Defunctionalized Interpreters for Call-by-Need Evaluation Olivier - PowerPoint PPT Presentation

Defunctionalized Interpreters for Call-by-Need Evaluation Olivier Danvy, University of Aarhus Kevin Millikin, Google Johan Munk, Arctic Lake Systems Ian Zerny, University of Aarhus Sendai, Japan FLOPS 2010

Motivation Formal semantics: why? ◮ Understanding linguistic features ◮ Proving programs correct or equivalent ◮ Proving language properties ◮ Proving implementations, analyses or transformations correct Formal semantics: which kind? ◮ Denotational? ◮ Operational? Big step? Small step? ◮ Axiomatic? Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 2 / 18

Foundations of this work Semantic artifacts can be inter-derived mechanically, and the inter-derivation is worthwhile: ◮ it can yield simpler semantics, and ◮ it can yield new semantics. Here: call by need. Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 3 / 18

Call-by-need evaluation ◮ Demand-driven computation ◮ Memoization of intermediate results Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 4 / 18

Semantics for call-by-need evaluation ◮ Store-based: results are saved in the global store ◮ Storeless: results are saved in the term itself Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 5 / 18

Syntactic theories of call by need ◮ Different opinions — the POPL’95 affair ◮ The call-by-need lambda-calculus by Ariola and Felleisen (JFP’97) ◮ The call-by-need lambda-calculus by Maraist, Odersky and Wadler (JFP’98) Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 6 / 18

Syntactic theories of call by need ◮ Different opinions — the POPL’95 affair ◮ The call-by-need lambda-calculus by Ariola and Felleisen (JFP’97) ◮ The call-by-need lambda-calculus by Maraist, Odersky and Wadler (JFP’98) ◮ Appearances can be deceiving: The standard reduction is common to both Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 6 / 18

Syntactic theories of call by need ◮ Different opinions — the POPL’95 affair ◮ The call-by-need lambda-calculus by Ariola and Felleisen (JFP’97) ◮ The call-by-need lambda-calculus by Maraist, Odersky and Wadler (JFP’98) ◮ Appearances can be deceiving: The standard reduction is common to both ◮ It is our starting point Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 6 / 18

Outline 1. The call-by-need λ -calculus 2. Deriving an abstract machine and a natural semantics Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 7 / 18

The call-by-name λ let -calculus Terms ∋ T ::= x | λ x . T | T T | let x be T in T Values ∋ V ::= λ x . T Eval Cont ∋ E ::= [ ] | E T | let x be T in E Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 8 / 18

The call-by-name λ let -calculus Terms ∋ T ::= x | λ x . T | T T | let x be T in T Values ∋ V ::= λ x . T Eval Cont ∋ E ::= [ ] | E T | let x be T in E ( λ z . z z ) (( λ y . y ) ( λ x . x )) �→ name Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 8 / 18

The call-by-name λ let -calculus Terms ∋ T ::= x | λ x . T | T T | let x be T in T Values ∋ V ::= λ x . T Eval Cont ∋ E ::= [ ] | E T | let x be T in E ( λ z . z z ) (( λ y . y ) ( λ x . x )) �→ name let z be ( λ y . y ) ( λ x . x ) in z z �→ name Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 8 / 18

The call-by-name λ let -calculus Terms ∋ T ::= x | λ x . T | T T | let x be T in T Values ∋ V ::= λ x . T Eval Cont ∋ E ::= [ ] | E T | let x be T in E ( λ z . z z ) (( λ y . y ) ( λ x . x )) �→ name let z be ( λ y . y ) ( λ x . x ) in z z �→ name let z be ( λ y . y ) ( λ x . x ) in (( λ y . y ) ( λ x . x )) z �→ name Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 8 / 18

The call-by-name λ let -calculus Terms ∋ T ::= x | λ x . T | T T | let x be T in T Values ∋ V ::= λ x . T Eval Cont ∋ E ::= [ ] | E T | let x be T in E ( λ z . z z ) (( λ y . y ) ( λ x . x )) �→ name let z be ( λ y . y ) ( λ x . x ) in z z �→ name let z be ( λ y . y ) ( λ x . x ) in (( λ y . y ) ( λ x . x )) z �→ name let z be ( λ y . y ) ( λ x . x ) in ( let y be λ x . x in y ) z �→ name Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 8 / 18

