A Non-Uniformly C-Productive Sequence & Non-Constructive - PowerPoint PPT Presentation

A Non-Uniformly C-Productive Sequence & Non-Constructive Disjunctions John Case 1 Michael Ralston 1 Yohji Akama 2 1 Computer & Information Sciences University of Delaware Newark, DE USA Email: { case , mralston } @udel.edu 2 Mathematical Institute Tohoku University Sendai, Japan Email: akama@m.tohoku.ac.jp Revision of Talk at Asian Logic Conference 2013 , Guangzhou, China

For Your Speed Reading Pleasure & Quick Impression ( .. ⌣ ) 1 Introduction Motivation Basic Definition & Relevant Theorem Proof of Theorem 2 Characterizing the Index Set Cases Uniform C-Productivity of S q , q ∈ M The Characterization Another Corollary of the Characterization 3 Further Examples & Future Work 4 References Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 2 / 11

Introduction Motivation Introduction & Motivation Let ϕ be an acceptable programming system/numbering of the partial computable functions: N = { 0 , 1 , 2 , . . . } → N , where, for p ∈ N , ϕ p is the partial computable function computed by program p of the ϕ -system; such numberings are characterized as intercompilable with standard general purpose programming formalisms such as a full TM formalism; let W p = domain( ϕ p ) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q , { x | ϕ x = ϕ q } is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain( ϕ q ) ∞ vs. not ∞ , a Π 0 2 -LEM [ABHK04]. A student asked why the proof involved cases. The answer given straightaway to the student was that his teacher didn’t know how else to do it. (. . ⌣ ) The present paper provides, among other things, a better answer: any proof that, for each q , { x | ϕ x = ϕ q } is not c.e. provably must involve some such non-constructivity. Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 3 / 11

Introduction Motivation Introduction & Motivation Let ϕ be an acceptable programming system/numbering of the partial computable functions: N = { 0 , 1 , 2 , . . . } → N , where, for p ∈ N , ϕ p is the partial computable function computed by program p of the ϕ -system; such numberings are characterized as intercompilable with standard general purpose programming formalisms such as a full TM formalism; let W p = domain( ϕ p ) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q , { x | ϕ x = ϕ q } is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain( ϕ q ) ∞ vs. not ∞ , a Π 0 2 -LEM [ABHK04]. A student asked why the proof involved cases. The answer given straightaway to the student was that his teacher didn’t know how else to do it. (. . ⌣ ) The present paper provides, among other things, a better answer: any proof that, for each q , { x | ϕ x = ϕ q } is not c.e. provably must involve some such non-constructivity. Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 3 / 11

Introduction Motivation Introduction & Motivation Let ϕ be an acceptable programming system/numbering of the partial computable functions: N = { 0 , 1 , 2 , . . . } → N , where, for p ∈ N , ϕ p is the partial computable function computed by program p of the ϕ -system; such numberings are characterized as intercompilable with standard general purpose programming formalisms such as a full TM formalism; let W p = domain( ϕ p ) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q , { x | ϕ x = ϕ q } is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain( ϕ q ) ∞ vs. not ∞ , a Π 0 2 -LEM [ABHK04]. A student asked why the proof involved cases. The answer given straightaway to the student was that his teacher didn’t know how else to do it. (. . ⌣ ) The present paper provides, among other things, a better answer: any proof that, for each q , { x | ϕ x = ϕ q } is not c.e. provably must involve some such non-constructivity. Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 3 / 11

Introduction Motivation Introduction & Motivation Let ϕ be an acceptable programming system/numbering of the partial computable functions: N = { 0 , 1 , 2 , . . . } → N , where, for p ∈ N , ϕ p is the partial computable function computed by program p of the ϕ -system; such numberings are characterized as intercompilable with standard general purpose programming formalisms such as a full TM formalism; let W p = domain( ϕ p ) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q , { x | ϕ x = ϕ q } is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain( ϕ q ) ∞ vs. not ∞ , a Π 0 2 -LEM [ABHK04]. A student asked why the proof involved cases. The answer given straightaway to the student was that his teacher didn’t know how else to do it. (. . ⌣ ) The present paper provides, among other things, a better answer: any proof that, for each q , { x | ϕ x = ϕ q } is not c.e. provably must involve some such non-constructivity. Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 3 / 11

Introduction Motivation Introduction & Motivation Let ϕ be an acceptable programming system/numbering of the partial computable functions: N = { 0 , 1 , 2 , . . . } → N , where, for p ∈ N , ϕ p is the partial computable function computed by program p of the ϕ -system; such numberings are characterized as intercompilable with standard general purpose programming formalisms such as a full TM formalism; let W p = domain( ϕ p ) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q , { x | ϕ x = ϕ q } is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain( ϕ q ) ∞ vs. not ∞ , a Π 0 2 -LEM [ABHK04]. A student asked why the proof involved cases. The answer given straightaway to the student was that his teacher didn’t know how else to do it. (. . ⌣ ) The present paper provides, among other things, a better answer: any proof that, for each q , { x | ϕ x = ϕ q } is not c.e. provably must involve some such non-constructivity. Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 3 / 11

Introduction Motivation Introduction & Motivation Let ϕ be an acceptable programming system/numbering of the partial computable functions: N = { 0 , 1 , 2 , . . . } → N , where, for p ∈ N , ϕ p is the partial computable function computed by program p of the ϕ -system; such numberings are characterized as intercompilable with standard general purpose programming formalisms such as a full TM formalism; let W p = domain( ϕ p ) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q , { x | ϕ x = ϕ q } is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain( ϕ q ) ∞ vs. not ∞ , a Π 0 2 -LEM [ABHK04]. A student asked why the proof involved cases. The answer given straightaway to the student was that his teacher didn’t know how else to do it. (. . ⌣ ) The present paper provides, among other things, a better answer: any proof that, for each q , { x | ϕ x = ϕ q } is not c.e. provably must involve some such non-constructivity. Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 3 / 11

Introduction Motivation Introduction & Motivation Let ϕ be an acceptable programming system/numbering of the partial computable functions: N = { 0 , 1 , 2 , . . . } → N , where, for p ∈ N , ϕ p is the partial computable function computed by program p of the ϕ -system; such numberings are characterized as intercompilable with standard general purpose programming formalisms such as a full TM formalism; let W p = domain( ϕ p ) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q , { x | ϕ x = ϕ q } is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain( ϕ q ) ∞ vs. not ∞ , a Π 0 2 -LEM [ABHK04]. A student asked why the proof involved cases. The answer given straightaway to the student was that his teacher didn’t know how else to do it. (. . ⌣ ) The present paper provides, among other things, a better answer: any proof that, for each q , { x | ϕ x = ϕ q } is not c.e. provably must involve some such non-constructivity. Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 3 / 11