (bound) in (32b). Sometimes we

enclose the recursion terms in extra brackets to sep-

arate them from the surrounding text, for example as

in (32b).

Generalized, Restricted Variables. For any given

1. type term C : TYPE, such that

ARGR(C) = {T

1

: arg

l

, . . . , T

k

: arg

k

}

(k ≥ 1)

(34)

2. memory (recursion) variables q

j

∈ MV

T

j

(that is,

q

j

: T

j

), for j = 1, . . . , k (k > 1), which are not nec-

essarily pairwise different and such that

(a) the proposition term

({T

1

: arg

1

: q

1

, . . . , T

k

: arg

k

: q

k

} : C) (35)

abbreviated as (q

1

, . . . , q

k

: C), is acyclic, i.e.,

no q

j

( j = 1, . . . , k) occurs freely in C, and

(b) {q

k

1

, . . . , q

k

l

} is the set of the pairwise differ-

ent variables, such that:

{q

k

1

, . . . , q

k

l

} = { q

1

, . . . , q

k

}

the expression in (36a), abbreviated as (36b) and

(36c), is a term of type PAR

SET

. We call it a memory

network (or a restricted memory net).

{q

k

1

, . . . , q

k

l

} suchthat { (q

1

, . . . , q

k

: C),

(q

1,1

, . . . , q

1,l

1

: C

1

), . . . ,

(q

m,1

, . . . , q

m,l

m

: C

m

)}

where {p

1

:

= A

1

, . . . ,

p

n

:

= A

n

}

(36a)

{q

k

1

, . . . , q

k

l

} suchthat { (q

1

, . . . , q

k

: C),

(

−→

q :

−→

C )}

where {

−→

p

:

=

−→

A }

(36b)

PUaNLP 2016 - Special Session on Partiality, Underspeciﬁcation, and Natural Language Processing

350

{q

k

1

, . . . , q

k

l

} s.th. { (q

1

, . . . , q

k

: C),

(

−→

q :

−→

C )}

{

−→

p

:

=

−→

A }

(36c)

We call the type term C the major constraint of the

generalized memory network in (36a). The term (36a)

denotes a complex situation-theoretic object, which is

a generalized constrained parameter.

For abbreviation, sometimes, the two constants

operators where and suchthat can be collapsed by us-

ing just one of them. In the special case when k = 1,

the type C has just one argument role, ARGR(C) =

{T : arg }. In such a case, we can identify the single-

ton set {q} with the parameter q : T, and the terms in

(37a) and (37b):

{q} s.th. { (q : C), (

−→

q :

−→

C )}{

−→

p

:

=

−→

A } (37a)

≡ q s.th. { (q : C), (

−→

q :

−→

C )}{

−→

p

:

=

−→

A } (37b)

In the special case when n, m = 1, we have the gener-

alized memory variable (38a), which can be abbrevi-

ated by the expression (38b).

{q

k

1

, . . . , q

k

l

} s.th. { (q

1

, . . . , q

k

: C)} (38a)

{q

k

1

, . . . , q

k

l

}

(q

1

,...,q

k

:C)

: PAR

SET

(38b)

The term {q

k

1

, . . . , q

k

l

}

(q

1

,...,q

k

:C)

: PAR provides

formalization and generalization of the more simple,

situation theoretic restricted parameters q

T

that are

singletons, i.e., l = 1. For restricted parameters, as

singletons, see, e.g., (Barwise and Perry, 1983) and

(Loukanova, 2014). A generalized parameter des-

ignated by a L

ST

GP

term of the form (36a) is a com-

plex, parametric object that corresponds to a ﬁnite

set of parameters. The term (36a) represents a partly

known complex of entities, or a complex of entities

that are under-development, and which are simulta-

neously restricted and “bundled” together by the con-

straint (q

1

, . . . , q

k

: C).

5 RELATIONS BETWEEN

SITUATIONS WITH

UNDERSPECIFICATION

As a typical example of a complex term that includes

most of the formal concepts introduced in the ﬁrst

part of the paper, we use the semantic representa-

tion of the verb “stop” in two of its typical usages.

