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A Non-Uniformly C-Productive Sequence & Non-Constructive Disjunctions

John Case1 Michael Ralston1 Yohji Akama2

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 Sq, 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, ϕp 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 Wp = domain(ϕp) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q, {x | ϕx = ϕq} is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain(ϕq) ∞ vs. not ∞, a Π0

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 | ϕx = ϕq} 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, ϕp 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 Wp = domain(ϕp) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q, {x | ϕx = ϕq} is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain(ϕq) ∞ vs. not ∞, a Π0

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 | ϕx = ϕq} 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, ϕp 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 Wp = domain(ϕp) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q, {x | ϕx = ϕq} is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain(ϕq) ∞ vs. not ∞, a Π0

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 | ϕx = ϕq} 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, ϕp 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 Wp = domain(ϕp) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q, {x | ϕx = ϕq} is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain(ϕq) ∞ vs. not ∞, a Π0

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 | ϕx = ϕq} 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, ϕp 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 Wp = domain(ϕp) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q, {x | ϕx = ϕq} is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain(ϕq) ∞ vs. not ∞, a Π0

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 | ϕx = ϕq} 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, ϕp 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 Wp = domain(ϕp) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q, {x | ϕx = ϕq} is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain(ϕq) ∞ vs. not ∞, a Π0

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 | ϕx = ϕq} 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, ϕp 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 Wp = domain(ϕp) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q, {x | ϕx = ϕq} is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain(ϕq) ∞ vs. not ∞, a Π0

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 | ϕx = ϕq} 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, ϕp 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 Wp = domain(ϕp) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q, {x | ϕx = ϕq} is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain(ϕq) ∞ vs. not ∞, a Π0

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 | ϕx = ϕq} 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, ϕp 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 Wp = domain(ϕp) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q, {x | ϕx = ϕq} is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain(ϕq) ∞ vs. not ∞, a Π0

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 | ϕx = ϕq} 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, ϕp 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 Wp = domain(ϕp) [Rog58, Rog67]. The first author taught a recursion theorem proof by cases that, for each q, {x | ϕx = ϕq} is not computably enumerable (c.e.). The non-constructive disjunction of cases [Bro81] used was ≈ domain(ϕq) ∞ vs. not ∞, a Π0

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 | ϕx = ϕq} 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 Basic Definition & Relevant Theorem

Basic Definition & Relevant Theorem

Recall: the completely productive (abbr: c-productive) sets (⊆ N) are the effectively non-c.e. sets, i.e., the sets S st (∃ computable f )(∀y)[f (y) ∈ ((S − Wy) ∪ (Wy − S))] [Pos44, Myh55, Dek55, Rog67]. Idea: f (y) is a counterexample to S = Wy. Definition A set sequence Sq, q = 0, 1, 2, . . . is uniformly c-productive iff there is a computable f so that, for all q, y, f (q, y) is a counterexample to Sq = Wy. Relevance: A completely constructive proof that each Sq is not c.e. would entail the Sqs forming a uniformly c-productive sequence. Let Eq = {x | ϕx = ϕq}. Then: Theorem The set sequence Eq, q = 0, 1, 2, . . . is not uniformly c-productive.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 4 / 11

Introduction Basic Definition & Relevant Theorem

Basic Definition & Relevant Theorem

Recall: the completely productive (abbr: c-productive) sets (⊆ N) are the effectively non-c.e. sets, i.e., the sets S st (∃ computable f )(∀y)[f (y) ∈ ((S − Wy) ∪ (Wy − S))] [Pos44, Myh55, Dek55, Rog67]. Idea: f (y) is a counterexample to S = Wy. Definition A set sequence Sq, q = 0, 1, 2, . . . is uniformly c-productive iff there is a computable f so that, for all q, y, f (q, y) is a counterexample to Sq = Wy. Relevance: A completely constructive proof that each Sq is not c.e. would entail the Sqs forming a uniformly c-productive sequence. Let Eq = {x | ϕx = ϕq}. Then: Theorem The set sequence Eq, q = 0, 1, 2, . . . is not uniformly c-productive.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 4 / 11

Introduction Basic Definition & Relevant Theorem

Basic Definition & Relevant Theorem

Recall: the completely productive (abbr: c-productive) sets (⊆ N) are the effectively non-c.e. sets, i.e., the sets S st (∃ computable f )(∀y)[f (y) ∈ ((S − Wy) ∪ (Wy − S))] [Pos44, Myh55, Dek55, Rog67]. Idea: f (y) is a counterexample to S = Wy. Definition A set sequence Sq, q = 0, 1, 2, . . . is uniformly c-productive iff there is a computable f so that, for all q, y, f (q, y) is a counterexample to Sq = Wy. Relevance: A completely constructive proof that each Sq is not c.e. would entail the Sqs forming a uniformly c-productive sequence. Let Eq = {x | ϕx = ϕq}. Then: Theorem The set sequence Eq, q = 0, 1, 2, . . . is not uniformly c-productive.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 4 / 11

Introduction Basic Definition & Relevant Theorem

Basic Definition & Relevant Theorem

Recall: the completely productive (abbr: c-productive) sets (⊆ N) are the effectively non-c.e. sets, i.e., the sets S st (∃ computable f )(∀y)[f (y) ∈ ((S − Wy) ∪ (Wy − S))] [Pos44, Myh55, Dek55, Rog67]. Idea: f (y) is a counterexample to S = Wy. Definition A set sequence Sq, q = 0, 1, 2, . . . is uniformly c-productive iff there is a computable f so that, for all q, y, f (q, y) is a counterexample to Sq = Wy. Relevance: A completely constructive proof that each Sq is not c.e. would entail the Sqs forming a uniformly c-productive sequence. Let Eq = {x | ϕx = ϕq}. Then: Theorem The set sequence Eq, q = 0, 1, 2, . . . is not uniformly c-productive.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 4 / 11

Introduction Basic Definition & Relevant Theorem

Basic Definition & Relevant Theorem

Recall: the completely productive (abbr: c-productive) sets (⊆ N) are the effectively non-c.e. sets, i.e., the sets S st (∃ computable f )(∀y)[f (y) ∈ ((S − Wy) ∪ (Wy − S))] [Pos44, Myh55, Dek55, Rog67]. Idea: f (y) is a counterexample to S = Wy. Definition A set sequence Sq, q = 0, 1, 2, . . . is uniformly c-productive iff there is a computable f so that, for all q, y, f (q, y) is a counterexample to Sq = Wy. Relevance: A completely constructive proof that each Sq is not c.e. would entail the Sqs forming a uniformly c-productive sequence. Let Eq = {x | ϕx = ϕq}. Then: Theorem The set sequence Eq, q = 0, 1, 2, . . . is not uniformly c-productive.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 4 / 11

