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First-order Quasi-canonical Proof Systems Speaker: Yotam Dvir Yotam Dvir Arnon Avron Tel-Aviv University, Israel 2019-09-04 1/32 Existential Information Processing A Motivating Example 2/32 Background Belnap [1977]: 3/32 Background

  1. Legal Valuations & Subtitutions Definition An M -legal A -valuation is a function v : clFrm → V s.t.: 9/32

  2. Legal Valuations & Subtitutions Definition An M -legal A -valuation is a function v : clFrm → V s.t.: A. If ϕ A ∼ ϕ ′ , then v [ ϕ ] = v [ ϕ ′ ] 9/32

  3. Legal Valuations & Subtitutions Definition An M -legal A -valuation is a function v : clFrm → V s.t.: A. If ϕ A ∼ ϕ ′ , then v [ ϕ ] = v [ ϕ ′ ] Q. v [ Q z 1 . . . z k ( ψ 1 , . . . ψ n )] ∈ ˜ Q [ 9/32

  4. Legal Valuations & Subtitutions Definition An M -legal A -valuation is a function v : clFrm → V s.t.: A. If ϕ A ∼ ϕ ′ , then v [ ϕ ] = v [ ϕ ′ ] Q. v [ Q z 1 . . . z k ( ψ 1 , . . . ψ n )] ∈ ˜ Q [ {� v [ ψ 1 { a 1 / z 1 , . . . a k / z k } ] , . . . v [ ψ n { a 1 / z 1 , . . . a k / z k } ] � | a 1 , . . . a k ∈ Dom A} ] 9/32

  5. Legal Valuations & Subtitutions Definition An M -legal A -valuation is a function v : clFrm → V s.t.: A. If ϕ A ∼ ϕ ′ , then v [ ϕ ] = v [ ϕ ′ ] Q. v [ Q z 1 . . . z k ( ψ 1 , . . . ψ n )] ∈ ˜ Q [ {� v [ ψ 1 { a 1 / z 1 , . . . a k / z k } ] , . . . v [ ψ n { a 1 / z 1 , . . . a k / z k } ] � | a 1 , . . . a k ∈ Dom A} ] Definition An A -source is a QM 3 r -legal A -valuation 9/32

  6. Legal Valuations & Subtitutions Definition An M -legal A -valuation is a function v : clFrm → V s.t.: A. If ϕ A ∼ ϕ ′ , then v [ ϕ ] = v [ ϕ ′ ] Q. v [ Q z 1 . . . z k ( ψ 1 , . . . ψ n )] ∈ ˜ Q [ {� v [ ψ 1 { a 1 / z 1 , . . . a k / z k } ] , . . . v [ ψ n { a 1 / z 1 , . . . a k / z k } ] � | a 1 , . . . a k ∈ Dom A} ] Definition An A -source is a QM 3 r -legal A -valuation Definition A substitution is a function σ : Var → clTrm 9/32

  7. Legal Valuations & Subtitutions Definition An M -legal A -valuation is a function v : clFrm → V s.t.: A. If ϕ A ∼ ϕ ′ , then v [ ϕ ] = v [ ϕ ′ ] Q. v [ Q z 1 . . . z k ( ψ 1 , . . . ψ n )] ∈ ˜ Q [ {� v [ ψ 1 { a 1 / z 1 , . . . a k / z k } ] , . . . v [ ψ n { a 1 / z 1 , . . . a k / z k } ] � | a 1 , . . . a k ∈ Dom A} ] Definition An A -source is a QM 3 r -legal A -valuation Definition A substitution is a function σ : Var → clTrm Definition A , v , σ | = C if v [ σ [ C ]] ∈ D (induces ⊢ M ) 9/32

  8. Processing Information from Sources ◮ Different sources may disagree 10/32

  9. Processing Information from Sources ◮ Different sources may disagree, e.g. ψ Alice Bob ψ Carol �− − − → 1 �− − − → 0 �− − − → U ψ 10/32

  10. Processing Information from Sources ◮ Different sources may disagree, e.g. ψ Alice Bob ψ Carol �− − − → 1 �− − − → 0 �− − − → U ψ ◮ Different sources may possess complementary info 10/32

  11. Processing Information from Sources ◮ Different sources may disagree, e.g. ψ Alice Bob ψ Carol �− − − → 1 �− − − → 0 �− − − → U ψ ◮ Different sources may possess complementary info, e.g. ψ Alice ψ ∧ θ Alice �− − − → 1 �− − − → U Bob Bob �− − − → 1 ψ ∧ θ �− − − → U θ 10/32

