` e : T. Gradual Typing. ` e X. Ronald Garcia University of British Columbia

Transcription

1 <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> ` e : T ` e X Gradual Typing Ronald Garcia University of British Columbia 1

2 Static vs. Dynamic? static early error detection enforced discipline dynamic rapid prototyping flexible idioms 2

3 Gradual Typing! gradual early error detection enforced discipline 3 rapid prototyping flexible idioms programmer-controlled!

4 Outline Motivating Example (In Two Acts) Gradual Typing For All! Typing in Small Pieces Meat Strands and Related Works 4

5 Motivating Example Act 1: A New Type 5

6 A Dynamic Language 6

7 A Dynamic Language Standard ML 7

8 A Dynamic Language Robin Milner Standard ML 8

9 Standard ML: dynamically typed? datatype nat = Zero Succ of nat case x : nat of Zero )... Succ y )... 5 Adapted from Jafery and Dunfield [POPL17] 9

10 Standard ML: dynamically typed? datatype nat = Zero Succ of nat case x : nat of Zero )... Succ y )... But the Definition requires compilers to accept nonexhaustive matches: case x : nat of Succ y )... 5 Adapted from Jafery and Dunfield [POPL17] 10

11 Standard ML: dynamically typed? datatype nat = Zero Succ of nat case x : nat of Zero )... Succ y )... But the Definition requires compilers to accept nonexhaustive matches: case x : nat of Succ y )... If x = Zero, then the exception Match is raised. This nonexhaustive match is fine, if we know that x will never be Zero. 5 Adapted from Jafery and Dunfield [POPL17] 11

12 Static Types! 12

13 Standard ML: dynamically typed? datatype nat = Zero Succ of nat case x : nat of Zero )... Succ y )... But the Definition requires compilers to accept nonexhaustive matches: case x : nat of Succ y )... If x = Zero, then the exception Match is raised. This nonexhaustive match is fine, if we know that x will never be Zero. 5 Adapted from Jafery and Dunfield [POPL17] 13

14 Standard ML: dynamically typed? datatype nat = Zero Succ of nat case x : nat of Zero )... Succ y )... But the Definition requires compilers to accept nonexhaustive matches: case x : nat of Succ y )... If x = Zero, then the exception Match is raised. This nonexhaustive match is fine, if we know that x will never be Zero. Frank Pfenning 5 Adapted from Jafery and Dunfield [POPL17] 13

15 Standard ML: dynamically typed? datatype nat = Zero Succ of nat case x : nat of Zero )... Succ y )... But the Definition requires compilers to accept nonexhaustive matches: A widely employed style of programming, s which impose no discipline of types rocedures which work well on objects of case x : nat of Succ y )... If x = Zero, then the exception Match is raised. This nonexhaustive match is fine, if we know that x will never be Zero. Frank Pfenning Well, actually Milner [1978] said that (about LISP). 5 Adapted from Jafery and Dunfield [POPL17] 13

16 Standard ML: dynamically typed? datatype nat = Zero Succ of nat case x : nat of Zero )... Succ y )... But the Definition requires compilers to accept nonexhaustive matches: A widely employed style of programming, s which impose no discipline of types rocedures which work well on objects of case x : nat of Succ y )... This nonexhaustive match is fine, if we know that x will never be Zero. Frank Pfenning a wide variety (e.g., on lists of atoms, integers, or lists). Such flexibility is almost essential in this style If x of = programming; Zero, then the unfortunately exception one Match often pays is raised. a price for it in the time taken to find rather inscrutable bugs-anyone who mistakenly applies CDR to an atom in LISP, and finds himself absurdly adding a property list to an integer, will know the Well, actually Milner [1978] said that (about LISP) too 5 Adapted from Jafery and Dunfield [POPL17] 13

17 Refined Standard ML Datasort refinements [Freeman & Pfenning 1991, Davies 2005,... ] push the knowledge that x is not Zero into the type system. case x : nonzero of Succ y )... This is exhaustive, because x has datasort nonzero. 6 Frank Pfenning Adapted from Jafery and Dunfield [POPL17] 14

18 Refined Standard ML Datasort refinements [Freeman & Pfenning 1991, Davies 2005,... ] push the knowledge that x is not Zero into the type system. case x : nonzero of Succ y )... This is exhaustive, because x has datasort nonzero. 6 Frank Pfenning Adapted from Jafery and Dunfield [POPL17] 15

19 Refined Standard ML Datasort refinements [Freeman & Pfenning 1991, Davies 2005,... ] push the knowledge that x is not Zero into the type system. case x : nonzero of Succ y )... This is exhaustive, because x has datasort nonzero.? 6 Frank Pfenning Adapted from Jafery and Dunfield [POPL17] 16

20 Outline Motivating Example (In Two Acts) STILL Gradual Typing For All! Typing in Small Pieces Meat Strands and Related Works 17

21 Motivating Example Act 2: Adoption 18

22 19

23 20

24 21

25 22

26 No Refined ML! 23

27 Paucity of RML Code SML Application1 Refined ML Application0 Application2 Library1 Library2 *Figures not drawn to scale 24

28 SML Application1 Refined ML Application0 Application2 Library1 Library2 *Figures not drawn to scale 25

29 SML Application1 Application2 Refined ML Application0 Application1 Library1 Library2 *Figures not drawn to scale 26

30 SML Application1 Application2 Refined ML Application0 Application1 Library1 Library2 *Figures not drawn to scale 27

31 SML Application1 Application2 Refined ML Application0 Application1 Library1 Library2 *Figures not drawn to scale 28

32 SML Application1 Application2 Refined ML Application0 Application1 Library1 Library2 *Figures not drawn to scale 29 Wholesale Migration?!?

33 SML Application1 Application2 Refined ML Application0 Application1 Library1 Library2 *Figures not drawn to scale 30 Wholesale Migration?!?

34 SML Application1 Application2 Refined ML Application0 Application1 Library1 Library2 *Figures not drawn to scale 31 Wholesale Migration?!?

35 SML Application1 Application2 Refined ML Application0 Application1 Library1 Library2 Library1 *Figures not drawn to scale 32 Wholesale Migration?!?

36 SML Application1 Application2 Library1 Library2 Refined ML Application0 Application1 Application2 Library1 *Figures not drawn to scale 33 Wholesale Migration?!?

37 SML Application1 Application2 Library1 Library2 Refined ML Application0 Application1 Application2 Library1 *Figures not drawn to scale 34

38 SML Application1 Application2 Library1 Library2 Refined ML Application0 Application1 Application2 Library1 *Figures not drawn to scale 35 Must We Assimilate?

39 SML Application1 Application2 Library1 Library2 Refined ML Application0 Application1 Application2 Library1 *Figures not drawn to scale 36

40 SML Application1 Application2 Library1 Library2 Refined ML Application0 Application1 Application2 Library1 Library2 Gradual Migration *Figures not drawn to scale 37

41 SML Application1 Application2 Library1 Library2 Refined ML Application0 Application1 Application2 Library1 Library2 Gradual Migration *Figures not drawn to scale 38

42 SML Application1 Application2 Library1 Library2 *Figures not drawn to scale 39 Refined ML Application0 Application1 Application2 Library1 Library2 Gradual Migration SML Code (& Guarantees) Refined ML Code (& Guarantees) Interoperating!

43 <latexit sha1_base64="184rnpzv8as3a91sfrw5waynu5g=">aaacc3icbva9swnben2lxzf+rs1tlgtbktyjogxqqssi5gnyicxtjsms3btjdy8qjvq2/hubc0vs/qn2/hv3khsa+gdg8d4mm/ocwhbtxpfbya2tb2xu5bclo7t7+wffw6ogjhlfsm4iealwabofd7fuubhyihwcdaq2g9fn5jfhqdspwgczibejyrdypmdgrnqtlvxbkbkop+6bhll0jzhhfkc+gyibsvcjabdydivudhsveatsjgvuusuvvxexrgjomact254bm04kynamcfrwe40xsbemsg1pcbj1j539mqwnvunrfqrshybo1n8tkuitjzkwndmpetnlxp+8dml6v52uh3fimgtzrf1eubprlbja4wqzernlgclub6vscaqysfevbaje8surphfe8dykd39rrl4v4site1iiz8qjl6rk7kin1akjj+szvji358l5cd6dj3lrzlnmhjm/cd5/aetlmzc=</latexit> <latexit sha1_base64="184rnpzv8as3a91sfrw5waynu5g=">aaacc3icbva9swnben2lxzf+rs1tlgtbktyjogxqqssi5gnyicxtjsms3btjdy8qjvq2/hubc0vs/qn2/hv3khsa+gdg8d4mm/ocwhbtxpfbya2tb2xu5bclo7t7+wffw6ogjhlfsm4iealwabofd7fuubhyihwcdaq2g9fn5jfhqdspwgczibejyrdypmdgrnqtlvxbkbkop+6bhll0jzhhfkc+gyibsvcjabdydivudhsveatsjgvuusuvvxexrgjomact254bm04kynamcfrwe40xsbemsg1pcbj1j539mqwnvunrfqrshybo1n8tkuitjzkwndmpetnlxp+8dml6v52uh3fimgtzrf1eubprlbja4wqzernlgclub6vscaqysfevbaje8surphfe8dykd39rrl4v4site1iiz8qjl6rk7kin1akjj+szvji358l5cd6dj3lrzlnmhjm/cd5/aetlmzc=</latexit> <latexit sha1_base64="184rnpzv8as3a91sfrw5waynu5g=">aaacc3icbva9swnben2lxzf+rs1tlgtbktyjogxqqssi5gnyicxtjsms3btjdy8qjvq2/hubc0vs/qn2/hv3khsa+gdg8d4mm/ocwhbtxpfbya2tb2xu5bclo7t7+wffw6ogjhlfsm4iealwabofd7fuubhyihwcdaq2g9fn5jfhqdspwgczibejyrdypmdgrnqtlvxbkbkop+6bhll0jzhhfkc+gyibsvcjabdydivudhsveatsjgvuusuvvxexrgjomact254bm04kynamcfrwe40xsbemsg1pcbj1j539mqwnvunrfqrshybo1n8tkuitjzkwndmpetnlxp+8dml6v52uh3fimgtzrf1eubprlbja4wqzernlgclub6vscaqysfevbaje8surphfe8dykd39rrl4v4site1iiz8qjl6rk7kin1akjj+szvji358l5cd6dj3lrzlnmhjm/cd5/aetlmzc=</latexit> <latexit sha1_base64="184rnpzv8as3a91sfrw5waynu5g=">aaacc3icbva9swnben2lxzf+rs1tlgtbktyjogxqqssi5gnyicxtjsms3btjdy8qjvq2/hubc0vs/qn2/hv3khsa+gdg8d4mm/ocwhbtxpfbya2tb2xu5bclo7t7+wffw6ogjhlfsm4iealwabofd7fuubhyihwcdaq2g9fn5jfhqdspwgczibejyrdypmdgrnqtlvxbkbkop+6bhll0jzhhfkc+gyibsvcjabdydivudhsveatsjgvuusuvvxexrgjomact254bm04kynamcfrwe40xsbemsg1pcbj1j539mqwnvunrfqrshybo1n8tkuitjzkwndmpetnlxp+8dml6v52uh3fimgtzrf1eubprlbja4wqzernlgclub6vscaqysfevbaje8surphfe8dykd39rrl4v4site1iiz8qjl6rk7kin1akjj+szvji358l5cd6dj3lrzlnmhjm/cd5/aetlmzc=</latexit> SML Application1 Application2 Refined ML Application0 Application1 Library1 Library2 ` e X Application2 Library1 Library2 *Figures not drawn to scale 40 Gradual Migration SML Code (& Guarantees) Refined ML Code (& Guarantees) Interoperating!