We have chosen this verb since it is a typical repre-

sentative of a class of verbs that exhibit several spe-

ciﬁc semantic phenomena. Firstly, the verb changes

its semantics depending whether it co-occurs with a

present participle clause as its complement, or with

inﬁnitival clause. Secondly, in both usages, the se-

mantics of sentences having such a verb as their head

verb, involves shared arguments and underspeciﬁca-

tion, which we can express with type constraints and

recursion (memory) locations that represent seman-

tic parameters. A grammatical analysis that includes

syntax-semantics interface of phrases with such verbs

is not in the space of this paper. We do not purport any

full semantic analysis of such verbs that take clausal

complements. The section offers an example for a

possible semantic representation, with the purpose of

the demonstration of the formal language L

ST

GP

and sit-

uational concepts.

We follow a tradition to represent lexical items

that have the same orthography, but differ either in

their lexical syntax or in semantics, as alternative lex-

ical items with subscripts. Here we render the verb

“stop” to different relation constants distinguished

by their subscripts, i.e., stop

1

and stop

2

. We give

T

namedJohn

as an example of a complex term for a type,

which includes, in the scope of the λ-abstraction, an

assignment (39d) that satisﬁes the constraint in (39c),

as a condition for well-formedness. I.e., we assume

that in a relevant application system, the date veriﬁes

the proposition (s |= name, John, l, 1 ). Note

that while we use linear notation in order to safe typ-

ing, and as a tradition from the predominant writ-

ing systems, there is no speciﬁc order over the λ-

abstractions and the argument roles associated with

relations, types, and functions in Situation Theory and

in the terms of its language L

ST

GP

.

T

namedJohn

≡ (39a)

λ{s, m, l}

(s |= named, m, n, l, 1 ) (39b)

s.th. {(n : λ{N}(s |= name, N, l, 1 ))} (39c)

where {n

:

= John}

(39d)

Let T

per

and T

namedJohn

be the following types:

T

ANIMATE

≡

λ{s, l, m}(s |= animate, m, l, 1 )

(40)

T

per

≡ λ{s, m, l}(s |= person, m, l, 1 ) (41)

Let A be the type of activities by animate actors i,

abstracted from speciﬁc activities, actors, space-time

locations, and polarity, i.e., abstraction over polarity

denoted by a pure variable p:

A ≡ λ{p

1

, i, l, p}(s |= activity, (42a)

REL : arg

1

: p

1

, (42b)

T

ANIMATE

(s

1

, l) : actor : i, (42c)

LOC : Loc : l, POL : Pol : p ) (42d)

A Formalization of Generalized Parameters in Situated Information

351

and d be the relation between individuals i, locations

l, and polarities p, such that i is or is not (expressed

by p) drinking tea at l:

d ≡ λ{i, l, p} drink, drinker : i, (43a)

liquid : tea, (43b)

Loc : l, Pol : p (43c)

Now, we can represent the interpretation of an utter-

ance of the sentence “John stopped drinking tea.” by

rendering it into the L

ST

GP

-term T

1

, as in (44a).

John stopped drinking tea.

render

−−−→ T

1

(44a)

John stopped to drink tea.

render

−−−→ T

2

(44b)

where T

1

, T

2

are the terms in (45a)–(45h) and (46a)–

(46k), respectively:

T

1

≡ (s

d

|= stop

1

, A

0

: arg

0

: j,

A

1

: arg

1

: a

1

,

Loc : l

d

, Pol : 1 )

(45a)

s.th. (s

d

, j, l

d

: T

per

), (45b)

(s

d

, j, l

d

: T

namedJohn

), (45c)

(a

1

, j, l

b

, 1 : A), (45d)

(s

d

|= a

1

, [i] : j, [l] : l

b

, [p] : 1 ),

(45e)

(s

d

|= a

1

, [i] : j, [l] : l

a

, [p] : 0 ),

(45f)

