bounded arithmetic in free logic
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Presentation at CTFM (Computability Theory and Foundation of Mathematics)TRANSCRIPT
Bounded Arithmetic in Free Logic
Yoriyuki Yamagata CTFM, 2013/02/20
Bussβs theories π2π β’ Language of Peano Arithmetic + β#β
β a # b = 2 π β |π| β’ BASIC axioms β’ PIND
π΄ π₯2 , Ξ β Ξ,π΄(π₯)
π΄ 0 , Ξ β Ξ,π΄(π‘)
where π΄ π₯ β Ξ£ππ, i.e. has π-alternations of bounded quantifiers βπ₯ β€ π‘,βπ₯ β€ π‘.
PH and Bussβs theories π2π
π21 = π22 = π23 = β¦ Implies
π = β‘(ππ) = β‘(Ξ£2π) = β¦
We can approach (non) collapse of PH from (non) collapse of hierarchy of Bussβs theories
(PH = Polynomial Hierarchy)
Our approach
β’ Separate π2π by GΓΆdel incompleteness theorem β’ Use analogy of separation of πΌΞ£π
Separation of πΌΞ£π
πΌΞ£3 β’ Con(IΞ£2)
πΌΞ£2 β’ Con IΞ£2
β¦
πΌΞ£1
β
β
Consistency proof inside π2π β’ Bounded Arithmetics generally are not
capable to prove consistency. β π2 does not prove consistency of Q (Paris, Wilkie) β π2 does not prove bounded consistency of π21 (PudlΓ‘k)
β π2π does not prove consistency the π΅ππ fragement of π2β1 (Buss and IgnjatoviΔ)
Buss and IgnjatoviΔ(1995)
β¦
β
π23 β’ π΅3b β Con(π2β1)
π22 β’ π΅2b β Con(π2β1)
π21 β’ π΅1b β Con(π2β1)
β
Whereβ¦
β’ π΅ππ β πΆπΆπΆ π β consistency of π΅ππ βproofs β π΅ππ βproofs : the proofs by π΅ππ-formule β π΅ππ:Ξ£0π(Ξ£ππ)β¦ Formulas generated from Ξ£ππ by
Boolean connectives and sharply bounded quantifiers.
β’ π2β1 β Induction free fragment of π2π
Ifβ¦
π2π β’ π΅ib β Con π2β1 , j > i
Then, Bussβs hierarchy does not collapse.
Consistency proof of π2β1 inside π2π
Problem β’ No truth definition, because β’ No valuation of terms, because
β’ The values of terms increase exponentially β’ E.g. 2#2#2#2#2#...#2
In π2π world, terms do not have values a priori. β’ Thus, we must prove the existence of values in proofs. β’ We introduce the predicate πΈ which signifies existence of
values.
Our result(2012)
β¦
β
π25 β’ 3 β Con(π2β1πΈ)
π24 β’ 2 β Con(π2β1πΈ)
π23 β’ 1 β Con(π2β1πΈ)
β
Whereβ¦
β’ π β πΆπΆπΆ π β consistency of π-normal proofs β π-normal proofs : the proofs by π-normal formulas β π-normal formulas: Formulas in the form: βπ₯1 β€ π‘1βπ₯2 β€ π‘2 β¦ππ₯π β€ π‘πππ₯π+1 β€ π‘π+1 .π΄(β¦ ) Where π΄ is quantifier free
Whereβ¦
β’ π2β1πΈ β Induction free fragment of π2ππΈ β have predicate πΈ which signifies existence of