Introduction Basic Definition & Relevant Theorem

Basic Definition & Relevant Theorem

Recall: the completely productive (abbr: c-productive) sets (⊆ N) are the effectively non-c.e. sets, i.e., the sets S st (∃ computable f )(∀y)[f (y) ∈ ((S − Wy) ∪ (Wy − S))] [Pos44, Myh55, Dek55, Rog67]. Idea: f (y) is a counterexample to S = Wy. Definition A set sequence Sq, q = 0, 1, 2, . . . is uniformly c-productive iff there is a computable f so that, for all q, y, f (q, y) is a counterexample to Sq = Wy. Relevance: A completely constructive proof that each Sq is not c.e. would entail the Sqs forming a uniformly c-productive sequence. Let Eq = {x | ϕx = ϕq}. Then: Theorem The set sequence Eq, q = 0, 1, 2, . . . is not uniformly c-productive.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 4 / 11

Introduction Proof of Theorem

Proof of Theorem

Suppose for contradiction that f is a computable function so that, for all q, y, f (q, y) is a counterexample to Eq (i.e., {x | ϕx = ϕq}) = Wy. By the Double Recursion Theorem there are programs q0, y0 each of which creates a copy of itself and the other and each uses its copies together with an algorithm for f to compute f (q0, y0) with: ϕq0 = ϕf (q0,y0) & Wy0 = {f (q0, y0)}. Since f (q0, y0) ∈ ({x | ϕx = ϕq0} ∩ Wy0), f (q0, y0) fails to be a counterexample to Eq0 = Wy0. ⇒⇐ By contrast, {q, x | ϕx = ϕq} is, of course, c-productive, Why? Insightful answer: Let q1 be st, say, ϕq1 = λy (1). Then, easily, K ≤1{x | ϕx = ϕq1}≤1 {q, x | ϕx = ϕq} — entailing [Rog67] the c-productivity of {x | ϕx = ϕq1} & {q, x | ϕx = ϕq}.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 5 / 11

Introduction Proof of Theorem

Proof of Theorem

Suppose for contradiction that f is a computable function so that, for all q, y, f (q, y) is a counterexample to Eq (i.e., {x | ϕx = ϕq}) = Wy. By the Double Recursion Theorem there are programs q0, y0 each of which creates a copy of itself and the other and each uses its copies together with an algorithm for f to compute f (q0, y0) with: ϕq0 = ϕf (q0,y0) & Wy0 = {f (q0, y0)}. Since f (q0, y0) ∈ ({x | ϕx = ϕq0} ∩ Wy0), f (q0, y0) fails to be a counterexample to Eq0 = Wy0. ⇒⇐ By contrast, {q, x | ϕx = ϕq} is, of course, c-productive, Why? Insightful answer: Let q1 be st, say, ϕq1 = λy (1). Then, easily, K ≤1{x | ϕx = ϕq1}≤1 {q, x | ϕx = ϕq} — entailing [Rog67] the c-productivity of {x | ϕx = ϕq1} & {q, x | ϕx = ϕq}.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 5 / 11

Introduction Proof of Theorem

Proof of Theorem

Suppose for contradiction that f is a computable function so that, for all q, y, f (q, y) is a counterexample to Eq (i.e., {x | ϕx = ϕq}) = Wy. By the Double Recursion Theorem there are programs q0, y0 each of which creates a copy of itself and the other and each uses its copies together with an algorithm for f to compute f (q0, y0) with: ϕq0 = ϕf (q0,y0) & Wy0 = {f (q0, y0)}. Since f (q0, y0) ∈ ({x | ϕx = ϕq0} ∩ Wy0), f (q0, y0) fails to be a counterexample to Eq0 = Wy0. ⇒⇐ By contrast, {q, x | ϕx = ϕq} is, of course, c-productive, Why? Insightful answer: Let q1 be st, say, ϕq1 = λy (1). Then, easily, K ≤1{x | ϕx = ϕq1}≤1 {q, x | ϕx = ϕq} — entailing [Rog67] the c-productivity of {x | ϕx = ϕq1} & {q, x | ϕx = ϕq}.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 5 / 11

Introduction Proof of Theorem

Proof of Theorem

Suppose for contradiction that f is a computable function so that, for all q, y, f (q, y) is a counterexample to Eq (i.e., {x | ϕx = ϕq}) = Wy. By the Double Recursion Theorem there are programs q0, y0 each of which creates a copy of itself and the other and each uses its copies together with an algorithm for f to compute f (q0, y0) with: ϕq0 = ϕf (q0,y0) & Wy0 = {f (q0, y0)}. Since f (q0, y0) ∈ ({x | ϕx = ϕq0} ∩ Wy0), f (q0, y0) fails to be a counterexample to Eq0 = Wy0. ⇒⇐ By contrast, {q, x | ϕx = ϕq} is, of course, c-productive, Why? Insightful answer: Let q1 be st, say, ϕq1 = λy (1). Then, easily, K ≤1{x | ϕx = ϕq1}≤1 {q, x | ϕx = ϕq} — entailing [Rog67] the c-productivity of {x | ϕx = ϕq1} & {q, x | ϕx = ϕq}.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 5 / 11

Introduction Proof of Theorem

Proof of Theorem

Suppose for contradiction that f is a computable function so that, for all q, y, f (q, y) is a counterexample to Eq (i.e., {x | ϕx = ϕq}) = Wy. By the Double Recursion Theorem there are programs q0, y0 each of which creates a copy of itself and the other and each uses its copies together with an algorithm for f to compute f (q0, y0) with: ϕq0 = ϕf (q0,y0) & Wy0 = {f (q0, y0)}. Since f (q0, y0) ∈ ({x | ϕx = ϕq0} ∩ Wy0), f (q0, y0) fails to be a counterexample to Eq0 = Wy0. ⇒⇐ By contrast, {q, x | ϕx = ϕq} is, of course, c-productive, Why? Insightful answer: Let q1 be st, say, ϕq1 = λy (1). Then, easily, K ≤1{x | ϕx = ϕq1}≤1 {q, x | ϕx = ϕq} — entailing [Rog67] the c-productivity of {x | ϕx = ϕq1} & {q, x | ϕx = ϕq}.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 5 / 11