  12. Processing Information from Sources ◮ Different sources may disagree, e.g. ψ Alice Bob ψ Carol �− − − → 1 �− − − → 0 �− − − → U ψ ◮ Different sources may possess complementary info, e.g. ψ Alice ψ ∧ θ Alice �− − − → 1 �− − − → U Bob Bob �− − − → 1 ψ ∧ θ �− − − → U θ Processing information from a set of A -sources S : 10/32

  13. Processing Information from Sources ◮ Different sources may disagree, e.g. ψ Alice Bob ψ Carol �− − − → 1 �− − − → 0 �− − − → U ψ ◮ Different sources may possess complementary info, e.g. ψ Alice ψ ∧ θ Alice �− − − → 1 �− − − → U Bob Bob �− − − → 1 ψ ∧ θ �− − − → U θ Processing information from a set of A -sources S : 1. A gatherer g : clFrm → P [ { 0 , 1 } ] collects all of the claims existentially, i.e. b ∈ g [ ϕ ] iff ∃ s ∈ S s.t. s [ ϕ ] = b 10/32

  14. Processing Information from Sources ◮ Different sources may disagree, e.g. ψ Alice Bob ψ Carol �− − − → 1 �− − − → 0 �− − − → U ψ ◮ Different sources may possess complementary info, e.g. ψ Alice ψ ∧ θ Alice �− − − → 1 �− − − → U Bob Bob �− − − → 1 ψ ∧ θ �− − − → U θ Processing information from a set of A -sources S : 1. A gatherer g : clFrm → P [ { 0 , 1 } ] collects all of the claims existentially, i.e. b ∈ g [ ϕ ] iff ∃ s ∈ S s.t. s [ ϕ ] = b 2. A processor d : clFrm → P [ { 0 , 1 } ] is effectively induced: ◮ Starting with d = g 10/32

  15. Processing Information from Sources ◮ Different sources may disagree, e.g. ψ Alice Bob ψ Carol �− − − → 1 �− − − → 0 �− − − → U ψ ◮ Different sources may possess complementary info, e.g. ψ Alice ψ ∧ θ Alice �− − − → 1 �− − − → U Bob Bob �− − − → 1 ψ ∧ θ �− − − → U θ Processing information from a set of A -sources S : 1. A gatherer g : clFrm → P [ { 0 , 1 } ] collects all of the claims existentially, i.e. b ∈ g [ ϕ ] iff ∃ s ∈ S s.t. s [ ϕ ] = b 2. A processor d : clFrm → P [ { 0 , 1 } ] is effectively induced: ◮ Starting with d = g ◮ Then taking closure under ‘integrity conditions’ 10/32

  16. Processing Information from Sources ◮ Different sources may disagree, e.g. ψ Alice Bob ψ Carol �− − − → 1 �− − − → 0 �− − − → U ψ ◮ Different sources may possess complementary info, e.g. ψ Alice ψ ∧ θ Alice �− − − → 1 �− − − → U Bob Bob �− − − → 1 ψ ∧ θ �− − − → U θ Processing information from a set of A -sources S : 1. A gatherer g : clFrm → P [ { 0 , 1 } ] collects all of the claims existentially, i.e. b ∈ g [ ϕ ] iff ∃ s ∈ S s.t. s [ ϕ ] = b 2. A processor d : clFrm → P [ { 0 , 1 } ] is effectively induced: ◮ Starting with d = g ◮ Then taking closure under ‘integrity conditions’, e.g. ◮ 1 ∈ � a ∈ Dom d [ ϕ ( a )] = ⇒ 1 ∈ d [ ∃ x ϕ ( x )] 10/32