(s

d

|= l

b

≺ l

d

), (s

d

|= l

d

≺ l

a

)} (45g)

where {a

1

:

= d} (45h)

T

2

≡ (46a)

(s

d

|= stop

2

, A

0

: arg

0

: j, (46b)

A

1

: arg

1

: a

1

, A

2

: arg

2

: a

2

, (46c)

Loc : l

d

, Pol : 1 ) (46d)

s.th. (s

d

, j, l

d

: T

per

), (s

d

, j, l

d

: T

namedJohn

), (46e)

(a

1

, j, l

b

, 1 : A), (a

2

, j, l

a

, 1 : A), (46f)

(s

d

|= a

1

, [i] : j, [l] : l

b

, [p] : 1 ), (46g)

(s

d

|= a

1

, [i] : j, [l] : l

a

, [p] : 0 ), (46h)

(s

d

|= intends, j, P, l

d

, 1 ), (46i)

(s

d

|= l

b

≺ l

d

), (s

d

|= l

d

≺ l

a

)} (46j)

where (46k)

{a

2

:

= d, (46l)

P

:

= (a

2

: λ{A}(s

f

|= A, [i] : j,

[l] : l

a

, [p] : 1 ))}

(46m)

6 FUTURE WORK

Our target is development of Situation Theory and

its formal languages, like L

ST

GP

introduced in this pa-

per, for enhanced theoretical developments and ap-

plications. By (36a), we introduced a new kind

of L

ST

GP

terms, which extend the formal language in

(Loukanova, 2015). Informally said, by the denota-

tional and algorithmic semantics (the formal appara-

tus of which is outside the subject of this paper) of the

L

ST

GP

terms, we also extend the informational coverage

introduced in the original Situation Theory, e.g., (Bar-

wise and Perry, 1983), (Devlin, 2008), (Seligman and

Moss, 2011), and (Loukanova, 2014). Denotationally,

the terms (36a) designate complex sets of parameters,

which, via the binding operators where and suchthat,

are memory networks of constrained parametric ob-

jects. Information carried by terms such as (32a), re-

spectively (32b), and (36a) is computed according to

an algorithm, by mutual recursion, represented by the

recursion system of assignments, via the scope of the

operator where. The scope of the operator suchthat

introduces constraints over the parameters and the al-

gorithm.

Generalized recursion terms, introduced here, pro-

vide a formal introduction of more complex restricted

parameters than the ones in classic Situation Theory.

The assignment sequences in such terms formalize,

and generalize, the situation theoretic notion of pa-

rameters and their anchoring to more speciﬁc objects.

The Acyclicity Constraint 1 provides a formal tech-

nique for computational representation of information

with acyclic algorithmic steps.

This paper demonstrates the informational rich-

ness and ﬁne-granularity of Situation Theory and

its potentials for applications via formal languages.

E.g., the formal language L

ST

GP

provides formal tech-

niques for (1) investigation of informational richness

of mathematical model-theory of Situation Theory,

and (2) useful applications that usually rely on formal

language, via syntax-semantic interface between for-

mal syntax and informational structures (Loukanova,

2010). Subjects such as formal reduction calculi, the-

ory of L

ST

GP

, distinctions between denotational and al-

gorithmic semantics of L

ST

GP

in Situation Theoretical

models are beyond the scope of this paper. Such top-

ics will be presented in future work. Furthermore, we

investigate in details the differences in expressiveness

between the formal language of Situation Theory in-

troduced in (Loukanova, 2015) and the language L

ST

GP

introduced in this paper, which is richer. E.g., the

restricted recursion terms in (32b) generalize the re-

cursion terms introduced in (Loukanova, 2015). The

language in (Loukanova, 2015) provides a basis for

PUaNLP 2016 - Special Session on Partiality, Underspeciﬁcation, and Natural Language Processing

352

proving proper inclusion in it of the functional, formal

language introduced in (Moschovakis, 2006), which

is different, future work.

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