Defining A Formal Semantics For The Rosetta Specification Language

1
Defining A Formal Semantics For The Rosetta
Specification Language
2
Overview

• Introduction
• Definitions
• Basic Semantics
• Denoting Units of Semantics
• State based model
• Event based model
• Ongoing work
• Conclusion

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Next

• Introduction
• Definitions
• Basic Semantics
• Denoting Units of Semantics
• State based model
• Event based model
• Ongoing Work
• Conclusion

4
Introduction

• Rosetta systems level design language
• Supports different computation models
• Addresses integration of information across
  domains
• Uses domains to define semantics of models
  there is no unique semantics for all domains
• All languages have syntax and semantics
• Rosetta has a fixed syntax, but different
  semantics
• Use of denotational functions for interpreting
  Rosetta syntax
• Decomposition of semantics into modules - each
  module has a different set of denotational
  functions from another

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Next

• Introduction
• Definitions
• Basic Semantics
• Denoting Units of Semantics
• State based model
• Event based model
• Ongoing Work
• Conclusion

6
Definitions

• Model of computation
• Distinctive paradigm for computation,
  communication,
• Semantic domain
• Set of mathematical objects used to model agents
• Unifying semantic domain
• Semantic domain that can be used to represent a
  variety of different computation paradigms
• Unit of semantics
• A unifying semantic domain
• Denotational semantics
• Use of valuation functions to map syntactic
  constructs to abstract values denoted

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Next

• Introduction
• Definitions
• Basic Semantics
• Denoting Units of Semantics
• State based model
• Event based model
• Ongoing Work
• Conclusion

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Basic Semantics Semantic modules
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Basic Semantics Denotational semantics

• Denotation in the sense of The Scott-Strachey
  Approach to Programming Language
• Define a semantic function that maps symbols from
  a language to a known object
• E.g. Let N be such a function for numerals
• N?1? 1
• N?one? 1
• where ? and ? are semantic brackets, and the
  element enclosed by them is an object from the
  language being defined

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Basic Semantics Evaluation functions

• Following Allisons approach (A Practical
  Introduction to denotational semantics), we
  identify 4 semantic functions
• Expression evaluation function given a store,
  maps an expression to a value
• E??? Store ? Values
• Constant evaluation function maps constants and
  literals to their values in the abstract value
  space
• V constants ? Values
• Function evaluation function maps binary
  operators to the corresponding function
• O??? Values ? Values ? Values
• Term evaluation function maps terms to a
  boolean function
• T Terms ? Store ? boolFnc
• where boolFncVars ? boolean is a function of any
  number of parameters to a boolean value

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Basic Semantics Specific evaluation functions

• E??? st (st_value ?)
• E?v? st V?v?
• E?? ? ?? st O??? ?E??? st, E??? st?
• O?? ??v1, v2?. If v1 v2 then True else False
  
• univ ? univ ? boolean
• where
• ? is an expression ? represents identifiers
• st is the store v represents constants
• ? is a binary operator (st_value x) indicates
  value of x in store st
• ??param?.bodytypeFnc is a lambda expression of
  type typeFnc

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Basic semantics Term evaluation examples

• Assuming store st contains
• x ? 2
• myFnc ? ?xint . x 1
• T?x2? st
• ? O?? (E?x? st) V?2?
• ? (??v1, v2?. If v1 v2 then True else False) 2
  2
• ? if (2 2) then True else False
• ? True

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Basic semantics Term evaluation examples

• T?(myFnc 1) 2? st
• ? O?? ((?xint . (x 1) V?1?) V?2?)
• ? O?? ((?xint . (x 1) 1) 2
• ? True
• T?(undefFnc x) 2? st
• ? O?? ((E?undefFnc? st) 2) 2
• ? if ((E?undefFnc? st) 2) 2 then True else
  False
• For a facet/model to be consistent, terms always
  need to be true
• ? ((E?undefFnc? st) 2) 2
• a boolean equation with one unknown
  (E?undefFnc? st) is the result of the evaluation