Introduction Proof of Theorem

Proof of Theorem

Suppose for contradiction that f is a computable function so that, for all q, y, f (q, y) is a counterexample to Eq (i.e., {x | ϕx = ϕq}) = Wy. By the Double Recursion Theorem there are programs q0, y0 each of which creates a copy of itself and the other and each uses its copies together with an algorithm for f to compute f (q0, y0) with: ϕq0 = ϕf (q0,y0) & Wy0 = {f (q0, y0)}. Since f (q0, y0) ∈ ({x | ϕx = ϕq0} ∩ Wy0), f (q0, y0) fails to be a counterexample to Eq0 = Wy0. ⇒⇐ By contrast, {q, x | ϕx = ϕq} is, of course, c-productive, Why? Insightful answer: Let q1 be st, say, ϕq1 = λy (1). Then, easily, K ≤1{x | ϕx = ϕq1}≤1 {q, x | ϕx = ϕq} — entailing [Rog67] the c-productivity of {x | ϕx = ϕq1} & {q, x | ϕx = ϕq}.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 5 / 11

Introduction Proof of Theorem

Proof of Theorem

Suppose for contradiction that f is a computable function so that, for all q, y, f (q, y) is a counterexample to Eq (i.e., {x | ϕx = ϕq}) = Wy. By the Double Recursion Theorem there are programs q0, y0 each of which creates a copy of itself and the other and each uses its copies together with an algorithm for f to compute f (q0, y0) with: ϕq0 = ϕf (q0,y0) & Wy0 = {f (q0, y0)}. Since f (q0, y0) ∈ ({x | ϕx = ϕq0} ∩ Wy0), f (q0, y0) fails to be a counterexample to Eq0 = Wy0. ⇒⇐ By contrast, {q, x | ϕx = ϕq} is, of course, c-productive, Why? Insightful answer: Let q1 be st, say, ϕq1 = λy (1). Then, easily, K ≤1{x | ϕx = ϕq1}≤1 {q, x | ϕx = ϕq} — entailing [Rog67] the c-productivity of {x | ϕx = ϕq1} & {q, x | ϕx = ϕq}.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 5 / 11

Characterizing the Index Set Cases Uniform C-Productivity of Sq, q ∈ M

Uniform C-Productivity of Sq, q ∈ M

An index set is a set of ϕ-programs M so that, for some class of (1-argument) partial computable functions S, M = {p | ϕp ∈ S}. Example (complementary) index sets include those implicit in the non-constructive disjunction of cases mentioned early on above: Minf = {q | domain(ϕq) ∞} vs. Mfin = {q | domain(ϕq) not ∞}. For any index set M, we define the subsequence of sets Sq, q ∈ M, to be uniformly c-productive iff, for some partial computable η, for any q ∈ M & any y, η(q, y)↓ to a counterexample to Sq = Wy. For Sq = Eq, when η just above exists, it can be taken to be total. By a pair of Kleene Parametric Recursion Theorem arguments, for each M ∈ {Minf, Mfin}, the corresponding subsequence Eq, q ∈ M, is uniformly c-productive. These two proofs each involve Σ0

1-LEM,

strictly subsumed by Π0

2-LEM above [ABHK04].

This provides a now-known-to-be necessarily non-constructive proof that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 6 / 11

Characterizing the Index Set Cases Uniform C-Productivity of Sq, q ∈ M

Uniform C-Productivity of Sq, q ∈ M

An index set is a set of ϕ-programs M so that, for some class of (1-argument) partial computable functions S, M = {p | ϕp ∈ S}. Example (complementary) index sets include those implicit in the non-constructive disjunction of cases mentioned early on above: Minf = {q | domain(ϕq) ∞} vs. Mfin = {q | domain(ϕq) not ∞}. For any index set M, we define the subsequence of sets Sq, q ∈ M, to be uniformly c-productive iff, for some partial computable η, for any q ∈ M & any y, η(q, y)↓ to a counterexample to Sq = Wy. For Sq = Eq, when η just above exists, it can be taken to be total. By a pair of Kleene Parametric Recursion Theorem arguments, for each M ∈ {Minf, Mfin}, the corresponding subsequence Eq, q ∈ M, is uniformly c-productive. These two proofs each involve Σ0

1-LEM,

strictly subsumed by Π0

2-LEM above [ABHK04].

This provides a now-known-to-be necessarily non-constructive proof that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 6 / 11

Characterizing the Index Set Cases Uniform C-Productivity of Sq, q ∈ M

Uniform C-Productivity of Sq, q ∈ M

An index set is a set of ϕ-programs M so that, for some class of (1-argument) partial computable functions S, M = {p | ϕp ∈ S}. Example (complementary) index sets include those implicit in the non-constructive disjunction of cases mentioned early on above: Minf = {q | domain(ϕq) ∞} vs. Mfin = {q | domain(ϕq) not ∞}. For any index set M, we define the subsequence of sets Sq, q ∈ M, to be uniformly c-productive iff, for some partial computable η, for any q ∈ M & any y, η(q, y)↓ to a counterexample to Sq = Wy. For Sq = Eq, when η just above exists, it can be taken to be total. By a pair of Kleene Parametric Recursion Theorem arguments, for each M ∈ {Minf, Mfin}, the corresponding subsequence Eq, q ∈ M, is uniformly c-productive. These two proofs each involve Σ0

1-LEM,

strictly subsumed by Π0

2-LEM above [ABHK04].

This provides a now-known-to-be necessarily non-constructive proof that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 6 / 11

Characterizing the Index Set Cases Uniform C-Productivity of Sq, q ∈ M

Uniform C-Productivity of Sq, q ∈ M

An index set is a set of ϕ-programs M so that, for some class of (1-argument) partial computable functions S, M = {p | ϕp ∈ S}. Example (complementary) index sets include those implicit in the non-constructive disjunction of cases mentioned early on above: Minf = {q | domain(ϕq) ∞} vs. Mfin = {q | domain(ϕq) not ∞}. For any index set M, we define the subsequence of sets Sq, q ∈ M, to be uniformly c-productive iff, for some partial computable η, for any q ∈ M & any y, η(q, y)↓ to a counterexample to Sq = Wy. For Sq = Eq, when η just above exists, it can be taken to be total. By a pair of Kleene Parametric Recursion Theorem arguments, for each M ∈ {Minf, Mfin}, the corresponding subsequence Eq, q ∈ M, is uniformly c-productive. These two proofs each involve Σ0

1-LEM,

strictly subsumed by Π0

2-LEM above [ABHK04].

This provides a now-known-to-be necessarily non-constructive proof that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 6 / 11

Characterizing the Index Set Cases Uniform C-Productivity of Sq, q ∈ M

Uniform C-Productivity of Sq, q ∈ M

An index set is a set of ϕ-programs M so that, for some class of (1-argument) partial computable functions S, M = {p | ϕp ∈ S}. Example (complementary) index sets include those implicit in the non-constructive disjunction of cases mentioned early on above: Minf = {q | domain(ϕq) ∞} vs. Mfin = {q | domain(ϕq) not ∞}. For any index set M, we define the subsequence of sets Sq, q ∈ M, to be uniformly c-productive iff, for some partial computable η, for any q ∈ M & any y, η(q, y)↓ to a counterexample to Sq = Wy. For Sq = Eq, when η just above exists, it can be taken to be total. By a pair of Kleene Parametric Recursion Theorem arguments, for each M ∈ {Minf, Mfin}, the corresponding subsequence Eq, q ∈ M, is uniformly c-productive. These two proofs each involve Σ0

1-LEM,

strictly subsumed by Π0

2-LEM above [ABHK04].