values β’ Such logic is called Free logic
π2ππΈ(Language)
Predicates β’ =,β€,πΈ
Function symbols β’ Finite number of polynomial functions
Formulas β’ Atomic formula, negated atomic formula β’ π΄ β¨ π΅,π΄ β§ π΅ β’ Bounded quantifiers
π2ππΈ(Axioms)
β’ πΈ-axioms β’ Equality axioms β’ Data axioms β’ Defining axioms β’ Auxiliary axioms
Idea behind axiomsβ¦
β π = π
Because there is no guarantee of πΈπ Thus, we add πΈπ in the antecedent
πΈπ β π = π
E-axioms
β’ πΈπΈ π1, β¦ ,ππ β πΈππ β’ π1 = π2 β πΈππ β’ π1 β π2 β πΈππ β’ π1 β€ π2 β πΈππ β’ Β¬π1β€ π2 β πΈππ
Equality axioms
β’ πΈπ β π = π
β’ πΈπΈ οΏ½βοΏ½ , οΏ½βοΏ½ = π β πΈ οΏ½βοΏ½ = πΈ π
Data axioms
β’ β πΈπΈ β’ πΈπ β πΈπ 0π β’ πΈπ β πΈπ 1π
Defining axioms
πΈ π’ π1 ,π2, β¦ , ππ = π‘(π1, β¦ , ππ)
πΈπ1, β¦ ,πΈππ,πΈπ‘ π1, β¦ , ππ β πΈ π’ π1 ,π2, β¦ , ππ = π‘(π1, β¦ , ππ)
π’ π = 0,π, π 0π, π 1π
Auxiliary axioms
π = π β π#π = π#π
πΈπ#π,πΈπ#π, π = |π| β π#π = π#π
PIND-rule
where π΄ is an Ξ£ππ-formulas
Bootstrapping π2ππΈ
I. π2ππΈ β’ Tot(πΈ) for any πΈ, π β₯ 0 II. π2ππΈ β’ BASICβ, equality axioms β III. π2ππΈ β’ predicate logic β IV. π2ππΈ β’ Ξ£ππ βPINDβ
Theorem (Consistency)
π2π+2 β’ i β Con(π2β1πΈ)
Valuation trees
a#a+b=19
a#a=16 b=3
a=2
Ο-valuation tree bounded by 19 Ο(a)=2, Ο(b)=3
π£ π#π + π ,π β19 19 π£ π‘ ,π βπ’ π is Ξ£1π
Bounded truth definition (1)
β’ π π’, π‘1 = π‘2 , π βdef βπ β€ π’, π£ π‘1 ,π βπ’ π β§ π£ π‘1 ,π βπ’ π
β’ π π’, π1 β§ π2 ,π βdef π π’, π1 , π β§ π π’, π2 , π β’ π π’, π1 β¨ π2 ,π βdef π π’, π1 , π β¨ π π’, π2 ,π
Bounded truth definition (2)
β’ π π’, βπ₯ β€ π‘,π(π₯) ,π βdef βπ β€ π’, π£ π‘ , π βπ’ π β§
βπ β€ π,π π’, π π₯ ,π π₯ β¦ π β’ π π’, βπ₯ β€ π‘,π(π₯) , π βdef
βπ β€ π’, π£ π‘ , π βπ’ π β§ βπ β€ π,π(π’, π π₯ ,π[π₯ β¦ π])
Remark: If π is Ξ£ππ,π π’, π is Ξ£π+1π
induction hypothesis
π’: enough large integer π: node of a proof of 0=1 Ξπ β Ξπ: the sequent of node π π: assignment π π β€ π’ βπ’β² β€ π’ β π, { βπ΄ β Ξπ π π’β², π΄ , π β
[βπ΅ β Ξr,π(π’β² β π, π΅ , π)]}
Conjecture
β’ π2β1πΈ is weak enough β π2π+2 can prove π-consistency of π2β1πΈ
β’ While π2β1πΈ is strong enough β π2ππΈ can interpret π2π
β’ Conjecture π2β1πΈ is a good candidate to separate π2π and π2π+2.
Future works
β’ Long-term goal π2π β’ πβCon(π2β1πΈ)?
β’ Short-term goal β Simplify π2ππΈ
Publications
β’ Bounded Arithmetic in Free Logic Logical Methods in Computer Science Volume 8, Issue 3, Aug. 10, 2012