Operational Semantics

1
Operational Semantics

• Mooly Sagiv
• http//www.math.tau.ac.il/sagiv/courses/pa.html
• Tel Aviv University
• 640-6706
• Textbook Semantics with Applications
• Chapter 2
• H. Nielson and F. Nielsonhttp//www.daimi.au.dk/
  bra8130/Wiley_book/wiley.html

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Outline

• Why formal semantics?
• Possible formal semantics
• A Simple programming language While
• Natural Operational Semantics for While
• Structural Operational Semantics for While
• Equivalence Result
• Extensions to While
• Abort
• Non determinism
• Parallel constructs
• Blocks and procedures

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Syntax vs. Semantics

• The pattern of formation of sentences or phrases
  in a language
• Examples
• Regular expressions
• Context free grammars

• The study or science of meaning in language
• Examples
• Interpreter
• Compiler
• Better mechanisms will be given today

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Benefits of Formal Semantics

• Programming language design
• hard- to-define hard-to-implementhard-to-use
• resolve ambiguities
• Programming language implementation
• Programming language understanding
• Program correctness
• Program equivalence
• Compiler Correctness
• Correctness of Static Analysis
• Design of Static Analysis
• Automatic generation of interpreter
• But probably not
• Automatic compiler generation

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Alternative Formal Semantics

• Operational Semantics
• The meaning of the program is described
  operationally
• Natural Operational Semantics
• Structural Operational Semantics
• Denotational Semantics
• The meaning of the program is an input/output
  relation
• Mathematically challenging but complicated
• Axiomatic Semantics
• Logical axioms
• The meaning of the program are observed properties

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int fact(int x) int z, y z 1 y x
while (ygt0) z z y y
y 1 return z
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int fact(int x) int z, y z 1 y x
while (ygt0) z z y y
y 1 return z
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int fact(int x) int z, y z 1 y x
while (ygt0) z z y y
y 1 return z
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int fact(int x) int z, y z 1 y x
while (ygt0) z z y y
y 1 return z
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Denotational Semantics
int fact(int x) int z, y z 1 y x
while (ygt0) z z y y
y 1 return z
?x. if x 0 then 1 else x f(x -1)
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Axiomatic Semantics
xn int fact(int x) int z, y z
1 xn ? z1 y x xn ? z1 ? yn while
(ygt0) xn ? y ?0 ? zn! / y! xn
? y gt0 ? zn! / y! z z y
xn ? ygt0 ? zn!/(y-1)! y y 1
xn ? y ?0 ? zn!/(y-1)! return
z xn ? zn!
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Static Analysis

• Automatic derivation of static properties which
  hold on every execution leading to a
  programlocation

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Example Static Analysis Problem

• Find variables with constant value at a given
  program location
• Example program

int p(int x) return x x void main() int
z if (getc()) z p(6) 8 else z p(5)
7 printf (z)
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Abstract (Conservative) interpretation
abstract representation
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Example rule of signs

• Safely identify the sign of variables at every
  program location
• Abstract representation P, N, ?
• Abstract (conservative) semantics of

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Abstract (conservative) interpretation
ltN, Ngt
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Example rule of signs (cont)

• Safely identify the sign of variables at every
  program location
• Abstract representation P, N, ?
• ?(C) if all elements in C are positive
  then return P
  else if all elements in C are negative
  then return N
  else return ?
• ?(a) if (aP) then
  return0, 1, 2,
  else if (aN) return -1, -2, -3, ,
  else return Z

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Benefits of Operational Semanticsfor Static
Analysis

• Correctness (soundness) of the analysis
• The compiler will never change the meaning of the
  program
• Establish the right mindset
• Design the analysis
• Becomes familiar with mathematical notations used
  in programming languages

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The While Programming Language

• Abstract syntaxS x a skip S1 S2
  if b then S1 else S2 while b do S
• Use parenthesizes for precedence
• Informal Semantics
• skip behaves like no-operation
• Import meaning of arithmetic and Boolean
  operations

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Example While Program
y 1 while ?(x1) do ( y y x x x -
1 )
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General Notations

• Syntactic categories
• Var the set of program variables
• Aexp the set of arithmetic expressions
• Bexp the set of Boolean expressions
• Stm set of program statements
• Semantic categories
• Natural values N0, 1, 2,
• Truth values Tff, tt
• States State Var ? N
• Lookup in a state s s x
• Update of a state s s x ? 5