This provides a now-known-to-be necessarily non-constructive proof that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 6 / 11

Characterizing the Index Set Cases Uniform C-Productivity of Sq, q ∈ M

Uniform C-Productivity of Sq, q ∈ M

An index set is a set of ϕ-programs M so that, for some class of (1-argument) partial computable functions S, M = {p | ϕp ∈ S}. Example (complementary) index sets include those implicit in the non-constructive disjunction of cases mentioned early on above: Minf = {q | domain(ϕq) ∞} vs. Mfin = {q | domain(ϕq) not ∞}. For any index set M, we define the subsequence of sets Sq, q ∈ M, to be uniformly c-productive iff, for some partial computable η, for any q ∈ M & any y, η(q, y)↓ to a counterexample to Sq = Wy. For Sq = Eq, when η just above exists, it can be taken to be total. By a pair of Kleene Parametric Recursion Theorem arguments, for each M ∈ {Minf, Mfin}, the corresponding subsequence Eq, q ∈ M, is uniformly c-productive. These two proofs each involve Σ0

1-LEM,

strictly subsumed by Π0

2-LEM above [ABHK04].

This provides a now-known-to-be necessarily non-constructive proof that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 6 / 11

Characterizing the Index Set Cases Uniform C-Productivity of Sq, q ∈ M

Uniform C-Productivity of Sq, q ∈ M

An index set is a set of ϕ-programs M so that, for some class of (1-argument) partial computable functions S, M = {p | ϕp ∈ S}. Example (complementary) index sets include those implicit in the non-constructive disjunction of cases mentioned early on above: Minf = {q | domain(ϕq) ∞} vs. Mfin = {q | domain(ϕq) not ∞}. For any index set M, we define the subsequence of sets Sq, q ∈ M, to be uniformly c-productive iff, for some partial computable η, for any q ∈ M & any y, η(q, y)↓ to a counterexample to Sq = Wy. For Sq = Eq, when η just above exists, it can be taken to be total. By a pair of Kleene Parametric Recursion Theorem arguments, for each M ∈ {Minf, Mfin}, the corresponding subsequence Eq, q ∈ M, is uniformly c-productive. These two proofs each involve Σ0

1-LEM,

strictly subsumed by Π0

2-LEM above [ABHK04].

This provides a now-known-to-be necessarily non-constructive proof that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 6 / 11

Characterizing the Index Set Cases The Characterization

The Characterization

There are divisions into non-constructive (top level) disjunctions besides the above example for proving that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e. {q | domain(ϕq) is not recursive} vs. its complement also works, but it’s Π0

3-LEM, more non-constructive than Π0 2-LEM above [ABHK04].

Let Fx, x ∈ N, be a canonical indexing [Rog67, MY78] of all/only the finite functions: N → N. We have for our characterization: Theorem For any index set M and corresponding subsequence of sets Eq = {x | ϕx = ϕq}, q ∈ M, the subsequence is uniformly c-productive iff (∃ c.e. A ⊆ M)(∀x)(∃y ∈ A)[ϕy ⊇ Fx]. Wolog: y effective in x. The disjoint index sets partitioning N, {q | domain(ϕq) = ∅} (c.e.)

• vs. its complement, {q | domain(ϕq) = ∅}, do not work — since,

by our above characterization and Rice-Shapiro [Rog67], for any = ∅ c.e. index set M, Eq, q ∈ M, is not uniformly c-productive.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 7 / 11

Characterizing the Index Set Cases The Characterization

The Characterization

There are divisions into non-constructive (top level) disjunctions besides the above example for proving that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e. {q | domain(ϕq) is not recursive} vs. its complement also works, but it’s Π0

3-LEM, more non-constructive than Π0 2-LEM above [ABHK04].

Let Fx, x ∈ N, be a canonical indexing [Rog67, MY78] of all/only the finite functions: N → N. We have for our characterization: Theorem For any index set M and corresponding subsequence of sets Eq = {x | ϕx = ϕq}, q ∈ M, the subsequence is uniformly c-productive iff (∃ c.e. A ⊆ M)(∀x)(∃y ∈ A)[ϕy ⊇ Fx]. Wolog: y effective in x. The disjoint index sets partitioning N, {q | domain(ϕq) = ∅} (c.e.)

• vs. its complement, {q | domain(ϕq) = ∅}, do not work — since,

by our above characterization and Rice-Shapiro [Rog67], for any = ∅ c.e. index set M, Eq, q ∈ M, is not uniformly c-productive.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 7 / 11

Characterizing the Index Set Cases The Characterization

The Characterization

There are divisions into non-constructive (top level) disjunctions besides the above example for proving that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e. {q | domain(ϕq) is not recursive} vs. its complement also works, but it’s Π0

3-LEM, more non-constructive than Π0 2-LEM above [ABHK04].

Let Fx, x ∈ N, be a canonical indexing [Rog67, MY78] of all/only the finite functions: N → N. We have for our characterization: Theorem For any index set M and corresponding subsequence of sets Eq = {x | ϕx = ϕq}, q ∈ M, the subsequence is uniformly c-productive iff (∃ c.e. A ⊆ M)(∀x)(∃y ∈ A)[ϕy ⊇ Fx]. Wolog: y effective in x. The disjoint index sets partitioning N, {q | domain(ϕq) = ∅} (c.e.)

• vs. its complement, {q | domain(ϕq) = ∅}, do not work — since,

by our above characterization and Rice-Shapiro [Rog67], for any = ∅ c.e. index set M, Eq, q ∈ M, is not uniformly c-productive.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 7 / 11

Characterizing the Index Set Cases The Characterization

The Characterization

There are divisions into non-constructive (top level) disjunctions besides the above example for proving that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e. {q | domain(ϕq) is not recursive} vs. its complement also works, but it’s Π0

3-LEM, more non-constructive than Π0 2-LEM above [ABHK04].

Let Fx, x ∈ N, be a canonical indexing [Rog67, MY78] of all/only the finite functions: N → N. We have for our characterization: Theorem For any index set M and corresponding subsequence of sets Eq = {x | ϕx = ϕq}, q ∈ M, the subsequence is uniformly c-productive iff (∃ c.e. A ⊆ M)(∀x)(∃y ∈ A)[ϕy ⊇ Fx]. Wolog: y effective in x. The disjoint index sets partitioning N, {q | domain(ϕq) = ∅} (c.e.)

