EXTENDING X-MACHINES TO SUPPORT REPRESENTATION OF

SPATIAL 2-D AGENTS

Isidora Petreska

, Petros Kefalas

, Marian Gheorghe

and I. Stamatopoulou

South East European Research Centre (SEERC), 24 Proxenou Koromila Str., Thessaloniki 54622, Greece

CITY College, International Faculty of the University of Shefﬁeld, 3 Leontos Sofou Str., Thessaloniki 54626, Greece

University of Shefﬁeld, Dept. of Computer Science Regent Court, 211 Portobello Str., Shefﬁeld S1 4DP, UK

Keywords:

Formal modelling, X-machines, Spatial agents.

Abstract:

Starting with the notion of modelling biologically inspired agents, this paper focuses on their spatial charac-

teristics. It will be demonstrated that one of the most prominent formalisms in modelling the behaviour of bio-

logical colonies, X-machines, cannot provide a neat and effective way to modelling spatial agents (i.e. agents

distributed and move through a physical space). We introduce a X-machines variation that besides facilitating

formal modelling, will provide grounds towards visual animation of these systems. This approach resulted

into a novel progression, Spatial X-machines, without retracting the legacy characteristics of X-machines such

as testing and veriﬁcation strategies. Unlike other formalisms that go behind the concept of treating the agent’s

behaviour as one uniform component, Spatial X-machines tend to draw a separation between different types

of agent’s behaviour.

1 INTRODUCTION

Formal modelling is considered as one of the most

essential stages in Multi-agent system (MAS) devel-

opment and it can be carried out with many different

methods and techniques (Kefalas et al., 2002). There

are varieties of formal methods in agent-oriented

engineering, and a number of approaches towards

modelling biological phenomena have been devel-

oped (Kefalas et al., 2002). X-machines (XM) and

Communicating X-machines (CXM) are targeted into

representing the behaviour of biological colonies,

which in turn can be characterised as spatial agents.

Spatial agents can be deﬁned as collections of agents

distributed and move through a physical space. They

have incomplete knowledge of the environment and

can change their direction and position within the en-

vironment (move in an n-dimensional space). When

it comes to data modelling of spatial agents, there are

three key features in their development (Y. Bdard and

Caron, 1992):

• Conceptual data model, which acts as a represen-

tation of the reality as it is, deﬁned by the users;

• Logical data model, which deﬁnes how the con-

ceptual model will be implemented; and

• The physical data model or the computer code of

the application.

Targeting the conceptual data model, there are dif-

ferent approaches for modelling spatial phenomena

of biological systems: process algebra can be ap-

plied to develop a calculus of processes that could de-

scribe the spatial geometric transformations (Cardelli

and Gardner, 2010). Membrane computing can

also be utilised by introducing geometric informa-

tion (Romero-Campero et al., 2009) or population P

systems (Petreska and Kefalas, 2011). Yet another

agent-based approach is the intracellular NF-κB sig-

nalling pathway for modelling spatial information in

predictive complex biological systems (Pogson et al.,

2008). However, the combination of biological agents

and spatial data modelling still remains an active re-

search ﬁeld. With this work we focus on modelling

spatial agents with the XM approach.

On the other hand, visual animation as an informal

veriﬁcation technique, helps in discovering the ﬂaws

of the formally unveriﬁable properties within a bio-

logically inspired systems (such as their position or

diction with respect to the environment). We discuss

that attempts to formally verify such properties would

result into a combinatorial explosion. Moreover, vi-

sual animation provides means to facilitate the com-

munication gap between the formal experts and the

Petreska I., Kefalas P., Gheorghe M. and Stamatopoulou I..

EXTENDING X-MACHINES TO SUPPORT REPRESENTATION OF SPATIAL 2-D AGENTS.

DOI: 10.5220/0003744600540061

In Proceedings of the 4th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2012), pages 54-61

ISBN: 978-989-8425-96-6

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

scientists (which in turn have no formal background)

by providing an immediate feedback understandable

to both of the teams. And ﬁnally, visual animation

can often lead to detecting some emergent behaviour

within a system. Therefore, this work opens the ques-

tion about how to automatically visualise a given XM

model.