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Example State Manipulations

• x?1, y?7, z?16 y
• x?1, y?7, z?16 t
• x?1, y?7, z?16x?5
• x?1, y?7, z?16x?5 x
• x?1, y?7, z?16x?5 y

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Semantics of arithmetic expressions

• Assume that arithmetic expressions are
  side-effect free
• A? Aexp ? State ? N
• Defined by induction on the syntax tree
• A? n ? s n
• A? x ? s s x
• A? e1 e2 ? s A? e1 ? s A ? e2 ? s
• A? e1 e2 ? s A? e1 ? s A ? e2 ? s
• A? ( e1 ) ? s A? e1 ? s --- not needed
• A? - e1 ? s -A ? e1 ? s

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Semantics of Boolean expressions

• Assume that Boolean expressions are side-effect
  free
• B? Bexp ? State ? T
• Defined by induction on the syntax tree
• B? true ? s tt
• B? false ? s ff
• B? x ? s s x
• B? e1 e2 ? s
• B? e1 ? e2 ? s
• B? e1 ?e2 ? s

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Natural Operational Semantics

• Describe the overall effect of program
  constructs
• Ignore non terminating computations

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Natural Semantics

• Notations
• ltS, sgt - the program statement S is executed on
  input state s
• s representing a terminal (final) state
• For every statement S, write meaning rulesltS, igt
  ? oIf the statement S is executed on an input
  state i, it terminates and yields an output state
  o
• The meaning of a program P on an input state s is
  the set of outputs states o such that ltP, igt ? o
• The meaning of compound statements is defined
  using the meaning of immediate constituent
  statements

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Natural Semantics for While
assns ltx a, sgt ? sx ?A?a?s skipns ltskip,
sgt ? s
axioms
rules
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Natural Semantics for While(More rules)
whilettns ltS , sgt ? s, ltwhile b do S, sgt ?
s ltwhile b do S, sgt ? s
if B?b?stt
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Simple Examples

• Let s0 be the state which assigns zero to all
  program variables
• Assignments assns ltx x1, s0gt ? s0x ?1
• Skip statementskipns ltskip, s0gt ? s0
• Composition

compns ltskip ,s0gt ? s0, ltx x1, s0gt ? s0x
?1 ltskip x x 1, s0gt ?s0x ?1
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Simple Examples (Cont)

• Let s0 be the state which assigns zero to all
  program variables
• if-construct

ifttns ltskip ,s0gt ? s0 ltif x0
then skip else x x 1, s0gt ?s0
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A Derivation Tree

• A proof that ltS, sgt ?s
• The root of tree is ltS, sgt ?s
• Leaves are instances of axioms
• Internal nodes rules
• Immediate children match rule premises
• Simple Example

ltskip x x 1, s0gt ?s0x ?1gt
ltskip, s0gt ?s0
lt x x 1, s0gt ?s0x ?1gt
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An Example Derivation Tree
lt(x x1 y x1) z y), s0gt ?s0x ?1y
?2z ?2
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Top Down Evaluation of Derivation Trees

• Given a program S and an input state s
• Find an output state s such that ltS, sgt ?s
• Start with the root and repeatedly apply rules
  until the axioms are reached
• Inspect different alternatives in order
• In While s and the derivation tree is unique

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Example of Top Down Tree Construction

• Input state s such that s x 3
• Factorial program

y 1 while ?(x1) do (y y x x x - 1)
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Program Termination

• Given a statement S and input s
• S terminates on s if there exists a state s such
  thatltS, sgt ? s
• S loops on s if there is no state s such that
  ltS, sgt ? s
• Given a statement S
• S always terminates if for every input state s, S
  terminates on s
• S always loops if for every input state s, S
  loops on s

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Properties of Natural Semantics

• Equivalence of program constructs
• skip skip is semantically equivalent to
  skip
• ((S1 S2) S3) is semantically equivalent to
  (S1 ( S2 S3))
• (x 5 y x 8) is semantically
  equivalent to(x 5 y 40)
• Deterministic
• If ltS, sgt ? s1 and ltS, sgt ? s2 then s1s2

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Semantic Equivalence

• S1 and S2 are semantically equivalent if for all
  s and sltS1, sgt ? s if and only if ltS2, sgt ? s
  
• Simple examplewhile b do Sis semantically
  equivalent toif b then (S while b do S) else
  skip

