SLIDE 1
15-411: Dynamic Semantics
Jan Hoffmann
SLIDE 2 Dynamic Semantics
- Static semantics: definition of valid programs
- Dynamic semantics: definition of how programs are executed
- So far: Dynamic semantics is given in English on lab handouts
- This only works since you know how C programs should behave
- Sometimes needed to consult the reference compiler
- A description in English will always be ambiguous
➡ Need precise ways of defining the meaning of programs
SLIDE 3 Types of (Formal) Dynamic Semantics
- Denotational Semantics: Abstract and elegant.
- Each part of a program is associated with a denotation (math. object)
- For example: a procedure is associated with a mathematical function
- Axiomatic Semantics: Strongly related to program logic.
- Gives meaning to phrases using logical axioms
- The meaning is identical to the set of properties that can be proved
- Operational Semantics: Describes how programs are executed
- Related to interpretation and abstract machines
- Most popular and flexible form of semantics
SLIDE 4 Types of (Formal) Dynamic Semantics
- Denotational Semantics: Abstract and elegant.
- Each part of a program is associated with a denotation (math. object)
- For example: a procedure is associated with a mathematical function
- Axiomatic Semantics: Strongly related to program logic.
- Gives meaning to phrases using logical axioms
- The meaning is identical to the set of properties that can be proved
- Operational Semantics: Describes how programs are executed
- Related to interpretation and abstract machines
- Most popular and flexible form of semantics
Dana Scott
SLIDE 5 Types of (Formal) Dynamic Semantics
- Denotational Semantics: Abstract and elegant.
- Each part of a program is associated with a denotation (math. object)
- For example: a procedure is associated with a mathematical function
- Axiomatic Semantics: Strongly related to program logic.
- Gives meaning to phrases using logical axioms
- The meaning is identical to the set of properties that can be proved
- Operational Semantics: Describes how programs are executed
- Related to interpretation and abstract machines
- Most popular and flexible form of semantics
Dana Scott Tony Hoare
SLIDE 6 Types of (Formal) Dynamic Semantics
- Denotational Semantics: Abstract and elegant.
- Each part of a program is associated with a denotation (math. object)
- For example: a procedure is associated with a mathematical function
- Axiomatic Semantics: Strongly related to program logic.
- Gives meaning to phrases using logical axioms
- The meaning is identical to the set of properties that can be proved
- Operational Semantics: Describes how programs are executed
- Related to interpretation and abstract machines
- Most popular and flexible form of semantics
Dana Scott Tony Hoare Bob Harper
SLIDE 7 Operational Semantics
- Many different styles
- Natural semantics (or big-step semantics)
- Structural operational semantics
- Substructural operational semantics
- Abstract machine (or small-step semantics)
- We will use small-step semantics
- Very general: can describe non-termination, concurrency, …
- Low-level and elaborate
SLIDE 8 Operational Semantics
- Many different styles
- Natural semantics (or big-step semantics)
- Structural operational semantics
- Substructural operational semantics
- Abstract machine (or small-step semantics)
- We will use small-step semantics
- Very general: can describe non-termination, concurrency, …
- Low-level and elaborate
Frank Pfenning
SLIDE 9
Evaluating Expressions
SLIDE 10
Continuations
Want to model a single evaluation step
e → e0
SLIDE 11
Continuations
Want to model a single evaluation step For example: ((4 + 5) ∗ 10 + 2) → (9 ∗ 10 + 2)
e → e0
SLIDE 12
Continuations
Want to model a single evaluation step For example: ((4 + 5) ∗ 10 + 2) → (9 ∗ 10 + 2)
e → e0
How can we find the right place at which to make the step?
SLIDE 13
Continuations
Want to model a single evaluation step For example: ((4 + 5) ∗ 10 + 2) → (9 ∗ 10 + 2)
e → e0
How can we find the right place at which to make the step? Use a continuation K:
e B K
“Evaluate expression e and pass the result to K” The continuation has a ‘hole’ for the result value of e.
SLIDE 14
Continuations
Want to model a single evaluation step For example: ((4 + 5) ∗ 10 + 2) → (9 ∗ 10 + 2)
e → e0
How can we find the right place at which to make the step? Use a continuation K:
e B K
“Evaluate expression e and pass the result to K” The continuation has a ‘hole’ for the result value of e. A stack of partial computations.
SLIDE 15
Evaluation Rules: Addition
e1 + e2 B K − → e1 B (_ + e2 , K)
SLIDE 16
Evaluation Rules: Addition
e1 + e2 B K − → e1 B (_ + e2 , K)
First evaluate e1.
SLIDE 17
Evaluation Rules: Addition
e1 + e2 B K − → e1 B (_ + e2 , K)
First evaluate e1. Plug the result here.
SLIDE 18
Evaluation Rules: Addition
e1 + e2 B K − → e1 B (_ + e2 , K)
First evaluate e1. Plug the result here.
c1 B (_ + e2 , K) − → e2 B (c1 + _ , K)
SLIDE 19
Evaluation Rules: Addition
e1 + e2 B K − → e1 B (_ + e2 , K)
First evaluate e1. Plug the result here.
c1 B (_ + e2 , K) − → e2 B (c1 + _ , K)
e is a constant.
