Cobham Recursive Set Functions Moritz M uller Kurt G odel - - PowerPoint PPT Presentation

Cobham Recursive Set Functions Moritz M¨ uller Kurt G¨

• del Research Center, Vienna, Austria.

joint with A. Beckmann, S. Buss, S.-D. Friedman and N. Thapen BASICS 2015 Summer School Logic Summer School in China 2015 Zhejiang Normal University

Computations on arbitrary sets There are many equivalent definitions of the class of recursive func- tions on the natural numbers. Different definitions have different uses while the equivalence of all the notions provides evidence for Church’s thesis, the thesis that the concept of recursive function is the most reasonable explication of our intuitive notion of effectively calculable function. As the various definitions are lifted to domains

• ther than the integers (e. g., admissible sets) some of the equiva-

lences break down. This break-down provides us with a laboratory for the study of recursion theory. Barwise, 1975

Computations on arbitrary sets There are many equivalent definitions of the class of recursive func- tions on the natural numbers. Different definitions have different uses while the equivalence of all the notions provides evidence for Church’s thesis, the thesis that the concept of recursive function is the most reasonable explication of our intuitive notion of effectively calculable function. As the various definitions are lifted to domains

• ther than the integers (e. g., admissible sets) some of the equiva-

lences break down. This break-down provides us with a laboratory for the study of recursion theory. Barwise, 1975 A computable function over N is one Recursion theoretic view

• btainable from certain initial functions by means
• f composition, primitive recursion and the µ-operator.

Definability theoretic view Σ1-definable in the language of arithmetic.

Primitive recursive set functions (Jensen, Karp 1971) are obtained from initial functions constant 0 = ∅ projections pair x, y → {x, y} union x → x conditional cond∈(x, y, u, v) := if u ∈ v then x else y by composition and ǫ-recursion if g(z, y, x) is PRSF, then so is f(y, x) = g({f(u, x) : u ∈ y}, y, x)

Primitive recursive set functions (Jensen, Karp 1971) are obtained from initial functions constant 0 = ∅ projections pair x, y → {x, y} union x → x conditional cond∈(x, y, u, v) := if u ∈ v then x else y by composition and ǫ-recursion if g(z, y, x) is PRSF, then so is f(y, x) = g({f(u, x) : u ∈ y}, y, x) Generalizes primitive recursive computations to arbitrary sets Goal Similarly generalize polynomial time computations to arbitrary sets Need Recursion theoretic definition of PTIME

Cobham’s definition of polynomial time 1965 The polynomial time function over N are those obtained from initial functions constant 0, projections, successors s0(x) = 2x and s1(x) = 2x + 1 smash x#y = 2|x| · |y| where |x| = ⌈log(x + 1)⌉ by composition and limited recursion on notation if h, g0, g1, t are polynomial time, then so is f(0, x) = h( x) f(sb(y), x) = gb(f(y, x), y, x) where b ∈ {0, 1}

Cobham’s definition of polynomial time 1965 The polynomial time function over N are those obtained from initial functions constant 0, projections, successors s0(x) = 2x and s1(x) = 2x + 1 smash x#y = 2|x| · |y| where |x| = ⌈log(x + 1)⌉ by composition and limited recursion on notation if h, g0, g1, t are polynomial time, then so is f(0, x) = h( x) f(sb(y), x) = gb(f(y, x), y, x) where b ∈ {0, 1} provided f(y, x) ≤ t(y, x) for all y, x.

Cobham’s definition of polynomial time 1965 The polynomial time function over N are those obtained from initial functions constant 0, projections, successors s0(x) = 2x and s1(x) = 2x + 1 smash x#y = 2|x| · |y| where |x| = ⌈log(x + 1)⌉ by composition and limited recursion on notation if h, g0, g1, t are polynomial time, then so is f(0, x) = h( x) f(sb(y), x) = gb(f(y, x), y, x) where b ∈ {0, 1} provided f(y, x) ≤ t(y, x) for all y, x. Equivalent proviso:

• t a #-term: built from variables, 1 = s1(0) and #.
• |f(y, x1, x2 . . .)| ≤ p(|y|, |x1|, |x2| . . .) for some polynomial p

