A denotational framework for comparing models of computation

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A denotational framework for comparing models of
computation
Daniele Gasperini
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Abstract

• Aim compare different model of computation and
  their certain essential features
• How use a denotational framework

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Main characteristics

• Denotational rather than operational
• Mathematical approach
• Main actors
• Processes
• Signals
• Events

Tagged signal model
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SIGNALS 1

• Event e tag value
• Set of values V
• Set of tags T
• Tags model time, precedence relationships,
  synchronization points
• Values represent operands and result of
  computation
• Signal s set of events subset of T x V

e ? T x V
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SIGNALS 2 Characteristics notations

• Functional signals
• e1 (t, v1) ? s and e2 (t, v2) ? s then v1 v2
• S the set of all signals
• SN collection/tuple of N signals
• ? empty signal (no events)
• ? collection/tuple of N empty signals
• ? ? S and ? ? SN
• bottom absence of value

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PROCESSES

• P process subset of SN for some N
• s behavior of a process
• s ? SN and s ? P
• ? P set of behaviors

• Composing processes
• Inputs and outputs
• Determinacy

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Composing processes 1

• P set of behaviors s
• Composing P1 x P2 x Pn

Given M processes in SN a process Q composed of
these processes is given by Where P is the
collection of processes
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Composing processes 2 projections
Process P set of behaviors s
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Inputs and outputs
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Determinacy
A process is determinate if for any input A ? B
it has exactly one behavior or exactly no
behaviors. Otherwise is nondeterminate.
functional functional functional ? preserve
Depends on the tag system and the process
Functional source processes with exacly one
behavior functional ? preserve
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Tags systems

• The central role of a tag system is to establish
  ordering among events.

Ordering relation among events properties
reflective
transitive
antisymmetric
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Timed models of computation

• The set of tags T is a totally ordered set
• Tag timestamp

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Flavors of timed models

• Metric time
• Continuous time
• Discrete event
• Discrete event simulators
• Synchronous and discrete-time systems
• Sequential systems

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Metric time

• T totally ordered
• T Abelian group
• closed to addition
• zero
• for every t ? T there is another element -t ? T
  such that t (-t) 0
• T has a metric

Directly modeling physical systems
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Continuous time

• T totally ordered
• T is a continuum
• T(s) T for each signal s in any tuple s that
  satisfies the system

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Discrete event

• T totally ordered
• The set of tags must be a discrete subset of T
• The set of tags must be order isomorphic

The timestamps that appear in any behavior can be
enumerated in chronological order.
Many simulators (most digital circuit simulators)
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Discrete events simulators

• Same characteristics of discrete event
• Events explicitly include timestamp
• The simulator operates by keeping a list of
  events sorted by timestamp
• Delta time (affects semantic and ensure strict
  causality)

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Synchronous and discrete-time systems

• Two events are synchronous if they have the same
  tag

Two signals are synchronous if all events in one
signal are synchronous with an event in the other
signal and vice versa
A process is synchronous if every signal in any
behavior of the process is synchronous with every
other signal in the behavior
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Sequential systems

• Single signal s
• The set of tags in the signal are totally ordered

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Untimed models of computation

• Tags are partially ordered
• Partial order is present in many models of
  computations because denotes
• Causality
• Synchronization

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Rendezvous of sequential processes

• Sequential processes reach a particular point at
  which they must verify that another process has
  reached a corresponding point before proceeding

T(e1) T(e2) T(e3)
Communicating sequential processes and the
calculus of communicating systems
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Kahn process networks

• Processes communicate via channels, which are
  one-way unbounded FIFO queues with a single
  reader and a single writer.

FIFO
T(s) is totally ordered for each signal s
Imposes an ordering on the events
Be S(s) denote the sequence of values in s
Two signals s and s sequence equivalent S(s)
S(s)
A process is sequence functional if given a set
of equivalent tuples of input signals all
possible outputs are sequence equivalent
Sequence determinacy
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Dataflow

• Special case of Kahn process networks
• A dataflow process is a Kahn process that is also
  sequential
• Self loop signal is called firing signal

consumed by ei 1
ei e ei 1
e
produced by ei
ei e ei 1
e
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Petri Nets

• Similar to dataflow
• Events within signals need not be ordered

f s2 ? s1 such that f(e) lt e for all e ? s2
A firing (an event in s2) has a corresponding
token (an event in s1) with a smaller tag
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Heterogeneous systems

• V divided into subsets ? carry the notion of data
  types
• T divided ? separately model parts of
  heterogeneous system (continuous time, discrete
  event and dataflow subsystems)

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The Role of Tags in composition of processes

• Aim connect tagged signal model to well-known
  results in semantics
• Two cases
• Discrete event systems
• Kahn process networks

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Discrete event systems
Cantor metric
Causality

• Synchronous events
• Feedback loops

Distance of signals
The n-tuples of signals ? metric space
Casual two possible outputs differ no earlier
than the inputs that produced them
Strictly casual two possible outputs differ
later than the inputs that produced them (or not
at all)
Three form of causality
Delta casual there is a delay of at least ?
before any output of a process can be produced in
reaction to an input event
If a process is functional and delta casual then
the feedback loop has exactly one behavior
Simulators (VHDL, Verilog)
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Monotonicity and continuity in Kahn process
networks
Causality
a natural partial ordering for signals that
contains a totally ordered set of events
Prefix order
S(s) sequence of values in the signal s
Monotonicity
Continuity
Determinacy of feedback compositions
Constructive procedure for finding one unique
behavior
A continuous process is monotonic, not all
monotonic functions are continuous
The feedback loops have an alternative
interpretation of the composition Q that brings
the sequence to be determinate (fixed point
theorem)