Moein Farshchian Ph.D Candidate of Cell and Molecular Biology.
Neveen Shlayan Ph.D. Candidate
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Transcript of Neveen Shlayan Ph.D. Candidate
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Formal Languages and Automata Theory Applied to Transportation Engineering Problem of
Incident Management
Neveen ShlayanPh.D. Candidate
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Outline
• Introduction• Set Theory Review• Formal Languages• Grammar • Automata Theory• Incident Management Problem• Specification and Verification• Conclusions
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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
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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
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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
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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
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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
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Even More Operations
• Iterations:• Mirror Image:
;;; 20 ababPPabP
;; 1 baPabP
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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)
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Language Examples
• Examples
*}|{
,...}1,0|{
},,{},{
13
2
1
VPPPL
ibaL
baLbaV
ii
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Grammar
• Definition
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Grammar Example
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Automata
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Incident Management Process
Incident Occurs
Emergency Responders (ER) Contacted
ER Arrive to the Scene
Incident Cleared
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Challenges In Current Incident Management Process
CommunicationCoordination
Increase in Clearance Time
Economical, Safety, Environmental, and Social Impacts
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Formal Language Theory used in Incident Management
Define a formal Language
Process FSM Model
Properties Specification
Liveness and Safety
Process Debugging
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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
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Finite State Process (FSP) Model
Labeled Transition Systems (LTS) Diagrams
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Sequence Properties
• Safety: “nothing bad happens”
• Liveness: “something good eventually happens”
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System Verification
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Thanks For Coming