• vs. its complement, {q | domain(ϕq) = ∅}, do not work — since,

by our above characterization and Rice-Shapiro [Rog67], for any = ∅ c.e. index set M, Eq, q ∈ M, is not uniformly c-productive.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 7 / 11

Characterizing the Index Set Cases The Characterization

The Characterization

There are divisions into non-constructive (top level) disjunctions besides the above example for proving that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e. {q | domain(ϕq) is not recursive} vs. its complement also works, but it’s Π0

3-LEM, more non-constructive than Π0 2-LEM above [ABHK04].

Let Fx, x ∈ N, be a canonical indexing [Rog67, MY78] of all/only the finite functions: N → N. We have for our characterization: Theorem For any index set M and corresponding subsequence of sets Eq = {x | ϕx = ϕq}, q ∈ M, the subsequence is uniformly c-productive iff (∃ c.e. A ⊆ M)(∀x)(∃y ∈ A)[ϕy ⊇ Fx]. Wolog: y effective in x. The disjoint index sets partitioning N, {q | domain(ϕq) = ∅} (c.e.)

• vs. its complement, {q | domain(ϕq) = ∅}, do not work — since,

by our above characterization and Rice-Shapiro [Rog67], for any = ∅ c.e. index set M, Eq, q ∈ M, is not uniformly c-productive.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 7 / 11

Characterizing the Index Set Cases The Characterization

The Characterization

There are divisions into non-constructive (top level) disjunctions besides the above example for proving that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e. {q | domain(ϕq) is not recursive} vs. its complement also works, but it’s Π0

3-LEM, more non-constructive than Π0 2-LEM above [ABHK04].

Let Fx, x ∈ N, be a canonical indexing [Rog67, MY78] of all/only the finite functions: N → N. We have for our characterization: Theorem For any index set M and corresponding subsequence of sets Eq = {x | ϕx = ϕq}, q ∈ M, the subsequence is uniformly c-productive iff (∃ c.e. A ⊆ M)(∀x)(∃y ∈ A)[ϕy ⊇ Fx]. Wolog: y effective in x. The disjoint index sets partitioning N, {q | domain(ϕq) = ∅} (c.e.)

• vs. its complement, {q | domain(ϕq) = ∅}, do not work — since,

by our above characterization and Rice-Shapiro [Rog67], for any = ∅ c.e. index set M, Eq, q ∈ M, is not uniformly c-productive.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 7 / 11

Characterizing the Index Set Cases The Characterization

The Characterization

There are divisions into non-constructive (top level) disjunctions besides the above example for proving that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e. {q | domain(ϕq) is not recursive} vs. its complement also works, but it’s Π0

3-LEM, more non-constructive than Π0 2-LEM above [ABHK04].

Let Fx, x ∈ N, be a canonical indexing [Rog67, MY78] of all/only the finite functions: N → N. We have for our characterization: Theorem For any index set M and corresponding subsequence of sets Eq = {x | ϕx = ϕq}, q ∈ M, the subsequence is uniformly c-productive iff (∃ c.e. A ⊆ M)(∀x)(∃y ∈ A)[ϕy ⊇ Fx]. Wolog: y effective in x. The disjoint index sets partitioning N, {q | domain(ϕq) = ∅} (c.e.)

• vs. its complement, {q | domain(ϕq) = ∅}, do not work — since,

by our above characterization and Rice-Shapiro [Rog67], for any = ∅ c.e. index set M, Eq, q ∈ M, is not uniformly c-productive.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 7 / 11

Characterizing the Index Set Cases The Characterization

The Characterization

There are divisions into non-constructive (top level) disjunctions besides the above example for proving that, for each q ∈ N, Eq = {x | ϕx = ϕq} is not c.e. {q | domain(ϕq) is not recursive} vs. its complement also works, but it’s Π0

3-LEM, more non-constructive than Π0 2-LEM above [ABHK04].

Let Fx, x ∈ N, be a canonical indexing [Rog67, MY78] of all/only the finite functions: N → N. We have for our characterization: Theorem For any index set M and corresponding subsequence of sets Eq = {x | ϕx = ϕq}, q ∈ M, the subsequence is uniformly c-productive iff (∃ c.e. A ⊆ M)(∀x)(∃y ∈ A)[ϕy ⊇ Fx]. Wolog: y effective in x. The disjoint index sets partitioning N, {q | domain(ϕq) = ∅} (c.e.)

• vs. its complement, {q | domain(ϕq) = ∅}, do not work — since,

by our above characterization and Rice-Shapiro [Rog67], for any = ∅ c.e. index set M, Eq, q ∈ M, is not uniformly c-productive.

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 7 / 11

Characterizing the Index Set Cases Another Corollary of the Characterization

Another Corollary of the Characterization

Above, we’ve looked at proving each Eq is not c.e., with the proof’s top level non-constructivity confined to disjunctions with two disjuncts as to which of two partitioning index sets contains q. How about such disjunctions but with numbers of such irreducible disjuncts more than 2? Thanks to our characterization: Corollary For each positive m, there are pairwise disjoint, non-trivial index sets M0, . . . , Mm unioning to N such that:

For each i ≤ m, the subsequence Eq = {x | ϕx = ϕq}, q ∈ Mi, is uniformly c-productive, but For each i, j ≤ m st i = j, the subsequence Eq, q ∈ (Mi ∪ Mj), is not uniformly c-productive.

A simple extension of just above Corollary’s proof yields the m = ω, infinitary, irreducible disjunctions case too. (. . ⌣)

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 8 / 11

Characterizing the Index Set Cases Another Corollary of the Characterization

Another Corollary of the Characterization

Above, we’ve looked at proving each Eq is not c.e., with the proof’s top level non-constructivity confined to disjunctions with two disjuncts as to which of two partitioning index sets contains q. How about such disjunctions but with numbers of such irreducible disjuncts more than 2? Thanks to our characterization: Corollary For each positive m, there are pairwise disjoint, non-trivial index sets M0, . . . , Mm unioning to N such that:

For each i ≤ m, the subsequence Eq = {x | ϕx = ϕq}, q ∈ Mi, is uniformly c-productive, but For each i, j ≤ m st i = j, the subsequence Eq, q ∈ (Mi ∪ Mj), is not uniformly c-productive.

A simple extension of just above Corollary’s proof yields the m = ω, infinitary, irreducible disjunctions case too. (. . ⌣)

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 8 / 11

Characterizing the Index Set Cases Another Corollary of the Characterization

Another Corollary of the Characterization

Above, we’ve looked at proving each Eq is not c.e., with the proof’s top level non-constructivity confined to disjunctions with two disjuncts as to which of two partitioning index sets contains q. How about such disjunctions but with numbers of such irreducible disjuncts more than 2? Thanks to our characterization: Corollary For each positive m, there are pairwise disjoint, non-trivial index sets M0, . . . , Mm unioning to N such that:

For each i ≤ m, the subsequence Eq = {x | ϕx = ϕq}, q ∈ Mi, is uniformly c-productive, but For each i, j ≤ m st i = j, the subsequence Eq, q ∈ (Mi ∪ Mj), is not uniformly c-productive.

