Bounded Arithmetic in Free Logic Yoriyuki Yamagata CTFM, 2013/02/20 - PowerPoint PPT Presentation

Bounded Arithmetic in Free Logic Yoriyuki Yamagata CTFM, 2013/02/20

𝑗 Buss’s theories 𝑇 2 • Language of Peano Arithmetic + “#” – a # b = 2 𝑏 ⋅ | 𝑐 | • BASIC axioms • PIND 𝑦 𝐵 , Γ → Δ , 𝐵 ( 𝑦 ) 2 𝐵 0 , Γ → Δ , 𝐵 ( 𝑢 ) 𝑐 , i.e. has 𝑗 -alternations of where 𝐵 𝑦 ∈ Σ 𝑗 bounded quantifiers ∀𝑦 ≤ 𝑢 , ∃𝑦 ≤ 𝑢 .

𝑗 PH and Buss’s theories 𝑇 2 1 = 𝑇 2 2 = 𝑇 2 3 = … 𝑇 2 Implies 𝑞 ) = … 𝑄 = □ ( 𝑂𝑄 ) = □ ( Σ 2 We can approach (non) collapse of PH from (non) collapse of hierarchy of Buss’s theories (PH = Polynomial Hierarchy)

Our approach 𝑗 by Gödel incompleteness theorem • Separate 𝑇 2 • Use analogy of separation of 𝐽Σ 𝑗

Separation of 𝐽Σ 𝑗 … 𝐽Σ 3 ⊢ Con(I Σ 2 ) ⊆ 𝐽Σ 2 ⊢ Con I Σ 2 ⊆ 𝐽Σ 1

𝑗 Consistency proof inside 𝑇 2 • Bounded Arithmetics generally are not capable to prove consistency. – 𝑇 2 does not prove consistency of Q (Paris, Wilkie) – 𝑇 2 does not prove bounded consistency of 1 (Pudlák) 𝑇 2 𝑗 does not prove consistency the 𝐶 𝑗 𝑐 fragement – 𝑇 2 −1 (Buss and Ignjatović ) of 𝑇 2

Buss and Ignjatović (1995) … 3 ⊢ 𝐶 3 b − Con( 𝑇 2 −1 ) 𝑇 2 ⊆ 2 ⊢ 𝐶 2 b − Con( 𝑇 2 −1 ) 𝑇 2 ⊆ 1 ⊢ 𝐶 1 b − Con( 𝑇 2 −1 ) 𝑇 2

Where… 𝑐 − 𝐷𝐷𝐷 𝑈 • 𝐶 𝑗 𝑐 − proofs – consistency of 𝐶 𝑗 𝑐 − proofs : the proofs by 𝐶 𝑗 𝑐 -formule – 𝐶 𝑗 𝑐 : Σ 0 𝑐 ) … Formulas generated from Σ 𝑗 𝑐 by 𝑐 ( Σ 𝑗 – 𝐶 𝑗 Boolean connectives and sharply bounded quantifiers. −1 • 𝑇 2 𝑗 – Induction free fragment of 𝑇 2

If… 𝑘 ⊢ 𝐶 i b − Con 𝑇 2 −1 , j > i 𝑇 2 Then, Buss’s hierarchy does not collapse.

−1 inside 𝑇 2 𝑗 Consistency proof of 𝑇 2 Problem • No truth definition, because • No valuation of terms, because • The values of terms increase exponentially • E.g. 2#2#2#2#2#...#2 𝑗 world, terms do not have values a priori . In 𝑇 2 • Thus, we must prove the existence of values in proofs. • We introduce the predicate 𝐹 which signifies existence of values.