  17. Processing Information from Sources ◮ Different sources may disagree, e.g. ψ Alice Bob ψ Carol �− − − → 1 �− − − → 0 �− − − → U ψ ◮ Different sources may possess complementary info, e.g. ψ Alice ψ ∧ θ Alice �− − − → 1 �− − − → U Bob Bob �− − − → 1 ψ ∧ θ �− − − → U θ Processing information from a set of A -sources S : 1. A gatherer g : clFrm → P [ { 0 , 1 } ] collects all of the claims existentially, i.e. b ∈ g [ ϕ ] iff ∃ s ∈ S s.t. s [ ϕ ] = b 2. A processor d : clFrm → P [ { 0 , 1 } ] is effectively induced: ◮ Starting with d = g ◮ Then taking closure under ‘integrity conditions’, e.g. ◮ 1 ∈ � a ∈ Dom d [ ϕ ( a )] = ⇒ 1 ∈ d [ ∃ x ϕ ( x )] ◮ 0 ∈ � a ∈ Dom d [ ϕ ( a )] = ⇒ 0 ∈ d [ ∃ x ϕ ( x )] 10/32

  18. Processing Information from Sources ◮ Different sources may disagree, e.g. ψ Alice Bob ψ Carol �− − − → 1 �− − − → 0 �− − − → U ψ ◮ Different sources may possess complementary info, e.g. ψ Alice ψ ∧ θ Alice �− − − → 1 �− − − → U Bob Bob �− − − → 1 ψ ∧ θ �− − − → U θ Processing information from a set of A -sources S : 1. A gatherer g : clFrm → P [ { 0 , 1 } ] collects all of the claims existentially, i.e. b ∈ g [ ϕ ] iff ∃ s ∈ S s.t. s [ ϕ ] = b 2. A processor d : clFrm → P [ { 0 , 1 } ] is effectively induced: ◮ Starting with d = g ◮ Then taking closure under ‘integrity conditions’, e.g. ◮ 1 ∈ � a ∈ Dom d [ ϕ ( a )] = ⇒ 1 ∈ d [ ∃ x ϕ ( x )] ◮ 0 ∈ � a ∈ Dom d [ ϕ ( a )] = ⇒ 0 ∈ d [ ∃ x ϕ ( x )] ◮ b ∈ d [ θ ] = ⇒ 1 − b ∈ d [ ¬ θ ] 10/32

  19. Processing Information from Sources ◮ Different sources may disagree, e.g. ψ Alice Bob ψ Carol �− − − → 1 �− − − → 0 �− − − → U ψ ◮ Different sources may possess complementary info, e.g. ψ Alice ψ ∧ θ Alice �− − − → 1 �− − − → U Bob Bob �− − − → 1 ψ ∧ θ �− − − → U θ Processing information from a set of A -sources S : 1. A gatherer g : clFrm → P [ { 0 , 1 } ] collects all of the claims existentially, i.e. b ∈ g [ ϕ ] iff ∃ s ∈ S s.t. s [ ϕ ] = b 2. A processor d : clFrm → P [ { 0 , 1 } ] is effectively induced: ◮ Starting with d = g ◮ Then taking closure under ‘integrity conditions’, e.g. ◮ 1 ∈ � a ∈ Dom d [ ϕ ( a )] = ⇒ 1 ∈ d [ ∃ x ϕ ( x )] ◮ 0 ∈ � a ∈ Dom d [ ϕ ( a )] = ⇒ 0 ∈ d [ ∃ x ϕ ( x )] ◮ b ∈ d [ θ ] = ⇒ 1 − b ∈ d [ ¬ θ ] ◮ etc. 10/32

  20. Processors are Characterized by a GNmatrix Recall ⊥ = {} f = { 0 } t = { 1 } ⊤ = { 0 , 1 } Definition QM 4 {⊥ , f , t , ⊤} , { t , ⊤} , QO 4 , where QO 4 � � E = E is given by: E ˜ X ∃ [ X ] ˜ ∨ ⊥ ⊤ a ¬ a ˜ f t {⊥} {⊥ , t } ⊥ {⊥ , t } {⊥ , t } { t } { t } ⊥ {⊥} {⊥ , f } {⊥ , t } { t } {⊥ , t } { f , ⊤} { t } {⊤} f f { f } { f , ⊤} { f } { t } { t } { t } { t } { f , ⊤} {⊤} t t ⊤ {⊤} ⊤ { t } {⊤} { t } {⊤} {⊤} {⊤} { t } else ∧ and ˜ ˜ ∀ are defined dually 11/32