Starting with a detailed discussion on XMs, later

extended by a case study (Sec. 2), we demonstrate

the drawbacks of XMs when it comes to modelling

the spatial characteristic of an agent. This provides

grounds to introducing a formal variation called Spa-

tial X-machines (

XMs, Sec. 3). This structure

will be carefully deliberated, followed by discussion

about their support towards veriﬁcation and validation

(Sec. 4). And ﬁnally, the paper will be concluded with

a discussion on whether

XMs provide means to sup-

port visual animation strategies.

2 BACKGROUND ON

X-MACHINES: FROM

DEFINITION TO PRACTICE

An XM resembles a Final State Machine (FSM) with

the power of being more expressive (Kefalas et al.,

2002). It is achieved due to the addition of a mem-

ory, and functions operating on the inputs and mem-

ory values. Considering stream XMs (a representative

class of XMs), the memory is a typed tuple of values

which appears to enhance the modelling of complex

data structures. The control, on the other hand, can

be visualised by utilising a diagrammatic approach.

Thus stream XMs have the ability of modelling both

the data (held in the memory) and the control. The

processing of the data is modelled by transitions be-

tween states, represented with functions. A function

receives the memory values together with an input,

performs changes on these memory values and pro-

duces an output. Based on the current state, the mem-

ory changes after an application of a function and the

output of that function, the stream XM evaluates the

next state.

Formally, a stream XM can be described as an 8-

tuple XM = (Σ, Γ, Q, M, Φ, F, q

, m

), such that (Ke-

falas and Kapeti, 2000; Kefalas et al., 2003b):

• Σ and Γ are input and output sets of symbols;

• Q is a ﬁnite set of states;

• M is an n-tuple called memory;

• Φ is a ﬁnite set of partial functions that map an

input and a memory state to an output and a new

memory state, φ: Σ × M → Γ × M;

• F is a function that determines the next state, given

a state and a function from the type Φ,

F: Q × Φ → Q; and

• q

and m

are the initial state and memory respec-

tively.

With the focus on the practical development of

communicating systems, the output of an X-machine

function can become input to a function of another

X-machine. This way a structure known as Commu-

nicating X-machine (CXM) is being formed, provid-

ing a way to deal with agents communication (Kefalas

et al., 2003a; Kefalas, 2002).

2.1 Case Study: A Foraging Agent

Let us consider the followingexample of an agent that

randomly moves in 2-D space, picks up an object it

encounters and carries it back to the base (Fig. 1).

Clearly, although this is a very simple example, there

could be quite a few solutions (from a very abstract

to more detailed one). Likewise, Fig. 2 pictures alter-

native different XM models that can be considered as

solutions to the foraging agent problem.

agent

block

base

Figure 1: The foraging agent example.

Table 1 demonstrates three ways of modelling the

foraging agent problem (the numbering a), b) and c)

corresponds to the numbering in Fig. 2). The ﬁrst so-

lution a) is a very abstract representation that does not

even takes into consideration the position (the coordi-

nates) of the agent. The fact that XMs are generic

and do not impose modelling of a position, in such

an example might result into an incomplete model. A

more detailed representation can be derived from the

second solution b) from Table 1 (it can be noted that

the representation is a design choice, for instance the

memory variables that correspond to positions are in-

tegers). Yet again, this representation is way too com-

plex and probably more difﬁcult for understanding.

Finally, the last representation c), is the best solution

in terms of completeness and complexity. More com-

prehensive speciﬁcation can be found as (Petreska,

2011).