A simple extension of just above Corollary’s proof yields the m = ω, infinitary, irreducible disjunctions case too. (. . ⌣)

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 8 / 11

Characterizing the Index Set Cases Another Corollary of the Characterization

Another Corollary of the Characterization

Above, we’ve looked at proving each Eq is not c.e., with the proof’s top level non-constructivity confined to disjunctions with two disjuncts as to which of two partitioning index sets contains q. How about such disjunctions but with numbers of such irreducible disjuncts more than 2? Thanks to our characterization: Corollary For each positive m, there are pairwise disjoint, non-trivial index sets M0, . . . , Mm unioning to N such that:

For each i ≤ m, the subsequence Eq = {x | ϕx = ϕq}, q ∈ Mi, is uniformly c-productive, but For each i, j ≤ m st i = j, the subsequence Eq, q ∈ (Mi ∪ Mj), is not uniformly c-productive.

A simple extension of just above Corollary’s proof yields the m = ω, infinitary, irreducible disjunctions case too. (. . ⌣)

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 8 / 11

Characterizing the Index Set Cases Another Corollary of the Characterization

Another Corollary of the Characterization

Above, we’ve looked at proving each Eq is not c.e., with the proof’s top level non-constructivity confined to disjunctions with two disjuncts as to which of two partitioning index sets contains q. How about such disjunctions but with numbers of such irreducible disjuncts more than 2? Thanks to our characterization: Corollary For each positive m, there are pairwise disjoint, non-trivial index sets M0, . . . , Mm unioning to N such that:

For each i ≤ m, the subsequence Eq = {x | ϕx = ϕq}, q ∈ Mi, is uniformly c-productive, but For each i, j ≤ m st i = j, the subsequence Eq, q ∈ (Mi ∪ Mj), is not uniformly c-productive.

A simple extension of just above Corollary’s proof yields the m = ω, infinitary, irreducible disjunctions case too. (. . ⌣)

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 8 / 11

Characterizing the Index Set Cases Another Corollary of the Characterization

Another Corollary of the Characterization

Above, we’ve looked at proving each Eq is not c.e., with the proof’s top level non-constructivity confined to disjunctions with two disjuncts as to which of two partitioning index sets contains q. How about such disjunctions but with numbers of such irreducible disjuncts more than 2? Thanks to our characterization: Corollary For each positive m, there are pairwise disjoint, non-trivial index sets M0, . . . , Mm unioning to N such that:

For each i ≤ m, the subsequence Eq = {x | ϕx = ϕq}, q ∈ Mi, is uniformly c-productive, but For each i, j ≤ m st i = j, the subsequence Eq, q ∈ (Mi ∪ Mj), is not uniformly c-productive.

A simple extension of just above Corollary’s proof yields the m = ω, infinitary, irreducible disjunctions case too. (. . ⌣)

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 8 / 11

Further Examples & Future Work

Further Examples & Future Work

We see that Eq = {x | ϕx = ϕq}. Clearly, for each q st domain(ϕq) = ∅, Eq is c.e. Let Mne = {q | domain(ϕq)=∅}. Then, by a Σ0

1-LEM recursion

theorem argument, Eq, q ∈ Mne, is uniformly c-productive — as witnessed by a total computable function. The sequence {x | Wx = Wq}, q ∈ N, and its complementary sequence satisfy results like the Eqs & Eqs mutatis mutandis, e.g., with Fx replaced by Rogers’ [Rog67] Dx. We can prove that some index set S (a possible Sq) is neither c.e. nor c-productive! Then S must be c-productive. Let ψ be an efficiently numbered, “natural”, subrecursive

• progr. syst. for a “closed” class ≥ LinearTime [RC94], e.g., based on

clocked, multi-tape TMs. Let Cq = {x | ψx = ψq}. Each Cq is c.e., but, by a Σ0

1-LEM subrecursive recursion theorem argument, Cq,

q ∈ N, is uniformly c-productive, with LinTime (Royer) witness. Future Reverse Math: are LEM upper-bonds above also lower?

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 9 / 11

Further Examples & Future Work

Further Examples & Future Work

We see that Eq = {x | ϕx = ϕq}. Clearly, for each q st domain(ϕq) = ∅, Eq is c.e. Let Mne = {q | domain(ϕq)=∅}. Then, by a Σ0

1-LEM recursion

theorem argument, Eq, q ∈ Mne, is uniformly c-productive — as witnessed by a total computable function. The sequence {x | Wx = Wq}, q ∈ N, and its complementary sequence satisfy results like the Eqs & Eqs mutatis mutandis, e.g., with Fx replaced by Rogers’ [Rog67] Dx. We can prove that some index set S (a possible Sq) is neither c.e. nor c-productive! Then S must be c-productive. Let ψ be an efficiently numbered, “natural”, subrecursive

• progr. syst. for a “closed” class ≥ LinearTime [RC94], e.g., based on

clocked, multi-tape TMs. Let Cq = {x | ψx = ψq}. Each Cq is c.e., but, by a Σ0

1-LEM subrecursive recursion theorem argument, Cq,

q ∈ N, is uniformly c-productive, with LinTime (Royer) witness. Future Reverse Math: are LEM upper-bonds above also lower?

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 9 / 11

Further Examples & Future Work

Further Examples & Future Work

We see that Eq = {x | ϕx = ϕq}. Clearly, for each q st domain(ϕq) = ∅, Eq is c.e. Let Mne = {q | domain(ϕq)=∅}. Then, by a Σ0

1-LEM recursion

theorem argument, Eq, q ∈ Mne, is uniformly c-productive — as witnessed by a total computable function. The sequence {x | Wx = Wq}, q ∈ N, and its complementary sequence satisfy results like the Eqs & Eqs mutatis mutandis, e.g., with Fx replaced by Rogers’ [Rog67] Dx. We can prove that some index set S (a possible Sq) is neither c.e. nor c-productive! Then S must be c-productive. Let ψ be an efficiently numbered, “natural”, subrecursive

• progr. syst. for a “closed” class ≥ LinearTime [RC94], e.g., based on

clocked, multi-tape TMs. Let Cq = {x | ψx = ψq}. Each Cq is c.e., but, by a Σ0

1-LEM subrecursive recursion theorem argument, Cq,

q ∈ N, is uniformly c-productive, with LinTime (Royer) witness. Future Reverse Math: are LEM upper-bonds above also lower?