44 <latexit sha1_base64="184rnpzv8as3a91sfrw5waynu5g=">aaacc3icbva9swnben2lxzf+rs1tlgtbktyjogxqqssi5gnyicxtjsms3btjdy8qjvq2/hubc0vs/qn2/hv3khsa+gdg8d4mm/ocwhbtxpfbya2tb2xu5bclo7t7+wffw6ogjhlfsm4iealwabofd7fuubhyihwcdaq2g9fn5jfhqdspwgczibejyrdypmdgrnqtlvxbkbkop+6bhll0jzhhfkc+gyibsvcjabdydivudhsveatsjgvuusuvvxexrgjomact254bm04kynamcfrwe40xsbemsg1pcbj1j539mqwnvunrfqrshybo1n8tkuitjzkwndmpetnlxp+8dml6v52uh3fimgtzrf1eubprlbja4wqzernlgclub6vscaqysfevbaje8surphfe8dykd39rrl4v4site1iiz8qjl6rk7kin1akjj+szvji358l5cd6dj3lrzlnmhjm/cd5/aetlmzc=</latexit> <latexit sha1_base64="184rnpzv8as3a91sfrw5waynu5g=">aaacc3icbva9swnben2lxzf+rs1tlgtbktyjogxqqssi5gnyicxtjsms3btjdy8qjvq2/hubc0vs/qn2/hv3khsa+gdg8d4mm/ocwhbtxpfbya2tb2xu5bclo7t7+wffw6ogjhlfsm4iealwabofd7fuubhyihwcdaq2g9fn5jfhqdspwgczibejyrdypmdgrnqtlvxbkbkop+6bhll0jzhhfkc+gyibsvcjabdydivudhsveatsjgvuusuvvxexrgjomact254bm04kynamcfrwe40xsbemsg1pcbj1j539mqwnvunrfqrshybo1n8tkuitjzkwndmpetnlxp+8dml6v52uh3fimgtzrf1eubprlbja4wqzernlgclub6vscaqysfevbaje8surphfe8dykd39rrl4v4site1iiz8qjl6rk7kin1akjj+szvji358l5cd6dj3lrzlnmhjm/cd5/aetlmzc=</latexit> <latexit sha1_base64="184rnpzv8as3a91sfrw5waynu5g=">aaacc3icbva9swnben2lxzf+rs1tlgtbktyjogxqqssi5gnyicxtjsms3btjdy8qjvq2/hubc0vs/qn2/hv3khsa+gdg8d4mm/ocwhbtxpfbya2tb2xu5bclo7t7+wffw6ogjhlfsm4iealwabofd7fuubhyihwcdaq2g9fn5jfhqdspwgczibejyrdypmdgrnqtlvxbkbkop+6bhll0jzhhfkc+gyibsvcjabdydivudhsveatsjgvuusuvvxexrgjomact254bm04kynamcfrwe40xsbemsg1pcbj1j539mqwnvunrfqrshybo1n8tkuitjzkwndmpetnlxp+8dml6v52uh3fimgtzrf1eubprlbja4wqzernlgclub6vscaqysfevbaje8surphfe8dykd39rrl4v4site1iiz8qjl6rk7kin1akjj+szvji358l5cd6dj3lrzlnmhjm/cd5/aetlmzc=</latexit> <latexit sha1_base64="184rnpzv8as3a91sfrw5waynu5g=">aaacc3icbva9swnben2lxzf+rs1tlgtbktyjogxqqssi5gnyicxtjsms3btjdy8qjvq2/hubc0vs/qn2/hv3khsa+gdg8d4mm/ocwhbtxpfbya2tb2xu5bclo7t7+wffw6ogjhlfsm4iealwabofd7fuubhyihwcdaq2g9fn5jfhqdspwgczibejyrdypmdgrnqtlvxbkbkop+6bhll0jzhhfkc+gyibsvcjabdydivudhsveatsjgvuusuvvxexrgjomact254bm04kynamcfrwe40xsbemsg1pcbj1j539mqwnvunrfqrshybo1n8tkuitjzkwndmpetnlxp+8dml6v52uh3fimgtzrf1eubprlbja4wqzernlgclub6vscaqysfevbaje8surphfe8dykd39rrl4v4site1iiz8qjl6rk7kin1akjj+szvji358l5cd6dj3lrzlnmhjm/cd5/aetlmzc=</latexit> SML Application1 Application2 Refined ML Application0 Application1 Library1 Library2 ` e X Application2 Library1 Library2 Free! *Figures not drawn to scale 41 Gradual Migration SML Code (& Guarantees) Refined ML Code (& Guarantees) Interoperating!

45 <latexit sha1_base64="184rnpzv8as3a91sfrw5waynu5g=">aaacc3icbva9swnben2lxzf+rs1tlgtbktyjogxqqssi5gnyicxtjsms3btjdy8qjvq2/hubc0vs/qn2/hv3khsa+gdg8d4mm/ocwhbtxpfbya2tb2xu5bclo7t7+wffw6ogjhlfsm4iealwabofd7fuubhyihwcdaq2g9fn5jfhqdspwgczibejyrdypmdgrnqtlvxbkbkop+6bhll0jzhhfkc+gyibsvcjabdydivudhsveatsjgvuusuvvxexrgjomact254bm04kynamcfrwe40xsbemsg1pcbj1j539mqwnvunrfqrshybo1n8tkuitjzkwndmpetnlxp+8dml6v52uh3fimgtzrf1eubprlbja4wqzernlgclub6vscaqysfevbaje8surphfe8dykd39rrl4v4site1iiz8qjl6rk7kin1akjj+szvji358l5cd6dj3lrzlnmhjm/cd5/aetlmzc=</latexit> <latexit sha1_base64="184rnpzv8as3a91sfrw5waynu5g=">aaacc3icbva9swnben2lxzf+rs1tlgtbktyjogxqqssi5gnyicxtjsms3btjdy8qjvq2/hubc0vs/qn2/hv3khsa+gdg8d4mm/ocwhbtxpfbya2tb2xu5bclo7t7+wffw6ogjhlfsm4iealwabofd7fuubhyihwcdaq2g9fn5jfhqdspwgczibejyrdypmdgrnqtlvxbkbkop+6bhll0jzhhfkc+gyibsvcjabdydivudhsveatsjgvuusuvvxexrgjomact254bm04kynamcfrwe40xsbemsg1pcbj1j539mqwnvunrfqrshybo1n8tkuitjzkwndmpetnlxp+8dml6v52uh3fimgtzrf1eubprlbja4wqzernlgclub6vscaqysfevbaje8surphfe8dykd39rrl4v4site1iiz8qjl6rk7kin1akjj+szvji358l5cd6dj3lrzlnmhjm/cd5/aetlmzc=</latexit> <latexit sha1_base64="184rnpzv8as3a91sfrw5waynu5g=">aaacc3icbva9swnben2lxzf+rs1tlgtbktyjogxqqssi5gnyicxtjsms3btjdy8qjvq2/hubc0vs/qn2/hv3khsa+gdg8d4mm/ocwhbtxpfbya2tb2xu5bclo7t7+wffw6ogjhlfsm4iealwabofd7fuubhyihwcdaq2g9fn5jfhqdspwgczibejyrdypmdgrnqtlvxbkbkop+6bhll0jzhhfkc+gyibsvcjabdydivudhsveatsjgvuusuvvxexrgjomact254bm04kynamcfrwe40xsbemsg1pcbj1j539mqwnvunrfqrshybo1n8tkuitjzkwndmpetnlxp+8dml6v52uh3fimgtzrf1eubprlbja4wqzernlgclub6vscaqysfevbaje8surphfe8dykd39rrl4v4site1iiz8qjl6rk7kin1akjj+szvji358l5cd6dj3lrzlnmhjm/cd5/aetlmzc=</latexit> <latexit sha1_base64="184rnpzv8as3a91sfrw5waynu5g=">aaacc3icbva9swnben2lxzf+rs1tlgtbktyjogxqqssi5gnyicxtjsms3btjdy8qjvq2/hubc0vs/qn2/hv3khsa+gdg8d4mm/ocwhbtxpfbya2tb2xu5bclo7t7+wffw6ogjhlfsm4iealwabofd7fuubhyihwcdaq2g9fn5jfhqdspwgczibejyrdypmdgrnqtlvxbkbkop+6bhll0jzhhfkc+gyibsvcjabdydivudhsveatsjgvuusuvvxexrgjomact254bm04kynamcfrwe40xsbemsg1pcbj1j539mqwnvunrfqrshybo1n8tkuitjzkwndmpetnlxp+8dml6v52uh3fimgtzrf1eubprlbja4wqzernlgclub6vscaqysfevbaje8surphfe8dykd39rrl4v4site1iiz8qjl6rk7kin1akjj+szvji358l5cd6dj3lrzlnmhjm/cd5/aetlmzc=</latexit> Optional Typing SML Application1 Application2 Refined ML Application0 Application1 Library1 Library2 ` e X Application2 Library1 Library2 Free! *Figures not drawn to scale 42 Gradual Migration SML Code (& Guarantees) Refined ML Code (& Guarantees) Interoperating!

46 SML Application1 Application2 Library1 Library2 *Figures not drawn to scale 43 Refined ML Application0 Application1 Application2 Library1 Library2 Gradual Migration SML Code (& Guarantees) Refined ML Code (& Guarantees) Interoperating!

47 SML Application1 Application2 Library1 Library2 Refined ML Application0 Application1 Application2 Library1 Library2 ` e : T *Figures not drawn to scale 44 Gradual Migration SML Code (& Guarantees) Refined ML Code (& Guarantees) Interoperating!

48 SML Application1 Application2 Library1 Library2 Refined ML Application0 Application1 Application2 Library1 Library2 COUGH! ` e : T *Figures not drawn to scale 45 Gradual Migration SML Code (& Guarantees) Refined ML Code (& Guarantees) Interoperating!

49 SML Application1 Application2 Library1 Library2 Refined ML Application0 Application1 Application2 Library1 Library2 ` e : T *Figures not drawn to scale 46 Gradual Migration SML Code (& Guarantees) Refined ML Code (& Guarantees) Interoperating!

50 SML Application1 Application2 Library1 Library2 Refined ML Application0 Application1 Application2 Library1??? Library2 ` e : T Gradual Migration The Challenge! SML Code (& Guarantees) Refined ML Code (& Guarantees) Interoperating! *Figures not drawn to scale 47

51 SML Sums of Uncertainty: Refinements Go Gradual Application1 Khurram A. Jafery Joshua Dunfield University of British Columbia Vancouver, Canada Refined ML Application0 ` e : T Application2 Library1 Library2 Application1 Application2 Library1??? Library2 *Figures not drawn to scale POPL17 Gradual Migration The Challenge! SML Code (& Guarantees) Refined ML Code (& Guarantees) Interoperating! 48

52 SML Sums of Uncertainty: Refinements Go Gradual Application1 Khurram A. Jafery Joshua Dunfield University of British Columbia Vancouver, Canada Refined ML Application0 ` e : T Application2 Library1 Library2 Application1 Application2 Library1??? Library2 *Figures not drawn to scale POPL17 Gradual Migration The Challenge! SML Code (& Guarantees) Refined ML Code (& Guarantees) Interoperating! 49

53 Outline Motivating Example (In Two Acts) Gradual Typing For All! Typing in Small Pieces Meat Strands and Related Works 50

54 Type Spectrum ML Types Refinement Types 51

55 Type Spectrum Untyped ML Types Refinement Types 52

56 Type Spectrum Untyped ML Types Refinement Types 53

57 Type Spectrum Untyped ML Types Refinement Types Languages Typedness Typier 54

58 Type Spectrum Untyped ML Types Refinement Types Gradual Typing 55

59 Type Spectrum Untyped ML Types Refinement Types Gradual Typing 56

60 Type Spectrum Untyped ML Types Refinement Types Gradual Typing 57

61 Type Spectrum Untyped ML Types Refinement Types Gradual Typing Gradual Typing is a Relative Concept! 57

62 Type Spectrum Untyped ML Types Refinement Types Gradual Typing Gradual Typing is a Relative Concept! 57

63 Type Spectrum I always assumed gradual types were to help those poor schmucks using untyped languages to migrate to typed languages. I now realise that I am one of the poor schmucks. My recent research involves session types, a linear type system that declares protocols for sending Untyped ML Types Refinement Types Gradual Typing Gradual Typing is a Relative Concept! [Wadler SNAPL2015] 57