Set composition and set smash Set composition x ⊙ y :=

• y

if x = 0 {u ⊙ y : u ∈ x} if x = 0

Set composition and set smash Set composition x ⊙ y :=

• y

if x = 0 {u ⊙ y : u ∈ x} if x = 0 Set smash x#y := y ⊙ {u#y : u ∈ x}

Set composition and set smash Set composition x ⊙ y :=

• y

if x = 0 {u ⊙ y : u ∈ x} if x = 0 Set smash x#y := y ⊙ {u#y : u ∈ x} #-term t( x) built from variables x, 1={0}, ⊙ and #

• There are polynomials p, q such that for all

x rk(t(x1, x2 . . .)) ≤ p(rk(x1), rk(x2) . . .) |tc(t(x1, x2 . . .))| ≤ q(|tc(x1)|, |tc(x2)| . . .)

Set composition and set smash Set composition x ⊙ y :=

• y

if x = 0 {u ⊙ y : u ∈ x} if x = 0 Set smash x#y := y ⊙ {u#y : u ∈ x} #-term t( x) built from variables x, 1={0}, ⊙ and #

• There are polynomials p, q such that for all

x rk(t(x1, x2 . . .)) ≤ p(rk(x1), rk(x2) . . .) |tc(t(x1, x2 . . .))| ≤ q(|tc(x1)|, |tc(x2)| . . .) Intuition #-terms play the role of polynomial length bounds. Intuition consider only hereditarily finite x: finite Mostowski graph,

• ne can compute f(x) = g({f(u) : u ∈ x}, x) with oracle g

in parallel time ≈ rk(x) and total work ≈ |tc(x)|.

Bounding relation A single-valued embedding of x into y is τ : tc(x) → tc(y) st if u = v, then τ(u) = τ(v) if u ∈ v, then τ(u) ∈ tc(τ(v)) Example if x ⊆ y, then the identity is such an embedding.

Bounding relation A single-valued embedding of x into y is τ : tc(x) → tc(y) st if u = v, then τ(u) = τ(v) if u ∈ v, then τ(u) ∈ tc(τ(v)) Example if x ⊆ y, then the identity is such an embedding. A (multi-valued) embedding of x into y is τ : tc(x) → P(tc(y)) \ {0} st if u = v, then τ(u) ∩ τ(v) = 0 if u ∈ v and v′ ∈ τ(v), then there exists u′ ∈ τ(u) such that u′ ∈ tc(v′)

Bounding relation A single-valued embedding of x into y is τ : tc(x) → tc(y) st if u = v, then τ(u) = τ(v) if u ∈ v, then τ(u) ∈ tc(τ(v)) Example if x ⊆ y, then the identity is such an embedding. A (multi-valued) embedding of x into y is τ : tc(x) → P(tc(y)) \ {0} st if u = v, then τ(u) ∩ τ(v) = 0 if u ∈ v and v′ ∈ τ(v), then there exists u′ ∈ τ(u) such that u′ ∈ tc(v′)

• Then rk(x) ≤ rk(y) and |tc(x)| ≤ |tc(y)|.

Intuition Then x is structurally no more complex than y.

Cobham recursive set functions are obtained from initial functions constant 0, projections, pair {x, y}, union x, cond∈(x, y, u, v), smash x#y by composition and Cobham recursion if g, τ, t are CRSF, then so is f(y, x) = g({f(u, x) : u ∈ y}, y, x)

Cobham recursive set functions are obtained from initial functions constant 0, projections, pair {x, y}, union x, cond∈(x, y, u, v), smash x#y by composition and Cobham recursion if g, τ, t are CRSF, then so is f(y, x) = g({f(u, x) : u ∈ y}, y, x) provided τ(·, y, x) : f(y, x) t(y, x) for all y, x. (i.e. u → τ(u, y, x) is an embedding of f(y, x) into t(y, x))

Cobham recursive set functions are obtained from initial functions constant 0, projections, pair {x, y}, union x, cond∈(x, y, u, v), smash x#y by composition and Cobham recursion if g, τ, t are CRSF, then so is f(y, x) = g({f(u, x) : u ∈ y}, y, x) provided τ(·, y, x) : f(y, x) t(y, x) for all y, x. (i.e. u → τ(u, y, x) is an embedding of f(y, x) into t(y, x))

• equivalent: demand t to be a #-term.
• equivalent: allow “impredicative” τ(·, y,

x, f(y, x)).