  21. Processors are Characterized by a GNmatrix Recall ⊥ = {} f = { 0 } t = { 1 } ⊤ = { 0 , 1 } Definition QM 4 {⊥ , f , t , ⊤} , { t , ⊤} , QO 4 , where QO 4 � � E = E is given by: E ˜ X ∃ [ X ] ˜ ∨ ⊥ ⊤ a ¬ a ˜ f t {⊥} {⊥ , t } ⊥ {⊥ , t } {⊥ , t } { t } { t } ⊥ {⊥} {⊥ , f } {⊥ , t } { t } {⊥ , t } { f , ⊤} { t } {⊤} f f { f } { f , ⊤} { f } { t } { t } { t } { t } { f , ⊤} {⊤} t t ⊤ {⊤} ⊤ { t } {⊤} { t } {⊤} {⊤} {⊤} { t } else ∧ and ˜ ˜ ∀ are defined dually Theorem The function S �→ d is onto the set of QM 4 E -legal A -valuations 11/32

  22. ¬ -GNmatrix Definition A GNmatix M = �V , D , O� for L is a ¬ -GNmatix if: ◮ V ⊆ {⊥ , f , t , ⊤} ◮ D = V ∩ { t , ⊤} 12/32

  23. ¬ -GNmatrix Definition A GNmatix M = �V , D , O� for L is a ¬ -GNmatix if: ◮ V ⊆ {⊥ , f , t , ⊤} ◮ D = V ∩ { t , ⊤} ◮ (support of ¬ ϕ reflects opposition to ϕ ) ◮ x ∈ { f , ⊤} ⇒ ¬ x ⊆ { t , ⊤} = ˜ ◮ x ∈ {⊥ , t } ⇒ ¬ x ⊆ {⊥ , f } = ˜ 12/32

  24. ¬ -GNmatrix Definition A GNmatix M = �V , D , O� for L is a ¬ -GNmatix if: ◮ V ⊆ {⊥ , f , t , ⊤} ◮ D = V ∩ { t , ⊤} ◮ (support of ¬ ϕ reflects opposition to ϕ ) ◮ x ∈ { f , ⊤} ⇒ ¬ x ⊆ { t , ⊤} = ˜ ◮ x ∈ {⊥ , t } ⇒ ¬ x ⊆ {⊥ , f } = ˜ Example ◮ FOUR (Dunn-Belnap) ◮ QM 4 E (The processor GNmatrix) 12/32

  25. From the Example to the General Case ◮ A ¬ -GNmatrix M induces a logic ⊢ M . ◮ Desire: analytic proof system for it ◮ Turns out that ¬ -GNmatrices correspond to quasi-canonical Gentzen-type proof systems 13/32

  26. Quasi-canonical Gentzen-type Proof Systems 14/32

  27. Quasi-canonical Systems – Intro For sets of formulas Γ , ∆ a construct Γ ⇒ ∆ is called a sequent We use the usual notational devices, e.g. Γ , A ⇒ means Γ ∪ { A } ⇒ ∅ Intuition: Γ ⇒ ∆ says “if everything in Γ then something in ∆ ” 15/32

  28. Quasi-canonical Systems – Intro For sets of formulas Γ , ∆ a construct Γ ⇒ ∆ is called a sequent We use the usual notational devices, e.g. Γ , A ⇒ means Γ ∪ { A } ⇒ ∅ Intuition: Γ ⇒ ∆ says “if everything in Γ then something in ∆ ” A quasi-canonical Gentzen-type Proof System is a system for deriving sequents which consists of: ◮ A fixed set of structural rules (common to all) ◮ Quasi-canonical logical rules 15/32

  29. Structural Rules 16/32

  30. Structural Rules (A) X α ∼ X ′ X ⇒ X ′ 16/32

  31. Structural Rules (A) X α ∼ X ′ X ⇒ X ′ Γ ⇒ ∆ (W) Γ ′ , Γ ⇒ ∆ , ∆ ′ 16/32

  32. Structural Rules (A) X α ∼ X ′ X ⇒ X ′ Γ ⇒ ∆ (W) Γ ′ , Γ ⇒ ∆ , ∆ ′ Γ ′ ⇒ ∆ , X X , Γ ⇒ ∆ ′ (C) Γ ′ , Γ ⇒ ∆ , ∆ ′ 16/32

  33. Structural Rules (A) X α ∼ X ′ X ⇒ X ′ Γ ⇒ ∆ (W) Γ ′ , Γ ⇒ ∆ , ∆ ′ Γ ′ ⇒ ∆ , X X , Γ ⇒ ∆ ′ (C) Γ ′ , Γ ⇒ ∆ , ∆ ′ Γ ⇒ ∆ (S) Γ { t 1 / x 1 , . . . t m / x m } ⇒ ∆ { t 1 / x 1 , . . . t m / x m } 16/32