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Deterministic Semantics for While

• If ltS, sgt ? s1 and ltS, sgt ? s2 then s1s2
• 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 statement S is defined as a
  partial function from State to State
• Sns Stm ? (State ? State)
• Sns ?S?s s if ltS, sgt ?s and otherwise Sns
  ?S?s is undefined
• Examples
• Sns ?skip?s s
• Sns ?x 1?s s x ?1
• Sns ?while true do skip?s undefined

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Structural Operational Semantics

• Emphasizes the individual steps
• Usually more suitable for analysis
• For every statement S, write meaning rules ltS, igt
  ? ?If the first step of executing the statement
  S on an input state i leads to ?
• Two possibilities for ?
• ? ltS, sgt The execution of S is not completed,
  S is the remaining computation which need to be
  performed on s
• ? o The execution of S has terminated with a
  final state o
• ? 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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Structural Semantics for While
asssos ltx a, sgt ? sx ?A?a?s skipsos
ltskip, sgt ? s
axioms
rules
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Structural Semantics for Whileif construct
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Structural Semantics for Whilewhile construct
whilesos ltwhile b do S, sgt ?
ltif b then (S while b do S) else skip, sgt

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Derivation Sequences

• A finite derivation sequence starting at ltS,
  sgt?0, ?1, ?2 , ?k such that
• ?0ltS, sgt
• ?i ? ?i1
• ?k is either stuck configuration or a final state
• An infinite derivation sequence starting at ltS,
  sgt?0, ?1, ?2 such that
• ?0ltS, sgt
• ?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 s0 such that s0 x 5 and s0 y 7
• S (zx x y) y z

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Factorial Program

• Input state s such that s x 3

y 1 while ?(x1) do (y y x x x - 1)
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Program Termination

• Given a statement S and input s
• S terminates on s if there exists a finite
  derivation sequence starting at ltS, sgt
• S terminates successfully on s if there exists a
  finite derivation sequence starting at ltS, sgt
  leading to a final state
• S loops on s if there exists an infinite
  derivation sequence starting at ltS, sgt

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Properties of the Semantics

• S1 and S2 are semantically equivalent if
• for all s and sltS1, sgt ? s if and only if
  ltS2, sgt ?s
• there is an infinite derivation sequence starting
  at ltS1, sgt if and only if there is an infinite
  derivation sequence starting at ltS2, sgt
• Deterministic
• If ltS, sgt ? s1 and ltS, sgt ? s2 then s1s2
• The execution of S1 S2 on an input can be split
  into two parts
• execute S1 on s yielding a state s
• execute S2 on s

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Sequential Composition

• If ltS1 S2, sgt ?k s then there exists a state
  s and numbers k1 and k2 such that
• ltS1, sgt ?k1 s
• ltS2, sgt ?k2 s
• 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 statement S is defined as a
  partial function from State to State
• Ssos Stm ? (State ? State)
• Ssos?S?s s if ltS, sgt ?s and otherwise Ssos
  ?S?s is undefined

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An Equivalence Result

• For every statement S of the While language
• Snat?S? Ssos?S?

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Extensions to While

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

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The While Programming Language with Abort

• Abstract syntaxS x a skip S1 S2
  if b then S1 else S2 while b do S
  abort
• Abort terminates the execution
• No new rules are needed in natural and
  structural operational semantics
• Statements
• 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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The Programming Language with Non-Determinism
While

• Abstract syntaxS x a skip S1 S2
  if b then S1 else S2 while b do S S1
  or S2
• Either S1 or S2 is executed
• Example
• x 1 or (x 2 x x2)

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The While Programming Language with
Non-DeterminismNatural Semantics
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The While Programming Language with
Non-DeterminismStructural Semantics
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The While Programming Language with
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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The While Programming Language with Parallel
Constructs

• Abstract syntaxS x a skip S1 S2
  if b then S1 else S2 while b do S S1
  par S2
• All the interleaving of S1 or S2 are executed
• Example
• x 1 par (x 2 x x2)

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The While Programming Language with Parallel
ConstructsStructural Semantics
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The While Programming Language with 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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The While Programming Language with local
variables and procedures

• Abstract syntaxS x a skip S1 S2
  if b then S1 else S2 while b do S
  begin Dv Dp S end call pDv var x a
  Dv ?Dp proc p is S Dp ?

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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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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
• Thinking in concrete semantics is essential for a
  compiler writer