Bootstrapping CRSF

• Bounded replacement if g, τ, t are CRSF, then so is

f(x) = {g(u, x) : u ∈ x} provided τ(·, x) : f(x) t(x)

Bootstrapping CRSF

• Bounded replacement if g, τ, t are CRSF, then so is

f(x) = {g(u, x) : u ∈ x} provided τ(·, x) : f(x) t(x)

• Separation if g is CRSF, then so is f(x) = {u ∈ x : g(u) = 0};

Bootstrapping CRSF

• Bounded replacement if g, τ, t are CRSF, then so is

f(x) = {g(u, x) : u ∈ x} provided τ(·, x) : f(x) t(x)

• Separation if g is CRSF, then so is f(x) = {u ∈ x : g(u) = 0};

h(u) := if g(u) ∈ {0} then 0 else {u} Bounded replacement gives f(x) = { h(u) : u ∈ x} Proviso satisfied since f(x) ⊆ x.

Bootstrapping CRSF

• Bounded replacement if g, τ, t are CRSF, then so is

f(x) = {g(u, x) : u ∈ x} provided τ(·, x) : f(x) t(x)

• Separation if g is CRSF, then so is f(x) = {u ∈ x : g(u) = 0};

h(u) := if g(u) ∈ {0} then 0 else {u} Bounded replacement gives f(x) = { h(u) : u ∈ x} Proviso satisfied since f(x) ⊆ x.

• x, y, x ∪ y, x \ y, x ∩ y, x ⊙ y and tc(x) = x ∪ {tc(u) : u ∈ x} are CRSF.

Bootstrapping CRSF

• Bounded replacement if g, τ, t are CRSF, then so is

f(x) = {g(u, x) : u ∈ x} provided τ(·, x) : f(x) t(x)

• Separation if g is CRSF, then so is f(x) = {u ∈ x : g(u) = 0};

h(u) := if g(u) ∈ {0} then 0 else {u} Bounded replacement gives f(x) = { h(u) : u ∈ x} Proviso satisfied since f(x) ⊆ x.

• x, y, x ∪ y, x \ y, x ∩ y, x ⊙ y and tc(x) = x ∪ {tc(u) : u ∈ x} are CRSF.
• CRSF relations (i.e. char function is CRSF) are

closed under Boolean combinations and bounded quantification.

Bootstrapping CRSF

• Bounded replacement if g, τ, t are CRSF, then so is

f(x) = {g(u, x) : u ∈ x} provided τ(·, x) : f(x) t(x)

• Separation if g is CRSF, then so is f(x) = {u ∈ x : g(u) = 0};

h(u) := if g(u) ∈ {0} then 0 else {u} Bounded replacement gives f(x) = { h(u) : u ∈ x} Proviso satisfied since f(x) ⊆ x.

• x, y, x ∪ y, x \ y, x ∩ y, x ⊙ y and tc(x) = x ∪ {tc(u) : u ∈ x} are CRSF.
• CRSF relations (i.e. char function is CRSF) are

closed under Boolean combinations and bounded quantification. ∃u ∈ yR(u, x) has char function y, x → {χR(u, x) : u ∈ y}. Use Bounded replacement. Proviso: values are ⊆ 1

Example: the rank function Suffices to show rk+(x) := rk(x) + 1 is CRSF. Define rk+(x) := Succ {rk+(u) : u ∈ x}

• where Succ(x) := x ∪ {x}

by Cobham Recursion.

Example: the rank function Suffices to show rk+(x) := rk(x) + 1 is CRSF. Define rk+(x) := Succ {rk+(u) : u ∈ x}

• where Succ(x) := x ∪ {x}

by Cobham Recursion. Proviso: multi-valued embedding (into {x}) τ(α, x) := {u ∈ tc({x}) : rk(u) = α}.

Example: the rank function Suffices to show rk+(x) := rk(x) + 1 is CRSF. Define rk+(x) := Succ {rk+(u) : u ∈ x}

• where Succ(x) := x ∪ {x}

by Cobham Recursion. Proviso: multi-valued embedding (into {x}) τ(α, x) := {u ∈ tc({x}) : rk(u) = α}. Need: τ is CRSF. By Separation, it suffices to show: rk(x) ≤ rk(y) is CRSF.

