Models of Concurrency Mana, Pnueli

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Models of Concurrency(Mana, Pnueli)

• Marjan Sirjani
• University of Tehran
• Formal Methods Laboratory

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• Models of Concurrency
• Manna, chapter 1,2

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Chapter 1

• 1.1 The Generic Model
• 1.2 Model 1 Transition Diagrams
• 1.3 Model 2 Shared-Variables Text
• 1.4 Semantics of Shared-Variables Text
• 1.5 Structural Relations Between Statements
• 1.6 Behavioral Equivalence
• 1.7 Grouped Statements
• 1.8 Semaphore Statements
• 1.9 Region Statements
• 1.10 Model 3 Message-Passing Text
• 1.11 Model 4 Petri-Nets

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SystemsReactive or Transformational
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Transformational

• A transformational program produce a result at
  the end
• It can be considered as a function from an
  initial state to a final state
• Can be formulated by specifying the relation
  between the initial and final states
• predicate logic

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Reactive

• Goal maintain some ongoing interaction with the
  environment
• An OS
• Systems for controlling mechanical processes

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Reactive (cont.)

• Some reactive programs are not expected to
  terminate
• They cannot specified by a relation between
  initial and final states
• Must specified by their unending behavior
• Temporal logic instead of predicate logic

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Reactivity and Concurrency

• Reactivity and concurrency are closely related
• The program and its environment act
• Concurrently in reactive programs
• Sequentially in transformational programs
• Parallel processes should be analyzed as a
  reactive system
• Even if the whole program has a transformational
  role

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Communication and Coordination

• Communication and coordination play an important
  role in achieving concurrency
• Many models have been proposed

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Communication and Coordination (cont.)

• Communication
• Shared variables
• Message passing
• Remote procedure calls
• Coordination
• Semaphores
• Critical regions
• Monitors
• Handshaking
• Rendezvous
• Asynchronous transmission

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The Generic Model
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Why a generic model?

• A generic model for modeling reactive systems
• Uniform treatment of all models
• The theory of specification and verification of
  reactive systems will be formulated in the
  generic model

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The Generic Model

• Using an underlying first-order language with
  these elements
• V Vocabulary
• E Expressions
• A Assertions
• I - Interpretations

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Vocabulary

• A countable set of typed variables
• Having two forms
• Data variables
• Range over data domains used in programs, such as
  Booleans, integers, or lists.
• Control variables
• Indicate progress in the execution of a program,
  range over locations in the program.

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Expressions

• Expressions are constructed from
• variables of V
• constants
• such as 0, ?(empty list), ? (empty set)
• functions
• such as ,, U
• predicates
• such as gt, null, and ?
• over the appropriate domains (such as integers,
  lists, and sets)
• For example x3y hd(u) tl(v) A U B

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Assertions

• Assertions are constructed from
• Boolean expressions using boolean connectives and
  quantification(?,?) over some variables that
  appear in the expressions
• For example ?x (xgt0) ??y (x y.y)

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Interpretation

• An interpretation I ?I of a set of typed
  variables V ? V is a mapping that assigns to each
  variable y ? V a value Iy in the domain of y
• If I?T, we say I satisfies ? I ?
• (? is a boolean expression or more generally
  an assertion)

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Basic Transition System

• A basic transition system (?,?,?,?)
• Represents a reactive program.
• ?u1,,u2 ? V a finite set of flexible state
  variables
• ? a set of states.
• ? a finite set of transitions.
• ? an initial condition.

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State variables

• ?u1,,u2 ? V a finite set of flexible state
  variables
• Variables can be
• Data variables
• Explicitly declared and manipulated
• Control variables
• Represent progress in the execution of the
  program (label of a statement)

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Set of states

• ? - a set of states.
• Each state s in ? is an interpretation of ?,
  assigning to each variable u in ? a value over
  its domain, denoted by su
• A state s that satisfies an assertion ?, i.e., s
  ? , is sometimes referred to as ?state

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Transitions set

• ? - a finite set of transitions.
• Each transition ? in T represents a
  state-transforming action of the system
• It is defined as a function ? ? ? 2 ? that
  maps a state s in ? into the (possibly empty) set
  of states ?(s) that can be obtained by applying
  action ? to state s

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Initial condition

• ? - an initial condition.
• This assertion characterizes the states at which
  execution of the program can begin
• A state s that satisfies ?, i.e., s ? , is
  called an initial state

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The Transition Relation ??

