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"pageCreatedAt": "2016-06-15 12:19:26", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 4, "isEditorComment": false, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 198, "text": "[summary:Every [-effectively_computable] [-3jy] is [Turing_machine Turing-computable].]\n\nThe **Church-Turing thesis** (often abbreviated **CT thesis**) states:\n> Every effectively computable function is Turing-computable\n\nThat is, for every function which can be computed by physical means %%note:So no [ hypercomputers]%% there exists a [ Turing machine] which computes exactly that.\n\nThe Church-Turing thesis is not a definite mathematical statement, but an inductive statement which affirms that every sensible model of computation we can come up with is equivalent or at least [reduction reducible] to the model proposed by Turing. Thus, we cannot prove it in a mathematical sense, but we can [ gather evidence for it].\n\nFor example, this model was proven to coincide with Church's lambda calculus, another universal model of computation, and the equivalence between Church's lambda calculus and Turing's automatic machines is often taken as [4j7 evidence] that they correctly capture our intuition of \"effectively computable\".\n\nThere are many [ consequences] of the CT thesis for computer science in general, artificial intelligence, [11w epistemology], and other fields of knowledge.", "metaText": "", "isTextLoaded": true, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 2, "maintainerCount": 2, "userSubscriberCount": 0, "lastVisit": "", "hasDraft": false, "votes": [], "voteSummary": null, "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", 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Church-Turing thesis", "clickbait": "Why do we believe in CT thesis?", "textLength": 2069, "alias": "evidence_for_CT_thesis", "externalUrl": "", "sortChildrenBy": "likes", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "1yq", "editCreatedAt": "2016-06-20 19:59:25", "pageCreatorId": "2vh", "pageCreatedAt": "2016-06-17 19:17:43", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 3, "isEditorComment": false, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 115, "text": "As the Church-Turing thesis is not a [ proper mathematical sentence] we cannot prove it. However, we can collect [-3s] to increase our confidence in its correctness.\n\n#The inductive argument\nEvery computational model we have seen so far is [ reducible] to Turing's model.\n\nIndeed, the thesis was originally formulated independently by Church and Turing in reference to two different computational models ([Turing_machines] and [Lambda_Calculus] respectively). When they were shown to be [equivilance_relation equivalent] it was massive evidence in favor of both of them.\n\nA non-exhaustive list of models which can be shown to be reducible to Turing machines are:\n\n* [Lambda_calculus]\n* [Quantum_computation]\n* [Nondeterministic_Turing_machines Non-deterministic_Turing_machines]\n* [Register_machines]\n* The set of [-recursive_functions]\n\n#Lack of counterexamples\nPerhaps the strongest argument we have for the CT thesis is that there is not a widely accepted candidate to a counterexample of the thesis. This is unlike the [-4ct], where quantum computation stands as a likely counterexample.\n\nOne may wonder whether the computational models which use a source of [ randomness] (such as quantum computation or [-probabilistic_Turing_machines]) are a proper counterexample to the thesis: after all, Turing machines are fully [-deterministic], so they cannot simulate randomness.\n\nTo properly explain this issue we have to recall what it means for a quantum computer or a probabilistic Turing machine to compute something: we say that such a device computes a function $f$ if for every input $x$ the probability of the device outputting $f(x)$ is greater or equal to some arbitrary constant greater than $1/2$. Thus we can compute $f$ in a classical machine, for there exists always the possibility of simulating every possible outcome of the randomness to deterministically compute the probability distribution of the output, and output as a result the possible outcome with greater probability.