Example: the rank function Suffices to show rk+(x) := rk(x) + 1 is CRSF. Define rk+(x) := Succ

{rk+(u) : u ∈ x}

• where Succ(x) := x ∪ {x}

by Cobham Recursion. Proviso: multi-valued embedding (into {x}) τ(α, x) := {u ∈ tc({x}) : rk(u) = α}. Need: τ is CRSF. By Separation, it suffices to show: rk(x) ≤ rk(y) is CRSF. f(x, y) :=

• u ∈ tc({x}) : u ⊆ {f(x, v) : v ∈ y}
• is CRSF by Cobham recursion. Proviso: identity embeds into {x}.

Example: the rank function Suffices to show rk+(x) := rk(x) + 1 is CRSF. Define rk+(x) := Succ

{rk+(u) : u ∈ x}

• where Succ(x) := x ∪ {x}

by Cobham Recursion. Proviso: multi-valued embedding (into {x}) τ(α, x) := {u ∈ tc({x}) : rk(u) = α}. Need: τ is CRSF. By Separation, it suffices to show: rk(x) ≤ rk(y) is CRSF. f(x, y) :=

• u ∈ tc({x}) : u ⊆ {f(x, v) : v ∈ y}
• is CRSF by Cobham recursion. Proviso: identity embeds into {x}.

Then: rk(x) ≤ rk(y) ⇐ ⇒ x ∈ f(x, y).

Normal form for bounds Transitivity if τ0 : x y and τ1 : y z, then σ : x z for some σ.

Normal form for bounds Transitivity if τ0 : x y and τ1 : y z, then σ : x z for some σ. Monotonicity Let t(y, ..) be a #-term. Then if τ0 : x t(y, ..) and τ1 : y z, then σ : x t(z, ..) for some σ.

Normal form for bounds Transitivity if τ0 : x y and τ1 : y z, then σ : x z for some σ. Monotonicity Let t(y, ..) be a #-term. Then if τ0 : x t(y, ..) and τ1 : y z, then σ : x t(z, ..) for some σ. Bounding Theorem if f( x) is CRSF, then τ(·, x) : f( x) t( x) for some #-term t( x), τ in CRSF.

Normal form for bounds Transitivity if τ0 : x y and τ1 : y z, then σ : x z for some σ. Monotonicity Let t(y, ..) be a #-term. Then if τ0 : x t(y, ..) and τ1 : y z, then σ : x t(z, ..) for some σ. Bounding Theorem if f( x) is CRSF, then τ(·, x) : f( x) t( x) for some #-term t( x), τ in CRSF. Example Neither x → P(x) nor x → |x| is CRSF. Proof: if f(x) is CRSF then rk(f(x)) is polynomially bounded in rk(x), and |tc(f(x))| is polynomially bounded in |tc(x)|.

Closure results Unbounded replacement if g is CRSF, then so is f(x) = {g(u, x) : u ∈ x}.

Closure results Unbounded replacement if g is CRSF, then so is f(x) = {g(u, x) : u ∈ x}. Example x × y is CRSF. Proof: two applications of unbounded replacement: x × y := {{u} × y : u ∈ x} {u} × y := {u, v : v ∈ y}

Closure results Unbounded replacement if g is CRSF, then so is f(x) = {g(u, x) : u ∈ x}. Example x × y is CRSF. Proof: two applications of unbounded replacement: x × y := {{u} × y : u ∈ x} {u} × y := {u, v : v ∈ y}

• Course of values recursion

if g, τ, t are CRSF, then so is f(x) := g({u, f(u) : u ∈ tc(x)}, x) provided τ(·, x) : f(x) t(x) for all x.

CRSF and PTIME view PTIME as a class of functions on binary strings b0 · · · bn−1 ∈ {0, 1}∗. Code by sets: ν(b0 · · · bn−1) := {i < n : bi = 1} ∪ {n}. F : V → V represents f : {0, 1}∗ → {0, 1}∗ if F(ν(b0 · · · bn−1)) = ν(f(b0 · · · bn−1))

CRSF and PTIME view PTIME as a class of functions on binary strings b0 · · · bn−1 ∈ {0, 1}∗. Code by sets: ν(b0 · · · bn−1) := {i < n : bi = 1} ∪ {n}. F : V → V represents f : {0, 1}∗ → {0, 1}∗ if F(ν(b0 · · · bn−1)) = ν(f(b0 · · · bn−1)) Theorem A string function is PTIME iff it is represented by some CRSF function.

