Operational Semantics

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Operational Semantics

• Semantics with Applications
• Chapter 2
• H. Nielson and F. Nielsonhttp//www.daimi.au.dk/
  bra8130/Wiley_book/wiley.html

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Outline

• Natural Semantics of IMP
• Properties of the Natural Semantics
• Structural Operational Semantics for IMP
• Equivalence Result
• Extensions to IMP
• Abort
• Non determinism
• Parallel constructs
• Blocks and procedures

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Assignment Rule
n
YX
?n/X(Y)
?(Y)
Y?X
X?1, Y ?2, Z?15/X X?5, Y ?2, Z?1
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Natural Semantics (IMP)
ltskip, ? gt ? ?
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Natural Semantics (IMP)
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Semantic Equivalence

• com1 and com2 are semantically equivalent if for
  all ? and ?ltcom1, ? gt ? ? if and only if
  ltcom2, ? gt ? ?
• Simple examplewhile b do comis semantically
  equivalent toif b then (com while b do com)
  else skip

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Properties of Natural Semantics

• Equivalence of program constructs
• skip com is semantically equivalent to com
• com skip is semantically equivalent to com
• ((com1 com2) com3) is semantically
  equivalent to (com1 ( com2 com3))
• (X 5 Y X 8) is semantically
  equivalent to(X 5 Y 40)
• Deterministic
• If ltcom, ?gt ? ?1 and ltcom, ? gt ? ?2 then ?1 ?2

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Deterministic Semantics for IMP

• If ltcom, ?gt ? ? 1 and ltcom, ? gt ? ? 2then ?1?2
• The proof uses induction on the shape of
  derivation trees
• Prove that the property holds for all simple
  derivation trees by showing it holds for axioms
• Prove that the property holds for all composite
  trees
• For each rule assume that the property holds for
  its premises (induction hypothesis) and prove it
  holds for the conclusion of the rule

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The Semantic Function Sns

• The meaning of a command com is defined as a
  partial function from State to State
• Sns Com ? (State ? State)
• Sns ?com?(?) ? if ltcom, ?gt ?? and otherwise
  Sns ?com? (?) is undefined
• Examples
• Sns ?skip?(?) ?
• Sns ?X1?(?) ? 1/X
• Sns ?while true do skip?(?) undefined

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Structural Operational Semantics

• Emphasizes the individual steps
• Usually more suitable for static analysis
• For every command S, write meaning rules ltcom, ?gt
  ? ?If the first step of executing the command
  com on ? leads to ?
• Two possibilities for ?
• ? ltcom, ?gt
• The execution of com is not completed, com is
  the remaining computation to be performed on ?
• ? ?
• The execution of com has terminated with a final
  state ?
• ? is a stuck configuration when there are no
  transitions
• The meaning of a program P on an input state s is
  the set of final states that can be executed in
  arbitrary finite steps

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SOS (IMP)
ltskip, ? gt ? ?
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SOS (IMP)
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SOS (IMP)
ltwhile b do com, ?gt ? ltif b then (comwhile b
do com) else skip, ?gt
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Derivation Sequences

• A finite derivation sequence starting at ltcom,
  ?gt?0, ?1, ?2 , ?k such that
• ?0ltcom, ?gt
• ?i ? ?i1
• ?k is either stuck configuration or a final state
• An infinite derivation sequence starting at
  ltcom, ?gt?0, ?1, ?2 such that
• ?0ltcom, ?gt
• ?i ? ?i1
• ?0 ?i ?i in i steps
• ?0 ? ?i in finite number of steps
• For each step there is a derivation tree

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Example

• Let ?0 such that ?0 (X) 5 and ?0 (Y) 7
• com (ZX X Y) Y Z

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Factorial Program

• Input state s such that ?(X) 3

Y 1 while ?(X1) do Y Y X X X - 1
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Program Termination

• Given a command com and input ?
• com terminates on ? if there exists a finite
  derivation sequence starting at ltcom, ? gt
• com terminates successfully on ? if there exists
  a finite derivation sequence starting at ltcom, ?gt
  leading to a final state
• com loops on ? if there exists an infinite
  derivation sequence starting at ltcom, ?gt