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 9 / 11

Further Examples & Future Work

Further Examples & Future Work

We see that Eq = {x | ϕx = ϕq}. Clearly, for each q st domain(ϕq) = ∅, Eq is c.e. Let Mne = {q | domain(ϕq)=∅}. Then, by a Σ0

1-LEM recursion

theorem argument, Eq, q ∈ Mne, is uniformly c-productive — as witnessed by a total computable function. The sequence {x | Wx = Wq}, q ∈ N, and its complementary sequence satisfy results like the Eqs & Eqs mutatis mutandis, e.g., with Fx replaced by Rogers’ [Rog67] Dx. We can prove that some index set S (a possible Sq) is neither c.e. nor c-productive! Then S must be c-productive. Let ψ be an efficiently numbered, “natural”, subrecursive

• progr. syst. for a “closed” class ≥ LinearTime [RC94], e.g., based on

clocked, multi-tape TMs. Let Cq = {x | ψx = ψq}. Each Cq is c.e., but, by a Σ0

1-LEM subrecursive recursion theorem argument, Cq,

q ∈ N, is uniformly c-productive, with LinTime (Royer) witness. Future Reverse Math: are LEM upper-bonds above also lower?

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 9 / 11

Further Examples & Future Work

Further Examples & Future Work

We see that Eq = {x | ϕx = ϕq}. Clearly, for each q st domain(ϕq) = ∅, Eq is c.e. Let Mne = {q | domain(ϕq)=∅}. Then, by a Σ0

1-LEM recursion

theorem argument, Eq, q ∈ Mne, is uniformly c-productive — as witnessed by a total computable function. The sequence {x | Wx = Wq}, q ∈ N, and its complementary sequence satisfy results like the Eqs & Eqs mutatis mutandis, e.g., with Fx replaced by Rogers’ [Rog67] Dx. We can prove that some index set S (a possible Sq) is neither c.e. nor c-productive! Then S must be c-productive. Let ψ be an efficiently numbered, “natural”, subrecursive

• progr. syst. for a “closed” class ≥ LinearTime [RC94], e.g., based on

clocked, multi-tape TMs. Let Cq = {x | ψx = ψq}. Each Cq is c.e., but, by a Σ0

1-LEM subrecursive recursion theorem argument, Cq,

q ∈ N, is uniformly c-productive, with LinTime (Royer) witness. Future Reverse Math: are LEM upper-bonds above also lower?

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 9 / 11

Further Examples & Future Work

Further Examples & Future Work

We see that Eq = {x | ϕx = ϕq}. Clearly, for each q st domain(ϕq) = ∅, Eq is c.e. Let Mne = {q | domain(ϕq)=∅}. Then, by a Σ0

1-LEM recursion

theorem argument, Eq, q ∈ Mne, is uniformly c-productive — as witnessed by a total computable function. The sequence {x | Wx = Wq}, q ∈ N, and its complementary sequence satisfy results like the Eqs & Eqs mutatis mutandis, e.g., with Fx replaced by Rogers’ [Rog67] Dx. We can prove that some index set S (a possible Sq) is neither c.e. nor c-productive! Then S must be c-productive. Let ψ be an efficiently numbered, “natural”, subrecursive

• progr. syst. for a “closed” class ≥ LinearTime [RC94], e.g., based on

clocked, multi-tape TMs. Let Cq = {x | ψx = ψq}. Each Cq is c.e., but, by a Σ0

1-LEM subrecursive recursion theorem argument, Cq,

q ∈ N, is uniformly c-productive, with LinTime (Royer) witness. Future Reverse Math: are LEM upper-bonds above also lower?

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 9 / 11

Further Examples & Future Work

Further Examples & Future Work

We see that Eq = {x | ϕx = ϕq}. Clearly, for each q st domain(ϕq) = ∅, Eq is c.e. Let Mne = {q | domain(ϕq)=∅}. Then, by a Σ0

1-LEM recursion

theorem argument, Eq, q ∈ Mne, is uniformly c-productive — as witnessed by a total computable function. The sequence {x | Wx = Wq}, q ∈ N, and its complementary sequence satisfy results like the Eqs & Eqs mutatis mutandis, e.g., with Fx replaced by Rogers’ [Rog67] Dx. We can prove that some index set S (a possible Sq) is neither c.e. nor c-productive! Then S must be c-productive. Let ψ be an efficiently numbered, “natural”, subrecursive

• progr. syst. for a “closed” class ≥ LinearTime [RC94], e.g., based on

clocked, multi-tape TMs. Let Cq = {x | ψx = ψq}. Each Cq is c.e., but, by a Σ0

1-LEM subrecursive recursion theorem argument, Cq,

q ∈ N, is uniformly c-productive, with LinTime (Royer) witness. Future Reverse Math: are LEM upper-bonds above also lower?

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 9 / 11

Further Examples & Future Work

Further Examples & Future Work

We see that Eq = {x | ϕx = ϕq}. Clearly, for each q st domain(ϕq) = ∅, Eq is c.e. Let Mne = {q | domain(ϕq)=∅}. Then, by a Σ0

1-LEM recursion

theorem argument, Eq, q ∈ Mne, is uniformly c-productive — as witnessed by a total computable function. The sequence {x | Wx = Wq}, q ∈ N, and its complementary sequence satisfy results like the Eqs & Eqs mutatis mutandis, e.g., with Fx replaced by Rogers’ [Rog67] Dx. We can prove that some index set S (a possible Sq) is neither c.e. nor c-productive! Then S must be c-productive. Let ψ be an efficiently numbered, “natural”, subrecursive

• progr. syst. for a “closed” class ≥ LinearTime [RC94], e.g., based on

clocked, multi-tape TMs. Let Cq = {x | ψx = ψq}. Each Cq is c.e., but, by a Σ0

1-LEM subrecursive recursion theorem argument, Cq,

q ∈ N, is uniformly c-productive, with LinTime (Royer) witness. Future Reverse Math: are LEM upper-bonds above also lower?

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 9 / 11

Further Examples & Future Work

Further Examples & Future Work

We see that Eq = {x | ϕx = ϕq}. Clearly, for each q st domain(ϕq) = ∅, Eq is c.e. Let Mne = {q | domain(ϕq)=∅}. Then, by a Σ0

1-LEM recursion

theorem argument, Eq, q ∈ Mne, is uniformly c-productive — as witnessed by a total computable function. The sequence {x | Wx = Wq}, q ∈ N, and its complementary sequence satisfy results like the Eqs & Eqs mutatis mutandis, e.g., with Fx replaced by Rogers’ [Rog67] Dx. We can prove that some index set S (a possible Sq) is neither c.e. nor c-productive! Then S must be c-productive. Let ψ be an efficiently numbered, “natural”, subrecursive

• progr. syst. for a “closed” class ≥ LinearTime [RC94], e.g., based on

clocked, multi-tape TMs. Let Cq = {x | ψx = ψq}. Each Cq is c.e., but, by a Σ0

1-LEM subrecursive recursion theorem argument, Cq,

q ∈ N, is uniformly c-productive, with LinTime (Royer) witness. Future Reverse Math: are LEM upper-bonds above also lower?