  34. Quasi-canonical Rules Canonical A { x / z } , ∆ A { t / z } ⇒ ∆ Γ ⇒ Γ , ( ⇒ ∀ ) ( ∀ ⇒ ) Γ ⇒ ∀ zA , ∆ Γ , ∀ zA ⇒ ∆ ( x is not free in the bottom sequent) 17/32

  35. Quasi-canonical Rules Quasi-Canonical Γ ⇒ ¬ A { x / z } , ∆ Γ , ¬ A { t / z } ⇒ ∆ ( ⇒ ¬∃ ) ( ¬∃ ⇒ ) Γ ⇒ ¬∃ zA , ∆ Γ , ¬∃ zA ⇒ ∆ ( x is not free in the bottom sequent) 17/32

  36. Quasi-canonical Rules Quasi-Canonical ∃ is � 1 , 1 � -ary { ⇒ ¬ p 1 ( v 1 ) } / ( ⇒ ¬∃ ) {¬ p 1 ( c 1 ) ⇒ } / ( ¬∃ ⇒ ) Γ ⇒ ¬ A { x / z } , ∆ Γ , ¬ A { t / z } ⇒ ∆ ( ⇒ ¬∃ ) ( ¬∃ ⇒ ) Γ ⇒ ¬∃ zA , ∆ Γ , ¬∃ zA ⇒ ∆ ( x is not free in the bottom sequent) 17/32

  37. Quasi-canonical Rules Quasi-Canonical ∃ is � 1 , 1 � -ary { ⇒ ¬ p 1 ( v 1 ) } / ( ⇒ ¬∃ ) {¬ p 1 ( c 1 ) ⇒ } / ( ¬∃ ⇒ ) Γ ⇒ ¬ A { x / z } , ∆ Γ , ¬ A { t / z } ⇒ ∆ ( ⇒ ¬∃ ) ( ¬∃ ⇒ ) Γ ⇒ ¬∃ zA , ∆ Γ , ¬∃ zA ⇒ ∆ ( x is not free in the bottom sequent) 17/32

  38. Quasi-canonical Rules Quasi-Canonical ∃ is � 1 , 1 � -ary { ⇒ ¬ p 1 ( v 1 ) } / ( ⇒ ¬∃ ) {¬ p 1 ( c 1 ) ⇒ } / ( ¬∃ ⇒ ) Γ ⇒ ¬ A { x / z } , ∆ Γ , ¬ A { t / z } ⇒ ∆ ( ⇒ ¬∃ ) ( ¬∃ ⇒ ) Γ ⇒ ¬∃ zA , ∆ Γ , ¬∃ zA ⇒ ∆ ( x is not free in the bottom sequent) 17/32

  39. Quasi-canonical Rules Quasi-Canonical ∃ is � 1 , 1 � -ary { ⇒ ¬ p 1 ( v 1 ) } / ( ⇒ ¬∃ ) {¬ p 1 ( c 1 ) ⇒ } / ( ¬∃ ⇒ ) Γ ⇒ ¬ A { x / z } , ∆ Γ , ¬ A { t / z } ⇒ ∆ ( ⇒ ¬∃ ) ( ¬∃ ⇒ ) Γ ⇒ ¬∃ zA , ∆ Γ , ¬∃ zA ⇒ ∆ ( x is not free in the bottom sequent) 17/32

  40. Conflicting Rules Definition A pair of rules of the following form are conflicting : ◮ Λ 1 / ( Q ⇒ ) and Λ 2 / ( ⇒ Q ) ◮ Λ 1 / ( ¬ Q ⇒ ) and Λ 2 / ( ⇒ ¬ Q ) 18/32

  41. Conflicting Rules Definition A pair of rules of the following form are conflicting : ◮ Λ 1 / ( Q ⇒ ) and Λ 2 / ( ⇒ Q ) ◮ Λ 1 / ( ¬ Q ⇒ ) and Λ 2 / ( ⇒ ¬ Q ) Example { ⇒ ¬ p 1 ( v 1 ) } / ( ⇒ ¬∃ ) and {¬ p 1 ( c 1 ) ⇒ } / ( ¬∃ ⇒ ) 18/32