CRSF and PTIME view PTIME as a class of functions on binary strings b0 · · · bn−1 ∈ {0, 1}∗. Code by sets: ν(b0 · · · bn−1) := {i < n : bi = 1} ∪ {n}. F : V → V represents f : {0, 1}∗ → {0, 1}∗ if F(ν(b0 · · · bn−1)) = ν(f(b0 · · · bn−1)) Theorem A string function is PTIME iff it is represented by some CRSF function. Proof idea ⊆: simulate limited recursion on notation in CRSF. Proof idea ⊇: let F(x) CRSF. Restrict to hereditarily finite x. Show Mostowski graph of x → Mostowski graph of F(x) is PTIME.

Related work Bellantoni-Cook 1992 recursion theoretic characterization of PTIME idea: arguments have two sorts, normal and safe: f(0, x/ y) = h( x/ y) f(sb(x), x/ y) = gb(x, x/f(x, x/ y), y) b ∈ {0, 1} The derivable functions f( x/) are precisely PTIME.

Related work Bellantoni-Cook 1992 recursion theoretic characterization of PTIME idea: arguments have two sorts, normal and safe: f(0, x/ y) = h( x/ y) f(sb(x), x/ y) = gb(x, x/f(x, x/ y), y) b ∈ {0, 1} The derivable functions f( x/) are precisely PTIME. Beckmann, Buss, Friedman 2015 define SRSF: safe recursive set functions

• n HF, SRSF is alternating exponential time with poly alternations.

Related work Bellantoni-Cook 1992 recursion theoretic characterization of PTIME idea: arguments have two sorts, normal and safe: f(0, x/ y) = h( x/ y) f(sb(x), x/ y) = gb(x, x/f(x, x/ y), y) b ∈ {0, 1} The derivable functions f( x/) are precisely PTIME. Beckmann, Buss, Friedman 2015 define SRSF: safe recursive set functions

• n HF, SRSF is alternating exponential time with poly alternations.

Arai 2015 defines PCSF: predicatively computable set functions. SRSF PCSF+ ⊇ PCSF is PTIME on HF

Related work Bellantoni-Cook 1992 recursion theoretic characterization of PTIME idea: arguments have two sorts, normal and safe: f(0, x/ y) = h( x/ y) f(sb(x), x/ y) = gb(x, x/f(x, x/ y), y) b ∈ {0, 1} The derivable functions f( x/) are precisely PTIME. Beckmann, Buss, Friedman 2015 define SRSF: safe recursive set functions

• n HF, SRSF is alternating exponential time with poly alternations.

Arai 2015 defines PCSF: predicatively computable set functions. SRSF PCSF+ ⊇ PCSF is PTIME on HF Theorem The functions f( x/) in PCSF+ are precisely CRSF.

Definability theoretic view α-recursion theory: computable means Σ1-definable on an admissible set. An admissible set is a model of KP: language {∈} Extensionality ∀u(u ∈ x ↔ u ∈ y) → x = y Pair ∃z∀u(u ∈ z ↔ u = x ∨ u = y) Union ∃z∀u(u ∈ z ↔ ∃v ∈ x u ∈ v) ∆0-Separation ∃z∀u(u ∈ z ↔ u ∈ x ∧ ϕ(u, x)) for ϕ ∈ ∆0 ∆0-Collection ∀u ∈ x∃v ϕ(u, v, x) → ∃z∀u ∈ x∃v ∈ z ϕ(u, v, x) for ϕ ∈ ∆0 Class Induction ∀y(∀u ∈ y ϕ(u, x) → ϕ(y, x)) → ϕ(x, x) for all ϕ

Definability theoretic view α-recursion theory: computable means Σ1-definable on an admissible set. An admissible set is a model of KP: language {∈, {·, ·}, ·} Extensionality (∀u(u ∈ x ↔ u ∈ y) → x = y) Pair (u ∈ {x, y} ↔ u = x ∨ u = y) defining axiom Union (u ∈ x ↔ ∃v ∈ x u ∈ v) defining axiom ∆0-Separation ∃z∀u(u ∈ z ↔ u ∈ x ∧ ϕ(u, x)) for ϕ ∈ ∆0 ∆0-Collection ∀u ∈ x∃v ϕ(u, v, x) → ∃z∀u ∈ x∃v ∈ z ϕ(u, v, x) for ϕ ∈ ∆0 Class Induction ∀y(∀u ∈ y ϕ(u, x) → ϕ(y, x)) → ϕ(x, x) for all ϕ

