Generalization and Formalization of Precisiation Language with

Applications to Human-Robot Interaction

Takehiko Nakama, Enrique Mu

˜

noz and Enrique Ruspini

European Center for Soft Computing, c/Gonzalo Guti

´

errez Quir

´

os, s/n, 33600 Mieres, Spain

Keywords:

Precisiated Natural Language, Precisiation Language, Formal Logic, Propositional Logic, Predicate Logic,

Quantiﬁcational Logic, Fuzzy Logic, Fuzzy Relation, Human-Robot Interaction.

Abstract:

We generalize and formalize precisiation language by establishing a formal logic as a generalized precisiation

language. Various syntactic structures in natural language are incorporated in the syntax of the formal logic so

that it can serve as a middle ground between the natural-language-based mode of human communication and

the low-level mode of machine communication. As regards the semantics, we establish the formal logic as a

many-valued logic, and fuzzy relations are employed to determine the truth values of propositions efﬁciently.

We discuss how the generalized precisiation language can facilitate human-robot interaction.

1 INTRODUCTION AND

SUMMARY

In his computational theory of perceptions (e.g.,

(Zadeh, 2001), (Zadeh, 2002), (Zadeh, 2004)), Zadeh

introduced the concept of precisiated natural lan-

guage (PNL), which refers to a set of natural-language

propositions that can be treated as objects of compu-

tation and deduction. The propositions in PNL are as-

sumed to describe human perceptions, and they allow

artiﬁcial intelligence to operate on and reason with

perception-based information, which is intrinsically

imprecise, uncertain, or vague.

Precisiation language is an integral part of this

framework. Each proposition in PNL is translated

into a precisiation language, which then expresses

it as a set or a sequence of computational ob-

jects that can be effectively processed by machines

(e.g., (Zadeh, 2001), (Zadeh, 2002), (Zadeh, 2004)).

Zadeh proposed a precisiation language in which each

proposition is a generalized constraint on a variable.

This precisation language is called a generalized-

constraint language.

Zadeh considered the primary function of natural

language as describing human perceptions, and his

PNL and precisiation language only deal with per-

ceptual propositions ((Zadeh, 2001), (Zadeh, 2002),

(Zadeh, 2004)). However, the importance of natu-

ral language is not limited to describing human per-

ceptions. For instance, using a natural language, we

describe not only perceptions but also actions. There-

fore, it is important, both theoretically and practically,

to extend PNL and precisiaton language to other types

of proposition. Generalized constraints in Zadeh’s

precisiation language are suitable for precisiating per-

ceptual propositions but not for precisiating action-

related propositions (See Section 3).

One of the major ﬁelds that require the precisi-

ation of action-related propositions in natural lan-

guage is robotics. Recently, many studies (e.g.,

(Marble et al., 2004), (Dias et al., 2006), (Dias

et al., 2008b), (Johnson et al., 2008), (Johnson and

Intlekofer, 2008)) have been conducted to develop

robotic systems in which humans and robots work

as true team members, requiring peer-to-peer human-

robot interaction. Such systems can be highly ef-

fective and efﬁcient in performing a wide range of

practical tasks—assistance to people with disabili-

ties (e.g., (Lacey and Dawson-Howe, 1998), (Shim

et al., 2004), (Feil-Seifer and Mataric, 2008),(Ku-

lyukin et al., 2006)), search and rescue (e.g., (Kitano

et al., 1999), (Casper and Murphy, 2003), (Dias et al.,

2008a), (Norbakhsh et al., 2005)), and space explo-

ration (e.g., (Wilcox and Nguyen, 1998), (Fong and

Thorpe, 2001), (Fong et al., 2005), (Ferketic et al.,

2006)), for instance. One of the major challenges

of developing these robotic systems is the increased

complexity of the human-robot interactions (e.g.,

(Goodrich and Schultz, 2007)). Although humans

prefer natural language as a communication medium,

it presents several major problems when used for

332

Nakama T., Muñoz E. and Ruspini E..

Generalization and Formalization of Precisiation Language with Applications to Human-Robot Interaction.

DOI: 10.5220/0004660703320343

In Proceedings of the 5th International Joint Conference on Computational Intelligence (SCA-2013), pages 332-343

ISBN: 978-989-8565-77-8

Copyright

c

2013 SCITEPRESS (Science and Technology Publications, Lda.)

human-robot communications; natural-language ex-

pressions tend to be notoriously underspeciﬁed, di-

verse, vague, or ambiguous, so they often lead to

errors that are hard to overcome (e.g., (Winograd

and Flores, 1986), (Tomassi, 1999), (Shneiderman,

2000), (Forsberg, 2003), (Gieselmann and Stenneken,

2006)). Low-level sensory and motor signals and ex-

ecutable code are easy for machines to interpret, but

they are cumbersome for humans and thus cannot,

on their own, create an effective human-robot inter-

face. Task descriptions or speciﬁcations for robotic

systems typically involve action-related propositions,

such as “bring the box to the room” and “keep the

robot in the building if it rains.” A generalized pre-

cisiation language that can precisiate not only per-

ceptual propositions but also action-related proposi-

tions can effectively mediate human-robot interaction

in robotic systems that employ a peer-to-peer commu-

nication mode.

Recently, Nakama et al. (Nakama et al., 2013)

have taken a ﬁrst step toward generalizing and formal-

izing precisiation language by establishing a formal

logic as a generalized precisiation language. Their

formal logic can generate inﬁnitely many precisiated

propositions, just as inﬁnitely many propositions can

be generated in natural language, while ensuring that

every proposition in the formal logic is precisiated.

In this paper, we further develop and elaborate on the

framework proposed in (Nakama et al., 2013).