64 Type Spectrum Untyped ML Types Refinement Types Most Gradual Typing Work 58

65 Type Spectrum Untyped ML Types Refinement Types Most Gradual Typing Work Static Language Dynamic Language 58 Application0 Application1 Application2 Library1 Library2

66 Type Spectrum Untyped ML Types Refinement Types Much Recent Gradual Typing Work! Static Language Dynamic Language 59 Application0 Application1 Application2 Library1 Library2

67 Outline Motivating Example (In Two Acts) Gradual Typing For All! Typing in Small Pieces Meat Strands and Related Works 60

68 Gradual Types? 61

69 Gradual Types? What does Gradual Typing have to do with Types? 61

70 Gradual Types 62

71 What are Types About? ` e : T 63

72 Typing Γ t : T x:t Γ Γ x : T (T-Var) Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 (T-Abs) Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 (T-App) 64

73 Typing Γ t : T x:t Γ Γ x : T (T-Var) Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 (T-Abs) Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 (T-App) Inductive Definition 64

74 Typing Γ t : T x:t Γ Γ x : T (T-Var) Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 (T-Abs) Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 (T-App) Inductive Definition Grammar on Steroids (i.e., data structure spec) 64

75 Typing Γ t : T x:t Γ Γ x : T (T-Var) Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 (T-Abs) Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 (T-App) Inductive Definition Grammar on Steroids (i.e., data structure spec) Informally ascribe behavioural meaning to it 64

76 Typing Γ t : T x:t Γ Γ x : T (T-Var) Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 (T-Abs) Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 (T-App) Inductive Definition Daniel Kahneman Grammar on Steroids (i.e., data structure spec) Informally ascribe behavioural meaning to it 64

77 Typing x:t Γ Γ x : T If you are shown a word Γ on t a : screen T in a language you know, you will read it. (T-Var) Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 (T-Abs) Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 (T-App) Inductive Definition Daniel Kahneman Grammar on Steroids (i.e., data Thinking, structure Fast spec) and Slow Informally ascribe behavioural meaning to it 64

78 Typing Γ t : T x:t Γ Γ x : T (T-Var) Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 (T-Abs) Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 (T-App) Inductive Definition Grammar on Steroids (i.e., data structure spec) Informally ascribe behavioural meaning to it 64

79 Typing Γ t : T If t1 turns T11s into T12s x:t Γ Γ x : T Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 (T-Var) (T-Abs) Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 (T-App) Inductive Definition Grammar on Steroids (i.e., data structure spec) Informally ascribe behavioural meaning to it 64

80 Typing and t2 yields T11s Γ t : T If t1 turns T11s into T12s x:t Γ Γ x : T Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 (T-Var) (T-Abs) Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 (T-App) Inductive Definition Grammar on Steroids (i.e., data structure spec) Informally ascribe behavioural meaning to it 64

81 Typing and t2 yields T11s Γ t : T If t1 turns T11s into T12s x:t Γ Γ x : T Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 (T-Var) (T-Abs) Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 Inductive Definition Grammar on Steroids (i.e., data structure spec) (T-App) then appropriately linking them yields T12s Informally ascribe behavioural meaning to it 64

82 Type Safety Theorem [Progress]: Suppose t is a closed, well-typed term (that is, t : T for some T). Then either t is a value or else there is some t with t t. Theorem [Preservation]: If Γ t : T and t t,thenγ t : T. Well-typed programs don t go wrong 65

83 Gradual Type Safety? Application0 Application1 Application2 Library1 Library2 66

84 <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> Now You ve Got Two Problems! Application0 Application1 Application2 Library1 Library2 ` e X ` e : T 67

85 <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> Now You ve Got Two Problems! Application0 Application1 Application2 Library1 Library2 ` e X ` e : T Two Typing Judgments = Two Behavioural Contracts 67

86 <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> Now You ve Got Two Problems! Application0 Application1 Application2 Library1 Library2 ` e X ` e : T Two Typing Judgments = Two Behavioural Contracts Conflicts Signal Runtime Errors (is that wrong?) 67

87 <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> Now You ve Got Two Problems! Application0 Application1 Application2 Library1 Library2 ` e X ` e : T What counts as appropriate linking? Two Typing Judgments = Two Behavioural Contracts Conflicts Signal Runtime Errors (is that wrong?) 67

88 Typing and t2 yields T11s Γ t : T If t1 turns T11s into T12s x:t Γ Γ x : T Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 (T-Var) (T-Abs) Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 (T-App) then appropriately linking them yields T12s 68

89 and t1 turns T11s into T12s Typing and t2 yields T11s x:t Γ Γ x : T Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 Γ t : T (T-Var) (T-Abs) Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 (T-App) If the surrounding context behaves as Γ says then appropriately linking them yields T12s 69

90 and t1 turns T11s into T12s Typing and t2 yields T11s x:t Γ Γ x : T Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 Γ t : T (T-Var) (T-Abs) If the surrounding context behaves as Γ says Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 Typing Judgments are about program fragments (T-App) then appropriately linking them yields T12s 69

91 and t1 turns T11s into T12s Typing and t2 yields T11s x:t Γ Γ x : T Γ, x:t 1 t 2 : T 2 Γ λx:t 1.t 2 : T 1 T 2 Γ t : T (T-Var) (T-Abs) If the surrounding context behaves as Γ says Γ t 1 : T 11 T 12 Γ t 2 : T 11 Γ t 1 t 2 : T 12 Typing Judgments are about program fragments (T-App) then appropriately linking them yields T12s Context Matters, at least intuitively 69

92 Type Safety Theorem [Progress]: Suppose t is a closed, well-typed term (that is, t : T for some T). Then either t is a value or else there is some t with t t. Theorem [Preservation]: If Γ t : T and t t,thenγ t : T. Well-typed programs don t go wrong 70

93 Type Safety Theorem [Progress]: Suppose t is a closed, well-typed term (that is, t : T for some T). Then either t is a value or else there is some t with t t. Theorem [Preservation]: If Γ t : T and t t,thenγ t : T. Well-typed (whole) programs don t go wrong 70

94 Type Safety Theorem [Progress]: Suppose t is a closed, well-typed term (that is, t : T for some T). Then either t is a value or else there is some t with t t. Theorem [Preservation]: If Γ t : T and t t,thenγ t : T. Well-typed (whole) programs don t go wrong Do program fragments go right? 70

95 Outline Motivating Example (In Two Acts) Gradual Typing For All! Typing in Small Pieces STILL Meat Strands and Related Works 71

96 Type Safety Theorem [Progress]: Suppose t is a closed, well-typed term (that is, t : T for some T). Then either t is a value or else there is some t with t t. Theorem [Preservation]: If Γ t : T and t t,thenγ t : T. Well-typed (whole) programs don t go wrong Do program fragments go right? 72

97 <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">aaacmnicbvbnaxrbeo1jnikbe1c9eilcah5kmakbxvpqg3qlkp2anwwp6anjnunugbprlizd/iyv+swbhpsgifd/hlmfhzxxqcpjvvdu10tlrtxh0bdga/vbw53dvuet/cchh0/at5/1ffe5sv1z6minuvsklauuk9y0kb2hstx108spc78/jedvyc95vtli4ivvuzlijtruf06yrrhwiywvqeuwh+qjgooqtdp0eyb4b+ewcgfpyg4ktjgr9qti2bq3o1eylqh3sbwmhbhg2bh9k2sfraxzlhq9h8zryamahsupad5kkk8lyku8ogfdlrryo3p58hyogiwdvhdnswxldxoirup9zkrn0ibp/f1vif7pg1acvx3vypyvk5wrrxmlgqty9aezcirzzxqc0qnmryan6fby0/kihpjuyfdj7zimozd+cti5fb+uy0+8ec/fkxgln+jufbjnoiuk+cpuxq/xm7govge/gt+r6fawnnku/khw5y+6c6dz</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> Semantic Soundness [Milner 78] discipline. A Semantic Soundness Theorem (based on a formal semantics for the language) states that well-type programs cannot go wrong and a Syntactic Soundness Theorem states that if fl accepts a program then it is well typed. We also discuss extending these THEOREM 1 (Semantic Soundness). If 7 respects jj and p 1 & is well typed then &[djt : 7. i.e., Ifif p ` d e : T then = p = e : d T. : [Milner 1978] 73

98 <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> Semantic Soundness [Milner 78] discipline. A Semantic Soundness Theorem (based on a formal semantics for the language) states that well-type programs cannot go wrong and a Syntactic Soundness Theorem states that if fl accepts a program then it is well typed. We also discuss extending these THEOREM 1 (Semantic Soundness). If 7 respects jj and p 1 & is well typed then &[djt : 7. i.e., Ifif p ` d e : T then = p = e : d T. : [Milner 1978] 73

99 <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> Semantic Soundness [Milner 78] discipline. A Semantic Soundness Theorem (based on a formal semantics for the language) states that well-type programs cannot go wrong and a Syntactic Soundness Theorem states that if fl accepts a program then it is well typed. We also discuss extending these THEOREM 1 (Semantic Soundness). If 7 respects jj and p 1 & is well typed then &[djt : 7. i.e., Ifif p ` d e : T then = p = e : d T. : [Milner 1978] 73

100 <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">aaacmnicbvbnaxrbeo1jnikbe1c9eilcah5kmakbxvpqg3qlkp2anwwp6anjnunugbprlizd/iyv+swbhpsgifd/hlmfhzxxqcpjvvdu10tlrtxh0bdga/vbw53dvuet/cchh0/at5/1ffe5sv1z6minuvsklauuk9y0kb2hstx108spc78/jedvyc95vtli4ivvuzlijtruf06yrrhwiywvqeuwh+qjgooqtdp0eyb4b+ewcgfpyg4ktjgr9qti2bq3o1eylqh3sbwmhbhg2bh9k2sfraxzlhq9h8zryamahsupad5kkk8lyku8ogfdlrryo3p58hyogiwdvhdnswxldxoirup9zkrn0ibp/f1vif7pg1acvx3vypyvk5wrrxmlgqty9aezcirzzxqc0qnmryan6fby0/kihpjuyfdj7zimozd+cti5fb+uy0+8ec/fkxgln+jufbjnoiuk+cpuxq/xm7govge/gt+r6fawnnku/khw5y+6c6dz</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">aaacmnicbvbnaxrbeo1jnikbe1c9eilcah5kmakbxvpqg3qlkp2anwwp6anjnunugbprlizd/iyv+swbhpsgifd/hlmfhzxxqcpjvvdu10tlrtxh0bdga/vbw53dvuet/cchh0/at5/1ffe5sv1z6minuvsklauuk9y0kb2hstx108spc78/jedvyc95vtli4ivvuzlijtruf06yrrhwiywvqeuwh+qjgooqtdp0eyb4b+ewcgfpyg4ktjgr9qti2bq3o1eylqh3sbwmhbhg2bh9k2sfraxzlhq9h8zryamahsupad5kkk8lyku8ogfdlrryo3p58hyogiwdvhdnswxldxoirup9zkrn0ibp/f1vif7pg1acvx3vypyvk5wrrxmlgqty9aezcirzzxqc0qnmryan6fby0/kihpjuyfdj7zimozd+cti5fb+uy0+8ec/fkxgln+jufbjnoiuk+cpuxq/xm7govge/gt+r6fawnnku/khw5y+6c6dz</latexit> Semantic Soundness [Milner 78] discipline. A Semantic Soundness Theorem (based on a formal semantics for the language) states that well-type programs cannot go wrong and a Syntactic Soundness Theorem states that if fl accepts a program then it is well typed. We also discuss extending these THEOREM 1 (Semantic Soundness). If 7 respects jj and p 1 & is well typed then &[djt : 7. i.e., Ifif p ` d e : T then = p = e : d T. : [Milner 1978] 73