Theories for primitive recursive computation KP1 is KP with Induction restricted to Σ1-formulas ϕ. f( x) is Σ1-definable in KP1 if there is a Σ1-formula ϕ( x, y) such that V | = ∀ x ϕ( x, f( x)) KP1 ⊢ ∀ x ∃=1y ϕ( x, y)

Theories for primitive recursive computation KP1 is KP with Induction restricted to Σ1-formulas ϕ. f( x) is Σ1-definable in KP1 if there is a Σ1-formula ϕ( x, y) such that V | = ∀ x ϕ( x, f( x)) KP1 ⊢ ∀ x ∃=1y ϕ( x, y) Rathjen 1992 A function is PRSF iff it is Σ1-definable in KP1.

Theories for primitive recursive computation KP1 is KP with Induction restricted to Σ1-formulas ϕ. f( x) is Σ1-definable in KP1 if there is a Σ1-formula ϕ( x, y) such that V | = ∀ x ϕ( x, f( x)) KP1 ⊢ ∀ x ∃=1y ϕ( x, y) Rathjen 1992 A function is PRSF iff it is Σ1-definable in KP1. Parsons 1970 A function over N is primitive recursive iff it is Σ1-definable in IΣ1. IΣ1 is Robinson’s Q plus ϕ(0, x) ∧ ∀y(ϕ(y, x) → ϕ(y + 1, x)) → ϕ(x, x) for ϕ ∈ Σ1

Theory for PTIME Language of arithmetic plus ⌊x/2⌋, |x| = ⌈log(x + 1)⌉, x#y = 2|x|·|y|. ∆b

0-formulas have only sharply bounded quantifiers ∃x ≤ |t|, ∀x ≤ |t|

Theory for PTIME Language of arithmetic plus ⌊x/2⌋, |x| = ⌈log(x + 1)⌉, x#y = 2|x|·|y|. ∆b

0-formulas have only sharply bounded quantifiers ∃x ≤ |t|, ∀x ≤ |t|

Σb

1-formulas have the form ∃x ≤ t ∆b 0.

‘there is x of polynomial length” Σb

1-definable sets (over N) are precisely those in NP.

Theory for PTIME Language of arithmetic plus ⌊x/2⌋, |x| = ⌈log(x + 1)⌉, x#y = 2|x|·|y|. ∆b

0-formulas have only sharply bounded quantifiers ∃x ≤ |t|, ∀x ≤ |t|

Σb

1-formulas have the form ∃x ≤ t ∆b 0.

‘there is x of polynomial length” Σb

1-definable sets (over N) are precisely those in NP.

S1

2 contains certain defining axioms for the symbols plus

ϕ(0, x) ∧ ∀y(ϕ(⌊y/2⌋, x) → ϕ(y, x)) → ϕ(x, x) for ϕ ∈ Σb

1

Theory for PTIME Language of arithmetic plus ⌊x/2⌋, |x| = ⌈log(x + 1)⌉, x#y = 2|x|·|y|. ∆b

0-formulas have only sharply bounded quantifiers ∃x ≤ |t|, ∀x ≤ |t|

Σb

1-formulas have the form ∃x ≤ t ∆b 0.

‘there is x of polynomial length” Σb

1-definable sets (over N) are precisely those in NP.

S1

2 contains certain defining axioms for the symbols plus

ϕ(0, x) ∧ ∀y(ϕ(⌊y/2⌋, x) → ϕ(y, x)) → ϕ(x, x) for ϕ ∈ Σb

1

Buss 1986 A function is PTIME iff it is Σb

1-definable in S1 2.

Theory for PTIME Language of arithmetic plus ⌊x/2⌋, |x| = ⌈log(x + 1)⌉, x#y = 2|x|·|y|. ∆b

0-formulas have only sharply bounded quantifiers ∃x ≤ |t|, ∀x ≤ |t|

Σb

1-formulas have the form ∃x ≤ t ∆b 0.

‘there is x of polynomial length” Σb

1-definable sets (over N) are precisely those in NP.

S1

2 contains certain defining axioms for the symbols plus

ϕ(0, x) ∧ ∀y(ϕ(⌊y/2⌋, x) → ϕ(y, x)) → ϕ(x, x) for ϕ ∈ Σb

1

Buss 1986 A function is PTIME iff it is Σb

1-definable in S1 2.