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 9 / 11

Further Examples & Future Work

Further Examples & Future Work

We see that Eq = {x | ϕx = ϕq}. Clearly, for each q st domain(ϕq) = ∅, Eq is c.e. Let Mne = {q | domain(ϕq)=∅}. Then, by a Σ0

1-LEM recursion

theorem argument, Eq, q ∈ Mne, is uniformly c-productive — as witnessed by a total computable function. The sequence {x | Wx = Wq}, q ∈ N, and its complementary sequence satisfy results like the Eqs & Eqs mutatis mutandis, e.g., with Fx replaced by Rogers’ [Rog67] Dx. We can prove that some index set S (a possible Sq) is neither c.e. nor c-productive! Then S must be c-productive. Let ψ be an efficiently numbered, “natural”, subrecursive

• progr. syst. for a “closed” class ≥ LinearTime [RC94], e.g., based on

clocked, multi-tape TMs. Let Cq = {x | ψx = ψq}. Each Cq is c.e., but, by a Σ0

1-LEM subrecursive recursion theorem argument, Cq,

q ∈ N, is uniformly c-productive, with LinTime (Royer) witness. Future Reverse Math: are LEM upper-bonds above also lower?

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 9 / 11

Further Examples & Future Work

Further Examples & Future Work

We see that Eq = {x | ϕx = ϕq}. Clearly, for each q st domain(ϕq) = ∅, Eq is c.e. Let Mne = {q | domain(ϕq)=∅}. Then, by a Σ0

1-LEM recursion

theorem argument, Eq, q ∈ Mne, is uniformly c-productive — as witnessed by a total computable function. The sequence {x | Wx = Wq}, q ∈ N, and its complementary sequence satisfy results like the Eqs & Eqs mutatis mutandis, e.g., with Fx replaced by Rogers’ [Rog67] Dx. We can prove that some index set S (a possible Sq) is neither c.e. nor c-productive! Then S must be c-productive. Let ψ be an efficiently numbered, “natural”, subrecursive

• progr. syst. for a “closed” class ≥ LinearTime [RC94], e.g., based on

clocked, multi-tape TMs. Let Cq = {x | ψx = ψq}. Each Cq is c.e., but, by a Σ0

1-LEM subrecursive recursion theorem argument, Cq,

q ∈ N, is uniformly c-productive, with LinTime (Royer) witness. Future Reverse Math: are LEM upper-bonds above also lower?

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 9 / 11

Further Examples & Future Work

Further Examples & Future Work

We see that Eq = {x | ϕx = ϕq}. Clearly, for each q st domain(ϕq) = ∅, Eq is c.e. Let Mne = {q | domain(ϕq)=∅}. Then, by a Σ0

1-LEM recursion

theorem argument, Eq, q ∈ Mne, is uniformly c-productive — as witnessed by a total computable function. The sequence {x | Wx = Wq}, q ∈ N, and its complementary sequence satisfy results like the Eqs & Eqs mutatis mutandis, e.g., with Fx replaced by Rogers’ [Rog67] Dx. We can prove that some index set S (a possible Sq) is neither c.e. nor c-productive! Then S must be c-productive. Let ψ be an efficiently numbered, “natural”, subrecursive

• progr. syst. for a “closed” class ≥ LinearTime [RC94], e.g., based on

clocked, multi-tape TMs. Let Cq = {x | ψx = ψq}. Each Cq is c.e., but, by a Σ0

1-LEM subrecursive recursion theorem argument, Cq,

q ∈ N, is uniformly c-productive, with LinTime (Royer) witness. Future Reverse Math: are LEM upper-bonds above also lower?

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 9 / 11

Further Examples & Future Work

Further Examples & Future Work

We see that Eq = {x | ϕx = ϕq}. Clearly, for each q st domain(ϕq) = ∅, Eq is c.e. Let Mne = {q | domain(ϕq)=∅}. Then, by a Σ0

1-LEM recursion

theorem argument, Eq, q ∈ Mne, is uniformly c-productive — as witnessed by a total computable function. The sequence {x | Wx = Wq}, q ∈ N, and its complementary sequence satisfy results like the Eqs & Eqs mutatis mutandis, e.g., with Fx replaced by Rogers’ [Rog67] Dx. We can prove that some index set S (a possible Sq) is neither c.e. nor c-productive! Then S must be c-productive. Let ψ be an efficiently numbered, “natural”, subrecursive

• progr. syst. for a “closed” class ≥ LinearTime [RC94], e.g., based on

clocked, multi-tape TMs. Let Cq = {x | ψx = ψq}. Each Cq is c.e., but, by a Σ0

1-LEM subrecursive recursion theorem argument, Cq,

q ∈ N, is uniformly c-productive, with LinTime (Royer) witness. Future Reverse Math: are LEM upper-bonds above also lower?

Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 9 / 11

References

References I

• Y. Akama, S. Berardi, S. Hayashi, and U. Kohlenbach.

An arithmetical hierarchy of the law of excluded middle and related principles. 19th Annual Symposium on Logic in Computer Science (LICS’04), pages 192–201, 2004. L.E.J. Brouwer. In D. van Dalen, editor, Brouwer’s Cambridge Lectures on Intuitionism. Cambridge University Press, 1981.

• J. Dekker.

Productive sets.

• Trans. of AMS, 78:129–149, 1955.
• M. Machtey and P. Young.

An Introduction to the General Theory of Algorithms. North Holland, New York, 1978.

• J. Myhill.

Creative sets.

• Z. Math. Logik Grundlagen Math, 1:97–108, 1955.
• E. Post.

Recursively enumerable sets of positive integers and their decision problems. Bulletin of the American Mathematical Society, 50:284–316, 1944.

• J. Royer and J. Case.

Subrecursive Programming Systems: Complexity and Succinctness. Research monograph in Progress in Theoretical Computer Science. Birkh¨ auser Boston, 1994. See www.cis.udel.edu/~case/RC94Errata.pdf for corrected errata. Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 10 / 11

References

References II

• H. Rogers.

G¨

• del numberings of partial recursive functions.

Journal of Symbolic Logic, 23:331–341, 1958.

• H. Rogers.

Theory of Recursive Functions and Effective Computability. McGraw Hill, New York, 1967. Reprinted, MIT Press, 1987. Case, Ralston, & Akama (UD & TU) A Non-Uniformly C-Productive Sequence ALC 2013, Guangzhou 11 / 11