SLIDE 20
Evaluation Rules: Addition
e1 + e2 B K − → e1 B (_ + e2 , K)
First evaluate e1. Plug the result here.
c1 B (_ + e2 , K) − → e2 B (c1 + _ , K)
e is a constant. Continuation is an addition.
SLIDE 21
Evaluation Rules: Addition
e1 + e2 B K − → e1 B (_ + e2 , K)
First evaluate e1. Plug the result here.
c1 B (_ + e2 , K) − → e2 B (c1 + _ , K)
e is a constant. Continuation is an addition. Continue with evaluating e2.
SLIDE 22
Evaluation Rules: Addition
e1 + e2 B K − → e1 B (_ + e2 , K)
First evaluate e1. Plug the result here.
c1 B (_ + e2 , K) − → e2 B (c1 + _ , K)
e is a constant. Continuation is an addition. Continue with evaluating e2. Plug the result here.
SLIDE 23
Evaluation Rules: Addition
e1 + e2 B K − → e1 B (_ + e2 , K)
First evaluate e1. Plug the result here.
c1 B (_ + e2 , K) − → e2 B (c1 + _ , K)
e is a constant. Continuation is an addition. Continue with evaluating e2. Plug the result here. c2 B (c1 + _ , K) − → c B K (c = c1 + c2 mod 232)
SLIDE 24
Evaluation Rules: Addition
e1 + e2 B K − → e1 B (_ + e2 , K)
First evaluate e1. Plug the result here.
c1 B (_ + e2 , K) − → e2 B (c1 + _ , K)
e is a constant. Continuation is an addition. Continue with evaluating e2. Plug the result here. Two constants c2 B (c1 + _ , K) − → c B K (c = c1 + c2 mod 232)
SLIDE 25
Evaluation Rules: Addition
e1 + e2 B K − → e1 B (_ + e2 , K)
First evaluate e1. Plug the result here.
c1 B (_ + e2 , K) − → e2 B (c1 + _ , K)
e is a constant. Continuation is an addition. Continue with evaluating e2. Plug the result here. Two constants Actual addition. c2 B (c1 + _ , K) − → c B K (c = c1 + c2 mod 232)
SLIDE 26 Evaluation Rules: Binary Operations
e1 e2 B K
e1 B (_ e2 , K) B B
B
e2 B (c1 _ , K) B B B
B
c B K (c = c1 c2 mod 232) Arithmetic operations are treated like addition Arithmetic is modulo 232 to match our x86 architecture
SLIDE 27 Evaluation Rules: Binary Operations
e1 e2 B K
e1 B (_ e2 , K) B B
B
e2 B (c1 _ , K) B B B
B
c B K (c = c1 c2 mod 232) Arithmetic operations are treated like addition Arithmetic is modulo 232 to match our x86 architecture What about effects?
SLIDE 28 Evaluation Rules: Binops with Effects
In case of an arithmetic exception: Abort the computation and
report and error
e1 ↵ e2 B K
e1 B (_ ↵ e2 , K) B B ↵ B
B ↵ c1 B (_ ↵ e2 , K)
e2 B (c1 ↵ _ , K) B B ↵
↵ c2 B (c1 ↵ _ , K)
c B K (c = c1 ↵ c2) B ↵
↵ c2 B (c1 ↵ _ , K)
exception(arith) (c1 ↵ c2 undefined)
There is no rule for further evaluating an exception.
SLIDE 29 Example Evaluation
((4 + 5) ⇤ 10) + 2 B ·
(4 + 5) ⇤ 10 B _ + 2
(4 + 5) ⇤ 10 B _ + 2
4 + 5 B _ ⇤ 10 , _ + 2
4 B _ + 5 , _ ⇤ 10 , _ + 2
5 B 4 + _ , _ ⇤ 10 , _ + 2
9 B _ ⇤ 10 , _ + 2
10 B 9 ⇤ _ , _ + 2
90 B _ + 2
2 B 90 + _
92 B ·
SLIDE 30 Example Evaluation