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Properties of the Semantics

• com1 and com2 are semantically equivalent if
• for all ? and ?ltcom1, ? gt ? ? if and only if
  ltcom2, ? gt ? ?
• there is an infinite derivation sequence starting
  at ltcom1, ? gt if and only if there is an
  infinite derivation sequence starting at ltcom2, ?
  gt
• Deterministic
• If ltcom, ? gt ? ?1 and ltcom, ?gt ? ? 2 then ?1 ?2

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Sequential Composition

• If ltcom1 com2, ? gt ?k ? then there exists a
  state ? and numbers k1 and k2 such that
• ltcom1, ? gt ?k1 ?
• ltcom2, ?gt ?k2 ?
• and k k1 k2
• The proof uses induction on the length of
  derivation sequences
• Prove that the property holds for all derivation
  sequences of length 0
• Prove that the property holds for all other
  derivation sequences
• Show that the property holds for sequences of
  length k1 using the fact it holds on all
  sequences of length k (induction hypothesis)

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The Semantic Function Ssos

• The meaning of a command com is defined as a
  partial function from State to State
• Ssos Com ? (State ? State)
• Ssos?com?? ? if ltcom, ?gt ?? and otherwise
  Ssos ?com?s is undefined

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An Equivalence Result

• For every command com of the IMP language
• Snat?com? Ssos?com?

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Extensions to IMP

• Abort command (like C exit)
• Non determinism
• Parallelism
• Local Variables
• Procedures
• Static Scope
• Dynamic scope

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IMP Abort

• Abstract syntaxcom X a skip com1
  com2 if b then com1 else com2
  while b do com abort
• Abort terminates the execution
• No new rules are needed in natural and
  structural operational semantics
• commands
• skip
• abort
• while true do skip

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Conclusion

• The natural semantics cannot distinguish between
  looping and abnormal termination (unless the
  states are modified)
• In the structural operational semantics looping
  is reflected by infinite derivations and abnormal
  termination is reflected by stuck configuration

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IMP Non-Determinism

• Abstract syntaxcom X a skip com1
  com2 if b then com1 else
  com2 while b do com
  com1 or com2
• Either com1 or com2 is executed
• Example
• X 1 or (X 2 X X2)

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IMPNon-DeterminismNatural Semantics
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IMP Non-DeterminismSOS
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IMP Non-DeterminismExamples

• X 1 or (X 2 X X2)
• (while true do skip) or (X 2 X X2)

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Conclusion

• In the natural semantics non-determinism will
  suppress looping if possible (mnemonic)
• In the structural operational semantics
  non-determinism does not suppress looping

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IMP Parallel Constructs

• Abstract syntaxcom X a skip com1
  com2 if b then com1 else com2
  while b do com com1 par com2
• All the interleavings of com1 or com2 are
  executed
• Example
• X 1 par (X 2 X X2)

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IMP Parallel ConstructsSOS
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IMP Parallel ConstructsNatural Semantics
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Conclusion

• In the natural semantics immediate constituent is
  an atomic entity so we cannot express
  interleaving of computations
• In the structural operational semantics we
  concentrate on small steps so interleaving of
  computations can be easily expressed

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IMP local variables

• Abstract syntaxcom X a skip com1
  com2 if b then com1 else com2
  while b do com begin Vars
  com endVars var X a Vars ?

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Example
begin var Y 1 X 1 begin
var X 2 Y X 1
end X Y X end
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Conclusions Local Variables

• The natural semantics can remember local states
• Need to introduce stack or heap into state of the
  structural semantics

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IMP local variables and procedures

• Abstract syntaxcom X a skip com1
  com2 if b then com1 else com2
  while b do com begin Vars
  Procs com end call pVars var X a Vars
  ?Procs proc p is com Procs ?

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Conclusions

• Structural operational semantics allows us to
  simulate low level computations without getting
  bugged into too many details
• Natural semantics allows to abstract more
• Local memory
• Non termination