{"id":395447,"date":"2017-06-05T16:14:56","date_gmt":"2017-06-05T23:14:56","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&p=395447"},"modified":"2022-01-04T07:33:34","modified_gmt":"2022-01-04T15:33:34","slug":"life-without-cons","status":"publish","type":"msr-video","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/video\/life-without-cons\/","title":{"rendered":"Life without CONS"},"content":{"rendered":"
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Can higher-order functional programs solve more problems than first-order programs? Answer: NO, since both program classes are Turing complete. The reason is that higher-order values can be simulated by first-order values: use function “closures” built by the list constructor “CONS”.<\/p>\n

Complexity theory characterizes the expressive power of “cons-free” programs with different data orders and control structures. We characterize exactly the problems of type [Bool]->Bool solvable by programs with respect to DATA: presence or absence of constructors, and data types (order 0, 1, or higher); and CONTROL: general recursion, tail recursion, primitive recursion. An instance for first-order cons-free programs: general recursion is more powerful than primitive recursion (e.g. “fold”) if and only if PTIME properly contains LOGSPACE. (This is a long-standing open problem in complexity.)<\/p>\n

Second-order cons-free programs are more powerful than first-order: a function of type [Bool]->Bool is in PTIME iff it is first-order cons-free computable; and in EXPTIME iff it is second-order cons-free computable. (PTIME is known to be a strictly smaller class that EXPTIME.)<\/p>\n

Further, deterministic cons-free programs characterize a complexity hierarchy. Programs with data orders 0, 1, 2, … can solve exactly these problems:<\/p>\n

PTIME < EXPTIME < EXP2TIME < … < ELEMENTARY,<\/p>\n

What happens if we add NON-determinism? Using some clever programming with first-order functions as data, Kop and Simonsen obtain a surprising result (2017), that this hierarchy “collapses” at second order:<\/p>\n

PTIME < ELEMENTARY = ELEMENTARY = … = ELEMENTARY<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/form>\n","protected":false},"excerpt":{"rendered":"

Can higher-order functional programs solve more problems than first-order programs? Answer: NO, since both program classes are Turing complete. The reason is that higher-order values can be simulated by first-order values: use function “closures” built by the list constructor “CONS”. Complexity theory characterizes the expressive power of “cons-free” programs with different data orders and control […]<\/p>\n","protected":false},"featured_media":395462,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[13561,13546],"msr-video-type":[206954],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-395447","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-research-area-algorithms","msr-research-area-computational-sciences-mathematics","msr-video-type-microsoft-research-talks","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/EaDN4Xu80ec","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/395447","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":4,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/395447\/revisions"}],"predecessor-version":[{"id":395465,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/395447\/revisions\/395465"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/media\/395462"}],"wp:attachment":[{"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/media?parent=395447"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=395447"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=395447"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=395447"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=395447"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=395447"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=395447"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=395447"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=395447"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=395447"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}