((4 + 5) ⇤ 10) + 2 B ·
(4 + 5) ⇤ 10 B _ + 2
(4 + 5) ⇤ 10 B _ + 2
4 + 5 B _ ⇤ 10 , _ + 2
4 B _ + 5 , _ ⇤ 10 , _ + 2
5 B 4 + _ , _ ⇤ 10 , _ + 2
9 B _ ⇤ 10 , _ + 2
10 B 9 ⇤ _ , _ + 2
90 B _ + 2
2 B 90 + _
92 B ·
SLIDE 31 Example Evaluation
((4 + 5) ⇤ 10) + 2 B ·
(4 + 5) ⇤ 10 B _ + 2
(4 + 5) ⇤ 10 B _ + 2
4 + 5 B _ ⇤ 10 , _ + 2
4 B _ + 5 , _ ⇤ 10 , _ + 2
5 B 4 + _ , _ ⇤ 10 , _ + 2
9 B _ ⇤ 10 , _ + 2
10 B 9 ⇤ _ , _ + 2
90 B _ + 2
2 B 90 + _
92 B ·
SLIDE 32 Example Evaluation
((4 + 5) ⇤ 10) + 2 B ·
(4 + 5) ⇤ 10 B _ + 2
(4 + 5) ⇤ 10 B _ + 2
4 + 5 B _ ⇤ 10 , _ + 2
4 B _ + 5 , _ ⇤ 10 , _ + 2
5 B 4 + _ , _ ⇤ 10 , _ + 2
9 B _ ⇤ 10 , _ + 2
10 B 9 ⇤ _ , _ + 2
90 B _ + 2
2 B 90 + _
92 B ·
SLIDE 33 Example Evaluation
((4 + 5) ⇤ 10) + 2 B ·
(4 + 5) ⇤ 10 B _ + 2
(4 + 5) ⇤ 10 B _ + 2
4 + 5 B _ ⇤ 10 , _ + 2
4 B _ + 5 , _ ⇤ 10 , _ + 2
5 B 4 + _ , _ ⇤ 10 , _ + 2
9 B _ ⇤ 10 , _ + 2
10 B 9 ⇤ _ , _ + 2
90 B _ + 2
2 B 90 + _
92 B ·
SLIDE 34 Example Evaluation
((4 + 5) ⇤ 10) + 2 B ·
(4 + 5) ⇤ 10 B _ + 2
(4 + 5) ⇤ 10 B _ + 2
4 + 5 B _ ⇤ 10 , _ + 2
4 B _ + 5 , _ ⇤ 10 , _ + 2
5 B 4 + _ , _ ⇤ 10 , _ + 2
9 B _ ⇤ 10 , _ + 2
10 B 9 ⇤ _ , _ + 2
90 B _ + 2
2 B 90 + _
92 B ·
SLIDE 35 Example Evaluation
((4 + 5) ⇤ 10) + 2 B ·
(4 + 5) ⇤ 10 B _ + 2
(4 + 5) ⇤ 10 B _ + 2
4 + 5 B _ ⇤ 10 , _ + 2
4 B _ + 5 , _ ⇤ 10 , _ + 2
5 B 4 + _ , _ ⇤ 10 , _ + 2
9 B _ ⇤ 10 , _ + 2
10 B 9 ⇤ _ , _ + 2
90 B _ + 2
2 B 90 + _
92 B ·
SLIDE 36 Example Evaluation
((4 + 5) ⇤ 10) + 2 B ·
(4 + 5) ⇤ 10 B _ + 2
(4 + 5) ⇤ 10 B _ + 2
4 + 5 B _ ⇤ 10 , _ + 2
4 B _ + 5 , _ ⇤ 10 , _ + 2
5 B 4 + _ , _ ⇤ 10 , _ + 2
9 B _ ⇤ 10 , _ + 2
10 B 9 ⇤ _ , _ + 2
90 B _ + 2
2 B 90 + _
92 B ·
SLIDE 37 Example Evaluation
((4 + 5) ⇤ 10) + 2 B ·
(4 + 5) ⇤ 10 B _ + 2
(4 + 5) ⇤ 10 B _ + 2
4 + 5 B _ ⇤ 10 , _ + 2
4 B _ + 5 , _ ⇤ 10 , _ + 2
5 B 4 + _ , _ ⇤ 10 , _ + 2
9 B _ ⇤ 10 , _ + 2
10 B 9 ⇤ _ , _ + 2
90 B _ + 2
2 B 90 + _
92 B ·
SLIDE 38 Example Evaluation
((4 + 5) ⇤ 10) + 2 B ·
(4 + 5) ⇤ 10 B _ + 2
(4 + 5) ⇤ 10 B _ + 2
4 + 5 B _ ⇤ 10 , _ + 2
4 B _ + 5 , _ ⇤ 10 , _ + 2
5 B 4 + _ , _ ⇤ 10 , _ + 2
9 B _ ⇤ 10 , _ + 2
10 B 9 ⇤ _ , _ + 2
90 B _ + 2
2 B 90 + _
92 B ·
SLIDE 39 Example Evaluation
((4 + 5) ⇤ 10) + 2 B ·
(4 + 5) ⇤ 10 B _ + 2
(4 + 5) ⇤ 10 B _ + 2
4 + 5 B _ ⇤ 10 , _ + 2
4 B _ + 5 , _ ⇤ 10 , _ + 2
5 B 4 + _ , _ ⇤ 10 , _ + 2
9 B _ ⇤ 10 , _ + 2
10 B 9 ⇤ _ , _ + 2
90 B _ + 2
2 B 90 + _
92 B ·
SLIDE 40 Evaluation Rules: End of and Evaluation
c B ·
value(c)
If we reach a constant and the empty continuation then we stop
SLIDE 41 Evaluation Rules: Boolean Expressions
True and false are also values
e1 && e2 B K
e1 B (_ && e2 , K) B B B
B false B (_ && e2 , K)
false B K B B B
B true B (_ && e2 , K)
e2 B K
(We could also use 1 and 0 but distinguishing helps detect errors.)