Definability S1

2 proves the defining equations of Cobham.

Witnessing If S1

2 ⊢ ∃y ϕ(

x, y), then S1

2 ⊢ ϕ(

x, f( x)) for some f ∈ PTIME.

Theory KP

1

language ∈, 0, 1, x, {x, y}, x × y, tc(x), x ⊙ y, x ⊙−1 y, x#y idea sharply bounded quantification corresponds to ∃x ∈ t, ∀x ∈ t

Theory KP

1

language ∈, 0, 1, x, {x, y}, x × y, tc(x), x ⊙ y, x ⊙−1 y, x#y idea sharply bounded quantification corresponds to ∃x ∈ t, ∀x ∈ t ∆0-formulas have only sharply bounded quantifiers Σ

1-formulas have the form ∃x t ∆0 for t a #-term

Theory KP

1

language ∈, 0, 1, x, {x, y}, x × y, tc(x), x ⊙ y, x ⊙−1 y, x#y idea sharply bounded quantification corresponds to ∃x ∈ t, ∀x ∈ t ∆0-formulas have only sharply bounded quantifiers Σ

1-formulas have the form ∃x t ∆0 for t a #-term

Extensionality ∀u(u ∈ x ↔ u ∈ y) → x = y defining axioms for the symbols ∆0-Separation ∃z∀u(u ∈ z ↔ u ∈ x ∧ ϕ(u, x)) for ϕ ∈ ∆0 ∆0-Collection ∀u ∈ x∃v ϕ(u, v, x) → ∃z∀u ∈ x∃v ∈ z ϕ(u, v, x) for ϕ ∈ ∆0 Σ

1-Induction ∀y(∀u ∈ y ϕ(u,

x) → ϕ(y, x)) → ϕ(x, x) for all ϕ ∈ Σ

1

Theory KP

1

language ∈, 0, 1, x, {x, y}, x × y, tc(x), x ⊙ y, x ⊙−1 y, x#y idea sharply bounded quantification corresponds to ∃x ∈ t, ∀x ∈ t ∆0-formulas have only sharply bounded quantifiers Σ

1-formulas have the form ∃x t ∆0 for t a #-term

Extensionality ∀u(u ∈ x ↔ u ∈ y) → x = y defining axioms for the symbols ∆0-Separation ∃z∀u(u ∈ z ↔ u ∈ x ∧ ϕ(u, x)) for ϕ ∈ ∆0 ∆0-Collection ∀u ∈ x∃v ϕ(u, v, x) → ∃z∀u ∈ x∃v ∈ z ϕ(u, v, x) for ϕ ∈ ∆0 Σ

1-Induction ∀y(∀u ∈ y ϕ(u,

x) → ϕ(y, x)) → ϕ(x, x) for all ϕ ∈ Σ

1

Intuition KP

1 is to KP1 as S1 2 is to IΣ1.

Goal prove Definability and Witnessing wrt CRSF.

The formula x y is ∃e e : x y where e : x y is e is the graph of a (multi-valued) embedding of x into y i.a.w, u → {v : u, v ∈ e} is an embedding of x into y.

The formula x y is ∃e e : x y where e : x y is e is the graph of a (multi-valued) embedding of x into y i.a.w, u → {v : u, v ∈ e} is an embedding of x into y. i.e. the ∆0-formula e ⊆ tc(x) × tc(y) ∧ ∀u ∈ tc(x) ∃v ∈ tc(y) u, v ∈ e ∧ ∀u, u′ ∈ tc(x) ∀v ∈ tc(y) (u = u′ ∧ u, v ∈ e → u′, v ∈ e) ∧ ∀u, u′ ∈ tc(x) ∀v′ ∈ tc(y) (u ∈ u′ ∧ u′, v′ ∈ e → ∃v ∈ tc(v′) u, v ∈ e).