The remainder of this paper is organized as fol-

lows. In Section 2, we examine the properties of for-

mal logic that are desirable for precisiating natural-

language expressions. The syntax of our formal logic

is explained in Section 3. In Section 4, we discuss

the generality of our framework. In Section 5, we ex-

amine how to add a deductive apparatus to our for-

mal logic so that we can infer and reason in it. In

Section 6, we develop a hierarchy of propositions that

enhances the expressive power and the interactivity of

our formal logic. The semantics of the formal logic is

explained in Section 7.

Since the precisiation of perceptual propositions

has been examined (e.g., (Zadeh, 2001), (Zadeh,

2004)), we focus on examining how to precisiate

action-related propositions in this paper. In (Mu

˜

noz

et al., 2013), we provide details on how to extend

our generalized precisiation language to perceptual

propositions. To explain our formal scheme, we will

ﬁrst appeal to ordinary practices and then move on

to formal considerations so that the reader can under-

stand it intuitively. To generate examples of ordinary

practice, we consider establishing task descriptions

for human-robot interaction. As mentioned earlier,

task descriptions inherently involve actions, so the

precisiation of task descriptions is an important step

toward generalizing precisiation language and PNL.

2 PROPERTIES OF FORMAL

LOGIC SUITABLE FOR

PRECISIATION OF

NATURAL-LANGUAGE

EXPRESSIONS

The application of formal logic to natural language

is a paradigm of logical analysis (Tomassi, 1999); it

provides genuine insight into the syntactic structures

of natural-language sentences and the consequential

characters of assertions expressed by them. This anal-

ysis is important for precisiating propositions in natu-

ral language.

In order for PNL to have high expressive power,

it is desirable that precisiation language can generate

inﬁnitely many precisiated propositions while ensur-

ing that every proposition in it is precisiated. Formal

logic achieves these properties by a recursive deﬁni-

tion of its syntax; it can generate inﬁnitely many well-

formed formulas while ensuring that every formula in

it is well-formed.

As in other formal logics, we can reason logically

in our formal logic by adding a deductive apparatus

to it; the resulting analytical machinery allows us to

determine when one sentence in the formal language

follows logically from other sentences. Thus our for-

mal logic precisiates the inference and the reasoning

in which humans engage using a natural language.

See Section 5.

Our scheme also reﬂects the theory of descrip-

tions in formal logic, which was introduced by Rus-

sell (Russell, 1984). He claimed that the reality con-

sists of logical atoms, which can be considered in-

decomposable, self-contained building blocks of all

propositions in formal logic, and that logical anal-

ysis ends when we arrive at logical atoms. In our

precisiation language, precisiation ends when we ar-

rive at logical atoms, which will be represented by

atomic propositions at the lowest level of a hierarchy

of propositions. See Section 6.

3 SYNTAX OF THE FORMAL

LOGIC

In this section, we describe how to form proposi-

tions in our formal language. To explain our scheme,

we will ﬁrst appeal to ordinary practices and then

GeneralizationandFormalizationofPrecisiationLanguagewithApplicationstoHuman-RobotInteraction

333

move on to formal considerations so that the reader

can understand it intuitively. To generate examples

of ordinary practice, we consider establishing task

descriptions for human-robot interaction, but keep

in mind that our scheme is not limited to precisi-

ating propositions that describe tasks. (Also notice

that, since tasks typically involve actions, we will be

extending PNL and precisiation language to action-

related propositions.) We will discuss the general-

ity of our formal logic in Section 4. In this for-

mal logic, each proposition has a syntactic form ob-

served in natural language. Our formal logic gen-

eralizes Zadeh’s generalized-constraint language by

incorporating multiple syntactic forms so that it can

deal with not only perceptual propositions but also

action-related propositions.

In Section 3.1, we describe the components of

such propositions. In Section 3.2, we describe how

to form an atomic proposition. In Section 3.3, we

describe how to form a compound proposition. In

Section 3.4, we provide a recursive deﬁnition of well-

formed formulas that allows our formal logic to gen-

erate inﬁnitely many well-formed formulas while en-

suring that every formula in it is well-formed.

In these sections, we will provide examples of

rather simple task descriptions, but our scheme can

be easily applied to robotic systems that require more

intricate task descriptions. See Section 4.

3.1 Component Sets

In our formal logic, we generate propositions using

elements in component sets. To provide concrete ex-

amples, we consider the sets S, V , O, A and C shown

in Table 1 as component sets.

In Section 3.2, we will explain how to generate

atomic propositions using elements in S, V , O, and

A. In this example, the elements in S are also in O

because they can not only perform a task but also re-

ceive an action in V . The element labeled as “null,”

called the null element, is included in O and A. In

Section 3.2, we will explain how the null element is

used in forming atomic propositions. In Section 3.3,

we will explain how to form compound propositions

using the connectives in C.

Although the component sets in Table 1 are rather

simple, they can be made as rich as necessary, and

other types of component sets can be incorporated in

our formal logic. See Section 4.

3.2 Atomic Propositions

In our formal logic, an atomic proposition is deﬁned

to be a tuple in the Cartesian product of component

Table 1: Examples of component sets.

set elements

S agents that can perform tasks

e.g., S = {robot1, robot2, user}

V verbs that characterize actions required by

tasks

e.g., V = { f ind, deliver, go, move, press}

O objects that may receive an action in V or

compose an adverbial phrase

e.g., O =

{

box, button, table, room1,

room2, robot1, robot2, user, null

}

A adverbial phrases that can be included in

task descriptions

e.g., A =

{

in γ, f rom γ, f rom γ

1

to γ

2

,

to γ, null | γ, γ

1

, γ

2

∈ O

}

C connectives that can be used to com-

bine multiple propositions in forming com-

pound propositions

e.g., C = {and, i f , or, then}

sets, and the Cartesian product speciﬁes each admis-

sible tuple structure. To develop formal propositions

that can be easily identiﬁed with natural-language

sentences, we employ tuple structures that reﬂect syn-

tactic structures observed in natural languages such

as English. For instance, using the component sets

described in Section 3.1, we can deﬁne each atomic

proposition in our formal logic to be an SVOA clause

(The S, V, O, and A in SVOA stand for subject, verb,

object, and adverbial phrase, respectively) by setting

the admissible tuple structure to S ×V × O × A. The

SVOA structure is observed in many languages, in-

cluding English, Russian, and Mandarin. Using the

null element in O and A, we can also generate SVO,

SVA, and SV clauses. See the following examples

of atomic propositions resulting from the component

sets in Table 1:

•

robot1

S

move

V

. (The actual form of this proposition

is

robot1

S

move

V

null

O

null

A

, but we will omit instances

of the null element to simplify the resulting

expressions.)