• Each transition ? is characterized by an
  assertion, called the transition relation
  ??(?,?)
• It relates the values of the state variables s to
  their values in a successor state s obtained by
  applying ? to s

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Transition relation

• ??(?,?) C? (?) ? (y1e1) ? ?(ykek)
• The transition relation consists of the following
  elements
• Enabling condition C? (?)
• Conjunction of modification statements (y1e1)
  ? ?(ykek)

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• Examples in pages 8,9

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Enabled and disabled transitions

• For a transition ? in ? and a state s in ? we
  say
• ? is enabled on s if ?(s)??
• ? is disabled on s if ?(s)?

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Idling and diligent transitions

• In the idling transition ?I models the behavior
  in which there is no change
• ??I T
• A state s is called terminal if the only
  transition that is enable on s is the idling
  transition ?I
• The transitions other than idling are called
  diligent

?I(s) s
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Computation

• Computation infinite sequence of states
• ? s0, s1, s2,
• A computation satisfies the following conditions
• Initiation the first state s0 is initial
• Consecution For each pair of consecutive states
  in ?, si1 ? ? for some ? in ?
• Diligence Either the sequence contains infinite
  diligent steps or it contains a terminal state

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• Computation prefix every finite prefix of a
  computation
• Reachable states it appears in some computation
  of the system

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Concrete models

• Model 1 Transition Diagram
• Model 2 Shared-Variables text
• Model 3 Message-Passing text
• Model 4 Petri Nets

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Transition Diagrams
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Model 1 Transition diagrams

• In this model, a program P has the following form
• PdeclarationP1 P2 Pm m?1
• Pi are processes
• Data variables Yy1, , yn n?1
• Declared at the head of the program
• Shared for all the processes

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Declarations

• Declarations appear at the head of the program
• mode var, ,var type where ?i
• mode in, local, out
• type
• basic ( int, char)
• structured ( array, list, set)
• assertion ?i imposes constraint on the initial
  values of some of the variables in this statement

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Example

• in k, n integer where 0?k?n
• local y1,y2 integer where y1n ? y21
• out b integer where b 1
• Data precondition of the program
• ? 0?k?n ? y1n ? y21 ? b1

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Processes

• Each process Pi is represented by a transition
  diagram (directed graph)
• Nodes locations
• For Pi Li li0, li1 , , liti
• One entry and zero or more exit locations
• Edges (atomic) instructions
• Guarded assignment
• c ? (y1, )(e1, )

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Program state

• State of a program
• Control variables Data variables
• Control variable ?i Pointing to the current
  location in process Pi
• Each ?i ranges over Li, the set of locations
  belonging to Pi

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Diagrams as Basic Transition Systems

• A basic transition system is a quadruple
• State variables
• States
• Transition
• Initial condition

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Diagram elements

• State variables
• All the data and control variables
• ? ?1, , ?m, y1, , yn
• States
• All the possible interpretations that assign to
  the state variables values over their respective
  domains.
• Domain of control variable ?i is the set of
  locations Li

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Diagram elements

• Transition
• Idling transition ?i is defined by transition
  relation ?i T
• Diligent transitions labeled edges that appear
  within the processes.

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Example
C ? yi ei
l
l
?

• is the edge.
• ?? (?i l) ? c ? (?il) ? (yi ei)

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Example (cont.)

• Initial condition
• Program P
• declaration where ?P1 Pm
• Initial condition
• ? ? ? /\i1m (?i loi)
• A process is enabled, or disabled on a state.

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Example Binomial coefficient
r4 (y2gtk)?
r0 (y2k)?
r1 ((y1y2)n)?
r2 b b div y2
r3 y2 y21
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• in k, n integer where 0?k?n
• local y1,y2 integer where y1n ? y21
• out b integer where b 1

r4 (y2gtk)?
r0 (y2k)?
r1 ((y1y2)n)?
r2 b b div y2
r3 y2 y21
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A process-deterministic program

• If every two guards c1 and c2 that label two
  edges departing from the same location are
  exclusive c1 and c2 is never true.
• In a process-deterministic program, each process
  has at most one transition that is enabled on any
  state.
• The computation is still not uniquely determined
  (several enabled transitions on a given state
  from different processes ).

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Representing Concurrency by Interleaving
Reduction of concurrency to nondeterminism
X0,Y0
X0,Y0
Y1
X1
X1
Y1
X1
Y1
Process P1
Process P2
Program B
Program A
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Scheduling

• The choice of the enabled transition to be
  executed next.
• A sequence of choices that leads to a complete
  computation is called a schedule.

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• The only restriction a schedule must obey is that
  as long as some process is enabled, some process
  must eventually be activated (implied by
  diligence requirement).
• Diligence either the sequence (computation)
  contains infinitely many diligent steps or it
  contains terminal state
• Excluding sequences in which even though some
  diligent transition is enabled, only idling steps
  are taken from some point.

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• Questions?