SLIDE 42 Evaluation Rules: Boolean Expressions
True and false are also values
e1 && e2 B K
e1 B (_ && e2 , K) B B B
B false B (_ && e2 , K)
false B K B B B
B true B (_ && e2 , K)
e2 B K
Notice the short- cutting. (We could also use 1 and 0 but distinguishing helps detect errors.)
SLIDE 43
Variables and Environments
How do we evaluate variable?
SLIDE 44
Variables and Environments
x B K − → ? How do we evaluate variable?
SLIDE 45
Variables and Environments
x B K − → ? How do we evaluate variable? Need to have a environment that maps variables to values
SLIDE 46
Variables and Environments
x B K − → ? How do we evaluate variable?
η ::= · | η, x 7! v
Need to have a environment that maps variables to values
SLIDE 47
Variables and Environments
x B K − → ? How do we evaluate variable?
η ::= · | η, x 7! v
Need to have a environment that maps variables to values The machine state consists now of an expression, a continuation, and an environment
SLIDE 48
Variables and Environments
x B K − → ? How do we evaluate variable?
η ::= · | η, x 7! v
Need to have a environment that maps variables to values
η ` e B K
The machine state consists now of an expression, a continuation, and an environment
SLIDE 49 Variables and Environments II
The rules we have seen so far just carry over
η ` e1 e2 B K
η ` e1 B (_ e2 , K) B B
`
` B
η ` e2 B (c1 _ , K) B B ` B
` B
η ` c B K (c = c1 c2 mod 232)
SLIDE 50 Variables and Environments II
The rules we have seen so far just carry over
η ` e1 e2 B K
η ` e1 B (_ e2 , K) B B
`
` B
η ` e2 B (c1 _ , K) B B ` B
` B
η ` c B K (c = c1 c2 mod 232)
η ` x B K
η ` η(x) B K
Variables are simply looked up
SLIDE 51 Variables and Environments II
The rules we have seen so far just carry over
η ` e1 e2 B K
η ` e1 B (_ e2 , K) B B
`
` B
η ` e2 B (c1 _ , K) B B ` B
` B
η ` c B K (c = c1 c2 mod 232)
η ` x B K
η ` η(x) B K
Variables are simply looked up We assume programs passed the semantic check and vars are always defined.
SLIDE 52 Variables and Environments II
The rules we have seen so far just carry over
η ` e1 e2 B K
η ` e1 B (_ e2 , K) B B
`
` B
η ` e2 B (c1 _ , K) B B ` B
` B
η ` c B K (c = c1 c2 mod 232)
η ` x B K
η ` η(x) B K
Variables are simply looked up We assume programs passed the semantic check and vars are always defined. The environment never changes when evaluating expressions
SLIDE 53
Executing Statements
SLIDE 54
Executing Statements I
η ` s I K
Executions of statements don’t pass values to the continuation Statements have usually an effect on the environment Machine configurations:
SLIDE 55
Executing Statements I
η ` s I K
Executions of statements don’t pass values to the continuation Statements have usually an effect on the environment Machine configurations: Continuations contain statements.
SLIDE 56 Executing Statements I
η ` s I K
Executions of statements don’t pass values to the continuation Statements have usually an effect on the environment Machine configurations: Continuations contain statements. Sequences: η ` seq(s1, s2) I K
η ` s1 I (s2 , K) ` I
` I
SLIDE 57 Executing Statements I
η ` s I K
Executions of statements don’t pass values to the continuation Statements have usually an effect on the environment Machine configurations: Continuations contain statements. Sequences: η ` seq(s1, s2) I K
η ` s1 I (s2 , K) ` I
` I Nops: `
1 2 I
`
1 I 2
η ` nop I (s , K)
η ` s I K
SLIDE 58 Executing Statements I
η ` s I K
Executions of statements don’t pass values to the continuation Statements have usually an effect on the environment Machine configurations: Continuations contain statements. Sequences: η ` seq(s1, s2) I K
η ` s1 I (s2 , K) ` I
` I Nops: `
1 2 I
`
1 I 2
η ` nop I (s , K)
η ` s I K A terminating execution ends with a nop.
SLIDE 59 Executing Statements II
Interaction with expressions is straightforward Assignments:
η ` assign(x, e) I K
η ` e B (assign(x, _) , K) ` B
7! ` I ` I
` B η ` v B (assign(x, _) , K)
η[x 7! v] ` nop I K
SLIDE 60 Executing Statements II
Interaction with expressions is straightforward Assignments:
η ` assign(x, e) I K
η ` e B (assign(x, _) , K) ` B
7! ` I ` I
` B η ` v B (assign(x, _) , K)
η[x 7! v] ` nop I K
SLIDE 61 Executing Statements II
Interaction with expressions is straightforward Assignments:
η ` assign(x, e) I K
η ` e B (assign(x, _) , K) ` B
7! ` I ` I
` B η ` v B (assign(x, _) , K)
η[x 7! v] ` nop I K
Update the environment with new mapping.