•

robot2

S

move

V

to room1

A

robot1

S

move

V

null

O

to room1

A

.

•

robot2

S

f ind

V

ball

O

.

•

robot1

S

deliver

V

box

O

f rom room1 to room2

A

.

For humans, these propositions (task descriptions)

IJCCI2013-InternationalJointConferenceonComputationalIntelligence

334

are easy to specify and understand. Meanwhile, the

structural and lexical constraints noticeably limit the

diversity and ﬂexibility of everyday language to en-

sure that robots can unambiguously interpret the re-

sulting propositions (i.e., the speciﬁed tasks can be

precisely interpreted and executed by robots).

Atomic propositions can be considered building

blocks of all propositions. As will be explained in

Section 6, we establish a hierarchy of propositions. At

the lowest level of the hierarchy, each atomic propo-

sition is directly associated with an indecomposable,

self-contained executable code, and atomic proposi-

tions at each level compose propositions at higher lev-

els.

There are several ways to deal with the undesir-

able or nonsensical atomic propositions that can be

formed in S × V × O × A. (Note that in formal log-

ics, there can be well-formed formulas that are self-

contradictory.) We can remove all such propositions

from the cartesian product to ensure that each re-

sulting atomic proposition is a precisiated proposi-

tion. (In this case, we abuse the notation and let

S × V × O × A denote the “cleaned” cartesian prod-

uct.) Also, we can consider them as always false so

that they will never be executed in practice (see Sec-

tion 7).

In our formal logic, the atomic propositions need

not be expressed as generalized constraints on vari-

ables. By incorporating the SV, SVO, SVA, and

SVOA structures in the syntax, we can precisiate

action-related propositions rather naturally and effec-

tively. Clearly, other syntactic structures can be incor-

porated in our formal logic; see Section 4.

3.3 Compound Propositions

In our formal logic, we generate each compound

proposition by combining multiple atomic proposi-

tions using one or more connectives in the component

set C. With the component sets in Table 1, we can

form the following compound propositions:

•

robot2

S

go

V

to room1

A

atomic proposition

i f

C

user

S

press

V

button

O

atomic proposition

.

•

robot1

S

go

V

to room1

A

atomic proposition

i f

C

user

S

call

V

robot1

O

atomic proposition

or

C

user

S

press

V

button

O

atomic proposition

!

. (1)

In formal logic, parentheses are used to indicate the

scope of each connective. In our examples, parenthe-

ses disambiguate the manner in which atomic tasks

are performed.

These compound propositions are still quite easy

for humans to specify and understand, and the syntac-

tic structures imposed on the clauses and the compo-

sitions ensure effective interpretation and execution

by robots. Note that these compound propositions

are precisiated propositions although they are not ex-

pressed as generalized constraints on variables. In

fact, it can be quite difﬁcult or ineffective to translate

them into generalized constraints.

In (1), the clause “user call robot1” or “user press

button” represents a condition that must be checked

in determining whether to send the robot to room1,

and the clause “robot1 go to room1” is an imperative.

There are many task descriptions that can be effec-

tively expressed as compound propositions consisting

of conditions and imperatives. In our scheme, task

descriptions consist of atomic propositions described

in Section 3.2, and each atomic proposition is either a

condition or an imperative. The type of each atomic

proposition in a compound task description is unam-

biguously determined by the logical connective that

connects it and by the location of the atomic propo-

sition relative to the connective. An atomic proposi-

tion that forms a subordinate clause immediately fol-

lowing the connective “if” is considered a condition,

whereas an atomic proposition that forms a clause

immediately preceding the connective is considered

an imperative. If the proposition is a condition, the

robotic system monitors the described condition. If it

is an imperative, the system executes the described ac-

tion provided that all the required conditions are sat-

isﬁed.

3.4 Recursive Deﬁnition of

Well-Formed Formulas

In order for a precisiation language to have high ex-

pressive power observed in natural language, it should

be able to generate inﬁnitely many precisiated propo-

sitions while ensuring that every proposition in it is

precisiated. As described in Section 2, we can attain

these properties in formal logic by recursively deﬁn-

ing its syntax; formally, our formal logic can generate

inﬁnitely many well-formed formulas while ensuring

that every formula in it is well-formed.

The syntax of the formal logic described in Sec-

tions 3.1–3.3 can be recursively deﬁned as follows:

1. Any x ∈ S ×V × O × A is an atomic well-formed

formula.

GeneralizationandFormalizationofPrecisiationLanguagewithApplicationstoHuman-RobotInteraction

335

2. If α and β are well-formed formulas, then α c β,

where c ∈ C, is also a well-formed formula.

3. Nothing else is a well-formed formula.

This recursive deﬁnition allows our formal logic

to generate inﬁnitely many precisiated propositions

while ensuring that every proposition in it is pre-

cisiated. As regards the examples described in Sec-

tions 3.1–3.3, with sufﬁciently rich component sets,

we can describe any task that must be performed by

the robotic system, and, using the syntax, we can

ensure that the robotic system does not operate on

any ill-formed task descriptions (i.e., task descriptions

that are not interpretable).