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Next

• Introduction
• Definitions
• Basic Semantics
• Denoting Units of Semantics
• State based model
• Event based model
• Ongoing Work
• Conclusion

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Denoting Units of Semantics State based

• Denotation of model is a relation over states
• Vocabulary of state based models
• States set of states, finite or infinite,
  denumerable or nondenumerable
• ? - the current state
• next the state transformation function
• _at_ - the apply function used for dereferencing
  labels
• Semantics of vocabulary
• States Var ? Values set of functions from
  variables to values
• ? States Var ? Values current state,
  particular function
• Next States ? States the transformation
  function
• _at_ Label ? States ? Values the apply function
• T Terms ? BoolFnc
• E ?? ? Values where ? is an expression

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Denoting Units of Semantics State based

• In Rosetta, no sequential execution
• x x 1 is expressed as x_at_next(s)x_at_s 1
• Change of state made explicit
• Elements of vocabulary apparent
• Definition of additional valuation functions
• O?_at_? ??v1, fnc? . fnc v1
• Label ? (Label ? univ) ? univ
• E?y_at_?? st E??? st y (st_value ?) y
• E?y_at_next(?)? st (st_value next) (st_value ?) y

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Next

• Introduction
• Definitions
• Basic Semantics
• Denoting Units of Semantics
• State based model
• Event based model
• Ongoing Work
• Conclusion

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Denoting Units of Semantics Event based

• Notion started with Lee and Sangiovanni-Vincentell
  is Tagged Signal model
• Denotation of a process is partial function or
  relation on signals
• Vocabulary
• Tags set of tags
• Values set of values
• Events set of events, i.e. set of T x V
• States set of states, each state uniquely
  mapping to an event
• Signals set of signals, each signal being a set
  of events
• _at_ - the apply function, dereferencing signals in
  states indexed by events

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Denoting Units of Semantics Event based

• Semantics associated with vocabulary
• Events Tags x Values set of pairs of tags
  and values
• previousEvt Events used as index to last state
  of process
• evt Events refers to current state and is
  event being processed
• Signals ?Events set of sets of events
• sig Signals Events particular set of
  events
• getEvent Signals ? Events next event to be
  processed
• States SignalLabel ? SignalValue states are
  dependent on events and are used to keep track of
  signals
• EventBasedState Events ? States each state
  uniquely referred to by an event
• _at_ SignalLabel ? Events ? ((Events ? States) ?
  SignalValue) the dereferencing of a signal
  label is obtained by first finding state
  corresponding to event and then applying state
  function to signal label to find signal value

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Denoting Units of Semantics Event based

• Modified Valuation function
• O?_at_? ??v1, evt? . (??fnc? . (fnc evt) v1)
• SignalLabels ?Events ? (Events ? States) ?
  SignalValue
• An example
• E?sig_at_evnt? st
• O?_at_? sig (E?evnt? st) (st_value EventBasedState)
  
• O?_at_? sig (st_value evnt) (st_value
  EventBasedState)
• (??fnc? . (fnc (st_value evnt)) sig) (st_value
  EventBasedState)
• ((st_value EventBasedState) (st_value evt)) sig

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Next

• Introduction
• Definitions
• Basic Semantics
• Denoting Units of Semantics
• State based model
• Event based model
• Ongoing Work
• Conclusion

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Ongoing work

• Where are we going with this ?
• Result of the denotation is a language (lambda)
  which allows us to do interpretation, evaluation
  and verification of a facet
• However, need to ensure correct and safe
  semantics
• Use of co-algebra as semantic basis for a facet
  model
• Show that

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Next

• Introduction
• Definitions
• Basic Semantics
• Denoting Units of Semantics
• State based model
• Event based model
• Ongoing Work
• Conclusion

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Conclusion

• Rosetta provides a framework where different
  semantics may be defined
• Use of domain specific denotational semantics
• Also denote a facet by a monad to allow for facet
  composition
• Moggi uses monads to modularize semantics
• Hudak and Liang use monad transformers
• Wadler uses monads to add interaction into a
  purely declarative language