EXTENDING X-MACHINES TO SUPPORT REPRESENTATION OF SPATIAL 2-D AGENTS

MEMORY = (Is_carrying_block, Is_at_base, Does_see_block)

carrying_nothing

has_seen_block

move_and_see_block

pick_block

move_and_see_nothing

at_base

move_and_be_at_base

move_and_not_be_at_base

carrying_block

leave_block

MEMORY = ((Xcurr,Ycurr), (Xbase ,Ybase),

P(Xblock xYblock))

a)

b)

MEMORY = ((Xcurr,Ycurr), (Xbase,Ybase),

CarryingBlockId)

search_and_see_block

search_for_block

search_for_base

leave_block

_at_base

c)

Output streamInput stream

leave_block

_at_base

Input stream Output stream Output stream Input stream

carrying_nothing

carrying_block

search_and_see_block

search_for_block

search_for_base

carrying_block

carrying_nothing

Figure 2: Examples of modelling the foraging agent example: a) very abstract representation b) more detailed, but complex

representation c) the best represented solution.

Table 1: Different ways to modelling the foraging agent example.

a) Q = {carrying nothing, has seen block, carrying block, at base}

M = {Is carrying block, Is at base, Does see block}, where Is carrying block, Is at base,

Does see block ∈ {true, false}

= {false, false, false}

= {carrying nothing}

Σ = {”move to a place w/o block”, ”move to a place with block”, ”pick block”, ”search for base”, ”move

to base”, ”leave block”}

Γ = { ”agent keeps moving empty”, ”agent detected block”, ”agent picked block”, ”agent searches for

base”, ”agent found base”, ”agent left block”}

b) Q = {carrying nothing, carrying block}

M = ((X

curr

, Y

curr

), (X

base

, Y

base

), P(X

block

× Y

block

), Hand) where X

curr

, Y

curr

, X

block

, Y

block

, X

base

, Y

base

∈

Z, Hand ∈ {full, empty}

= ((2, 3) , (0, 0), {(2, -3), (4, -6), (2, 1), (3, 5), (-1, 5)}, empty)

= {carrying nothing}

Σ = (X

new

, Y

new

), where X

new

, Y

new

∈ Z

Γ = { ”agent keeps moving empty”, ”agent detected and picked block”, ”agent searches for base”, ”agent

found base and left block”}

c) Q = {carrying nothing, carrying block}

M = ((X

curr

, Y

curr

), (Xbase, Ybase), CarryingBlockId) where X

curr

, Y

curr

, Xbase, Ybase ∈ Z, Carrying-

BlockId ∈ {block

, block

, ... block

} ∪ nil, n ∈ N

= ((2, 3), (0, 0), nil)

= {carrying nothing}

Σ = ((X

new

, Y

new

), BlockId), where X

new

, Y

new

∈ Z, BlockId ∈ {block

, block

, ... block

} ∪ {nil}, n ∈ N

Γ = { ”agent keeps moving empty”, ”agent detected and picked block”, ”agent searches for base”, ”agent

found base and left block”}

2.2 Shortcomings of XM for Spatial

Agents

The case study in the previous section demonstrated

that there might be different ways to modelling spatial

agents with the XM approach, even for the simplest

scenario. This lead towards identiﬁcation of the fol-

lowing shortcomings when modelling spatial agents

with XM is concerned:

• There might be many different solutions (even

for the simplest model) for representing the com-

monly found properties, such as the initial posi-

tion or the direction of a spatial agent. This makes

it more difﬁcult to read a given model (we have

to understand how the modeller decided to repre-

sent these properties) and even to create one (ev-

ery time the modeller has to think how to repre-

sent them).

• There are difﬁculties in simulating a given model

because there is not a standard way that deals with

ICAART 2012 - International Conference on Agents and Artificial Intelligence

manipulation and processing of the spatial prop-

erties like the initial position or the direction of a

spatial agent.

• The memory holds all data structures required, in-

cluding the position and the direction.

Initiated by these shortcomings, a question that

can be imposed is: How can we extend XMs to sup-

port spatial agent modelling natively and why? The

motivation behind this question can be further broad-

ened into the following aspects:

• The subset of MAS that deal with movement in

space is quite numerous, starting with biologically

inspired MAS, up to MAS used in many industrial

applications like robotics, etc.

• Different modellers might represent a spatial

agent’s basic characteristics, like position and di-

rection, in different ways.

• The current XM representation for a spatial

agent model does not directly map to an anima-

tion/simulation.