4 GENERALITY OF THE

FORMAL LOGIC

In Sections 3–3.4, we facilitated the exposition of our

formal logic by explaining it with rather simple com-

ponent sets and syntactic structures. Obviously, we

can easily extend our scheme to more sophisticated

component sets and syntactic structures so that our

formal logic can deal with highly complex actions

and perceptions. Each component set can be made as

large as necessary, and other component sets or clause

structures can be incorporated in the formal logic. For

instance, in addition to the SV, SVO, SVA, and SVOA

structures described and used in Sections 3–3.4 (and

in Section 6), we can also incorporate other com-

monly observed clause structures (see, for instance,

(Biber et al., 2002)), such as the SVC, SVOC, and

SVOO structures, in the syntax of atomic proposi-

tions. Furthermore, we can extend the clause struc-

tures so that a phrase can be used as the subject or the

object in an atomic proposition. Negation, a unary

logical connective, can certainly be incorporated in

the formal logic. We can also include Zadeh’s gen-

eralized constraints, which are suitable for express-

ing perceptual propositions, in our formal logic; each

generalized constraint can be considered an atomic

proposition that has the SVC structure, and it can be

combined with other propositions by connectives to

form a compound proposition.

We can establish not only a propositional logic

but also a quantiﬁcational logic, which fully incorpo-

rates quantiﬁers and predicates in well-formed formu-

las. Since propositions that describe perceptions often

include quantiﬁers (see, for instance, (Zadeh, 2002),

(Zadeh, 2004)), it is desirable to develop a quantiﬁca-

tional logic as a precisiation language that covers both

actions and perceptions.

5 INFERENCE AND REASONING

IN THE FORMAL LOGIC

As in other formal logics, we can infer and reason in

our formal logic by adding a deductive apparatus to

it. Syntactically, a set of inference rules can be con-

structed, and axioms can also be established. (The hi-

erarchy described in Section 6 represents non-logical,

domain-dependent axioms.) Typical induction- and

elimination-rules in formal logics, such as modus po-

nens and modus tollens, can be easily incorporated in

our formal logic. Consequently, we can form a se-

quent, which consists of a ﬁnite (possibly empty) set

of well-formed formulas (the premises) and a single

well-formed formula (the conclusion), and we can ex-

amine its provability (derivability) using proof theory;

we can determine if a conclusion follows logically

from a set of premises by examining whether there

is a proof of that conclusion from just those premises

in the formal logic.

As will be described in Section 7, we can employ

fuzzy relations to establish the semantics of our for-

mal logic. This semantics allows us to investigate

the truth conditions and the semantic validity of each

proposition or sequent. As in other formal logics,

comparative truth tables can be used to determine se-

mantic validity.

6 HIERARCHY OF

PROPOSITIONS

The importance of the hierarchy of propositions de-

scribed in this section is threefold. First, it enhances

the expressive power of the formal logic by build-

ing up its vocabulary while ensuring the precisiabil-

ity of each resulting proposition. Second, it fortiﬁes

the high interactivity of our formal logic by allowing

human-robot communications to take place at various

levels of detail. Third, it strengthens the deductive

apparatus of the formal logic by establishing domain-

dependent axioms that can be used for inference and

reasoning.

We will ﬁrst explain the hierarchy intuitively using

the task description scheme described in Section 3. A

complex task can be described rather concisely, i.e., it

can be expressed by an atomic proposition. In many

cases, naive users will prefer specifying a complex

task using an atomic proposition as opposed to a more

lengthy compound proposition. On the other hand, in

order for a robotic system to actually execute a com-

plex task, the task must be broken down into sim-

pler subtasks, and the manner in which the subtasks

IJCCI2013-InternationalJointConferenceonComputationalIntelligence

336

are performed must be speciﬁed. Consequently, the

atomic proposition describing a complex task can be

reexpressed as a compound proposition that consists

of atomic propositions describing the required sub-

tasks. Some of the subtasks may have to be further

decomposed in order to fully specify how to execute

them. Expert users may want to establish and com-

bine these subtasks carefully so that the robotic sys-

tem can perform the task effectively and efﬁciently.

Thus, we can precisiate an atomic proposition by

reexpressing it as a compound proposition consisting

of atomic propositions that precisiate it. This pro-

cess also leads to ﬂexibility in the level of detail.

In the task description example, the ﬂexibility gives

naive users an efﬁcient, user-friendly interface with

robots while giving expert users the power to cus-

tomize tasks. As a result, the hierarchy allows human-

robot interactions to take place at various levels of de-

tail, and it also helps determine the appropriate level

of detail for each human-robot interaction; the hierar-

chy determines whether, from the point of view of a

given user, a given task is “atomic” or “compound.”

Consider the following task description:

robot1

S

examine

V

patient1

O

in room1

A

. (2)

This atomic proposition can be reexpressed as a com-

pound proposition that consists of three atomic propo-

sitions representing subtasks that must be performed

to accomplish the task:

robot1

S

f ind

V

patient1

O

in room1

A

atomic proposition 1

then

C

robot1

S

check

V

patient1

O

atomic proposition 2

then

C

robot1

S

send

V

data

O

atomic proposition 3

. (3)

Atomic propositions 1 and 2 in (3) can also be re-

expressed as compound propositions that clarify how

they are performed; atomic proposition 1 in (3) can be

deﬁned as

robot1

S

go

V

to room1

A

atomic proposition

then

C

robot1

S

search

V

patient1

O

atomic proposition

,

(4)

and atomic proposition 2 in (3) can be deﬁned as

robot1

S

go

V

to patient1

A

atomic proposition

then

C

robot1

S

measure

V

heart rate

O

atomic proposition

and

C

robot1

S

measure

V

blood pressure

O

atomic proposition

!