Our result(2012) … 5 ⊢ 3 − Con( 𝑇 2 −1 𝐹 ) 𝑇 2 ⊆ 4 ⊢ 2 − Con( 𝑇 2 −1 𝐹 ) 𝑇 2 ⊆ 3 ⊢ 1 − Con( 𝑇 2 −1 𝐹 ) 𝑇 2

Where… • 𝑗 − 𝐷𝐷𝐷 𝑈 – consistency of 𝑗 -normal proofs – 𝑗 -normal proofs : the proofs by 𝑗 -normal formulas – 𝑗 -normal formulas: Formulas in the form: ∃𝑦 1 ≤ 𝑢 1 ∀𝑦 2 ≤ 𝑢 2 … 𝑅𝑦 𝑗 ≤ 𝑢 𝑗 𝑅𝑦 𝑗+1 ≤ 𝑢 𝑗+1 . 𝐵 (… ) Where 𝐵 is quantifier free

Where… −1 𝐹 • 𝑇 2 𝑗 𝐹 – Induction free fragment of 𝑇 2 – have predicate 𝐹 which signifies existence of values • Such logic is called Free logic

𝑗 𝐹 (Language) 𝑇 2 Predicates • =, ≤ , 𝐹 Function symbols • Finite number of polynomial functions Formulas • Atomic formula, negated atomic formula • 𝐵 ∨ 𝐶 , 𝐵 ∧ 𝐶 • Bounded quantifiers

𝑗 𝐹 (Axioms) 𝑇 2 • 𝐹 -axioms • Equality axioms • Data axioms • Defining axioms • Auxiliary axioms

Idea behind axioms… → 𝑏 = 𝑏 Because there is no guarantee of 𝐹𝑏 Thus, we add 𝐹𝑏 in the antecedent 𝐹𝑏 → 𝑏 = 𝑏

E-axioms • 𝐹𝐹 𝑏 1 , … , 𝑏 𝑜 → 𝐹𝑏 𝑘 • 𝑏 1 = 𝑏 2 → 𝐹𝑏 𝑘 • 𝑏 1 ≠ 𝑏 2 → 𝐹𝑏 𝑘 • 𝑏 1 ≤ 𝑏 2 → 𝐹𝑏 𝑘 • ¬ 𝑏 1 ≤ 𝑏 2 → 𝐹𝑏 𝑘

Equality axioms • 𝐹𝑏 → 𝑏 = 𝑏 • 𝐹𝐹 𝑏 ⃗ , 𝑏 ⃗ = 𝑐 → 𝐹 𝑏 ⃗ = 𝐹 𝑐

Data axioms • → 𝐹𝐹 • 𝐹𝑏 → 𝐹𝑡 0 𝑏 • 𝐹𝑏 → 𝐹𝑡 1 𝑏

Defining axioms 𝐹 𝑣 𝑏 1 , 𝑏 2 , … , 𝑏 𝑜 = 𝑢 ( 𝑏 1 , … , 𝑏 𝑜 ) 𝑣 𝑏 = 0, 𝑏 , 𝑡 0 𝑏 , 𝑡 1 𝑏 𝐹𝑏 1 , … , 𝐹𝑏 𝑜 , 𝐹𝑢 𝑏 1 , … , 𝑏 𝑜 → 𝐹 𝑣 𝑏 1 , 𝑏 2 , … , 𝑏 𝑜 = 𝑢 ( 𝑏 1 , … , 𝑏 𝑜 )

Auxiliary axioms 𝑏 = 𝑐 ⊃ 𝑏 # 𝑑 = 𝑐 # 𝑑 𝐹𝑏 # 𝑑 , 𝐹𝑐 # 𝑑 , 𝑏 = | 𝑐 | → 𝑏 # 𝑑 = 𝑐 # 𝑑

PIND-rule 𝑐 -formulas where 𝐵 is an Σ 𝑗

𝑗 𝐹 Bootstrapping 𝑇 2 𝑗 𝐹 ⊢ Tot( 𝐹 ) for any 𝐹 , 𝑗 ≥ 0 I. 𝑇 2 𝑗 𝐹 ⊢ BASIC ∗ , equality axioms ∗ II. 𝑇 2 𝑗 𝐹 ⊢ predicate logic ∗ III. 𝑇 2 𝑐 −PIND ∗ 𝑗 𝐹 ⊢ Σ 𝑗 IV. 𝑇 2