SLIDE 62 Executing Statements III
Conditionals:
η ` if(e, s1, s2) I K
η ` e B (if(_, s1, s2) , K) ` B
` I ` I
` B η ` true B (if(_, s1, s2), K)
η ` s1 I K η ` false B (if( , s , s ), K)
η ` s I K ` B
` I η ` false B (if(_, s1, s2), K)
η ` s2 I K
SLIDE 63 Executing Statements IV
Loops:
η ` while(e, s) I K
SLIDE 64 Executing Statements IV
Loops:
η ` while(e, s) I K
while(e, s) ⌘ if(e, seq(s, while(e, s)), nop)
Not that the following statements are equivalent:
SLIDE 65 Executing Statements IV
Loops:
η ` while(e, s) I K
η ` while(e, s) I K
η ` if(e, seq(s, while(e, s)), nop) I K while(e, s) ⌘ if(e, seq(s, while(e, s)), nop) Not that the following statements are equivalent:
SLIDE 66 Executing Statements IV
Loops:
η ` while(e, s) I K
η ` while(e, s) I K
η ` if(e, seq(s, while(e, s)), nop) I K while(e, s) ⌘ if(e, seq(s, while(e, s)), nop) Not that the following statements are equivalent:
transitions s0
s1
s2
· · · without final states are just nop I · and except
Non-termination: We can make an infinite number of steps without reaching a final state
SLIDE 67 Executing Statements V
Assertions: η ` assert(e) I K
η ` e B (assert(_), K) B I ` I
` B η ` true B (assert(_), K)
η ` nop I K B η ` true B (assert(_), K)
η ` nop I K η ` false B (assert(_), K)
exception(abort)
SLIDE 68 Executing Statements V
Assertions: η ` assert(e) I K
η ` e B (assert(_), K) B I ` I
` B η ` true B (assert(_), K)
η ` nop I K B η ` true B (assert(_), K)
η ` nop I K η ` false B (assert(_), K)
exception(abort) η ` decl(x, τ, s) I K
η[x 7! nothing] ` s I K Declarations:
SLIDE 69 Executing Statements V
Assertions: η ` assert(e) I K
η ` e B (assert(_), K) B I ` I
` B η ` true B (assert(_), K)
η ` nop I K B η ` true B (assert(_), K)
η ` nop I K η ` false B (assert(_), K)
exception(abort) η ` decl(x, τ, s) I K
η[x 7! nothing] ` s I K Declarations: If C0 had shadowing then we would have to save and restore the previous value of x.
SLIDE 70 Executing Statements V
Assertions: η ` assert(e) I K
η ` e B (assert(_), K) B I ` I
` B η ` true B (assert(_), K)
η ` nop I K B η ` true B (assert(_), K)
η ` nop I K η ` false B (assert(_), K)
exception(abort) Final states:
just nop I · and · · · and exception(E), caused by a failing η ` decl(x, τ, s) I K
η[x 7! nothing] ` s I K Declarations: If C0 had shadowing then we would have to save and restore the previous value of x.
SLIDE 71 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 72 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 73 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 74 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 75 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 76 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 77 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 78 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 79 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 80 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 81 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 82 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 83 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 84 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 85 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 86 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 87 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 88 Example: Infinite Loop
[x7!1] ` while(x > 0, s) I ·
[x7!1] ` if(x>0, seq(s, while(x>0, s)), nop) I ·
[x7!1] ` x > 0 B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` x B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 1 B _ > 0; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` 0 B 1 > _; if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` true B if(_, seq(s, while(x > 0, s)), nop)
[x7!1] ` seq(s, while(x > 0, s)) I ·
[x7!1] ` assign(x, x + 1)) I while(x > 0, assign(x, x + 1))
[x7!1] ` x + 1 B assign(x, _)); while(x > 0, s)
[x7!1] ` x B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B _ + 1; assign(x, _)); while(x > 0, s)
[x7!1] ` 1 B 1 + _; assign(x, _)); while(x > 0, s)
[x7!1] ` 2 B assign(x, _)); while(x > 0, s)
[x7!2] ` nop I while(x > 0, s)
[x7!2] ` while(x > 0, s) I · · · ·
statement while(x > 0, assign(x, x + 1)) execution, we obtion the following execution; and η = [x7!1]. execution; where s ⌘
SLIDE 89
Functions
SLIDE 90
Function Calls
What needs to happen at a function call?
SLIDE 91 Function Calls
What needs to happen at a function call?
- Evaluate the arguments in left-to-right order
SLIDE 92 Function Calls
What needs to happen at a function call?
- Evaluate the arguments in left-to-right order
- Save the environment of the caller to continue the execution after the
function call
SLIDE 93 Function Calls
What needs to happen at a function call?
- Evaluate the arguments in left-to-right order
- Save the environment of the caller to continue the execution after the
function call
- Save the continuation of the callee
SLIDE 94 Function Calls
What needs to happen at a function call?
- Evaluate the arguments in left-to-right order
- Save the environment of the caller to continue the execution after the
function call
- Save the continuation of the callee
- Execute the body in a new environment that maps the formal
parameters to the argument values
SLIDE 95 Function Calls
What needs to happen at a function call?