. (5)

Therefore, using (4)–(5) and atomic proposition 3 in

(3), we can reexpress (2) as

robot1

S

go

V

to room1

A

atomic proposition

then

C

robot1

S

search

V

patient1

O

atomic proposition

then

C

robot1

S

go

V

to patient1

A

atomic proposition

then

C

robot1

S

measure

V

heart rate

O

atomic proposition

and

C

robot1

S

measure

V

blood pressure

O

atomic proposition

!

then

C

robot1

S

send

V

data

O

atomic proposition

. (6)

Figure 1 visualizes this hierarchy, which consists of

three levels (levels 0, 1, and 2). For simplicity,

each atomic proposition is represented by its verb;

for instance, the atomic proposition at the highest

level (level 2), “Robot1 examine patient1 in room1,”

is represented by “examine.” The task expressed

by the atomic proposition at level 2 is described in

more detail at the intermediate level (level 1), where

the atomic propositions that involve the verbs “ﬁnd,”

“check,” and “send” describe the subtasks that consti-

tute the task. These subtasks are described in more

detail at the lowest level (level 0), where they are ex-

pressed by the atomic propositions that involve the

verbs “go,” “search,” “measure,” and “send.”

The hierarchy clearly shows how atomic proposi-

tion (2) at level 2 is precisiated. At level 0, we have

atomic propositions that are not decomposable; each

of them is directly associated with a self-contained ex-

ecutable code that is run to perform the corresponding

task. Thus, atomic propositions at level 0 can be con-

GeneralizationandFormalizationofPrecisiationLanguagewithApplicationstoHuman-RobotInteraction

337

Find Check Send

Examine

Level 2

Level 1

Level 0

MeasureGoSearch Send

Figure 1: Hierarchical task description. At the highest level

(level 2), the task is expressed as atomic proposition (2),

represented by “examine.” At the intermediate level (level

1), the task is expressed as compound proposition (3),

which consists of atomic propositions represented by “ﬁnd,”

“check,” and “send.” At the lowest level (level 0), the task

is expressed as compound proposition (6), which consists

of atomic propositions represented by “go,” “search,” “mea-

sure,” and “send.”

sidered logical atoms described in Section 2, and they

precisiate each proposition at higher levels.

The suitability of a given proposition depends on

the level of granularity required for it. As regards the

task description scheme, naive users will most likely

prefer describing tasks at level 2, thus preferring (2).

For expert users, there may be situations where they

prefer specifying a given task step by step or reconﬁg-

uring its subtasks according to various circumstances;

in such cases, interacting with robots at level 1 using

(3) or at level 0 using (6) will be desirable. Thus,

the hierarchy allows a variety of users to interact with

robots at various levels of detail.

The hierarchy of propositions can be characterized

more formally as follows. Let S

i

, V

i

, O

i

, A

i

, and C

i

denote the component sets for level i of the hierarchy

(i ≥ 0). The elements in these sets reﬂect the degree

of detail suitable for level i. Then at level i, we estab-

lish a formal logic with these component sets, as de-

scribed in Section 3. Atomic propositions in S

0

×V

0

×

O

0

× A

0

are the logical atoms and the building blocks

of all propositions; each of them is indecomposable

and directly associated with a self-contained compu-

tational unit. We will call such a computational unit as

a computational atom. For each i ≥ 1, every atomic

proposition in S

i

× V

i

× O

i

× A

i

can be decomposed

into atomic propositions in S

i−1

×V

i−1

×O

i−1

×A

i−1

.

Figure 2 visualizes a typical form of this hierarchy.

For each i and j, p

(i)

j

denotes an atomic proposition at

level i, and e

j

denotes a computational atom associ-

ated with p

(0)

j

. In the ﬁgure, p

(0)

j

is connected to e

j

for

each j, and for each i ≥ 1 and j, p

(i)

j

is connected to

atomic propositions at level i − 1 that precisiate p

(i)

j

.

The ﬁgure shows that, for instance, p

(2)

2

can be ex-

pressed as a compound proposition consisting of two

Computational atoms

Figure 2: Hierarchy of propositions. At level i, the formal

logic described in Sections 3–3.4 is established with com-

ponent sets S

i

, V

i

, O

i

, A

i

, and C

i

. Atomic propositions in

S

0

× V

0

× O

0

× A

0

are the logical atoms and the building

blocks of all propositions, and each of them is directly as-

sociated with a computational atom. For each i ≥ 1, every

atomic proposition at level i is connected to atomic propo-

sitions at level i − 1 that precisiate it.

atomic propositions (p

(1)

1

and p

(1)

4

) at level 1 and also

as a compound proposition consisting of four atomic

propositions (p

(0)

1

, p

(0)

2

, p

(0)

3

, and p

(0)

4

) at level 0.

The hierarchy clearly shows the deﬁnition of

each proposition by expressing it in terms of precisi-

ated propositions at lower levels. Thus, non-logical,

domain-dependent axioms result from the hierarchy,

and they can be used for inference and reasoning in

the formal logic.

Different levels of granularity may require differ-

ent component sets, but the same syntactic structure

is enforced at all levels. Using the hierarchy, we can

ensure that all the resulting propositions remain pre-

cisiated at each level, and we can attain ﬂexibility in

the level of detail.

7 SEMANTICS OF THE FORMAL

LOGIC

The semantics of formal logic speciﬁes how to de-

termine the truth value of each proposition. In two-

valued logics, for instance, the truth value is either

1 (true) or 0 (false). As described by Zadeh (e.g.,

(Zadeh, 2001), (Zadeh, 2004)), this bivalence is not

suitable for PNL, so we develop a many-valued se-

mantics for our formal logic. The meaning of the truth

value depends on the context. For the task description

scheme described in Sections 3–6, for instance, one

can evaluate each proposition and let its truth value

reﬂect the feasibility of the corresponding task speci-

ﬁcation; 1 indicates that the task certainly can be car-

ried out whereas 0 indicates that it certainly cannot

be. In this case, it is more realistic and practical to let

IJCCI2013-InternationalJointConferenceonComputationalIntelligence

338

the degree of feasibility take on not only the values 0

and 1 but also other values between 0 and 1. In real-

world problems, it can be highly practical to evaluate

the feasibility of a task description before any seri-

ous attempt is made to execute it. If robotic systems

interact with a variety of users, including users who

have no knowledge of these systems, some users may

describe tasks that are virtually impossible to accom-

plish. It is more desirable to disregard such highly in-

feasible tasks immediately than to waste resources by

attempting to realize them. Also, the user may want to

be informed of the degree of feasibility of the task that

he speciﬁes before the system attempts to perform it.