  42. Coherence Definition A quasi-canonical system is coherent if for every pair of conflicting rules Λ 1 / T 1 and Λ 2 / T 2 the set Λ 1 ⋒ Λ 2 is inconsistent (i.e. ⇒ is derivable using only ( C ) and ( S ) ) 19/32

  43. Coherence Definition A quasi-canonical system is coherent if for every pair of conflicting rules Λ 1 / T 1 and Λ 2 / T 2 the set Λ 1 ⋒ Λ 2 is inconsistent (i.e. ⇒ is derivable using only ( C ) and ( S ) ) Example ⇒ ¬ p ( v 1 ) (S) ⇒ ¬ p ( c 1 ) ¬ p ( c 1 ) ⇒ (C) ⇒ ( { ⇒ ¬ p 1 ( v 1 ) } / ( ⇒ ¬∃ ) and {¬ p 1 ( c 1 ) ⇒ } / ( ¬∃ ⇒ ) ) 19/32

  44. Coherence Definition A quasi-canonical system is coherent if for every pair of conflicting rules Λ 1 / T 1 and Λ 2 / T 2 the set Λ 1 ⋒ Λ 2 is inconsistent (i.e. ⇒ is derivable using only ( C ) and ( S ) ) Example ⇒ ¬ p ( v 1 ) (S) ⇒ ¬ p ( c 1 ) ¬ p ( c 1 ) ⇒ (C) ⇒ ( { ⇒ ¬ p 1 ( v 1 ) } / ( ⇒ ¬∃ ) and {¬ p 1 ( c 1 ) ⇒ } / ( ¬∃ ⇒ ) ) Theorem Coherence is decidable 19/32

  45. Strong Cut-elimination Example From assumptions { ⇒ ¬ p ( x ) , ¬ p ( c ) ⇒ } deriving ⇒ ⇒ ¬ p ( x ) ¬ p ( c ) ⇒ ( ⇒ ¬∃ ) ( ¬∃ ⇒ ) ⇒ ¬∃ xp ( x ) ¬∃ xp ( x ) ⇒ (C) ⇒ 20/32

  46. Strong Cut-elimination Example From assumptions { ⇒ ¬ p ( x ) , ¬ p ( c ) ⇒ } deriving ⇒ ⇒ ¬ p ( x ) ¬ p ( c ) ⇒ ( ⇒ ¬∃ ) ( ¬∃ ⇒ ) ⇒ ¬∃ xp ( x ) ¬∃ xp ( x ) ⇒ (C) ⇒ � ⇒ ¬ p ( x ) (S) ⇒ ¬ p ( c ) ¬ p ( c ) ⇒ (C) ⇒ 20/32

  47. Strong Cut-elimination Example From assumptions { ⇒ ¬ p ( x ) , ¬ p ( c ) ⇒ } deriving ⇒ ⇒ ¬ p ( x ) ¬ p ( c ) ⇒ ( ⇒ ¬∃ ) ( ¬∃ ⇒ ) ⇒ ¬∃ xp ( x ) ¬∃ xp ( x ) ⇒ (C) ⇒ � ⇒ ¬ p ( x ) (S) ⇒ ¬ p ( c ) ¬ p ( c ) ⇒ (C) ⇒ The cut is eliminated and replaced by a cut on a substitution instance of a formula from the assumptions 20/32

  48. Strong Cut-elimination Example From assumptions { ⇒ ¬ p ( x ) , ¬ p ( c ) ⇒ } deriving ⇒ ⇒ ¬ p ( x ) ¬ p ( c ) ⇒ ( ⇒ ¬∃ ) ( ¬∃ ⇒ ) ⇒ ¬∃ xp ( x ) ¬∃ xp ( x ) ⇒ (C) ⇒ � ⇒ ¬ p ( x ) (S) ⇒ ¬ p ( c ) ¬ p ( c ) ⇒ (C) ⇒ The cut is eliminated and replaced by a cut on a substitution instance of a formula from the assumptions 20/32

  49. Strong Cut-elimination Example From assumptions { ⇒ ¬ p ( x ) , ¬ p ( c ) ⇒ } deriving ⇒ ⇒ ¬ p ( x ) ¬ p ( c ) ⇒ ( ⇒ ¬∃ ) ( ¬∃ ⇒ ) ⇒ ¬∃ xp ( x ) ¬∃ xp ( x ) ⇒ (C) ⇒ � ⇒ ¬ p ( x ) (S) ⇒ ¬ p ( c ) ¬ p ( c ) ⇒ (C) ⇒ The cut is eliminated and replaced by a cut on a substitution instance of a formula from the assumptions 20/32