Theorem (Consistency) 𝑗+2 ⊢ i − Con( 𝑇 2 −1 𝐹 ) 𝑇 2

Valuation trees ρ -valuation tree bounded by 19 ρ(a)=2, ρ(b)=3 a=2 a#a=16 b=3 a#a+b=19 𝑤 𝑏 # 𝑏 + 𝑐 , 𝜍 ↓ 19 19 𝑐 𝑤 𝑢 , 𝜍 ↓ 𝑣 𝑑 is Σ 1

Bounded truth definition (1) • 𝑈 𝑣 , 𝑢 1 = 𝑢 2 , 𝜍 ⇔ def ∃𝑑 ≤ 𝑣 , 𝑤 𝑢 1 , 𝜍 ↓ 𝑣 𝑑 ∧ 𝑤 𝑢 1 , 𝜍 ↓ 𝑣 𝑑 • 𝑈 𝑣 , 𝜚 1 ∧ 𝜚 2 , 𝜍 ⇔ def 𝑈 𝑣 , 𝜚 1 , 𝜍 ∧ 𝑈 𝑣 , 𝜚 2 , 𝜍 • 𝑈 𝑣 , 𝜚 1 ∨ 𝜚 2 , 𝜍 ⇔ def 𝑈 𝑣 , 𝜚 1 , 𝜍 ∨ 𝑈 𝑣 , 𝜚 2 , 𝜍

Bounded truth definition (2) • 𝑈 𝑣 , ∃𝑦 ≤ 𝑢 , 𝜚 ( 𝑦 ) , 𝜍 ⇔ def ∃𝑑 ≤ 𝑣 , 𝑤 𝑢 , 𝜍 ↓ 𝑣 𝑑 ∧ ∃𝑒 ≤ 𝑑 , 𝑈 𝑣 , 𝜚 𝑦 , 𝜍 𝑦 ↦ 𝑒 • 𝑈 𝑣 , ∀𝑦 ≤ 𝑢 , 𝜚 ( 𝑦 ) , 𝜍 ⇔ def ∃𝑑 ≤ 𝑣 , 𝑤 𝑢 , 𝜍 ↓ 𝑣 𝑑 ∧ ∀𝑒 ≤ 𝑑 , 𝑈 ( 𝑣 , 𝜚 𝑦 , 𝜍 [ 𝑦 ↦ 𝑒 ]) 𝑐 , 𝑈 𝑣 , 𝜚 is Σ 𝑗+1 𝑐 Remark: If 𝜚 is Σ 𝑗

induction hypothesis 𝑣 : enough large integer 𝑠 : node of a proof of 0=1 Γ 𝑠 → Δ 𝑠 : the sequent of node 𝑠 𝜍 : assignment 𝜍 𝑏 ≤ 𝑣 ∀𝑣 ′ ≤ 𝑣 ⊖ 𝑠 , { ∀𝐵 ∈ Γ 𝑠 𝑈 𝑣 ′ , 𝐵 , 𝜍 ⊃ [ ∃𝐶 ∈ Δ r , 𝑈 ( 𝑣 ′ ⊕ 𝑠 , 𝐶 , 𝜍 ) ]}

Conjecture −1 𝐹 is weak enough • 𝑇 2 𝑗+2 can prove 𝑗 -consistency of 𝑇 2 −1 𝐹 – 𝑇 2 −1 𝐹 is strong enough • While 𝑇 2 𝑗 𝐹 can interpret 𝑇 2 𝑗 – 𝑇 2 • Conjecture −1 𝐹 is a good candidate to separate 𝑇 2 𝑗 and 𝑇 2 𝑗+2 . 𝑇 2

Future works • Long-term goal 𝑗 ⊢ 𝑗−Con(𝑇 2 −1 𝐹 )? 𝑇 2 • Short-term goal 𝑗 𝐹 – Simplify 𝑇 2

Publications • Bounded Arithmetic in Free Logic Logical Methods in Computer Science Volume 8, Issue 3, Aug. 10, 2012