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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
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)
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