  50. Strong Cut-elimination Example From assumptions { ⇒ ¬ p ( x ) , ¬ p ( c ) ⇒ } deriving ⇒ ⇒ ¬ p ( x ) ¬ p ( c ) ⇒ ( ⇒ ¬∃ ) ( ¬∃ ⇒ ) ⇒ ¬∃ xp ( x ) ¬∃ xp ( x ) ⇒ (C) ⇒ � ⇒ ¬ p ( x ) (S) ⇒ ¬ p ( c ) ¬ p ( c ) ⇒ (C) ⇒ The cut is eliminated and replaced by a cut on a substitution instance of a formula from the assumptions 20/32

  51. Strong Cut-elimination Example From assumptions { ⇒ ¬ p ( x ) , ¬ p ( c ) ⇒ } deriving ⇒ ⇒ ¬ p ( x ) ¬ p ( c ) ⇒ ( ⇒ ¬∃ ) ( ¬∃ ⇒ ) ⇒ ¬∃ xp ( x ) ¬∃ xp ( x ) ⇒ (C) ⇒ � ⇒ ¬ p ( x ) (S) ⇒ ¬ p ( c ) ¬ p ( c ) ⇒ (C) ⇒ The cut is eliminated and replaced by a cut on a substitution instance of a formula from the assumptions 20/32

  52. Semantics Extended to Sequents Definition ◮ A , v , σ � C if v [ σ [ C ]] ∈ D 21/32

  53. Semantics Extended to Sequents Definition ◮ A , v , σ � C if v [ σ [ C ]] ∈ D ◮ A , v , σ � Γ ⇒ ∆ if there exists A ∈ Γ such that A , v , σ � A or there exists B ∈ ∆ such that A , v , σ � B 21/32

  54. Semantics Extended to Sequents Definition ◮ A , v , σ � C if v [ σ [ C ]] ∈ D ◮ A , v , σ � Γ ⇒ ∆ if there exists A ∈ Γ such that A , v , σ � A or there exists B ∈ ∆ such that A , v , σ � B if A , v , σ � Γ ′ ⇒ ∆ ′ for every Γ ′ ⇒ ∆ ′ ∈ Θ ◮ A , v , σ � Θ 21/32

  55. Semantics Extended to Sequents Definition ◮ A , v , σ � C if v [ σ [ C ]] ∈ D ◮ A , v , σ � Γ ⇒ ∆ if there exists A ∈ Γ such that A , v , σ � A or there exists B ∈ ∆ such that A , v , σ � B if A , v , σ � Γ ′ ⇒ ∆ ′ for every Γ ′ ⇒ ∆ ′ ∈ Θ ◮ A , v , σ � Θ if A ′ , v ′ , σ ′ � Θ implies A ′ , v ′ , σ ′ � Γ ⇒ ∆ ◮ Θ ⊢ M Γ ⇒ ∆ 21/32

  56. Semantics Extended to Sequents Definition ◮ A , v , σ � C if v [ σ [ C ]] ∈ D ◮ A , v , σ � Γ ⇒ ∆ if there exists A ∈ Γ such that A , v , σ � A or there exists B ∈ ∆ such that A , v , σ � B if A , v , σ � Γ ′ ⇒ ∆ ′ for every Γ ′ ⇒ ∆ ′ ∈ Θ ◮ A , v , σ � Θ if A ′ , v ′ , σ ′ � Θ implies A ′ , v ′ , σ ′ � Γ ⇒ ∆ ◮ Θ ⊢ M Γ ⇒ ∆ Definition M is strongly characteristic for a system G if Θ ⊢ M Γ ⇒ ∆ is equivalent to derivability in G of Γ ⇒ ∆ from assumptions Θ . 21/32

  57. Correspondence Theorem Definition A 4-quasi-canonical system is a quasi-canonical system without ( ¬ ⇒ ) rules and without ( ⇒ ¬ ) rules Theorem (Correspondence) If G is a coherent 4-quasi-canonical system, then: 22/32

  58. Correspondence Theorem Definition A 4-quasi-canonical system is a quasi-canonical system without ( ¬ ⇒ ) rules and without ( ⇒ ¬ ) rules Theorem (Correspondence) If G is a coherent 4-quasi-canonical system, then: ◮ There is an induced ¬ -GNmatrix M G that is strongly characteristic for G 22/32