• The current XM representation for a spatial agent

model is rather cumbersome/difﬁcult to code, and

in many situations it is also difﬁcult to be under-

stood.

• When it comes to verifying a spatial model with

XM, this will result into space explosion due to

the spatial information.

3 INTRODUCTION TO SPATIAL

X-MACHINES

XMs represent an extension of stream XMs by

deﬁning three new components and modifying some

existing ones in order to facilitate uniﬁcation with the

newly deﬁned components. The input and the output

set, the memory, the set of states and the next state re-

main intact, because these structures do not deal with

the spatial attributes that we intend to support. On

the other hand, the following components have been

introduced:

• A tuple containing the current position of the

agent and an integer that represents its current

direction. The current position determines the

agent’s location in its environment, and the direc-

tion represents its heading (such as south, north,

etc.).

• A set containing elementary operations. These

operations allow manipulation with the current

position tuple and the current direction.

XM is a 13-tuple

XM = (Σ, Γ, Q, q

, π, π

θ, θ

, M, m

, E, Φ, F) that can be formally deﬁned as

(See Fig 3):

• Σ is an input set of symbols;

• Γ is an output sets of symbols;

• Q is a ﬁnite set of states;

• q

is the initial state;

• M is an n-tuple called memory;

• m

is the initial memory;

• π is a tuple of the current position, i.e. (x, y) when

a 2D representation is considered;

• π

is the initial position;

• θ is an integer in the range 0 to 360, that repre-

sents a direction (integer values are used as a de-

sign choice);

• θ

is the initial direction;

• E is a set which contains elementary positioning

operations: e

such as e

: Π× Θ −→ Π× Θ, such

as direction, moving forward and moving to a spe-

ciﬁc position;

• Φ is a ﬁnite set of partial functions φ that map a

memory state, position, direction and set of inputs

to a new memory state, position, direction and set

of outputs:

φ: M × π × θ × Σ −→ M × π × θ × Γ; and

• F is a function that determines the next state, given

a state and a function from the type Φ, such as F:

Q × Φ → Q.

If we take a closer look at the memory M, it may

be noted that M is composed of M’ ˆ< π, θ >, where

M’ is a memory structure from the standard XM.

Moreover, talking about the set which contains ele-

mentary operations, we have currently deﬁned three

operations:

• change direction m - changes the spatial agent’s

direction to m, where m is of type θ and m ∈ Θ

(ex. change direction 60)

• move n forward n - moves forward for n units,

where n is an integer (ex. move n forward 3)

• move to position x y - moves to speciﬁc posi-

tion (x, y), where x is the x-coordinate, y is the

y-coordinate of the agent and (x, y) ∈ Π (ex.

move to position 126 43)

As it can be determined from the deﬁnition, a

XM in essence provides a separation of the be-

haviour within the system that deals with the move-

ment (and the other spatial attributes) from the rest

EXTENDING X-MACHINES TO SUPPORT REPRESENTATION OF SPATIAL 2-D AGENTS

MEMORY, POSITION, DIRECTION

m, π, θ m’, π’, θ’

input stream

output stream

Figure 3: An abstract example of a

XM.

of the behaviour. This way we establish a standard-

ised way to modelling motion, which is easily under-

standable and provides a direct mapping to an anima-

tion/simulation. And ﬁnally, this work is achieved by

maintaining an obvious equivalence with the standard

XM.

A deﬁnition language for modelling the foraging

agent problem (from Fig. 2) as a

XM, is presented

in (Petreska, 2011). Moreover, a discussion about the

speciﬁcation and the grammar for describing a

model, can be found in (Petreska, 2011) as well.

4 VERIFICATION, VALIDATION

AND ANIMATION OF

XMS

Formal veriﬁcation of spatial agents is an extremely

complex task. On one hand stands the fact that the

veriﬁcation process leads to combinatorial explosion,

because modelling these agents means modelling of

their spatial properties (such as position or direction).