When multiple options are available for performing a

speciﬁed task, the user may want to compare their de-

grees of feasibility before determining which option

to take.

To determine the truth value of each proposition

in our formal logic systematically and effectively, we

use fuzzy relations. A fuzzy relation is a generaliza-

tion of a classical (“crisp”) relation (see, for instance,

(Klir and Folger, 1988)). It is a mapping from a Carte-

sian product to the set of real numbers between 0 and

1. While a classical relation only expresses the pres-

ence or absence of some form of association between

the elements of factors in a Cartesian product, a fuzzy

relation can express various degrees or strengths of

association between them. (Hence a classical rela-

tion can be considered a “crisp” case of a fuzzy rela-

tion.) In our formal logic, each proposition consists of

pre-speciﬁed components, so a function that assigns a

truth value to each proposition can be represented by a

fuzzy relation on the Cartesian product of the compo-

nents. Using some of the operations deﬁned on fuzzy

relations (they are described in Appendix), we can

systematically and economically determine the truth

value of each proposition.

We will explain the semantics of our formal

logic using concrete examples of task descriptions for

human-robot interaction so that the reader can under-

stand it intuitively. To facilitate our exposition, we

consider very simple task descriptions resulting from

atomic propositions in S ×V × O. Notice that even in

this rather simple case, we need an efﬁcient scheme

for determining the truth value of each proposition.

For instance, if each of the component sets S, V , and

O contains ten elements, then there are 10

3

atomic

propositions in S ×V × O, and it may be impractical

to determine the truth values of all the atomic propo-

sitions individually. Moreover, if the component set C

consists of three connectives, then we can generate a

total of 3 · 10

6

compound propositions that consist of

two atomic propositions. In practice, it may be neces-

sary to promptly evaluate and compare the truth val-

ues of a large number of task descriptions represented

by such compound propositions in order to determine

which option to execute, so even this simple case re-

quires an efﬁcient, systematic scheme for examining

the truth conditions of propositions. (Also note that

inﬁnitely many propositions can be generated from

these component sets.)

In Section 7.1, we explain how to determine the

truth values of atomic propositions. In Section 7.2,

we explain how to determine the truth values of com-

pound propositions.

7.1 Truth Conditions of Atomic

Propositions

For concreteness, we consider establishing a fuzzy

relation on S × V × O, which is a mapping from

the Cartesian product to a totally ordered set called

a valuation set. In our formulation of many-

valued logic, the valuation set is the unit interval

[0, 1]. We consider the following component sets:

S = {robot1, robot2}, V = {recognize, hold}, O =

{ball, pen}. Table 2 shows the atomic propositions in

S ×V × O. Since the component sets considered here

Table 2: Atomic propositions resulting from

S = {robot1, robot2}, V = {recognize, hold}, and

O = {ball, pen}.

s ∈ S v ∈ V o ∈ O

robot1 recognize ball

robot1 recognize pen

robot1 hold ball

robot1 hold pen

robot2 recognize ball

robot2 recognize pen

robot2 hold ball

robot2 hold pen

are small, we only have eight atomic propositions in

the Cartesian product. However, as mentioned ear-

lier, the total number of atomic propositions becomes

quite large with large component sets, and we need

an efﬁcient, systematic scheme for determining the

truth conditions of these propositions. We establish

our scheme using three operations on fuzzy relations:

projection, cylindric extension, and cylindric closure.

These operations are explained in Appendix.

Suppose that the truth conditions (for concrete-

ness, we assume that they represent degrees of fea-

sibility) of these atomic propositions are determined

for a robotic system under the following conditions:

(a) Robot1 is equipped with a high-resolution cam-

era that enables it to recognize various objects, in-

339

cluding a ball and a pen. However, it does not

have any arm, so it cannot hold any object.

(b) Robot2 has an arm that enables it to hold various

objects, including a ball and a pen. However, it is

not equipped with a high-resolution camera, so it

is not fully capable of identifying objects.

(c) With the high-resolution camera, a ball is easier to

recognize compared to a pen.

(d) With the arm, a pen is easier to hold compared to

a ball.

Our strategy is to derive a fuzzy relation on S ×

V × O from fuzzy relations on S × V and on V × O.

Hence, we ﬁrst establish fuzzy relations on S ×V and

on V × O. Let R

S×V

: S ×V → [0, 1] denote a fuzzy

relation on S ×V . Based on conditions (a) and (b), we

set the values of R

S×V

as shown in Table 3. Recall

Table 3: Fuzzy relation R

S×V

on S ×V based on conditions

(a) and (b).

s ∈ S v ∈ V R

S×V

(s, v)

robot1 recognize .9

robot1 hold 0

robot2 recognize .2

robot2 hold .8

that a fuzzy relation expresses various degrees or

strengths of association between elements in compo-

nent sets. The value assigned to (robot1, recognize)

is relatively large (.9) because robot1 is equipped

with a high-resolution camera and is thus suitable for

recognizing objects, whereas the value assigned to

(robot1, hold) is zero because robot1 is not equipped

with an arm and is thus incapable of holding objects.

Similarly, the value assigned to (robot2, recognize) is

relatively small (.2) because robot2 is not equipped

with a high-resolution camera and is thus unsuitable

for recognizing objects, whereas the value assigned to

(robot2, hold) is relatively large (.8) because robot2

is equipped with an arm and is thus suitable for hold-

ing objects. Technically, the fuzzy relation R

S×V

is

considered the underlying fuzzy relation on S ×V ×O

projected onto S ×V (see Appendix for the operation

of projection).

