cs-4400-sp26

Lecture 18: Classical Logic and Continuations

Outline

• Review: Propositions as Types
  • Plait code
• First-class continuations
  • Example programs with letcc/throw

First-class continuations

Review: Abstract machines

States `s: k |> e, k <| v

State transition function ` s |—> s’ Stacks ` k ::= empty | k ; f Frames ` f ::= … (depends on the language)

Example: IfLang e ::= (e1 + e2) | num n | bool b | if e e1 e2 f ::= (_ + e2) | (n + _) | (if _ e1 e2)

Revew: stacks/continuations

So far, we have been using these as a meta-level tool for implementing operational semantics. Why?

• Make order of operations explicit
• Implementing certain features:
• Backtracking: making choices you want to undo
• Nondeterminism
• Early returning
• Exceptions – try/catch
• Tail recursion – limit recursion depth

Stacks/continuations as user-level feature

Internalizing meta-language features is a common thing to do in PL design when it comes to enriching expressiveness, enabling well-principled and expressive programming idioms. When we internalize a feature, sometimes we call that making the feature “first-class” (as opposed to a notion that is separate from expressions in the language). We will now explore this concept for the meta-language feature of stacks by introducing first-class continuations.

New type: `` tau ::= … | Cont tau

New expressions: `` e ::= … | cont(k) | throw e1 => e2 | letcc x. e

New frames: `` f ::= … | throw _ => e2 | throw v => _

Operational semantics (state transition rules):

 k |> cont(k')       |-----> k <| cont(k')
 k |> letcc x.e      |-----> k |> [cont(k)/x] e
 k |> throw e1 => e2 |-----> k; throw _ => e2 |> e1
 k; throw _ => e2 <| v |---> k; throw v => _ |> e2 
 k; throw v => _ <| cont(k') |---> k' <| v