Therefore, the veriﬁcation would require exploration

of a state space developed by the combination of all

agent positions evolved through time (Petreska et al.,

2011). On the other hand, there is the fact that the

emergent properties of the system should be known

in advance in order to be veriﬁed. The concept of

emergence can be explained as a pattern appearing in

the conﬁguration of the agents, at some instance dur-

ing the lifetime of the system. In biology or biology-

inspired agents the emergence can be observed in-

vivo (for example, line formation, ﬂocks, schools,

herds etc.). However, when it comes to artiﬁcial

agents, it is not always straightforward. Driven from

these two problems, it might be desirable to combine

several formal with informal techniques that would be

able to join forces towards the veriﬁcation of spatial

MAS (Petreska et al., 2011).

The models of an X-machine can be de-

scribed with the X-machine Deﬁnition Language

(XMDL (Kefalas and Kapeti, 2000)) and textual sim-

ulation. XMDL is a listing of deﬁnitions that match

the tuples of X-machine’s deﬁnition. Starting with the

diagram in Fig. 4, XMDL is facilitated with a parser

built using Deﬁnite Clause Grammars (DCG) nota-

tion (Kefalas et al., 2002), which apart from the syn-

tax errors, output as warnings any kind of logical er-

rors or omissions. The semantic analysis and the rules

for transformation are being checked by the compil-

ing component, with the aid of deﬁned rules under

which the speciﬁcation is translated into the equiva-

lent Prolog code. This Prolog code is after utilised

by an animation tool, which simulates the computa-

tion of an X-machine. Furthermore, the model check-

ing component deﬁnes a new logic, XmCTL (Eleft-

herakis et al., 2002), and with the implementation of

a model checking algorithms can determine whether

a property is true or false. And ﬁnally, XMs are

supported with automatic generation of test cases,

which is proved that ﬁnds all faults in the implemen-

tation (Ipate and Holcombe, 1998).

Figure 4: Veriﬁcation and validation of an X-machine.

Taking

XMs into consideration, the following

discussion will concentrate on investigating whether

they inherit the mentioned veriﬁcation and validation

techniques of XMs. An informal proof that a

XM is

equivalent to any XM could be derived by investigat-

ing:

• The memory M of a normal XM is equivalent to

the structure of memory M, position Π and direc-

tion Θ within a

XM. In other words, the posi-

tion and direction can either become members of

the memory tuple in a normal XM model, or they

can be excluded from the model without loss of

its integrity.

• Any function in a

XM model can be translated

into a function of the normal XM. More particu-

ICAART 2012 - International Conference on Agents and Artificial Intelligence

larly, the predeﬁned operations in a where state-

ment of a function in a

XM model can be omit-

ted or replaced with the standard XMDL syntax to

preserve the logic ﬂow.

Along these lines, by removing the newly deﬁned

components that in essence deﬁne a

XM, what we

get is still a completely valid skeleton of a normal

XM model. Therefore,

XMs tend to provide a stan-

dardised way to representing some properties of the

system, which could also be represented with an XM

model. Thus, the

XM deﬁnition can lead in easy for-

malisation, veriﬁcation (model checking), testing and

implementation (Fig. 5). The only condition imposed

would be not to test or model check the position (co-

ordinates) and direction properties, which in turn will

result into a state explosion.

Figure 5: Veriﬁcation and validation of a

X-machine.

Referring to the initial discussion of this chapter

for combining formal with informal techniques to-

wards the veriﬁcation of spatial MAS, we suggest that

visual animation can be exploited for detecting the

emergent properties of a system. In biological MAS,

animation becomes even more interesting because of

the spatial attributes of an agent, e.g. agents move

in an n-dimensional space. This raises the question:

Having a model of a system, how can we visualise it?

An animator as a form of simulation, is any kind of

program which given the code in the intermediate lan-

guage, implements an algorithm to facilitate the com-

putation of the model and its output though a textual

description (Wilensky, 1999; Stamatopoulou et al.,

2007). However, most of the animation techniques

share one major drawback, i.e. the outputs they pro-

duce are in a textual form and thus not even close to

the real visual perceptions on the system. Therefore,

we focus on a visual simulation platform, namely Net-

Logo (Wilensky, 1999; Wilensky, 1997).