- Evaluate the arguments in left-to-right order
- Save the environment of the caller to continue the execution after the
function call
- Save the continuation of the callee
- Execute the body in a new environment that maps the formal
parameters to the argument values
- Pass the return value the the environment of the caller
SLIDE 96
Call Stack
We need to keep track of continuations and environment in stack frames Call stack:
S ::= · | S , hη, Ki
SLIDE 97
Call Stack
We need to keep track of continuations and environment in stack frames Call stack: Environment
S ::= · | S , hη, Ki
SLIDE 98
Call Stack
We need to keep track of continuations and environment in stack frames Call stack: Environment Continuation
S ::= · | S , hη, Ki
SLIDE 99
Call Stack
We need to keep track of continuations and environment in stack frames Call stack: Environment Continuation
S ::= · | S , hη, Ki
Configurations:
S ; η ` e B K I ` B S ; η ` s I K
Evaluation Execution
SLIDE 100
Call Stack
We need to keep track of continuations and environment in stack frames Call stack: Environment Continuation
S ::= · | S , hη, Ki
Configurations:
S ; η ` e B K I ` B S ; η ` s I K
Evaluation Execution Existing rules can be lifted to the new configurations by
passing through the call stack
SLIDE 101
Rules for Function Calls
We only show the special case of 0 and 2 arguments
SLIDE 102
Rules for Function Calls
We only show the special case of 0 and 2 arguments n args is similar.
SLIDE 103 Rules for Function Calls
We only show the special case of 0 and 2 arguments n args is similar. S ; η ` f( ) B K
(S , hη, Ki) ; · ` s I · (given that is i · ` I · (given that f is defined as f( ){s}) No arguments:
SLIDE 104 Rules for Function Calls
We only show the special case of 0 and 2 arguments n args is similar. Store callee’s stack frame S ; η ` f( ) B K
(S , hη, Ki) ; · ` s I · (given that is i · ` I · (given that f is defined as f( ){s}) No arguments:
SLIDE 105 Rules for Function Calls
We only show the special case of 0 and 2 arguments n args is similar. Store callee’s stack frame Evaluate s in empty environment. S ; η ` f( ) B K
(S , hη, Ki) ; · ` s I · (given that is i · ` I · (given that f is defined as f( ){s}) No arguments:
SLIDE 106 Rules for Function Calls
We only show the special case of 0 and 2 arguments n args is similar. Store callee’s stack frame Evaluate s in empty environment. S ; η ` f( ) B K
(S , hη, Ki) ; · ` s I · (given that is i · ` I · (given that f is defined as f( ){s}) No arguments: S ; η ` f(e1, e2) B K
S ; η ` e1 B (f(_, e2) , K) B B `
1 2 B
`
1 B 2
S ; η ` c1 B (f(_, e2) , K)
S ; η ` e2 B (f(c1, _) , K) B Two arguments:
SLIDE 107 Rules for Function Calls
We only show the special case of 0 and 2 arguments n args is similar.
`
1 B 2
`
2 B 1
S ; η ` c2 B (f(c1, _) , K)
(S , hη, Ki) ; [x1 7! c1, x2 7! c2] ` s I · (given that is defined as h i 7! 7! ` I · (given that f is defined as f(x1, x2){s})
Store callee’s stack frame Evaluate s in empty environment. S ; η ` f( ) B K
(S , hη, Ki) ; · ` s I · (given that is i · ` I · (given that f is defined as f( ){s}) No arguments: S ; η ` f(e1, e2) B K
S ; η ` e1 B (f(_, e2) , K) B B `
1 2 B
`
1 B 2
S ; η ` c1 B (f(_, e2) , K)
S ; η ` e2 B (f(c1, _) , K) B Two arguments:
SLIDE 108 Rules for Returns
Need to restore continuation and environment and pass return value
S ; η ` return(e) I K
S ; η ` e B (return(_) , K) B B S ; η ` return(e) I K
S ; η ` e B (return( S , hη0, K0i ; η ` v B (return(_) , K)
S ; η0 ` v B K0
SLIDE 109 Rules for Returns
Need to restore continuation and environment and pass return value
S ; η ` return(e) I K
S ; η ` e B (return(_) , K) B B S ; η ` return(e) I K
S ; η ` e B (return( S , hη0, K0i ; η ` v B (return(_) , K)
S ; η0 ` v B K0
SLIDE 110 Rules for Returns
Need to restore continuation and environment and pass return value
S ; η ` return(e) I K
S ; η ` e B (return(_) , K) B B S ; η ` return(e) I K
S ; η ` e B (return( S , hη0, K0i ; η ` v B (return(_) , K)
S ; η0 ` v B K0
SLIDE 111 Rules for Returns
Need to restore continuation and environment and pass return value
S ; η ` return(e) I K
S ; η ` e B (return(_) , K) B B S ; η ` return(e) I K
S ; η ` e B (return( S , hη0, K0i ; η ` v B (return(_) , K)
S ; η0 ` v B K0
Special case: returning void
S , hη0, K0i ; η ` nop I ·
S ; η0 ` nothing B K0
SLIDE 112 Rules for Returns
Need to restore continuation and environment and pass return value Will only be reached by functions without return.
