Formal Languages and Automata Theory Applied to Transportation Engineering Problem of

Incident Management

Neveen ShlayanPh.D. Candidate

Outline

• Introduction• Set Theory Review• Formal Languages• Grammar • Automata Theory• Incident Management Problem• Specification and Verification• Conclusions

Formal and Automata Theory

• Has a fundamental role in the progress of computer science – Precise definition of syntax for programming

languages• Bring order to the Chaos

– Hardware and Software debugging

Set Theory Review (Notation)

• A={a, b, c} A is the set whose elements a, b, and c

• W equals the set of all x such that x is a

natural number

• Ø = { } Empty Set

Examples…..A={a, b, c, d} B={c, d, e, f, g}

• Union • Intersection • Complement of B relative to A

• Ac Complement of A

• A is a subset of B • Cartesian product, set of all

ordered pairs in the form (a, b)

• Function from A to B is a subset of

BA

BA

BA

BA

BA

}:{ NxxW

BA

Set Theory (Power Set)

• P(X) is the power set of X, the collection of all subsets of X.

• |X| is the number of elements in the set X |P(X)| = 2|X|

Examples..{1}, {1, 2}, {1, 2, 3}

Power Set of Natural numbers is uncountable

Operations over Symbols

• Finite Alphabet: V, A non-void set of arbitrary symbols (e.g. {a,b})– a and b here are called letters or symbols.

• Finite strings of letters are called words over V, e.g. ab, aab, baba etc.

• V*: The set of all words (obviously each has finite length)– is the empty word and is in V* for any V

More Operations

• Catenation: Joining of words…– e.g. abaa+abb=abaaabb;

• Associative but not commutative; i.e. in general, but (PQ)R=P(QR)

• V* is closed with respect to catenation, i.e. P and Q in V* implies PQ is in V*

• Unit :– We can define length function on words and

study properties etc.

QPPQ

PPP

Even More Operations

• Iterations:• Mirror Image:

;;; 20 ababPPabP

;; 1 baPabP

Language

• Language: An arbitrary set of words of V*, e.g. {a, ab, aa, aaa, aab,….}; – Finite or infinite– V* is countable infinite (denumerable)– Number of languages out of V* (i.e. how

many subsets, i.e. the size of powerset of V*) is uncountable (nondenumerable)

Language Examples

• Examples

*}|{

,...}1,0|{

},,{},{

13

2

1

VPPPL

ibaL

baLbaV

ii

Grammar

• Definition

Grammar Example

Automata

Incident Management Process

Incident Occurs

Emergency Responders (ER) Contacted

ER Arrive to the Scene

Incident Cleared

Challenges In Current Incident Management Process

CommunicationCoordination

Increase in Clearance Time

Economical, Safety, Environmental, and Social Impacts

Formal Language Theory used in Incident Management

Define a formal Language

Process FSM Model

Properties Specification

Liveness and Safety

Process Debugging

Software for Finite State Machine

• Labelled Transition System Analyzer v3.0 http://www.doc.ic.ac.uk/~jnm/book/

• Temporal Logic of Actions http://research.microsoft.com/en-us/um/people/lamport/tla/tools.html

Finite State Process (FSP) Model

Labeled Transition Systems (LTS) Diagrams

Sequence Properties

• Safety: “nothing bad happens”

• Liveness: “something good eventually happens”

System Verification

Thanks For Coming