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Transcript of Operational Semantics Mooly Sagiv Reference: Semantics with Applications Chapter 2 H. Nielson and F....
![Page 1: Operational Semantics Mooly Sagiv Reference: Semantics with Applications Chapter 2 H. Nielson and F. Nielson bra8130/Wiley_book/wiley.html.](https://reader030.fdocuments.us/reader030/viewer/2022032803/56649e245503460f94b11eb3/html5/thumbnails/1.jpg)
Operational SemanticsMooly Sagiv
Reference: Semantics with Applications
Chapter 2H. Nielson and F. Nielson
http://www.daimi.au.dk/~bra8130/Wiley_book/wiley.html
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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-implement=hard-to-use
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– The meaning of the program are observed properties
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int fact(int x) {
int z, y;
z = 1;
y= x
while (y>0) {
z = z * y ;
y = y – 1;
}
return z
}
[x3]
[x3, z, y]
[x3, z1, y]
[x3, z1, y3]
[x3, z1, y3]
[x3, z3, y3]
[x3, z3, y2]
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int fact(int x) {
int z, y;
z = 1;
y= x
while (y>0) {
z = z * y ;
y = y – 1;
}
return z
}
[x3, z3, y2]
[x3, z6, y2]
[x3, z6, y1]
[x3, z3, y2]
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int fact(int x) {
int z, y;
z = 1;
y= x
while (y>0) {
z = z * y ;
y = y – 1;
}
return z
}
[x3, z6, y1]
[x3, z6, y1]
[x3, z6, y0]
[x3, z6, y1]
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int fact(int x) {
int z, y;
z = 1;
y= x
while (y>0) {
z = z * y ;
y = y – 1;
}
return z
}
[x3, z6, y0]
[x3, z6, y0]
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int fact(int x) {
int z, y;
z = 1;
y= x;
while (y>0) {
z = z * y ;
y = y – 1;
}
return 6
}
[x3, z6, y0]
[x3, z6, y0]
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f=x. if x = 0 then 1 else x * f(x -1)
Denotational Semanticsint fact(int x) {
int z, y;
z = 1;
y= x ;
while (y>0) {
z = z * y ;
y = y – 1;
}
return z;
}
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{x=n}
int fact(int x) { int z, y;
z = 1;
{x=n z=1}
y= x
{x=n z=1 y=n}
while
{x=n y 0 z=n! / y!} (y>0) {
{x=n y >0 z=n! / y!}
z = z * y ;
{x=n y>0 z=n!/(y-1)!}
y = y – 1;
{x=n y 0 z=n!/y!}
} return z} {x=n z=n!}
Axiomatic Semantics
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Static Analysis
Automatic derivation of static properties which hold on every execution leading to a programlocation
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Abstract (Conservative) interpretation
abstract representation
Set of states
concretization
Abstractsemantics
statement s abstract representation
abstraction
Operational semantics
statement sSet of states
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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 *
*# P N
?
P P N ?
N N P ?
? ? ? ?
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abstraction
<P, N>
Abstract (conservative) interpretation
<N, N>
{…,<-88, -2>,…}
concretization
Abstractsemantics
x := x*#y
Operational semantics
x := x*y {…, <176, -2>…}
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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 (a==P) then return{0, 1, 2, … } else if (a==N)
return {-1, -2, -3, …, } else return Z
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Benefits of Operational Semantics
for Static Analysis
Correctness (soundness) of the analysis– The compiler will never change the meaning of the
program
– All impacts are identified
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 (x=1) 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 N={0, 1, 2, …}
– Truth values T={ff, 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
[x1, y7, z16] y = [x1, y7, z16] t = [x1, y7, z16][x5] = [x1, y7, z16][x5] x = [x1, y7, z16][x5] 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
Compositional Properties can be proved by structural induction
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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 e1 = e2 s =
– B e1 e2 s =
– B e1 e2 s =
tt if A e1 s = A e2 s
ff if A e1 s A e2 s
tt if B e1 s = tt and B e2 =tt
ff if B e1 s=ff or B e2 s=ff
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Natural Operational Semantics
Describe the “overall” effect of program constructs
Ignores non terminating computations
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Natural Semantics Notations
– <S, s> - the program statement S is executed on input state s
– s representing a terminal (final) state
For every statement S, write meaning rules<S, i> o“If 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 <P, i> o
The meaning of compound statements is defined using the meaning immediate constituent statements
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Natural Semantics for While[assns] <x := a, s> s[x Aas]
[skipns] <skip, s> s
[compns] <S1 , s> s’, <S2, s’> s’’
<S1; S2, s> s’’
[ifttns] <S1 , s> s’
<if b then S1 else S2, s> s’ if Bbs=tt
[ifffns] <S2 , s> s’
<if b then S1 else S2, s> s’ if Bbs=ff
axioms
rules
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Natural Semantics for While(More rules)
[whilettns] <S , s> s’, <while b do S, s’> s’’
<while b do S, s> s’’ if Bbs=tt
[whileffns]
<while b do S, s> s if Bbs=ff
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A Derivation Tree
A “proof” that <S, s> s’ The root of tree is <S, s> s’ Leaves are instances of axioms Internal nodes rules
– Immediate children match rule premises
Simple Example <skip; x := x +1, s0> s0[x 1]>
<skip, s0> s0 < x := x +1, s0> s0[x 1]>
compns
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An Example Derivation Tree<(x :=x+1; y :=x+1) ; z := y), s0> s0[x 1][y 2][z 2]
<x :=x+1; y :=x+1, s0> s0[x 1][y 2] <z :=y,s0[x 1][y 2]>s0[x1][y2][z 2]
<x :=x+1; s0> s0[x 1] <y :=x+1, s0[x 1]> s0[x 1][y 2]
compns
compns
assns assns
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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
<S, s> 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 = 2 Factorial program
<y := 1; while (x=1) do (y := y * x; x := x - 1), s> >
assns assns
<y :=1, s>
<W, > >compns
<(y := y * x ; x := x -1, s[y1]> >
