Continuation calculus at Term Rewriting Seminar

Continuation calculus
Bram Geron1 Herman Geuvers1,2
1Eindhoven University of Technology
2Radboud University Nijmegen
Term Rewriting Seminar, May 2013
Geron, Geuvers (TU/e, RU) Continuation calculus TERESE May 2013 1 / 22

Outline
1 First look
2 Deﬁnition
3 Long-term goal
4 Interfaces, not pattern matching
5 The importance of head reduction
6 Relation to lambda calculus
7 Conclusion

First look Rules, names, variables, terms
CC is a constrained term rewriting system
Comp.f .g.x
def
−→ f .(g.x)
Comp.AddOne.Fact.4 → AddOne.(Fact.4)
Rules
One deﬁnition per name
(max)
No pattern matching
Only head reduction
Consequences
Deterministic
Simple operational semantics
Suitable for modeling
continuations, hence exceptions

First look Rules, names, variables, terms
rule
Comp
name
(constant)
.f .g.x
variables
def
−→ f .(g.x)
Comp.AddOne.Fact.4
term
→ AddOne.(Fact.4)
term
Rules
One deﬁnition per name
(max)
No pattern matching
Only head reduction
Consequences
Deterministic
Simple operational semantics
Suitable for modeling
continuations, hence exceptions

First look Head reduction: continuation passing style
Functions ﬁll in the result in their continuation parameter
Comp.f .g.x
def
−→ f .(g.x)
Comp.AddOne.Fact.4 → AddOne.(Fact).4
Realistic names:
Fact.r.x r.x!
AddOne.r.x r.(x +1)
Fact.(AddOne.r).4 AddOne.r.24
r.25

First look Head reduction: continuation passing style
Functions ﬁll in the result in their continuation parameter
Comp.r.f .g.x
def
−→ g.(f .r).x
Comp.r.AddOne.Fact.4 → Fact.(AddOne.r).4
Realistic names:
Fact.r.x r.x!
AddOne.r.x r.(x +1)
Fact.(AddOne.r).4 AddOne.r.24
r.25

Definition
Definitions
There is an infinite set of names, indicated by a capital.
A term is either a name, or two terms combined by a dot.
We write a.b.c as shorthand for (a.b).c.
A program P consists of a set of rules, of the form
Name.variable.··· .variable
def
−→ term over those variables
That rule defines that name.
Each rule defines a different name.
If “N.x1 .··· .xk
def
−→ r” ∈ P, then we reduce N.t → r[t/x].
Term N.x1.··· .xl for l = k does not reduce at all.
There is only head reduction.

Long-term goal
Long-term goal
Formally modeling programming languages
Hopefully as a base for better languages
Operational semantics
Running time: #steps ∼ seconds CPU time
Warning: some unsubstantiated claims
Compositional
Facilitates proving properties
Deterministic by nature
Works with continuations
Explained later

Interfaces, not pattern matching
Outline
1 First look
2 Deﬁnition
3 Long-term goal
7 Conclusion

Outline
1 First look
2 Deﬁnition
3 Long-term goal
7 Conclusion

Natural numbers
Slightly similar to lambda calculus
Zero.z.s
def
−→ z
S.x.z.s
def
−→ s.x
Add.r.x.y
def
−→ y.(r.x).(Add.r.(S.x))
Algorithm:
x +0 = x
x +S(y) = S(x)+y
Add.r.(S.Zero).Zero → Zero.(r.(S.Zero)).(Add.r.(S.(S.Zero)))
→ r.(S.Zero)
Add.r.Zero.(S.Zero) → S.Zero.(r.Zero).(Add.r.(S.Zero))
→ Add.r.(S.Zero).Zero
→ Zero.(r.(S.Zero)).(Add.r.(S.(S.Zero)))
→ r.(S.Zero)

Natural numbers
Zero.z.s
def
−→ z
S.x.z.s
def
−→ s.x
Add.r.x.y
def
−→ y.(r.x).(Add.r.(S.x))
Algorithm:
x +0 = x
x +S(y) = S(x)+y
What can we feed to Add?
Zero
S.(··· .(S.Zero)···)
Something else?

Natural numbers
Zero.z.s
def
−→ z
S.x.z.s
def
−→ s.x
Add.r.x.y
def
−→ y.(r.x).(Add.r.(S.x))
Algorithm:
x +0 = x
x +S(y) = S(x)+y
What can we feed to Add?
Zero
S.(··· .(S.Zero)···)
Something else? Yes!

Natural numbers
Compatible naturals
Zero.z.s
def
−→ z
S.x.z.s
def
−→ s.x
Add.r.x.y
def
−→ y.(r.x).(Add.r.(S.x))
Algorithm:
x +0 = x
x +S(y) = S(x)+y
We deﬁne when a term t represents natural number n.
1 t represents 0 if ∀z,s : t.z.s z
Informally: t “behaves the same as” Zero
2 t represents n +1 if ∀z,s : t.z.s s.q, and q represents n
Informally: t “behaves the same as” S.q
With this, we can deﬁne LazyFact such that LazyFact.x represents (x!).
(Omitted.)