Analogously, based on conditions (c) and (d), we

set the values of a fuzzy relation R

V ×O

: V × O →

[0, 1] as shown in Table 4. The value assigned

to (recognize, ball) is larger than that assigned to

(recognize, pen) because with a high-resolution cam-

era, a ball is easier to recognize compared to a pen.

Similarly, the value assigned to (hold, pen) is larger

than that assigned to (hold, ball) because with an

arm, a pen is easier to hold compared to a ball. Tech-

nically, the fuzzy relation R

V ×O

is considered the un-

Table 4: Fuzzy relation R

V ×O

on V ×O based on conditions

(c) and (d).

v ∈ V o ∈ O R

V ×O

(v, o)

recognize ball .9

recognize pen .8

hold ball .7

hold pen .8

derlying fuzzy relation on S ×V × O projected onto

V × O (see Appendix for the operation of projection).

We establish a fuzzy relation R

S×V ×O

: S × V ×

O → [0, 1] by combining the fuzzy relations R

S×V

and R

V ×O

. Formally, we obtain R

S×V ×O

by ﬁrst ob-

taining the cylindric extensions of R

S×V

and R

V ×O

to S × V × O and then computing the cylindric clo-

sure of the two cylindric extensions (see Appendix

for cylindric extension and cylindric closure). First,

we obtain the cylindric extensions of R

S×V

and R

V ×O

to S ×V ×O. Table 5 shows the cylindric extension of

R

S×V

to S ×V ×O, which we denote by R

S×V ↑S×V×O

.

Table 5: Cylindric extension R

S×V ↑S×V×O

.

s ∈ S v ∈ V o ∈ O R

S×V ↑S×V ×O

(s, v, o)

robot1 recognize ball .9

robot1 recognize pen .9

robot1 hold ball 0

robot1 hold pen 0

robot2 recognize ball .2

robot2 recognize pen .2

robot2 hold ball .8

robot2 hold pen .8

This cylindric extension can be characterized as

maximizing nonspeciﬁcity in deriving a fuzzy rela-

tion on S ×V × O from a fuzzy relation on S ×V (see

Appendix). Similarly, Table 6 shows the cylindric ex-

tension of R

V ×O

to S × V × O, which we denote by

R

V ×O↑S×V ×O

.

Table 6: Cylindric extension R

V ×O↑S×V ×O

.

s ∈ S v ∈ V o ∈ O R

V ×O↑S×V ×O

(s, v, o)

robot1 recognize ball .9

robot1 recognize pen .8

robot1 hold ball .7

robot1 hold pen .8

robot2 recognize ball .9

robot2 recognize pen .8

robot2 hold ball .7

robot2 hold pen .8

This cylindric extension can be characterized as

maximizing nonspeciﬁcity in deriving a fuzzy relation

on S ×V ×O from a fuzzy relation on V ×O (see Ap-

pendix). Finally, we set R

S×V ×O

to the cylindric clo-

sure of R

S×V ↑S×V×O

and R

V ×O↑S×V ×O

on S × V × O

IJCCI2013-InternationalJointConferenceonComputationalIntelligence

340

(see Appendix for the operation of cylindric closure),

which is shown in Table 7.

Table 7: Cylindric closure R

S×V ×O

of R

S×V ↑S×V×O

and

R

V ×O↑S×V ×O

on S ×V ×O.

s ∈ S v ∈ V o ∈ O R

S×V ×O

(s, v, o)

robot1 recognize ball .9

robot1 recognize pen .8

robot1 hold ball 0

robot1 hold pen 0

robot2 recognize ball .2

robot2 recognize pen .2

robot2 hold ball .7

robot2 hold pen .8

Notice that the resulting truth values (degrees of

feasibility) assigned to the eight atomic propositions

reﬂect the conditions (a)–(d). For example, the truth

values clearly indicate that robot1 is highly capable

of recognizing objects (because it is equipped with a

high-resolution camera) but is incapable of holding

objects (because it does not have any arm). Sim-

ilarly, the truth values clearly indicate that robot2

is highly capable of holding objects (because it is

equipped with an arm) but is rather incapable of rec-

ognizing objects (because it is not equipped with a

high-resolution camera).

It is efﬁcient to derive a fuzzy relation on S ×V ×

O from fuzzy relations on S ×V and on V ×O. Again,

suppose that each of these component sets consists

of ten elements. Then a total of 10

3

atomic proposi-

tions result from them, and it may be time-consuming

to determine 10

3

truth values individually. With our

scheme, we can derive the 10

3

truth values by deter-

mining 2 · 10

2

values of the fuzzy relations R

S×V

and

R

V ×O

. This efﬁciency of the scheme becomes more

notable as the size of each component set or the num-

ber of component sets increases.

Another important strength of our scheme lies in

updating the truth values of the atomic propositions.

In practice, the values shown in Tables 3–4 will be

determined dynamically based on the conditions of

the robots. For instance, if the high-resolution camera

of robot1 becomes dysfunctional, then we will use the

fuzzy relation R

0

S×V

shown in Table 8 instead of the

fuzzy relation R

S×V

in Table 3 in computing the truth

values of the atomic propositions.

Notice that the value of R

0

(robot1, recognize) is

.2, reﬂecting the fact that robot1 can no longer use

its high-resolution camera to recognize objects. It is

easy to verify that Table 9 shows the cylindric closure

R

0

S×V ×O

of the cylindric extensions R

0

S×V ↑S×V×O

and

R

V ×O↑S×V ×O

.