101 <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">aaacmnicbvbnaxrbeo1jnikbe1c9eilcah5kmakbxvpqg3qlkp2anwwp6anjnunugbprlizd/iyv+swbhpsgifd/hlmfhzxxqcpjvvdu10tlrtxh0bdga/vbw53dvuet/cchh0/at5/1ffe5sv1z6minuvsklauuk9y0kb2hstx108spc78/jedvyc95vtli4ivvuzlijtruf06yrrhwiywvqeuwh+qjgooqtdp0eyb4b+ewcgfpyg4ktjgr9qti2bq3o1eylqh3sbwmhbhg2bh9k2sfraxzlhq9h8zryamahsupad5kkk8lyku8ogfdlrryo3p58hyogiwdvhdnswxldxoirup9zkrn0ibp/f1vif7pg1acvx3vypyvk5wrrxmlgqty9aezcirzzxqc0qnmryan6fby0/kihpjuyfdj7zimozd+cti5fb+uy0+8ec/fkxgln+jufbjnoiuk+cpuxq/xm7govge/gt+r6fawnnku/khw5y+6c6dz</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> Semantic Soundness [Milner 78] discipline. A Semantic Soundness Theorem (based on a formal semantics for the language) states that well-type programs cannot go wrong and a Syntactic Soundness Theorem states that if fl accepts a program then it is well typed. We also discuss extending these THEOREM 1 (Semantic Soundness). If 7 respects jj and p 1 & is well typed then &[djt : 7. i.e., Ifif p ` d e : T then = p = e : d T. : Data Structure Behavioural Invariant [Milner 1978] 74

102 <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">aaacmnicbvbnaxrbeo1jnikbe1c9eilcah5kmakbxvpqg3qlkp2anwwp6anjnunugbprlizd/iyv+swbhpsgifd/hlmfhzxxqcpjvvdu10tlrtxh0bdga/vbw53dvuet/cchh0/at5/1ffe5sv1z6minuvsklauuk9y0kb2hstx108spc78/jedvyc95vtli4ivvuzlijtruf06yrrhwiywvqeuwh+qjgooqtdp0eyb4b+ewcgfpyg4ktjgr9qti2bq3o1eylqh3sbwmhbhg2bh9k2sfraxzlhq9h8zryamahsupad5kkk8lyku8ogfdlrryo3p58hyogiwdvhdnswxldxoirup9zkrn0ibp/f1vif7pg1acvx3vypyvk5wrrxmlgqty9aezcirzzxqc0qnmryan6fby0/kihpjuyfdj7zimozd+cti5fb+uy0+8ec/fkxgln+jufbjnoiuk+cpuxq/xm7govge/gt+r6fawnnku/khw5y+6c6dz</latexit> Semantic Soundness discipline. A Semantic Soundness Theorem (based on a formal semantics for the language) states that well-type programs cannot go wrong and a Syntactic Soundness Theorem states that if fl accepts a program then it is well typed. We also discuss extending these THEOREM 1 (Semantic Soundness). If 7 respects jj and p 1 & is well typed then &[djt : 7. i.e., If if p` e d : : T then then = p e = : T. d : Syntax Semantics Every proof of type assignment says something meaningful about code [Milner 1978] 75

103 <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> Semantic Soundness discipline. A Semantic Soundness Theorem (based on a formal semantics for the language) states that well-type programs cannot go wrong and a Syntactic Soundness Theorem states that if fl accepts a program then it is well typed. We also discuss extending these THEOREM 1 (Semantic Soundness). If 7 respects jj and p 1 & is well typed then &[djt : 7. i.e., If if p` e d : : T then then = p e = : T. d : Compositional Reasoning Modular Reasoning [Milner 1978] 76

104 <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> Semantic Soundness discipline. A Semantic Soundness Theorem (based on a formal semantics for the language) states that well-type programs cannot go wrong and a Syntactic Soundness Theorem states that if fl accepts a program then it is well typed. We also discuss extending these THEOREM 1 (Semantic Soundness). If 7 respects jj and p 1 & is well typed then &[djt : 7. i.e., If if p` e d : : T then then = p e = : T. d : Syntax Semantics As a corollary, under the conditions of the theorem we have WIT # wrong, since wrong has no type. Whole-Program Payoff! [Milner 1978] 77

105 <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> Semantic Soundness implies discipline. A Semantic Soundness Theorem (based on a formal semantics for the language) states that well-type programs cannot go wrong and a Syntactic Soundness Theorem states that if fl accepts a program then it is well typed. We also discuss extending these THEOREM 1 (Semantic Soundness). If 7 respects jj and p 1 & is well typed then &[djt : 7. i.e., If if p` e d : : T then then = p e = : T. d : Syntax Semantics As a corollary, under the conditions of the theorem we have WIT # wrong, since wrong has no type. Whole-Program Payoff! [Milner 1978] 78

106 <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">aaacmnicbvbnaxrbeo1jnikbe1c9eilcah5kmakbxvpqg3qlkp2anwwp6anjnunugbprlizd/iyv+swbhpsgifd/hlmfhzxxqcpjvvdu10tlrtxh0bdga/vbw53dvuet/cchh0/at5/1ffe5sv1z6minuvsklauuk9y0kb2hstx108spc78/jedvyc95vtli4ivvuzlijtruf06yrrhwiywvqeuwh+qjgooqtdp0eyb4b+ewcgfpyg4ktjgr9qti2bq3o1eylqh3sbwmhbhg2bh9k2sfraxzlhq9h8zryamahsupad5kkk8lyku8ogfdlrryo3p58hyogiwdvhdnswxldxoirup9zkrn0ibp/f1vif7pg1acvx3vypyvk5wrrxmlgqty9aezcirzzxqc0qnmryan6fby0/kihpjuyfdj7zimozd+cti5fb+uy0+8ec/fkxgln+jufbjnoiuk+cpuxq/xm7govge/gt+r6fawnnku/khw5y+6c6dz</latexit> <latexit sha1_base64="xc7bemic1t8t5xixg+hzt35dagq=">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</latexit> Semantic Soundness implies discipline. A Semantic Soundness Theorem (based on a formal semantics for the language) states that well-type programs cannot go wrong and a Syntactic Soundness Theorem states that if fl accepts a program then it is well typed. We also discuss extending these THEOREM 1 (Semantic Soundness). If 7 respects jj and p 1 & is well typed then &[djt : 7. i.e., If if p` e d : : T then then = p e = : T. d : Syntax Semantics As a corollary, under the conditions of the theorem we have WIT # wrong, since wrong has no type. Whole-Program Payoff! [Milner 1978] 79

107 Discourse On The Method 80

108 Discourse On The Method 80

109 Discourse On The Method Payoff 80

110 Discourse On The Method Behavioural Invariant 80

111 Discourse On The Method Fragment Soundness is often(*) a Corollary Behavioural Invariant 80

112 <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> Syntactic Thinking Application0 Application1 Application2 Library1 Library2 ` e X ` e : T Two Typing Judgments = Two Behavioural Contracts Conflicts Signal Runtime Errors (go wrong!) 81

113 Semantic Thinking = e : T Application0 Application1 Application2 Library1 Library2 = e X (Semantically) Sound Gradual Typing Semantic Judgments Denote Behavioural Contracts Appropriate Linking Enforces Contracts 82

114 Outline Motivating Example (In Two Acts) Gradual Typing For All! Typing in Small Pieces Meat Strands and Related Works 83

115 Outline Motivating Example (In Two Acts) Gradual Typing For All! Typing in Small Pieces Meat Strands and Related Works 84

116 Gradual Typing 85

117 Gradual in which sense? 6.1 Gradual Typing Migratory typing is closely related to gradual typing [66, 67]. In the broad sense, the term gradual typing has come to describe any type system that allows some amount of dynamic typing. In the precise sense of Siek et al. [67], a gradual typing system includes: (1) a dynamic type that may be implicitly cast to any other type; (2) a relation between types that are equal up to occurrences [Greenman & Felleisen ICFP18] 86

118 Gradual in which sense? 6.1 Gradual Typing Migratory typing is closely related to gradual typing [66, 67]. In the broad sense, the term gradual typing has come to describe any type system that allows some amount of dynamic typing. In the precise sense of Siek et al. [67], a gradual typing system includes: (1) a dynamic type that may be implicitly cast to any other type; (2) a relation between types that are equal up to occurrences [Greenman & Felleisen ICFP18] 86

119 Gradual in which sense? 6.1 Gradual Typing Migratory typing is closely related to gradual typing [66, 67]. In the broad sense, the term gradual typing has come to describe any type system that allows some amount of dynamic typing. In the precise sense of Siek et al. [67], a gradual typing system includes: (1) a dynamic type that may be implicitly cast to any other type; (2) a relation between types that are equal up to occurrences [Greenman & Felleisen ICFP18] 86

120 Gradual Typing for Functional Languages Jeremy G. Siek University of Colorado Walid Taha Rice University Scheme

121 Gradual Typing for Functional Languages Jeremy G. Siek University of Colorado Walid Taha Rice University Rejected from ICFP

122 Gradual Typing for Functional Languages Jeremy G. Siek University of Colorado Walid Taha Rice University Rejected from ICFP 2006 >300 citations 89

123 Typing Gradually def f(x) = x + 2 def h(g) = g(1) h(f) Mixed Checking 90

124 Typing Gradually def f(x) = x + 2 def h(g) = g(1) Types might be inferred h(f) [Siek and Vachharajani DLS08] [Garcia and Cimini POPL15] Mixed Checking 91

125 Gradual Enforcement def f(x:bool) = x + 2 def h(g) = g(true) h(f) static error 92

126 <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> <latexit sha1_base64="s/o2rgsfr99gnxfhcxfwoiile6a=">aaab/hicbvbns8naen34wetxtecvi0xwvbirfe9fd3qs0c9oqplsnu3s3stsbgqh1l/ixymixv0h3vw3btsctpxbwoo9gwbmbslnsjvot7w2vrg5tv3ake/u7r8c2kfhbzvkktawsxgiuweoyllmw5pptruppcactjvb6g7md8zukpbetz2n1bcwifnecggj9e2kdw9capbgiaghpvggn/t21ak5c+bv4hakigo0+vaxfyykeztwhinspddjtt8bqrnhdfr2mkvticmy0j6hmqiq/mn8+ck+m0qio0saijweq78njicuykvgogxoovr2zuj/xi/t0bu/yxgaarqtxaio41gnejyedpmkrppceccsmvsxgyieok1ezrocu/zykmlf1fyn5j5evuu3rrwldijo0tly0rwqowfuqc1eui6e0st6s56sf+vd+li0rlnftax9gfx5a8zmk40=</latexit> Gradual Enforcement def f(x:bool) = x + 2 def h(g) = g(true) h(f) static error ` e : T 92

127 Gradual Enforcement def f(x:int) = x + 2 def h(g) = g(true) h(f) runtime error 93

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129 Refined Criteria for Gradual Typing ú Jeremy G. Siek 1, Michael M. Vitousek 2, Matteo Cimini 3, and John Tang Boyland Indiana University Bloomington, School of Informatics and Computing Gradual `G e : e T 94

130 Refined Criteria for Gradual Typing ú Jeremy G. Siek 1, Michael M. Vitousek 2, Matteo Cimini 3, and John Tang Boyland Indiana University Bloomington, School of Informatics and Computing Gradual `G e : e T Injection Static ` e : T Conservative Embedding 94

131 Refined Criteria for Gradual Typing ú Jeremy G. Siek 1, Michael M. Vitousek 2, Matteo Cimini 3, and John Tang Boyland Indiana University Bloomington, School of Informatics and Computing Gradual Compositional Translation `G e : e T Injection ` e X Dynamic Static ` e : T Conservative Embedding 94