Natural numbers
Compatible naturals
Zero.z.s
def
−→ z
S.x.z.s
def
−→ s.x
Add.r.x.y
def
−→ y.(r.x).(Add.r.(S.x))
Algorithm:
x +0 = x
x +S(y) = S(x)+y
(Omitted.)
Add.r.Zero.(LazyFact.(S.Zero)) → LazyFact.(S.Zero).(r.Zero).(Add.r.(S.Zero))
Add.r.(S.Zero).Zero
→ Zero.(r.(S.Zero)).(Add.r.(S.(S.Zero)))
→ r.(S.Zero)

Natural numbers
Compatible naturals
Zero.z.s
def
−→ z
S.x.z.s
def
−→ s.x
Add.r.x.y
def
−→ y.(r.x).(Add.r.(S.x))
Algorithm:
x +0 = x
x +S(y) = S(x)+y
(Omitted.)
Add.r.Zero.(LazyFact.(S.Zero)) r.(S.Zero)
Two types of function names
Call-by-value / eager: calculate result, ﬁll in in continuation
Call-by-name / lazy: compatible with Sn.Zero but delayed
computation

The importance of head reduction
Outline
1 First look
2 Deﬁnition
3 Long-term goal
7 Conclusion

Outline
1 First look
2 Deﬁnition
3 Long-term goal
7 Conclusion

Lambda calculus (normal order)
Reduction using two instructions on the top level
M N Push N on stack, continue in M
λx.M Pop N oﬀ stack, continue in M[N/x]
as presented in [Levy(2001)]

Back to modeling programs
Lambda calculus
λx.+ (f x) (g x)
M N Push N on stack,
continue in M
λx.M Pop N oﬀ stack,
continue in M[N/x]
Programming language
fun x → (f x) + (g x)
M N Apply function object
to argument
fun x → M Make function object
In “real languages”, functions throw exceptions and have other side eﬀects.

Back to modeling programs
Lambda calculus
λx.+ (f x) (g x)
M N Push N on stack,
continue in M
λx.M Pop N oﬀ stack,
continue in M[N/x]
Programming language
fun x → (f x) + (g x)
throws exception
now what?
M N Apply function object
to argument
fun x → M Make function object
In “real languages”, functions throw exceptions and have other side eﬀects.

Computational models with control
Lambda calculus + continuations (λC)
Reduction using four instructions on the top level
A M Empty the stack, continue in M
C M Empty the stack, continue in M (λx.S x)
where function S ‘restores’ the previous stack
(details omitted)
Can model exception-like facilities
CPS transformation to transform λC terms to λ terms

CPS transformation
λC
CPS transformation
−−−−−−−−−−−→ subset of λ, makes minimal use of stack
Two calculi in action:
λC calculus: M N, λx.M, A M, C M
Expressive
Subset of λ calculus: M N, λx.M
Simpler

CPS transformation
λC
CPS transformation
Expressive
Simpler
What is this subset, really?

CPS transformation
λC
CPS transformation
Expressive
Simpler
Can we describe it?

CPS transformation
λC
CPS transformation
Expressive
Simpler
Can we describe it?
An elegant model of computation, perhaps?

Lambda calculus vs. continuation calculus
λC
Four instructions on the top level
A M Empty the stack, continue in M
C M Empty the stack, continue in M (λx.S x)
where function S ‘restores’ the stack
One instruction on the top level, using program P
n.t1.··· .tk Empty the stack, continue in r[t/x]
if n.x1 .··· .xk
def
−→ r ∈ P

One instruction on the top level, using program P
n.t1.··· .tk Continue in r[t/x]
if n.x1 .··· .xk
def
−→ r ∈ P
No stack needed, because
n.t1.··· .tl does not reduce for l = k.
Subterms are not reduced

Relation to lambda calculus
Relation to lambda calculus
λ,λC
CPS transformation
−−−−−−−−−−−→ subset of λ
subset of λ
λx to rules∗
−−−−−−−−−−−−−−−−−−−−−−
unfold†
rules to λx
CC
∗ λx to rules: involves a supercombinator transformation
† Unfold rules to λx: ﬁrst apply a ﬁxed-point elimination to eliminate cyclic references
Catchphrase
CC is more limited than λ,
thus more suitable to model continuations

Conclusion
Conclusion
CC is deterministic
Only head reduction
As a consequence,
Suitable for control with continuations (exceptions)
CC is similar to a subset of lambda calculus
Allows both eager and lazy function terms
call-by-value vs. call-by-name

Appendix
Bibliography
P.B. Levy.
Call-by-push-value.
PhD thesis, Queen Mary, University of London, 2001.

Continuation calculus at Term Rewriting Seminar

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