NetLogo is a simulation platform for visual an-

imation of multi-agent systems regardless the num-

ber of agents, being supported by a functional lan-

guage that can represent an agent’s behaviour, as well

as an environment for creation of a graphical user in-

terface. NetLogo facilitates the veriﬁcation of a bio-

logical model in a way that simulation scenarios may

be executed, thus the expected behaviour of the sys-

tem could be compared to the visual outcome. This

platform was our initial choice due to its simplicity

and the legacy of work we have done so far in ex-

perimenting with Netlogo features and emergent bi-

ological phenomena. Similar but more advanced de-

velopment toolkits such as Repast (Collier and North,

2011) should not be excluded but could considered

too, as alternatives to visualisation.

Given an XM model, it is not always easy nor uni-

form to map its representation into the equivalent Net-

Logo code. This is due to the already discussed dis-

advantages in Sec. 3 that deal with the behaviour of

the system that represents the motion (and the other

spatial attributes). However,

XMs overcome this is-

sue, and thus enhance visual animation (the agent’s

position and direction can be interpreted into motion

within an animation platform). This feature opened

the horizon towards ideas for automation of the sim-

ulation scenarios for a

XM model, on which the au-

thors are currently working.

5 CONCLUSIONS AND FUTURE

WORK

Given X-machines, one might argue that the biologi-

cal agent models might be very abstract, i.e. there is

a freedom in the representation of a model. On the

other hand, certain knowledge is required for simulat-

ing a biological agent, for instance the initial position

or direction of the agent. This introduces difﬁculties

in simulating a given model, because an X-machine

does not specify how these knowledge will be mod-

elled. Thus we presented an idea of extending X-

machines into more speciﬁc formalism for modelling

spatial agents that move in space, called

XMs.

Further work could include extending

XMs in

a way to support other spatial agent properties and

more functions which will bring in more realistic

modelling of the spatial concept. Additionally, exam-

ples could be created towards simulation, testing and

model checking of an XM and a

XM model for a

critical comparison, pointing out their differences and

advantages.

Regarding simulation, a tool for automatic trans-

EXTENDING X-MACHINES TO SUPPORT REPRESENTATION OF SPATIAL 2-D AGENTS

lation of a

XM model into the NetLogo platform

for a visual animation is currently being developed

by the authors. Moreover, we currently work on a

framework towards the veriﬁcation of emergent be-

haviour of spatial MAS by utilising the

XMs ap-

proach (Petreska et al., 2011). This framework basi-

cally tries to support identifying emergent behaviour

by utilizing the tool for automatic translation (Pe-

treska et al., 2011). Initially, we propose that the for-

mal modelling should be accomplished with a formal-

ism that is able to clearly distinguish modelling of var-

ious types of behaviours (spatial or other behaviours),

such as

XM. This would make it possible to apply

different transformations facilitating further process-

ing.

At this point, there are two paths. On one hand,

the spatial behaviour can lead towards visual anima-

tion which will help detection of emergence (by utiliz-

ing NetLogo through the automatic translation tool).

On the other hand, the spatial behaviour should be

abstracted (together with the rest of the behaviours)

in order to lead towards simulation and logging of

time series data (Petreska et al., 2011). This might be

accomplished with a tool such as FLAME (M. Pog-

son and Holcombe, 2006; R. Smallwood and Walker,

2004), used to animate XM models. The next step

involves utilizing a tool for identifying patterns, such

as DAIKON (Michael et al., 1999). Therefore, all of

the patterns of behaviours together with the visual an-

imation would produce a set of desired properties. Fi-

nally, they can be veriﬁed in the original spatial agent

model by model checking.

Finally, the

XM can be suitably transformed into

an equivalent model in SPIN, PRISM or SMV (Holz-

mann, 1997; M.Kwiatkowska et al., 2001; McMillan,

1993). In this case, given a temporal formulae, all of

the desired properties could be veriﬁed upon the orig-

inal model.

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EXTENDING X-MACHINES TO SUPPORT REPRESENTATION OF SPATIAL 2-D AGENTS