The call-by-name λ let -calculus Terms ∋ T ::= x | λ x . T | T T | let x be T in T Values ∋ V ::= λ x . T Eval Cont ∋ E ::= [ ] | E T | let x be T in E ( λ z . z z ) (( λ y . y ) ( λ x . x )) �→ name let z be ( λ y . y ) ( λ x . x ) in z z �→ name let z be ( λ y . y ) ( λ x . x ) in (( λ y . y ) ( λ x . x )) z �→ name let z be ( λ y . y ) ( λ x . x ) in ( let y be λ x . x in y ) z �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in y z �→ name Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 8 / 18

The call-by-name λ let -calculus Terms ∋ T ::= x | λ x . T | T T | let x be T in T Values ∋ V ::= λ x . T Eval Cont ∋ E ::= [ ] | E T | let x be T in E ( λ z . z z ) (( λ y . y ) ( λ x . x )) �→ name let z be ( λ y . y ) ( λ x . x ) in z z �→ name let z be ( λ y . y ) ( λ x . x ) in (( λ y . y ) ( λ x . x )) z �→ name let z be ( λ y . y ) ( λ x . x ) in ( let y be λ x . x in y ) z �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in y z �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in ( λ x . x ) z �→ name Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 8 / 18

The call-by-name λ let -calculus Terms ∋ T ::= x | λ x . T | T T | let x be T in T Values ∋ V ::= λ x . T Eval Cont ∋ E ::= [ ] | E T | let x be T in E ( λ z . z z ) (( λ y . y ) ( λ x . x )) �→ name let z be ( λ y . y ) ( λ x . x ) in z z �→ name let z be ( λ y . y ) ( λ x . x ) in (( λ y . y ) ( λ x . x )) z �→ name let z be ( λ y . y ) ( λ x . x ) in ( let y be λ x . x in y ) z �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in y z �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in ( λ x . x ) z �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in let x be z in x �→ name Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 8 / 18

The call-by-name λ let -calculus Terms ∋ T ::= x | λ x . T | T T | let x be T in T Values ∋ V ::= λ x . T Eval Cont ∋ E ::= [ ] | E T | let x be T in E ( λ z . z z ) (( λ y . y ) ( λ x . x )) �→ name let z be ( λ y . y ) ( λ x . x ) in z z �→ name let z be ( λ y . y ) ( λ x . x ) in (( λ y . y ) ( λ x . x )) z �→ name let z be ( λ y . y ) ( λ x . x ) in ( let y be λ x . x in y ) z �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in y z �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in ( λ x . x ) z �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in let x be z in x �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in let x be z in z �→ name Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 8 / 18

The call-by-name λ let -calculus Terms ∋ T ::= x | λ x . T | T T | let x be T in T Values ∋ V ::= λ x . T Eval Cont ∋ E ::= [ ] | E T | let x be T in E ( λ z . z z ) (( λ y . y ) ( λ x . x )) �→ name let z be ( λ y . y ) ( λ x . x ) in z z �→ name let z be ( λ y . y ) ( λ x . x ) in (( λ y . y ) ( λ x . x )) z �→ name let z be ( λ y . y ) ( λ x . x ) in ( let y be λ x . x in y ) z �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in y z �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in ( λ x . x ) z �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in let x be z in x �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in let x be z in z �→ name let z be ( λ y . y ) ( λ x . x ) in let y be λ x . x in let x be z in ( λ y . y ) ( λ x . x ) . . . Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 8 / 18

The call-by-need λ let -calculus Terms ∋ T ::= x | λ x . T | T T | let x be T in T Values ∋ V ::= λ x . T Eval Cont ∋ E ::= [ ] | E T | let x be T in E | let x be E in E [ x ] Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 9 / 18

The call-by-need λ let -calculus Terms ∋ T ::= x | λ x . T | T T | let x be T in T Values ∋ V ::= λ x . T Eval Cont ∋ E ::= [ ] | E T | let x be T in E | let x be E in E [ x ] ( λ z . z z ) (( λ y . y ) ( λ x . x )) �→ need Olivier Danvy ( danvy@cs.au.dk ) Defunctionalized Interpreters for Call-by-Need Evaluation FLOPS’10 9 / 18