132 Refined Criteria for Gradual Typing ú Jeremy G. Siek 1, Michael M. Vitousek 2, Matteo Cimini 3, and John Tang Boyland Indiana University Bloomington, School of Informatics and Computing Gradual Compositional Translation `G e : e T Injection ` e X Dynamic Static ` e : T Mixed 94

133 Refined Criteria for Gradual Typing ú Common Language Runtime Jeremy G. Siek 1, Michael M. Vitousek 2, Matteo Cimini 3, and John Tang Boyland Indiana University Bloomington, School of Informatics and Computing Gradual Compositional Translation `G e : e T Injection ` e X Dynamic Static ` e : T Mixed 94

134 Common Language Runtime Gradual Type Structure Determines Appropriate Linking Gradual Compositional Translation `G e : e T Injection ` e X Dynamic Static ` e : T Mixed 94

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sha1_base64="gzhfofgrriyy9e+5yvwmkfe79hg=">aaacphicbvhbbtnaen24byq5jewrl1ujec9edoueqgstwgtipbsrtjxiek3xk3svvvjemsha+ln4hw/ib1i7lkpsrlrpzjwzo7c0l8jifp3pbdu7d+7e27sfpnj46pgtbm//zjqy4ddmrpriimuwpnawroeslvicmeolnkfldzv//h0kk4we4tqhqwilleacm/shwbc3okdh7+iqxterffwejsjztx8nosbobrc3oe9ao531oi7jdc8vaosswtujoxynjhuouiqqteoloenltocjh5opsfpx9f7r5z6s0bkp/nnim+i/gy4pa9cq9urf8njuc3xwf9ykxpnbqrm6lxe0vy40lyvfq+tf0ewuwfgupwc8el5xyi9zwtj6dyubzswopr1gy5r2gddrsojgeuxmkupkjqg/enedv3vnvdrmpzukm8ppgm+0tbp109ayesnovmp3vc8pxcjdeu6jrkrpfw8f5zy4oxze0sd+8rp/mmwpt0eekqpykstkdtkhh8kpgrnovuqn+uv+by+cz8hxyhwtdtptzloyycg3v5ge0ne=</latexit> <latexit 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sha1_base64="e/fgdird2qduj5yobra9dleugni=">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</latexit> <latexit sha1_base64="e/fgdird2qduj5yobra9dleugni=">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</latexit> Simple Gradual Types Static Types (Type) T ::= B T! T Gradual Types (GType) U ::=? B U! U Type GType 95

136 <latexit sha1_base64="vo7zcg0ps6w3mi0wjgcpwqwholm=">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</latexit> <latexit sha1_base64="vo7zcg0ps6w3mi0wjgcpwqwholm=">aaacm3icbvhbshxbeo2dxntjru0erwhcannjminc8uskvitbbwnzlews0tntuzbbl6g7jmbpzf/knfmv/e16xggzqwunp06d6rrlprqok+rvl3r0+mnttfwn+nnzfy83t7zfntttwq4jbqsxlzlziiwgeqqucflaycqxcjhpt5r4xxewthj9frcltbsbatevngggvmwfnnj3hzqbxm31k0hsgr0p0g70swdnv9s9nxwgvwo0csmcg6djirpplaouoy6zykhj+jznybygzgrcxlct1/rnyao6nta80ell/p/hmxjuofkgvayv3wqsir+kjsucfph4ocskqfpbqtnkujs0mz8wwgjhuqiacstcr5rfm8s4hi3fs2uckkadygmuboa403ddjqqkwmfd2mcipxdrd+umpp5rc6pbisvkso87beos/hrsjlwttf4dumrmlglmf4ayekudh3olq8e5d873b2kysl8c9i+ou8otkx2yr96sllwnr+qjosmjwokmv8hv8ifaju6iz9hprttqdtmvyzjfo38guten</latexit> <latexit sha1_base64="vo7zcg0ps6w3mi0wjgcpwqwholm=">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</latexit> <latexit sha1_base64="vo7zcg0ps6w3mi0wjgcpwqwholm=">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</latexit> <latexit sha1_base64="xtiahkmmxnzaftuzgfso48xqray=">aaaconicbvflb9naen6yvzgppndksiigcypsvalurcqlcgofbivfubret9jv9uhujinr1r+ek/yo/puuxuslksot9m183+y8ikokh0lypygephz0+mnw0/jz8xcvt4c7r06cqs2hjbtp7fnbheihiuobes4qc0wvek6l5zewp/0j1gmjf+cqgqlicy3mgjmmodlwo6o5u3b14qdhgmaz4sgzj53r+ydtwyj0djtbgfi8nlxwojfl5twktsqcemzrcalnnncoksaxbagtadvt4ka+67yh70okphnjw9niu+i/gz4p51aqcerf8nxtcm3wf9ykxvnnqre6qhe0vy00ryvfq9s10fjy4chxatbureiv8nnmgcewrhitjapfdbcsjdipeocalrlrqvh6fk/xocivrpr7tvttl7w51f3fdevxjxfa1tn46cayesfovcz01c4pxcjfbu6jsondudln49whjx/hatjoj3dh+wf94bbig/kwfcap+ut2yse5ihnhpca/yr/yn3oxfyuoo++30mjq57wmaxbln5tw1fm=</latexit> <latexit sha1_base64="xtiahkmmxnzaftuzgfso48xqray=">aaaconicbvflb9naen6yvzgppndksiigcypsvalurcqlcgofbivfubret9jv9uhujinr1r+ek/yo/puuxuslksot9m183+y8ikokh0lypygephz0+mnw0/jz8xcvt4c7r06cqs2hjbtp7fnbheihiuobes4qc0wvek6l5zewp/0j1gmjf+cqgqlicy3mgjmmodlwo6o5u3b14qdhgmaz4sgzj53r+ydtwyj0djtbgfi8nlxwojfl5twktsqcemzrcalnnncoksaxbagtadvt4ka+67yh70okphnjw9niu+i/gz4p51aqcerf8nxtcm3wf9ykxvnnqre6qhe0vy00ryvfq9s10fjy4chxatbureiv8nnmgcewrhitjapfdbcsjdipeocalrlrqvh6fk/xocivrpr7tvttl7w51f3fdevxjxfa1tn46cayesfovcz01c4pxcjfbu6jsondudln49whjx/hatjoj3dh+wf94bbig/kwfcap+ut2yse5ihnhpca/yr/yn3oxfyuoo++30mjq57wmaxbln5tw1fm=</latexit> <latexit sha1_base64="xtiahkmmxnzaftuzgfso48xqray=">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</latexit> <latexit sha1_base64="xtiahkmmxnzaftuzgfso48xqray=">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</latexit> Gradual Type Precision??!??! Bool Int!? U v U (Int!?)! Bool Int! Int!? (Int! Int)! Bool Int! Int! Int!? Static Type Information ordering relation... 96

137 <latexit sha1_base64="gomqq0h+5qmci7arfg9vr8fchma=">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</latexit> <latexit sha1_base64="gomqq0h+5qmci7arfg9vr8fchma=">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</latexit> <latexit sha1_base64="gomqq0h+5qmci7arfg9vr8fchma=">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</latexit> <latexit sha1_base64="gomqq0h+5qmci7arfg9vr8fchma=">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</latexit> Consistent Lifting(*) U 1 U 2 Gradual Type Consistency (*) Reformulation of original definition 97