\n\nThus, randomness is reducible to Turing machines, and the CT thesis holds.", "metaText": "", "isTextLoaded": true, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 2, "maintainerCount": 2, "userSubscriberCount": 0, "lastVisit": "", "hasDraft": false, "votes": [], "voteSummary": [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": { "edit": { "has": false, "reason": "You don't have 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Not sure I understand this. I have BS in...\"", "clickbait": "", "textLength": 116, "alias": "4mc", "externalUrl": "", "sortChildrenBy": "recentFirst", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "1", "editCreatedAt": "2016-06-20 04:32:10", "pageCreatorId": "1", "pageCreatedAt": "2016-06-20 04:32:10", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 0, "isEditorComment": true, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "To properly explain this issue we have to recall what it means for a quantum computer or a probabilistic Turing machine to compute something: we say that such a device computes a function $f$ if for every input $x$ the probability of the device outputting $f(x)$ is greater or equal to some arbitrary constant greater than $1/2$\\. Thus we can compute $f$ in a classical machine, for there exists always the possibility of simulating every possible outcome of the randomness to deterministically compute the probability distribution of the output, and output as a result the possible outcome with greater probability\\.", "anchorText": "Turing machine to compute something: we say that such a device computes a function $f$ if for every input $x$ the probability of the device outputting $f(x)$ is greater or equal to some arbitrary constant greater than $1/2$\\. T", "anchorOffset": 105, "mergedInto": "", "isDeleted": false, "viewCount": 104, "text": "Huh... Not sure I understand this. I have BS in CS, but don't remember running across this. Would love to read more.", "metaText": "", "isTextLoaded": true, "isSubscribedToDiscussion": false, "isSubscribedToUser": false, "isSubscribedAsMaintainer": false, "discussionSubscriberCount": 2, "maintainerCount": 2, "userSubscriberCount": 0, "lastVisit": "", "hasDraft": false, "votes": [], "voteSummary": null, "muVoteSummary": 0, "voteScaling": 0, "currentUserVote": -2, "voteCount": 0, "lockedVoteType": "", "maxEditEver": 0, "redLinkCount": 0, "lockedBy": "", "lockedUntil": "", "nextPageId": "", "prevPageId": "", "usedAsMastery": false, "proposalEditNum": 0, "permissions": { "edit": { "has": false, "reason": "You don't have domain permission to edit this page" }, "proposeEdit": { "has": true, "reason": "" }, "delete": { "has": false, "reason": "You don't have domain permission to delete this page" }, "comment": { "has": false, "reason": "You can't comment in this domain because you are not a member" }, "proposeComment": { "has": true, "reason": "" } }, "summaries": {}, "creatorIds": [], "childIds": [], "parentIds": [ "4j7" ], "commentIds": [ "4p2" ], "questionIds": [], "tagIds": [], "relatedIds": [], "markIds": [], "explanations": [], "learnMore": [], "requirements": [], "subjects": [], "lenses": [], "lensParentId": "", "pathPages": [], "learnMoreTaughtMap": {}, "learnMoreCoveredMap": {}, "learnMoreRequiredMap": {}, "editHistory": {}, "domainSubmissions": {}, "answers": [], "answerCount": 0, "commentCount": 0, "newCommentCount": 0, "linkedMarkCount": 0, "changeLogs": [], "feedSubmissions": [], "searchStrings": {}, "hasChildren": false, "hasParents": true, "redAliases": {}, "improvementTagIds": [], "nonMetaTagIds": [], "todos": [], "slowDownMap": null, "speedUpMap": null, "arcPageIds": null, "contentRequests": {} }, "4p2": { "likeableId": "2800", "likeableType": "page", "myLikeValue": 0, "likeCount": 1, "dislikeCount": 0, "likeScore": 1, "individualLikes": [ "1" ], "pageId": "4p2", "edit": 1, "editSummary": "", "prevEdit": 0, "currentEdit": 1, "wasPublished": true, "type": "comment", "title": "\"I'll try to cover probabilistic Turing machines...\"", "clickbait": "", "textLength": 108, "alias": "4p2", "externalUrl": "", "sortChildrenBy": "recentFirst", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "2vh", "editCreatedAt": "2016-06-21 18:35:47", "pageCreatorId": "2vh", "pageCreatedAt": "2016-06-21 18:35:47", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 0, "isEditorComment": true, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 97, "text": "I'll try to cover probabilistic Turing machines in other article once I have time. 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