Definability Σ

1-expansion of KP 1:

Add certain defining axioms ϕ( x, f( x)) for new symbols f( x) where ϕ( x, y) ∈ Σ

1, t(

x) #-term KP

1 ⊢ ∃≤1y ϕ(

x, y) KP

1 ⊢ ∀

x ∃y t( x) ϕ( x, y)

Definability Σ

1-expansion of KP 1:

Add certain defining axioms ϕ( x, f( x)) for new symbols f( x) where ϕ( x, y) ∈ Σ

1, t(

x) #-term KP

1 ⊢ ∃≤1y ϕ(

x, y) KP

1 ⊢ ∀

x ∃y t( x) ϕ( x, y) There exists such an expansion KP

1(Lcrsf) to language Lcrsf such that

• for all h(y1, . . . , yr), g1(

x), . . . , gr( x) ∈ Lcrsf there is f( x) ∈ Lcrsf such that KP

1(Lcrsf) ⊢ f(

x) = h(g1( x), . . . , gr( x))

Definability Σ

1-expansion of KP 1:

Add certain defining axioms ϕ( x, f( x)) for new symbols f( x) where ϕ( x, y) ∈ Σ

1, t(

x) #-term KP

1 ⊢ ∃≤1y ϕ(

x, y) KP

1 ⊢ ∀

x ∃y t( x) ϕ( x, y) There exists such an expansion KP

1(Lcrsf) to language Lcrsf such that

• for all h(y1, . . . , yr), g1(

x), . . . , gr( x) ∈ Lcrsf there is f( x) ∈ Lcrsf such that KP

1(Lcrsf) ⊢ f(

x) = h(g1( x), . . . , gr( x))

• for all g(z, x), τ(u, x) ∈ Lcrsf, #-terms t(x) there is f(x) ∈ Lcrsf such that

KP

1(Lcrsf) ⊢ τ(·, x) : g(f”(x), x) t(x) → f(x) = g(f”(x), x)

where f”(x) ∈ Lcrsf is such that KP

1(Lcrsf) ⊢ f”(x) = {f(y) : y ∈ x}.

Witnessing fails Witnessing Question if ϕ( x, y) ∈ ∆0 and KP

1(Lcrsf) ⊢ ∃y ϕ(

x, y), then there is a witnessing function f( x) ∈ Lcrsf such that KP

1(Lcrsf) ⊢ ϕ(

x, f( x)) ?

Witnessing fails Witnessing Question if ϕ( x, y) ∈ ∆0 and KP

1(Lcrsf) ⊢ ∃y ϕ(

x, y), then there is a witnessing function f( x) ∈ Lcrsf such that KP

1(Lcrsf) ⊢ ϕ(

x, f( x)) ? Counterexample KP

1 ⊢ ∃y(x = 0 → y ∈ x)

Witnessing function C satisfies (x = 0 → C(x) ∈ x). This Global Choice (GC). It implies (AC), not provable in ZF.

Adding (GC) KPC

1: add (GC) to KP 1

Get expansion KPC

1(LC crsf) ⊇ KP 1(Lcrsf).

Definability theorem holds for KPC

1(LC crsf).

Adding (GC) KPC

1: add (GC) to KP 1

Get expansion KPC

1(LC crsf) ⊇ KP 1(Lcrsf).

Definability theorem holds for KPC

1(LC crsf).

Witnessing Theorem if ϕ( x, y) ∈ ∆0 and KPC

1(LC crsf) ⊢ ∃y ϕ(

x, y), then there is a witnessing function f( x) ∈ LC

crsf such that

KPC

1(LC crsf) ⊢ ϕ(

x, f( x)).

Partial conservativity KPC

1(LC crsf) ⊇ KP 1(Lcrsf)

(AC) shows that extension is not conservative.

Partial conservativity KPC

1(LC crsf) ⊇ KP 1(Lcrsf)

(AC) shows that extension is not conservative. Theorem Assume KPC

1(LC crsf) ⊢ ψ, an Lcrsf-formula.

Then ψ is provable in KP

1(Lcrsf) plus the following local choice principles:

Partial conservativity KPC

1(LC crsf) ⊇ KP 1(Lcrsf)

(AC) shows that extension is not conservative. Theorem Assume KPC

1(LC crsf) ⊢ ψ, an Lcrsf-formula.

Then ψ is provable in KP

1(Lcrsf) plus the following local choice principles:

Well-Ordering ∀x∃α∃y ( y : α

bij

− → x ∧ ∀β, γ ∈ α (y(β) ∈ y(γ) → β ∈ γ) ) ∆0-Dependent Choice for ϕ ∈ ∆0 ∀x∃yϕ(x, y, x) → ∀α ∃z (Fct(z) ∧ dom(z) = α ∧ ∀β ∈ α ϕ(z ↿ β, z(β), x))

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