138 Consistent Lifting(*) 97 U 1 U 2 <latexit sha1_base64="gomqq0h+5qmci7arfg9vr8fchma=">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</latexit> <latexit sha1_base64="gomqq0h+5qmci7arfg9vr8fchma=">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</latexit> <latexit sha1_base64="gomqq0h+5qmci7arfg9vr8fchma=">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</latexit> <latexit sha1_base64="gomqq0h+5qmci7arfg9vr8fchma=">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</latexit> T 1 = T 2 <latexit sha1_base64="h6jf2x3rppsnrp+1dr0vkd0yqju=">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</latexit> <latexit sha1_base64="h6jf2x3rppsnrp+1dr0vkd0yqju=">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</latexit> <latexit sha1_base64="h6jf2x3rppsnrp+1dr0vkd0yqju=">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</latexit> <latexit sha1_base64="h6jf2x3rppsnrp+1dr0vkd0yqju=">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</latexit> v<latexit sha1_base64="heefdjg9uk2f47bil7+neabgksa=">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</latexit> l< ta xe ti s ah _1 ab es 46 "= eh Fe jd 9G Ku F2 74 ib 7L n+ ae gb SK =A >" AA CA Hn ci Vb bh ta At FE r2 Zb qi cz NX 6Y IG Em ec Bj KQ Vz Cu 40 We ku hi gd WK ac Vv F2 8m 2F 3V Ml sr R9 9l bb r6 9f VO ki tj Gd hd Jz zw 7O se NE ix PH p/ CB e9 lv bp 33 dg rb Z7 t2 Z7 b7 d4 VW a0 kb mo Th Xa bg gu Iu hi hc wr RW gi hm wn ck P3 va i7 of lz vw A3 QR TE WS Ke zt jm E6 jp N1 Y7 rm AO Hc 1O 42 7v Wc RP 5c 0C Eo ad 7O q/ ld ay 1a KZ MU Eg Xt ca Ax OV DH IX Ym qa ET Ls WB zv oo xo 4h Kp Bs XP F9 Fx zb S6 1R tn F/ YM dn /F xm Vy i1 k5 p5 SW r4 dv e5 gr 7/ xh di jp ei qo BK UE ye 0w UL EW qo Xo OE cx EA xo I8 ya 3w v2 be mu jh 0L wa Xq ly Qi HV Bn i0 tj GA qg I4 p5 R6 5W 0S q8 yl cl 6I 7I eq aq qa 07 Wn qo il /u HK Sz s1 K7 wt to Zx go S8 fr TV n2 D4 3P 30 SO 15 ox 5z sw uh 5c BG 13 7k fi Pf Q/ xo 02 9h gs 87 +g SO JQ U+ Oh Wy yd ot Ea UE +1 dk k/ XI fg Af +q tb e8 Up ng dz jk xs Ma g/ kj NA QM /< al et ix >t l< ta xe ti s ah _1 ab es 46 "= eh Fe jd 9G Ku F2 74 ib 7L n+ ae gb SK =A >" AA CA Hn ci Vb bh ta At FE r2 Zb qi cz NX 6Y IG Em ec Bj KQ Vz Cu 40 We ku hi gd WK ac Vv F2 8m 2F 3V Ml sr R9 9l bb r6 9f VO ki tj Gd hd Jz zw 7O se NE ix PH p/ CB e9 lv bp 33 dg rb Z7 t2 Z7 b7 d4 VW a0 kb mo Th Xa bg gu Iu hi hc wr RW gi hm wn ck P3 va i7 of lz vw A3 QR TE WS Ke zt jm E6 jp N1 Y7 rm AO Hc 1O 42 7v Wc RP 5c 0C Eo ad 7O q/ ld ay 1a KZ MU Eg Xt ca Ax OV DH IX Ym qa ET Ls WB zv oo xo 4h Kp Bs XP F9 Fx zb S6 1R tn F/ YM dn /F xm Vy i1 k5 p5 SW r4 dv e5 gr 7/ xh di jp ei qo BK UE ye 0w UL EW qo Xo OE cx EA xo I8 ya 3w v2 be mu jh 0L wa Xq ly Qi HV Bn i0 tj GA qg I4 p5 R6 5W 0S q8 yl cl 6I 7I eq aq qa 07 Wn qo il /u HK Sz s1 K7 wt to Zx go S8 fr TV n2 D4 3P 30 SO 15 ox 5z sw uh 5c BG 13 7k fi Pf Q/ xo 02 9h gs 87 +g SO JQ U+ Oh Wy yd ot Ea UE +1 dk k/ XI fg Af +q tb e8 Up ng dz jk xs Ma g/ kj NA QM /< al et ix >t l< ta xe ti s ah _1 ab es 46 "= eh Fe jd 9G Ku F2 74 ib 7L n+ ae gb SK =A >" AA CA Hn ci Vb bh ta At FE r2 Zb qi cz NX 6Y IG Em ec Bj KQ Vz Cu 40 We ku hi gd WK ac Vv F2 8m 2F 3V Ml sr R9 9l bb r6 9f VO ki tj Gd hd Jz zw 7O se NE ix PH p/ CB e9 lv bp 33 dg rb Z7 t2 Z7 b7 d4 VW a0 kb mo Th Xa bg gu Iu hi hc wr RW gi hm wn ck P3 va i7 of lz vw A3 QR TE WS Ke zt jm E6 jp N1 Y7 rm AO Hc 1O 42 7v Wc RP 5c 0C Eo ad 7O q/ ld ay 1a KZ MU Eg Xt ca Ax OV DH IX Ym qa ET Ls WB zv oo xo 4h Kp Bs XP F9 Fx zb S6 1R tn F/ YM dn /F xm Vy i1 k5 p5 SW r4 dv e5 gr 7/ xh di jp ei qo BK UE ye 0w UL EW qo Xo OE cx EA xo I8 ya 3w v2 be mu jh 0L wa Xq ly Qi HV Bn i0 tj GA qg I4 p5 R6 5W 0S q8 yl cl 6I 7I eq aq qa 07 Wn qo il /u HK Sz s1 K7 wt to Zx go S8 fr TV n2 D4 3P 30 SO 15 ox 5z sw uh 5c BG 13 7k fi Pf Q/ xo 02 9h gs 87 +g SO JQ U+ Oh Wy yd ot Ea UE +1 dk k/ XI fg Af +q tb e8 Up ng dz jk xs Ma g/ kj NA QM /< al et ix >t v<latexit sha1_base64="heefdjg9uk2f47bil7+neabgksa=">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</latexit> l< ta xe ti s ah _1 ab es 46 "= eh Fe jd 9G Ku F2 74 ib 7L n+ ae gb SK =A >" AA CA Hn ci Vb bh ta At FE r2 Zb qi cz NX 6Y IG Em ec Bj KQ Vz Cu 40 We ku hi gd WK ac Vv F2 8m 2F 3V Ml sr R9 9l bb r6 9f VO ki tj Gd hd Jz zw 7O se NE ix PH p/ CB e9 lv bp 33 dg rb Z7 t2 Z7 b7 d4 VW a0 kb mo Th Xa bg gu Iu hi hc wr RW gi hm wn ck P3 va i7 of lz vw A3 QR TE WS Ke zt jm E6 jp N1 Y7 rm AO Hc 1O 42 7v Wc RP 5c 0C Eo ad 7O q/ ld ay 1a KZ MU Eg Xt ca Ax OV DH IX Ym qa ET Ls WB zv oo xo 4h Kp Bs XP F9 Fx zb S6 1R tn F/ YM dn /F xm Vy i1 k5 p5 SW r4 dv e5 gr 7/ xh di jp ei qo BK UE ye 0w UL EW qo Xo OE cx EA xo I8 ya 3w v2 be mu jh 0L wa Xq ly Qi HV Bn i0 tj GA qg I4 p5 R6 5W 0S q8 yl cl 6I 7I eq aq qa 07 Wn qo il /u HK Sz s1 K7 wt to Zx go S8 fr TV n2 D4 3P 30 SO 15 ox 5z sw uh 5c BG 13 7k fi Pf Q/ xo 02 9h gs 87 +g SO JQ U+ Oh Wy yd ot Ea UE +1 dk k/ XI fg Af +q tb e8 Up ng dz jk xs Ma g/ kj NA QM /< al et ix >t l< ta xe ti s ah _1 ab es 46 "= eh Fe jd 9G Ku F2 74 ib 7L n+ ae gb SK =A >" AA CA Hn ci Vb bh ta At FE r2 Zb qi cz NX 6Y IG Em ec Bj KQ Vz Cu 40 We ku hi gd WK ac Vv F2 8m 2F 3V Ml sr R9 9l bb r6 9f VO ki tj Gd hd Jz zw 7O se NE ix PH p/ CB e9 lv bp 33 dg rb Z7 t2 Z7 b7 d4 VW a0 kb mo Th Xa bg gu Iu hi hc wr RW gi hm wn ck P3 va i7 of lz vw A3 QR TE WS Ke zt jm E6 jp N1 Y7 rm AO Hc 1O 42 7v Wc RP 5c 0C Eo ad 7O q/ ld ay 1a KZ MU Eg Xt ca Ax OV DH IX Ym qa ET Ls WB zv oo xo 4h Kp Bs XP F9 Fx zb S6 1R tn F/ YM dn /F xm Vy i1 k5 p5 SW r4 dv e5 gr 7/ xh di jp ei qo BK UE ye 0w UL EW qo Xo OE cx EA xo I8 ya 3w v2 be mu jh 0L wa Xq ly Qi HV Bn i0 tj GA qg I4 p5 R6 5W 0S q8 yl cl 6I 7I eq aq qa 07 Wn qo il /u HK Sz s1 K7 wt to Zx go S8 fr TV n2 D4 3P 30 SO 15 ox 5z sw uh 5c BG 13 7k fi Pf Q/ xo 02 9h gs 87 +g SO JQ U+ Oh Wy yd ot Ea UE +1 dk k/ XI fg Af +q tb e8 Up ng dz jk xs Ma g/ kj NA QM /< al et ix >t l< ta xe ti s ah _1 ab es 46 "= eh Fe jd 9G Ku F2 74 ib 7L n+ ae gb SK =A >" AA CA Hn ci Vb bh ta At FE r2 Zb qi cz NX 6Y IG Em ec Bj KQ Vz Cu 40 We ku hi gd WK ac Vv F2 8m 2F 3V Ml sr R9 9l bb r6 9f VO ki tj Gd hd Jz zw 7O se NE ix PH p/ CB e9 lv bp 33 dg rb Z7 t2 Z7 b7 d4 VW a0 kb mo Th Xa bg gu Iu hi hc wr RW gi hm wn ck P3 va i7 of lz vw A3 QR TE WS Ke zt jm E6 jp N1 Y7 rm AO Hc 1O 42 7v Wc RP 5c 0C Eo ad 7O q/ ld ay 1a KZ MU Eg Xt ca Ax OV DH IX Ym qa ET Ls WB zv oo xo 4h Kp Bs XP F9 Fx zb S6 1R tn F/ YM dn /F xm Vy i1 k5 p5 SW r4 dv e5 gr 7/ xh di jp ei qo BK UE ye 0w UL EW qo Xo OE cx EA xo I8 ya 3w v2 be mu jh 0L wa Xq ly Qi HV Bn i0 tj GA qg I4 p5 R6 5W 0S q8 yl cl 6I 7I eq aq qa 07 Wn qo il /u HK Sz s1 K7 wt to Zx go S8 fr TV n2 D4 3P 30 SO 15 ox 5z sw uh 5c BG 13 7k fi Pf Q/ xo 02 9h gs 87 +g SO JQ U+ Oh Wy yd ot Ea UE +1 dk k/ XI fg Af +q tb e8 Up ng dz jk xs Ma g/ kj NA QM /< al et ix >t if and only if For some T1 and T2 Gradual Type Consistency Static Type Equality (*) Reformulation of original definition

139 Static Checking static type equality gradual type consistency Int = Int Bool = Bool Int! Bool 6= Bool! Int extend Int Int Bool Bool Int! Bool 6 Bool! Int? Bool?! Bool Bool!? Consistency conservatively extends equality 98

140 Static Checking static type equality gradual type consistency Int = Int Bool = Bool Int! Bool 6= Bool! Int extend Int Int Bool Bool Int! Bool 6 Bool! Int? Bool?! Bool Bool!? Consistency conservatively extends equality 98

141 Static Checking static type equality gradual type consistency Int = Int Bool = Bool Int! Bool 6= Bool! Int extend Int Int Bool Bool Int! Bool 6 Bool! Int? Bool?! Bool Bool!? Consistency conservatively extends equality 98

142 Static Checking static type equality gradual type consistency Int = Int Bool = Bool Int! Bool 6= Bool! Int extend Int Int Bool Bool Int! Bool 6 Bool! Int? Bool?! Bool Bool!? Consistency conservatively extends equality 98