S ; η ` return(e) I K
S ; η ` e B (return(_) , K) B B S ; η ` return(e) I K
S ; η ` e B (return( S , hη0, K0i ; η ` v B (return(_) , K)
S ; η0 ` v B K0
Special case: returning void
S , hη0, K0i ; η ` nop I ·
S ; η0 ` nothing B K0
SLIDE 113 Rules for Returns
Need to restore continuation and environment and pass return value Will only be reached by functions without return.
S ; η ` return(e) I K
S ; η ` e B (return(_) , K) B B S ; η ` return(e) I K
S ; η ` e B (return( S , hη0, K0i ; η ` v B (return(_) , K)
S ; η0 ` v B K0
Special case: returning void
S , hη0, K0i ; η ` nop I ·
S ; η0 ` nothing B K0
Dummy value
SLIDE 114 Rules for Returns
Need to restore continuation and environment and pass return value Will only be reached by functions without return.
S ; η ` return(e) I K
S ; η ` e B (return(_) , K) B B S ; η ` return(e) I K
S ; η ` e B (return( S , hη0, K0i ; η ` v B (return(_) , K)
S ; η0 ` v B K0
Special case: returning void
S , hη0, K0i ; η ` nop I ·
S ; η0 ` nothing B K0
Dummy value Alternative: elaborate each function that returns void with
return(nothing) statements.
SLIDE 115
Execution of the Main Function
How can we execute a program?
SLIDE 116
Execution of the Main Function
How can we execute a program?
· ; · ` main( ) B · B (initial state) final state
SLIDE 117 Execution of the Main Function
How can we execute a program?
· ; · ` main( ) B · B (initial state) final state · · ` B · ; η ` c B ·
value(c) (final state)
SLIDE 118
Statics, Dynamics, and Safety
SLIDE 119 Overview of Machine States (Configurations)
ST
– Evaluating the expression e with the continuation K
– Evaluating the statement s with the continuation K
– Final state, return a value
– Final state, report an error
SLIDE 120 Overview of Machine States (Configurations)
ST
– Evaluating the expression e with the continuation K
– Evaluating the statement s with the continuation K
– Final state, return a value
– Final state, report an error What do we expect from the transitions?
SLIDE 121 Overview of Machine States (Configurations)
ST
– Evaluating the expression e with the continuation K
– Evaluating the statement s with the continuation K
– Final state, return a value
– Final state, report an error What do we expect from the transitions? There shouldn’t be more steps after reaching a final state
SLIDE 122 Overview of Machine States (Configurations)
ST
– Evaluating the expression e with the continuation K
– Evaluating the statement s with the continuation K
– Final state, return a value
– Final state, report an error What do we expect from the transitions? There shouldn’t be more steps after reaching a final state The language should be deterministic: there at most one transition
per state
SLIDE 123
Progress
There are many non-final states that don’t have transitions, e.g.
SLIDE 124
Progress
There are many non-final states that don’t have transitions, e.g. transition: S; η ` 42 B (if(_, s1, s2); K) states are called stuck – it is literally
SLIDE 125
Progress
There are many non-final states that don’t have transitions, e.g. transition: S; η ` 42 B (if(_, s1, s2); K) states are called stuck – it is literally and ·; · ` nop I · is what the program
SLIDE 126
Progress
Stuck states. There are many non-final states that don’t have transitions, e.g. transition: S; η ` 42 B (if(_, s1, s2); K) states are called stuck – it is literally and ·; · ` nop I · is what the program
SLIDE 127
Progress
Stuck states. There are many non-final states that don’t have transitions, e.g. transition: S; η ` 42 B (if(_, s1, s2); K) states are called stuck – it is literally and ·; · ` nop I · is what the program The behavior of these states is undefined.
SLIDE 128
Progress
Stuck states. There are many non-final states that don’t have transitions, e.g. transition: S; η ` 42 B (if(_, s1, s2); K) states are called stuck – it is literally and ·; · ` nop I · is what the program The behavior of these states is undefined. Programs that are well-defined according to the static semantics
should be free of undefined behavior. Central relationship between static and dynamic semantics:
SLIDE 129 Progress
Stuck states. There are many non-final states that don’t have transitions, e.g. transition: S; η ` 42 B (if(_, s1, s2); K) states are called stuck – it is literally and ·; · ` nop I · is what the program The behavior of these states is undefined. Programs that are well-defined according to the static semantics
should be free of undefined behavior. Central relationship between static and dynamic semantics:
Theorem 1 (No undefined behavior) If a program passes all the static semantics, and ·; · ` main( )
ST 1
. . .
ST n then either ST n is a final state or else ST n is not-stuck because there exists a state ST 0 such that ST n
ST 0.
SLIDE 130 Progress
Stuck states. There are many non-final states that don’t have transitions, e.g. transition: S; η ` 42 B (if(_, s1, s2); K) states are called stuck – it is literally and ·; · ` nop I · is what the program Well-typed programs don’t go wrong! The behavior of these states is undefined. Programs that are well-defined according to the static semantics
should be free of undefined behavior. Central relationship between static and dynamic semantics:
Theorem 1 (No undefined behavior) If a program passes all the static semantics, and ·; · ` main( )
ST 1
. . .
ST n then either ST n is a final state or else ST n is not-stuck because there exists a state ST 0 such that ST n
ST 0.