Comparing Tables 7 and 9, we can see that the up-

Table 8: Fuzzy relation R

0

S×V

on S ×V (reﬂecting a damage

to robot1’s high-resolution camera).

s ∈ S v ∈ V R

0

S×V

(s, v)

robot1 recognize .2

robot1 hold 0

robot2 recognize .2

robot2 hold .8

Table 9: Cylindric closure R

0

S×V ×O

of R

0

S×V ↑S×V×O

and

R

V ×O↑S×V ×O

on S×V ×O (reﬂecting a damage to robot1’s

high-resolution camera).

s ∈ S v ∈ V o ∈ O R

0

S×V ×O

(s, v, o)

robot1 recognize ball .2

robot1 recognize pen .2

robot1 hold ball 0

robot1 hold pen 0

robot2 recognize ball .2

robot2 recognize pen .2

robot2 hold ball .7

robot2 hold pen .8

dated truth values (shown in Table 9) reﬂect the con-

dition that the high-resolution camera of robot1 has

become dysfunctional. Notice that we have efﬁciently

updated the fuzzy relation on S×V ×O by just updat-

ing the fuzzy relation on S ×V . With our scheme, it

is possible to keep the truth values of a large number

of atomic propositions updated continually.

7.2 Truth Conditions of Compound

Propositions

Since we establish our formal logic as a many-valued

logic, we treat the connectives in C as logic primitives

of many-valued logic or fuzzy logic. Here we exam-

ine thee typical logical primitives: conjunction (rep-

resented by ”and” in C), disjunction (represented by

”or” in C), and implication (also called conditional,

represented by “if” in C).

In evaluating the truth value of a compound propo-

sition, conjunction is often implemented as a t-norm,

whereas disjunction is often implemented as a t-

conorm (e.g., (Klir and Folger, 1988), (H

´

ajek, 1998)).

Various forms of t-norm and t-conorm have been

proposed. Some of the frequently used t-norms are

the minimum t-norm, the product t-norm, and the

Łukasiewicz t-norm, and some of the frequently used

t-conorms are the maximum t-conorm, the probabilis-

tic sum, and the Łukasiewicz t-conorm. In practice,

the suitability of each of these t-norms or t-conorms

depends on what the truth value represents. Also,

there are various ways to implement implication in

evaluating the truth value of a compound proposition

341

(e.g., (Trillas and Alsina, 2012)). Some of the main

forms of implication are the material implication, the

conjunctive conditional, the residuated conditional

Sasaki hook, the Dishkant hook, and the Mamdani-

Larsen conditional. Again, in practice, the suitability

of each conditional depends on what the truth value

represents.

8 CONCLUSIONS

We have taken a ﬁrst step toward establishing a formal

logic as a generalized precisiation language, which

is essential for generalizing PNL. Various syntactic

structures in natural language can be incorporated in

our formal logic so that it precisiates not only percep-

tual propositions but also action-related propositions.

The syntax of the formal logic allows us to create in-

ﬁnitely many precisiated propositions while ensuring

that every proposition in it is precisiated. As in other

formal logics, we can infer and reason in our for-

mal logic. The resulting generalized precisiation lan-

guage serves as a middle ground between the natural-

language-based mode of human communication and

the low-level mode of machine communication and

thus signiﬁcantly facilitates human-machine interac-

tion.

ACKNOWLEDGEMENTS

This research is supported by the Spanish Ministry

of Economy and Competitiveness through the project

TIN2011-29824-C02-02 (ABSYNTHE).

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APPENDIX

We describe three operations on fuzzy relations that

are used in determining the truth conditions of atomic

propositions in our formal logic: projection, cylin-

dric extension, and cylindric closure. First, we es-

tablish notation. Let X

1

, X

2

, . . . , X

n

be sets, and let

X

1

× X

2

× ·· · × X

n

denote their Cartesian product. We

will also denote the Cartesian product by ×

i∈N

n

X

i

,

where N

n

denotes the set of integers 1 through n.

A fuzzy relation on ×

i∈N

n

X

i

is a function from the

Cartesian product to a totally ordered set, which is

called a valuation set. In our formulation, the unit in-

terval [0, 1] is used as a valuation set. Each n-tuple

(x

1

, x

2

, . . . , x

n

) in X

1

× X

2

× ·· · × X

n

(thus x

i

∈ X

i

for

each i ∈ N

n

) will also be denoted by (x

i

| i ∈ N

n

). Let

I ⊂ N

n

. A tuple y := (y

i

| i ∈ I) in Y := ×

i∈I

X

i

is said

to be a sub-tuple of x := (x

i

| i ∈ N

n

) in ×

i∈N

n

X

i

if

y

i

= x

i

for each i ∈ I, and we write y ≺ x to indicate

that y is a sub-tuple of x.

Let X := ×

i∈N

n

X

i

and Y := ×

i∈I

X

i

for some I ⊂

N

n

. Suppose that R : X → [0, 1] is a fuzzy relation on

X. Then a fuzzy relation R

0

: Y → [0, 1] is called the

projection of R on Y if for each y ∈ Y , we have

R

0

(y) = max

x∈X : y≺x

R(x).

We let R

↓Y

denote the projection of R on Y .

We continue with X := ×

i∈N

n

X

i

and Y := ×

i∈I

X

i

(I ⊂ N

n

). Let F : Y → [0, 1] be a fuzzy relation on

Y . A fuzzy relation F

0

: X → [0, 1] is said to be the

cylindric extension of F to X if for all x ∈ X , we have

F

0

(x) = F(y),

where y is the tuple in Y such that y ≺ x. We let F

↑X

denote the cylindric extension of F to X. The cylin-

dric extension F

↑X

of a fuzzy relation F : Y → [0, 1] is

the “largest” fuzzy relation on X such that its projec-

tion on Y equals F; if we let R denote the set of all

fuzzy relations R

0

: X → [0, 1] such that R

0

↓Y

= F, then

for all x ∈ X, we have

F

↑X

(x) = max{R

0

(x) | R

0

∈ R}.

For each j, let Y

j

:= ×

i∈I

j

X

i

, where I

j

⊂ N

n

. Let

R

( j)

: Y

j

→ [0, 1] denote a fuzzy relation on Y

j

. Then

a fuzzy relation F : X → [0, 1] is called the cylindric

closure of R

(1)

, R

(2)

, . . . , R

(m)

on X if for each x ∈ X,

F(x) = min

1≤ j≤m

R

( j)

↑X

(x).

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