143 Consistent Lifting(*) 99 v<latexit sha1_base64="heefdjg9uk2f47bil7+neabgksa=">aaacnhicbvhbattaef2rbziqzcxny6gimecejbqkzvuc04ewukihdgkwcavv2fm8f2v3lmrs9rl9bb6rf9ovikjtdgdhzjwzo7esenxihp/pbc9evlpb33gdbr7z2t7z7b4dwv0abkomhtaxgbuguiihchrwwrigmhnwkc3pav7ifozlwv3arqetswektzmj6epj1n7ymroacho124v7cwprc5c0oedao7/qdlyaa1zkumgetxacxavohdximyaqteslbwvzoooxh4pksbpx9fxfbz6sr1nt/fmyndf/mxyv1i5k5pws4rvd5erg/7hxidpjieoqkbeueyw0luweoqoxeoxcaeox8iayw32vebumhjl0awqxyliqvhnb0ijtaggq4i5p6rw5s08qlylci6i7qeqaaq70nwoqliu/khzs1s7ktwotxzog8srfvt2n4dp303os51xoz5wshuc5gb31k7iffp/qox20h9sg78g+osqj+uhoywdytoaeeu1+kd/kixgffaq+bt8epugnzdkjsxam/gjkanmq</latexit> l< ta xe ti s ah _1 ab es 46 "= eh Fe jd 9G Ku F2 74 ib 7L n+ ae gb SK =A >" AA CA Hn ci Vb bh ta At FE r2 Zb qi cz NX 6Y IG Em ec Bj KQ Vz Cu 40 We ku hi gd WK ac Vv F2 8m 2F 3V Ml sr R9 9l bb r6 9f VO ki tj Gd hd Jz zw 7O se NE ix PH p/ CB e9 lv bp 33 dg rb Z7 t2 Z7 b7 d4 VW a0 kb mo Th Xa bg gu Iu hi hc wr RW gi hm wn ck P3 va i7 of lz vw A3 QR TE WS Ke zt jm E6 jp N1 Y7 rm AO Hc 1O 42 7v Wc RP 5c 0C Eo ad 7O q/ ld ay 1a KZ MU Eg Xt ca Ax OV DH IX Ym qa ET Ls WB zv oo xo 4h Kp Bs XP F9 Fx zb S6 1R tn F/ YM dn /F xm Vy i1 k5 p5 SW r4 dv e5 gr 7/ xh di jp ei qo BK UE ye 0w UL EW qo Xo OE cx EA xo I8 ya 3w v2 be mu jh 0L wa Xq ly Qi HV Bn i0 tj GA qg I4 p5 R6 5W 0S q8 yl cl 6I 7I eq aq qa 07 Wn qo il /u HK Sz s1 K7 wt to Zx go S8 fr TV n2 D4 3P 30 SO 15 ox 5z sw uh 5c BG 13 7k fi Pf Q/ xo 02 9h gs 87 +g SO JQ U+ Oh Wy yd ot Ea UE +1 dk k/ XI fg Af +q tb e8 Up ng dz jk xs Ma g/ kj NA QM /< al et ix >t l< ta xe ti s ah _1 ab es 46 "= eh Fe jd 9G Ku F2 74 ib 7L n+ ae gb SK =A >" AA CA Hn ci Vb bh ta At FE r2 Zb qi cz NX 6Y IG Em ec Bj KQ Vz Cu 40 We ku hi gd WK ac Vv F2 8m 2F 3V Ml sr R9 9l bb r6 9f VO ki tj Gd hd Jz zw 7O se NE ix PH p/ CB e9 lv bp 33 dg rb Z7 t2 Z7 b7 d4 VW a0 kb mo Th Xa bg gu Iu hi hc wr RW gi hm wn ck P3 va i7 of lz vw A3 QR TE WS Ke zt jm E6 jp N1 Y7 rm AO Hc 1O 42 7v Wc RP 5c 0C Eo ad 7O q/ ld ay 1a KZ MU Eg Xt ca Ax OV DH IX Ym qa ET Ls WB zv oo xo 4h Kp Bs XP F9 Fx zb S6 1R tn F/ YM dn /F xm Vy i1 k5 p5 SW r4 dv e5 gr 7/ xh di jp ei qo BK UE ye 0w UL EW qo Xo OE cx EA xo I8 ya 3w v2 be mu jh 0L wa Xq ly Qi HV Bn i0 tj GA qg I4 p5 R6 5W 0S q8 yl cl 6I 7I eq aq qa 07 Wn qo il /u HK Sz s1 K7 wt to Zx go S8 fr TV n2 D4 3P 30 SO 15 ox 5z sw uh 5c BG 13 7k fi Pf Q/ xo 02 9h gs 87 +g SO JQ U+ Oh Wy yd ot Ea UE +1 dk k/ XI fg Af +q tb e8 Up ng dz jk xs Ma g/ kj NA QM /< al et ix >t l< ta xe ti s ah _1 ab es 46 "= eh Fe jd 9G Ku F2 74 ib 7L n+ ae gb SK =A >" AA CA Hn ci Vb bh ta At FE r2 Zb qi cz NX 6Y IG Em ec Bj KQ Vz Cu 40 We ku hi gd WK ac Vv F2 8m 2F 3V Ml sr R9 9l bb r6 9f VO ki tj Gd hd Jz zw 7O se NE ix PH p/ CB e9 lv bp 33 dg rb Z7 t2 Z7 b7 d4 VW a0 kb mo Th Xa bg gu Iu hi hc wr RW gi hm wn ck P3 va i7 of lz vw A3 QR TE WS Ke zt jm E6 jp N1 Y7 rm AO Hc 1O 42 7v Wc RP 5c 0C Eo ad 7O q/ ld ay 1a KZ MU Eg Xt ca Ax OV DH IX Ym qa ET Ls WB zv oo xo 4h Kp Bs XP F9 Fx zb S6 1R tn F/ YM dn /F xm Vy i1 k5 p5 SW r4 dv e5 gr 7/ xh di jp ei qo BK UE ye 0w UL EW qo Xo OE cx EA xo I8 ya 3w v2 be mu jh 0L wa Xq ly Qi HV Bn i0 tj GA qg I4 p5 R6 5W 0S q8 yl cl 6I 7I eq aq qa 07 Wn qo il /u HK Sz s1 K7 wt to Zx go S8 fr TV n2 D4 3P 30 SO 15 ox 5z sw uh 5c BG 13 7k fi Pf Q/ xo 02 9h gs 87 +g SO JQ U+ Oh Wy yd ot Ea UE +1 dk k/ XI fg Af +q tb e8 Up ng dz jk xs Ma g/ kj NA QM /< al et ix >t v<latexit sha1_base64="heefdjg9uk2f47bil7+neabgksa=">aaacnhicbvhbattaef2rbziqzcxny6gimecejbqkzvuc04ewukihdgkwcavv2fm8f2v3lmrs9rl9bb6rf9ovikjtdgdhzjwzo7esenxihp/pbc9evlpb33gdbr7z2t7z7b4dwv0abkomhtaxgbuguiihchrwwrigmhnwkc3pav7ifozlwv3arqetswektzmj6epj1n7ymroacho124v7cwprc5c0oedao7/qdlyaa1zkumgetxacxavohdximyaqteslbwvzoooxh4pksbpx9fxfbz6sr1nt/fmyndf/mxyv1i5k5pws4rvd5erg/7hxidpjieoqkbeueyw0luweoqoxeoxcaeox8iayw32vebumhjl0awqxyliqvhnb0ijtaggq4i5p6rw5s08qlylci6i7qeqaaq70nwoqliu/khzs1s7ktwotxzog8srfvt2n4dp303os51xoz5wshuc5gb31k7iffp/qox20h9sg78g+osqj+uhoywdytoaeeu1+kd/kixgffaq+bt8epugnzdkjsxam/gjkanmq</latexit> l< ta xe ti s ah _1 ab es 46 "= eh Fe jd 9G Ku F2 74 ib 7L n+ ae gb SK =A >" AA CA Hn ci Vb bh ta At FE r2 Zb qi cz NX 6Y IG Em ec Bj KQ Vz Cu 40 We ku hi gd WK ac Vv F2 8m 2F 3V Ml sr R9 9l bb r6 9f VO ki tj Gd hd Jz zw 7O se NE ix PH p/ CB e9 lv bp 33 dg rb Z7 t2 Z7 b7 d4 VW a0 kb mo Th Xa bg gu Iu hi hc wr RW gi hm wn ck P3 va i7 of lz vw A3 QR TE WS Ke zt jm E6 jp N1 Y7 rm AO Hc 1O 42 7v Wc RP 5c 0C Eo ad 7O q/ ld ay 1a KZ MU Eg Xt ca Ax OV DH IX Ym qa ET Ls WB zv oo xo 4h Kp Bs XP F9 Fx zb S6 1R tn F/ YM dn /F xm Vy i1 k5 p5 SW r4 dv e5 gr 7/ xh di jp ei qo BK UE ye 0w UL EW qo Xo OE cx EA xo I8 ya 3w v2 be mu jh 0L wa Xq ly Qi HV Bn i0 tj GA qg I4 p5 R6 5W 0S q8 yl cl 6I 7I eq aq qa 07 Wn qo il /u HK Sz s1 K7 wt to Zx go S8 fr TV n2 D4 3P 30 SO 15 ox 5z sw uh 5c BG 13 7k fi Pf Q/ xo 02 9h gs 87 +g SO JQ U+ Oh Wy yd ot Ea UE +1 dk k/ XI fg Af +q tb e8 Up ng dz jk xs Ma g/ kj NA QM /< al et ix >t l< ta xe ti s ah _1 ab es 46 "= eh Fe jd 9G Ku F2 74 ib 7L n+ ae gb SK =A >" AA CA Hn ci Vb bh ta At FE r2 Zb qi cz NX 6Y IG Em ec Bj KQ Vz Cu 40 We ku hi gd WK ac Vv F2 8m 2F 3V Ml sr R9 9l bb r6 9f VO ki tj Gd hd Jz zw 7O se NE ix PH p/ CB e9 lv bp 33 dg rb Z7 t2 Z7 b7 d4 VW a0 kb mo Th Xa bg gu Iu hi hc wr RW gi hm wn ck P3 va i7 of lz vw A3 QR TE WS Ke zt jm E6 jp N1 Y7 rm AO Hc 1O 42 7v Wc RP 5c 0C Eo ad 7O q/ ld ay 1a KZ MU Eg Xt ca Ax OV DH IX Ym qa ET Ls WB zv oo xo 4h Kp Bs XP F9 Fx zb S6 1R tn F/ YM dn /F xm Vy i1 k5 p5 SW r4 dv e5 gr 7/ xh di jp ei qo BK UE ye 0w UL EW qo Xo OE cx EA xo I8 ya 3w v2 be mu jh 0L wa Xq ly Qi HV Bn i0 tj GA qg I4 p5 R6 5W 0S q8 yl cl 6I 7I eq aq qa 07 Wn qo il /u HK Sz s1 K7 wt to Zx go S8 fr TV n2 D4 3P 30 SO 15 ox 5z sw uh 5c BG 13 7k fi Pf Q/ xo 02 9h gs 87 +g SO JQ U+ Oh Wy yd ot Ea UE +1 dk k/ XI fg Af +q tb e8 Up ng dz jk xs Ma g/ kj NA QM /< al et ix >t l< ta xe ti s ah _1 ab es 46 "= eh Fe jd 9G Ku F2 74 ib 7L n+ ae gb SK =A >" AA CA Hn ci Vb bh ta At FE r2 Zb qi cz NX 6Y IG Em ec Bj KQ Vz Cu 40 We ku hi gd WK ac Vv F2 8m 2F 3V Ml sr R9 9l bb r6 9f VO ki tj Gd hd Jz zw 7O se NE ix PH p/ CB e9 lv bp 33 dg rb Z7 t2 Z7 b7 d4 VW a0 kb mo Th Xa bg gu Iu hi hc wr RW gi hm wn ck P3 va i7 of lz vw A3 QR TE WS Ke zt jm E6 jp N1 Y7 rm AO Hc 1O 42 7v Wc RP 5c 0C Eo ad 7O q/ ld ay 1a KZ MU Eg Xt ca Ax OV DH IX Ym qa ET Ls WB zv oo xo 4h Kp Bs XP F9 Fx zb S6 1R tn F/ YM dn /F xm Vy i1 k5 p5 SW r4 dv e5 gr 7/ xh di jp ei qo BK UE ye 0w UL EW qo Xo OE cx EA xo I8 ya 3w v2 be mu jh 0L wa Xq ly Qi HV Bn i0 tj GA qg I4 p5 R6 5W 0S q8 yl cl 6I 7I eq aq qa 07 Wn qo il /u HK Sz s1 K7 wt to Zx go S8 fr TV n2 D4 3P 30 SO 15 ox 5z sw uh 5c BG 13 7k fi Pf Q/ xo 02 9h gs 87 +g SO JQ U+ Oh Wy yd ot Ea UE +1 dk k/ XI fg Af +q tb e8 Up ng dz jk xs Ma g/ kj NA QM /< al et ix >t if and only if For some T1 and T2 Consistent Subtyping Static Subtyping T 1 <: T 2 <latexit 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sha1_base64="dedvzfqzznzehl+so8ujgscjox8=">aaacphicbvflb9naen64pip5jexizuue4htzcaluqdolltguqdpkcyjw60m6yj6s3tfptpin9aq/ix/d2rvukjlsst/m983oky+lcjgkf3rr3r37dx7up4ofp3n67hl/chdmtgu5tlirxl7kzieugiyoumjfaygpxmj5vjpu+pmfyj0w+htuspgpttriitjdejr3b9myzhkcc0lr7erjpo8pk1hsgr0l0g4mswen80hpz4xhlqknxdlnpmls4swzi4jlqoosclayvmjlmaaomqi3823vnx0digvdgbuertpg/83wtdm3uxlqkoaxbpdrgv/jphuups680gwfoplnouulkrraliiwwgjhuqmacstcr5rfmss4hnxfw2uckkadyguuboa407dmrgvf4bnx7toek0t047qpqvfarhvbcvszvn2rbzydn46mkbec1qtdv82cuiz9z8apudtnudld49wfz+9gatjkv7wfhh51h9snl8kr8pak5am5jj/ikzkqttbkmvwiv6m30efoazs5kua9lucf2blo+19vhnsz</latexit> Consistent Lifting Int. Int Int 6. Bool Int. > >6. Int Int.??. Int 100

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Int Int 6. Bool Int. > >6. Int Int.??. Int 100

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Int Int 6. Bool Int. > unknown is not the top type >6. Int Int.??. Int 100

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sha1_base64="m0+btmm45d6j6r4fyyy1wd/jig0=">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</latexit> <latexit sha1_base64="m0+btmm45d6j6r4fyyy1wd/jig0=">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</latexit> Lift Typing RuLes Static Type System Gradual Type System ` t 1 : T 1 ` t 2 : T 2 T 1 = Int T 2 = Int ` t 1 : U 1 ` t 2 : U 2 U 1 Int U 2 Int ` t 1 + t 2 : Int ` t 1 + t 2 : Int 101

148 Dynamic Semantics Static Checking Gradual Language Type-Directed Translation Runtime Checking Instrumentation Language Cast Calculus 102