<W, > >
whilettns
whileffns
<y := y * x ; s[y1]> > <x := x - 1 , > >
compns
assns
s[y 1]
s[y 1]
s[y 2][x1]s[y 2]
s[y 2][x1
s[y 2][x1]
s[y 2][x1
s[y 2][x1]
s[y 2]
s[y 2][x1]
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Semantic Equivalence
S1 and S2 are semantically equivalent if for all s and s’<S1, s> s’ if and only if <S2, s> s’
Simple example“while b do S”is semantically equivalent to:“if b then (S ; while b do S) else skip”
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Deterministic Semantics for While
If <S, s> s1 and <S, s> s2 then s1=s2
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 Ss = s’ if <S, s> s’ and otherwise Sns Ss is undefined
Examples– Sns skips =s
– Sns x :=1s = s [x 1]
– Sns while true do skips = undefined
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Structural Operational Semantics Emphasizes the individual steps Usually more suitable for analysis For every statement S, write meaning rules <S, i>
“If the first step of executing the statement S on an input state i leads to ”
Two possibilities for = <S’, s’> 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] <x := a, s> s[x Aas]
[skipsos] <skip, s> s
[comp1sos] <S1 , s> <S’1, s’>
<S1; S2, s> < S’1; S2, s’>
axioms
rules
[comp2sos] <S1 , s> s’
<S1; S2, s> < S2, s’>
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Structural Semantics for Whileif construct
[ifttsos] <if b then S1 else S2, s> <S1, s> if Bbs=tt
[ifffos] <if b then S1 else S2, s> <S2, s> if Bbs=ff
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Structural Semantics for Whilewhile construct
[whilesos] <while b do S, s> <if b then (S; while b do S) else skip, s>
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Derivation Sequences A finite derivation sequence starting at <S, s>
0, 1, 2 …, k such that 0=<S, s>
i i+1
k is either stuck configuration or a final state
An infinite derivation sequence starting at <S, s>0, 1, 2 … such that 0=<S, s>
i i+1
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 = (z:=x; x := y); y := z
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Factorial Program Input state s such that s x = 3
y := 1; while (x=1) do (y := y * x; x := x - 1)<y :=1 ; W, s>
<W, s[y 1]>
<if (x =1) then skip else ((y := y * x ; x := x – 1); W), s[y 1]>
< ((y := y * x ; x := x – 1); W), s[y 1]>
<(x := x – 1 ; W), s[y 3]>
< W , s[y 3][x 2]>
<if (x =1) then skip else ((y := y * x ; x := x – 1); W), s[y 3][x 2]>
< ((y := y * x ; x := x – 1); W), s[y 3] [x 2] >
<(x := x – 1 ; W) , s[y 6] [x 2] >
< W, s[y 6][x 1]>
<if (x =1) then skip else ((y := y * x ; x := x – 1); W), s[y 6][x 1]>
<skip, s[y 6][x 1]> s[y 6][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 <S, s>
– S terminates successfully on s if there exists a finite derivation sequence starting at <S, s> leading to a final state
– S loops on s if there exists an infinite derivation sequence starting at <S, s>
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Properties of the Semantics S1 and S2 are semantically equivalent if:
– for all s and which is either final or stuck<S1, s> * if and only if <S2, s> *
– there is an infinite derivation sequence starting at <S1, s> if and only if there is an infinite derivation sequence starting at <S2, s>
Deterministic– If <S, s> * s1 and <S, s> * s2 then s1=s2
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 <S1; S2, s> k s’’ then there exists a state s’
and numbers k1 and k2 such that
– <S1, s> k1 s’
– <S2, s’> 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 k+1 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)
SsosSs = s’ if <S, s> *s’ and otherwise Ssos Ss is undefined
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An Equivalence Result
For every statement S of the While language– SnatS = SsosS
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Extensions to While
Abort statement (like C exit w/o return value) 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
– if x = 0 then abort else y := y / x
– 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 While Programming Language with Non-Determinism
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 := x+2)
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[or1ns] <S1 , s> s’
<S1 or S2, s> s’
The While Programming Language with Non-Determinism
Natural Semantics
[or2ns] <S2 , s> s’
<S1 or S2, s> s’
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The While Programming Language with Non-Determinism
Structural Semantics
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The While Programming Language with Non-Determinism
Examples
x := 1 or (x :=2 ; x := x+2) (while true do skip) or (x :=2 ; x := x+2)
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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 suppress not termination configuration
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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 := x+2)
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The While Programming Language with Parallel Constructs
Structural Semantics
[par1sos] <S1 , s> <S’1, s’>
<S1 par S2, s> < S’1par S2, s’> [par2
sos] <S1 , s> s’
<S1 par S2, s> < S2, s’> [par3
sos] <S2 , s> <S’2, s’>
<S1 par S2, s> < S1par S’2, s’> [par4
sos] <S2 , s> s’
<S1 par S2, s> < S1, s’>
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The While Programming Language with Parallel Constructs
Natural 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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Transition Systems
Low-level semantics Include program counter in the set of states The meaning of a program is a relation
Execution is a finite sequence of states
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Example1: y := 1;
while 2: (x=1) do (
3: y := y * x;
4: x := x - 1
)
5:
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Summary
SOS is powerful enough to describe imperative programs– Can define the set of traces
– Can represent program counter implicitly
– Handle gotos
Natural operational semantics is an abstraction Different semantics may be used to justify
different behaviors Thinking in concrete semantics is essential for a
compiler writer