Logical Properties of CPS Transforms Deepak Garg Fall, 2004

Introduction CPS = Continuation Passing Style Studied to simulate cbv with cbn We look at CPS transforms as proof- transformations CPS transforms embed classical logic into intuitionistic logic

Simply Typed -calculus Terms: Types (Logical Propositions):

Evaluation and Typing Rules Standard Typing Rules Small step, call by value evaluation

Abort (A) Operator

What we have so far … We have INTUITIONISTIC logic with ?

Control (C) Operator

What we have now … We have CLASSICAL logic! Why? Rule of double negation elimination.

Summary -calculus + A $ Intuitionistic logic with ? -calculus + A + C $ Classical logic

CPS Transforms

CPS Transform: History [1975] Plotkin. CPS studied formally. [1986] Felleisen et al. Extended to control operators (call/cc, C and A). [1993] Griffin. Typed CPS transforms. [2003] Wadler. Duality with CPS transforms.

CPS Transform: Properties Translate terms, types and proofs On terms: – No control (C) operator in transformed terms On types: – No double negation elimination in transformed proofs – Translates classical into intuitionistic logic!

CPS: Term Translation -calculus CPS translation Each translated term expects a continuation

CPS: Operational Interpretation Explicitly formalize evaluation in terms of continuations To evaluate (M N) in the continuation k, evaluate M in the continuation that binds its input to m and evaluates N in the continuation that binds its input to n and evaluates (m n) in the continuation k.

CPS: Type Translation Translation for types: Types:

CPS: Logical Interpretation

CPS: A-Operator k is thrown away like E[ ]

Soundness: A-Operator

CPS: C-Operator d is thrown away like E’[ ].

Soundness: C-Operator

CPS: Logical Interpretation

Summary of CPS

CPS as an Embedding

Two more theorems (unrelated to proof- terms):

Doing it all in Twelf

Representing Terms Use HOAS

Classical and Intuitionistic Terms The previous definition is not enough. We have to distinguish classical and intuitionistic logic Introduce two types of terms: – Classical: termc – Intuitionistic: termi

Classical and Intuitionistic Terms

Representing the CPS Transform Terms represented with HOAS No direct representation for variables How do we represent the following? Create a judgment:

More Trouble …

The solution We make the translation a hypothetical judgment Recall: We get:

The solution

Soundness Theorem

Problems: Soundness Theorem Input Coverage Problem:

Problem: Soundness Theorem From worlds (soundnessblock) Output External Can’t be changed (Needed for Induction) Make this an output?

Soundness Theorem Solution New Theorem:

Soundness Theorem? What have we shown? What we need …

Soundness Theorem? Are these theorems the same? To us they are Why? We know that given A, there is exactly one A’ such that A* = A’. We never told Twelf this fact So, in Twelf these are different theorems!

Telling Twelf about Uniqueness Can we tell Twelf that A* is unique? Not directly! There is no uniqueness check We can make an equality judgment

Telling Twelf about Uniqueness Now we prove a theorem

A Typing-soundness Theorem We also need the following theorem

True Soundness Using all our previous theorems, we can now prove the soundness with correct modes.

Extensions Can be extended to include conjunction and disjunction. We can also use a call-by-name transform – Gives a translation from classical to minimal logic – Very similar to Kolmogrov’s double negation translation