149 Common Language Runtime Gradual 103

150 Common Language Runtime Gradual Static Types (Type) Gradual Types (GType) T ::= B T! T U ::=? B U! U 103

151 Also Works as a Surface Language! Gradual Static Types (Type) Gradual Types (GType) T ::= B T! T U ::=? B U! U 103

152 Also Works as a Surface Language! Gradual Static Types (Type) T ::= B T! T Gradual Types (GType) U ::=? B U! U Much of the Literature is Written This Way 103

153 Refined Criteria for Gradual Typing ú Jeremy G. Siek 1, Michael M. Vitousek 2, Matteo Cimini 3, and John Tang Boyland Indiana University Bloomington, School of Informatics and Computing Gradual 104

154 Refined Criteria for Gradual Typing ú Jeremy G. Siek 1, Michael M. Vitousek 2, Matteo Cimini 3, and John Tang Boyland Indiana University Bloomington, School of Informatics and Computing Static and Dynamic Gradual Guarantee! Gradual 104

155 Refined Criteria for Gradual Typing ú Jeremy G. Siek 1, Michael M. Vitousek 2, Matteo Cimini 3, and John Tang Boyland Indiana University Bloomington, School of Informatics and Computing Static and Dynamic Gradual Guarantee! Gradual 104

156 Refined Criteria for Gradual Typing ú Jeremy G. Siek 1, Michael M. Vitousek 2, Matteo Cimini 3, and John Tang Boyland Indiana University Bloomington, School of Informatics and Computing Static and Dynamic Gradual Guarantee! Gradual Varying The Type Precision of a Program Monotonically Changes only static and dynamic type errors 104

157 Robust Theoretical Framework Dynamic Gradual Static 105

158 Robust Theoretical Framework Dynamic Gradual Static Unityped [Siek and Taha 06] Simple 105

159 Robust Theoretical Framework Dynamic Gradual Static Unityped Unityped [Siek and Taha 06] [Siek and Taha 08] Simple Subtyping 105

160 Robust Theoretical Framework Dynamic Gradual Static Unityped Unityped [Siek and Taha 06] [Siek and Taha 08] Simple Subtyping Unityped [Siek and Vachharajani 08] Hindley/Milner 105

161 Robust Theoretical Framework Dynamic Gradual Static Unityped Unityped [Siek and Taha 06] [Siek and Taha 08] Simple Subtyping Unityped Simple [Siek and Vachharajani 08] [Lehmann and Tanter 17] Hindley/Milner Refinement 105

162 Robust Theoretical Framework Dynamic Gradual Static Unityped Unityped [Siek and Taha 06] [Siek and Taha 08] Simple Subtyping Unityped Simple Simple [Siek and Vachharajani 08] [Lehmann and Tanter 17] [Bañados et al. 14] Hindley/Milner Refinement Type&Effect 105

163 Robust Theoretical Framework Dynamic Gradual Static Unityped Unityped [Siek and Taha 06] [Siek and Taha 08] Simple Subtyping Unityped Simple Simple [Siek and Vachharajani 08] [Lehmann and Tanter 17] [Bañados et al. 14] Hindley/Milner Refinement Type&Effect Simple [Toro et al. to appear] 105 Security

164 Robust Theoretical Framework Dynamic Gradual Static Unityped [Siek and Taha 06] Simple Unityped Unityped Simple Simple Simple [Siek and Taha 08] [Siek and Vachharajani 08] [Lehmann and Tanter 17] [Bañados et al. 14] [Toro et al. to appear] And More! 105 Subtyping Hindley/Milner Refinement Type&Effect Security

165 Challenge: Dynamics Static Checking Gradual Language Type-Directed Translation Runtime Checking Instrumentation Language Cast Calculus 106

166 Challenge: Dynamics Static Checking Gradual Language Given the name gradual typing, one might think that the most interesting aspect is the type system. It turns out that the dynamic semantics of gradually-typed languages is more complex than the static semantics, with many points in the Type-Directed design space [Wadler and Findler 2009; Siek et al. 2009] and many challenges concerning Translation [Siek and Garcia 2012] Runtime Checking Instrumentation Language Cast Calculus 106

167 Challenge: Dynamics Static Checking Gradual Language Type-Directed Translation Runtime Checking Instrumentation Language Cast Calculus 106

168 static type system & type safety proof interpretation of gradual types Abstracting Gradual Typing Ronald Garcia Alison M. Clark Software Practices Lab Department of Computer Science University of British Columbia, Canada Éric Tanter PLEIAD Laboratory Computer Science Department (DCC) University of Chile, Chile POPL 2016 gradual language TYPE SYSTEM DYNAMIC SEMANTICS 107

169 Breadth of AGT Applications of AGT so far records with subtyping gradual rows (à la row polymorphism) security typing effect typing refinement types set-theoretic types parametric polymorphism POPL 16 TOPLAS 18 ICFP 14 (statics) POPL 17 ICFP 17 (statics) ongoing work

170 Outline Motivating Example (In Two Acts) Gradual Typing For All! Typing in Small Pieces Meat Strands and Related Works 109

171 Gradual Typing for Functional Languages Jeremy G. Siek University of Colorado Walid Taha Rice University Scheme

172 Interlanguage Migration: From Scripts to Programs Sam Tobin-Hochstadt Northeastern University Boston, MA Matthias Felleisen Northeastern University Boston, MA DLS

173 Interlanguage Migration: From Scripts to Programs Sam Tobin-Hochstadt Northeastern University Boston, MA Matthias Felleisen Northeastern University Boston, MA DLS 2006 Dynamic 111

174 Interlanguage Migration: From Scripts to Programs Sam Tobin-Hochstadt Northeastern University Boston, MA Matthias Felleisen Northeastern University Boston, MA DLS 2006 Dynamic 111

175 Interlanguage Migration: From Scripts to Programs Sam Tobin-Hochstadt Northeastern University Boston, MA Matthias Felleisen Northeastern University Boston, MA DLS 2006 Dynamic Mixed 111

176 Interlanguage Migration: From Scripts to Programs Sam Tobin-Hochstadt Northeastern University Boston, MA Matthias Felleisen Northeastern University Boston, MA DLS 2006 Dynamic Mixed 111

177 Retrospective Migratory Typing: Ten Years Later Sam Tobin-Hochstadt, Matthias Felleisen, Robert Bruce Findler, Matthew Flatt, Ben Greenman, Andrew M. Kent, Vincent St-Amour, T. Stephen Strickland, Asumu Takikawa 1 1 PLT *@racket-lang.org Abstract In this day and age, many developers work on large, untyped code repositories. Even if they are the creators of the code, they notice that they have to figure out the equivalent of method signatures every time they work on old code. This step is time consuming and error prone. Ten years ago, the two lead authors outlined a linguistic solution to this problem. Specifically they proposed the creation of typed twins for untyped programming languages so that developers could migrate scripts from the untyped world to a typed one in an incremental manner. Their programmatic paper also spelled out three guiding design principles concerning the acceptance of grown idioms, the soundness of mixed-typed programs, and the units of migration. This paper revisits this idea of a migratory type system as implemented for Racket. It explains how the design principles have been used to produce the Typed Racket twin and presents an assessment of the project s status, highlighting successes and failures ACM Subject Classification D.3 Language Design, F.3.3 Type Structure 112

178 Ben Greenman Matthias Felleisen 113

179 Soft Typing PLDI 1991 A Practical Soft Type System for Scheme ANDREW K. WRIGHT NEC Research Institute and ROBERT CARTWRIGHT Rice University TOPLAS

180 Soft Typing PLDI 1991 A Practical Soft Type System for Scheme ANDREW K. WRIGHT NEC Research Institute and ROBERT CARTWRIGHT Rice University TOPLAS 1997 Idea: Use H/M Type Inference to Migrate Dynamic Programs 114

181 Set-Based Analysis PLDI96 TOPLAS99 Not Types! 115

182 Set-Based Analysis PLDI96 TOPLAS99 Not Types! 116

183 Set-Based Analysis William Bowman PLDI96 TOPLAS99 Not Types! 116

184 Migration By Inference The Ins and Outs of Gradual Type Inference Aseem Rastogi Stony Brook University Avik Chaudhuri Basil Hosmer Advanced Technology Labs, Adobe Systems POPL

185 Migration By Inference The Ins and Outs of Gradual Type Inference Aseem Rastogi Stony Brook University Avik Chaudhuri Basil Hosmer Advanced Technology Labs, Adobe Systems POPL 2012 Migrating Gradual Types JOHN PETER CAMPORA III, SHENG CHEN, MARTIN ERWIG, ERIC WALKINGSHAW, 117 University of Louisiana at Lafayette University of Louisiana at Lafayette Oregon State University Oregon State University

186 John Campora Sheng Chen Eric Walkingshaw 118

187 Dynamic Typing Dynamic typing: syntax and proof theory* Fritz Henglein** University of Copenhagen, Universitetsparken 1, 2100 Copenhagen 0, Denmark Received July 1992; revised March

188 Dynamic Typing Dynamic typing: syntax and proof theory* Fritz Henglein** University of Copenhagen, Universitetsparken 1, 2100 Copenhagen 0, Denmark Received July 1992; revised March

189 Influence Herman, et al. [TFP 2007] Siek, Garcia, Taha [ESOP 2008] Siek and Wadler [POPL 2010] Garcia [ICFP 2013] Siek et al. [PLDI 2015] 120

190 Fresh Influence Max New Amal Ahmed 121

191 Outline Motivating Example (In Two Acts) Gradual Typing For All! Typing in Small Pieces Meat Strands and Related Works 122

192 123

193 Gratitude 124

194 125

195 It Takes a Village Andy Lumsdaine Dan Friedman Frank Pfenning Amr Sabry 126

196 It Takes a Village Andy Lumsdaine Dan Friedman Frank Pfenning Amr Sabry 126

197 It Takes a Village Andy Lumsdaine Dan Friedman Frank Pfenning Amr Sabry 126

198 It Takes a Village Andy Lumsdaine Dan Friedman Frank Pfenning Amr Sabry 126

199 It Takes a Village Andy Lumsdaine Dan Friedman Frank Pfenning Amr Sabry 126

200 It Takes a Village Me Andy Lumsdaine Dan Friedman Frank Pfenning Amr Sabry 126

201 127

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205 Students 128

206 Students 128

207 Common Language Runtime Gradual Type Structure Determines Appropriate Linking Gradual `G e : e T Dynamic Static ` e X ` e : T Mixed 129

208 Dear Today Ron, You went one slide too far. Go back one slide. Yesterday Ron 130

209 Bonus Tracks 131

210 Common Language Runtime Gradual `G e : e T 132

211 Common Language Runtime Gradual `G e : e T Injection Static ` e : T 132

212 Common Language Runtime Gradual Compositional Translation `G e : e T Injection ` e X Dynamic Static ` e : T 132

213 Common Language Runtime Gradual Type Structure Determines Appropriate Linking Gradual Compositional Translation `G e : e T Injection ` e X Dynamic Static ` e : T Mixed 132

214 Refined Criteria for Gradual Typing ú Jeremy G. Siek 1, Michael M. Vitousek 2, Matteo Cimini 3, and John Tang Boyland Indiana University Bloomington, School of Informatics and Computing Static and Dynamic Gradual Guarantee! Gradual Varying The Type Precision of a Program Monotonically Changes only static and dynamic type errors 133

215 Blame 134

216 Theorems about Blame Tobin-Hochstadt and Felleisen 2006 Wadler and Findler 2008 Dimoulas et al. Dimoulas Takikawa 135

217 Racket Contract Blame point-in?: contract violation expected: real? given: #f in: the 2nd argument of (-> pict? real? real? boolean?) contract from: point-in-module blaming: top-level (assuming the contract is correct) Robby Findler 172 Findler 136

218 Wherein Shriram Unwittingly Writes My Blame Schpiel For Me 137

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222 Racket Contract Blame point-in?: contract violation expected: real? given: #f in: the 2nd argument of (-> pict? real? real? boolean?) contract from: point-in-module blaming: top-level (assuming the contract is correct) Robby Findler 172 Findler 141