SLIDE 131 Progress
Stuck states. There are many non-final states that don’t have transitions, e.g. transition: S; η ` 42 B (if(_, s1, s2); K) states are called stuck – it is literally and ·; · ` nop I · is what the program Well-typed programs don’t go wrong! The behavior of these states is undefined. Programs that are well-defined according to the static semantics
should be free of undefined behavior. Central relationship between static and dynamic semantics:
Theorem 1 (No undefined behavior) If a program passes all the static semantics, and ·; · ` main( )
ST 1
. . .
ST n then either ST n is a final state or else ST n is not-stuck because there exists a state ST 0 such that ST n
ST 0.
How to prove this?
SLIDE 132 Progress
Stuck states. There are many non-final states that don’t have transitions, e.g. transition: S; η ` 42 B (if(_, s1, s2); K) states are called stuck – it is literally and ·; · ` nop I · is what the program Well-typed programs don’t go wrong! The behavior of these states is undefined. Programs that are well-defined according to the static semantics
should be free of undefined behavior. Central relationship between static and dynamic semantics:
Theorem 1 (No undefined behavior) If a program passes all the static semantics, and ·; · ` main( )
ST 1
. . .
ST n then either ST n is a final state or else ST n is not-stuck because there exists a state ST 0 such that ST n
ST 0.
How to prove this? 15-312 next term.
SLIDE 133 Summary I
Expressions e ::= c | e1 e2 | true | false | e1 && e2 | x | f(e1, e2) | f( ) Statements s ::= nop | seq(s1, s2) | assign(x, e) | decl(x, τ, s) | if(e, s1, s2) | while(e, s) | return(e) | assert(e) Values v ::= c | true | false | nothing Environments η ::= · | η, x 7! c Stacks S ::= · | S , hη, Ki
φ ::= _ e | c _ | _ && e | f(_, e) | f(c, _) | s | assign(x, _) | if(_, s1, s2) | return(_) | assert(_) Continuations K ::= · | φ , K Exceptions E ::= arith | abort | mem
SLIDE 134 Summary I
Expressions e ::= c | e1 e2 | true | false | e1 && e2 | x | f(e1, e2) | f( ) Statements s ::= nop | seq(s1, s2) | assign(x, e) | decl(x, τ, s) | if(e, s1, s2) | while(e, s) | return(e) | assert(e) Values v ::= c | true | false | nothing Environments η ::= · | η, x 7! c Stacks S ::= · | S , hη, Ki
φ ::= _ e | c _ | _ && e | f(_, e) | f(c, _) | s | assign(x, _) | if(_, s1, s2) | return(_) | assert(_) Continuations K ::= · | φ , K Exceptions E ::= arith | abort | mem
All ops.
SLIDE 135 Summary: Expressions
S ; η ` e1 e2 B K
S ; η ` e1 B (_ e2 , K) S ; η ` c1 B (_ e2 , K)
S ; η ` e2 B (c1 _ , K) S ; η ` c2 B (c1 _ , K)
S ; η ` c B K (c = c1 c2) S ; η ` c2 B (c1 _ , K)
exception(arith) (c1 c2 undefined) S ; η ` e1 && e2 B K
S ; η ` e1 B (_ && e2 , K) S ; η ` false B (_ && e2 , K)
S ; η ` false B K S ; η ` true B (_ && e2 , K)
S ; η ` e2 B K S ; η ` x B K
S ; η ` η(x) B K
SLIDE 136 Summary: Statements
S ; η ` nop I (s , K)
S ; η ` s I K S ; η ` assign(x, e) I K
S ; η ` e B (assign(x, _) , K) S ; η ` c B (assign(x, _) , K)
S ; η[x 7! c] ` nop I K S ; η ` decl(x, τ, s) I K
S ; η[x 7! nothing] ` s I K S ; η ` assert(e) I K
S ; η ` e B (assert(_) , K) S ; η ` true B (assert(_) , K)
S ; η ` nop I K S ; η ` false B (assert(_) , K)
exception(abort) S ; η ` if(e, s1, s2) I K
S ; η ` e B (if(_, s1, s2) , K) S ; η ` true B (if(_, s1, s2), K)
S ; η ` s1 I K S ; η ` false B (if(_, s1, s2), K)
S ; η ` s2 I K S ; η ` while(e, s) I K
S ; η ` if(e, seq(s, while(e, s)), nop) I K
SLIDE 137 Summary: Functions
`
` S ; η ` f(e1, e2) B K
S ; η ` e1 B (f(_, e2) , K) S ; η ` c1 B (f(_, e2) , K)
S ; η ` e2 B (f(c1, _) , K) S ; η ` c2 B (f(c1, _) , K)
(S , hη, Ki) ; [x1 7! c1, x2 7! c2] ` s I · (given that f is defined as f(x1, x2){s}) S ; η ` f( ) B K
(S , hη, Ki) ; · ` s I · (given that f is defined as f( ){s}) S ; η ` return(e) I K
S ; η ` e B (return(_) , K) (S , hη0, K0i) ; η ` v B (return(_) , K)
S ; η0 ` v B K0 · ; η ` c B (return(_) , K)
value(c)