  59. Correspondence Theorem Definition A 4-quasi-canonical system is a quasi-canonical system without ( ¬ ⇒ ) rules and without ( ⇒ ¬ ) rules Theorem (Correspondence) If G is a coherent 4-quasi-canonical system, then: ◮ There is an induced ¬ -GNmatrix M G that is strongly characteristic for G ◮ G admits strong cut-elimination 22/32

  60. Existential Information Processing Gentzen-type Proof System 23/32

  61. Processors’ Gentzen Proof System Definition QG 4 EIP is the quasi-canonical system with the following rules (presented in application form): 24/32

  62. Processors’ Gentzen Proof System Definition QG 4 EIP is the quasi-canonical system with the following rules (presented in application form): Γ , ϕ ⇒ ∆ Γ ⇒ ϕ, ∆ ( ¬¬ ⇒ ) ( ⇒ ¬¬ ) Γ , ¬¬ ϕ ⇒ ∆ Γ ⇒ ¬¬ ϕ, ∆ 24/32

  63. Processors’ Gentzen Proof System Definition QG 4 EIP is the quasi-canonical system with the following rules (presented in application form): Γ , ϕ ⇒ ∆ Γ ⇒ ϕ, ∆ ( ¬¬ ⇒ ) ( ⇒ ¬¬ ) Γ , ¬¬ ϕ ⇒ ∆ Γ ⇒ ¬¬ ϕ, ∆ Γ ⇒ ϕ, ψ, ∆ ( ⇒ ∨ ) Γ ⇒ ϕ ∨ ψ, ∆ 24/32

  64. Processors’ Gentzen Proof System Definition QG 4 EIP is the quasi-canonical system with the following rules (presented in application form): Γ , ϕ ⇒ ∆ Γ ⇒ ϕ, ∆ ( ¬¬ ⇒ ) ( ⇒ ¬¬ ) Γ , ¬¬ ϕ ⇒ ∆ Γ ⇒ ¬¬ ϕ, ∆ Γ ⇒ ϕ, ψ, ∆ ( ⇒ ∨ ) Γ ⇒ ϕ ∨ ψ, ∆ Γ , ¬ ϕ, ¬ ψ ⇒ ∆ ( ¬∨ ⇒ ) Γ , ¬ ( ϕ ∨ ψ ) ⇒ ∆ 24/32

  65. Processors’ Gentzen Proof System Definition QG 4 EIP is the quasi-canonical system with the following rules (presented in application form): Γ , ϕ ⇒ ∆ Γ ⇒ ϕ, ∆ ( ¬¬ ⇒ ) ( ⇒ ¬¬ ) Γ , ¬¬ ϕ ⇒ ∆ Γ ⇒ ¬¬ ϕ, ∆ Γ ⇒ ϕ, ψ, ∆ ( ⇒ ∨ ) Γ ⇒ ϕ ∨ ψ, ∆ Γ , ¬ ϕ, ¬ ψ ⇒ ∆ Γ ⇒ ¬ ϕ, ∆ Γ ⇒ ¬ ψ, ∆ ( ¬∨ ⇒ ) ( ⇒ ¬∨ ) Γ , ¬ ( ϕ ∨ ψ ) ⇒ ∆ Γ ⇒ ¬ ( ϕ ∨ ψ ) , ∆ 24/32

  66. Processors’ Gentzen Proof System Definition QG 4 EIP is the quasi-canonical system with the following rules (presented in application form): Γ , ϕ ⇒ ∆ Γ ⇒ ϕ, ∆ ( ¬¬ ⇒ ) ( ⇒ ¬¬ ) Γ , ¬¬ ϕ ⇒ ∆ Γ ⇒ ¬¬ ϕ, ∆ Γ ⇒ ϕ, ψ, ∆ ( ⇒ ∨ ) Γ ⇒ ϕ ∨ ψ, ∆ Γ , ¬ ϕ, ¬ ψ ⇒ ∆ Γ ⇒ ¬ ϕ, ∆ Γ ⇒ ¬ ψ, ∆ ( ¬∨ ⇒ ) ( ⇒ ¬∨ ) Γ , ¬ ( ϕ ∨ ψ ) ⇒ ∆ Γ ⇒ ¬ ( ϕ ∨ ψ ) , ∆ Continued next slide... 24/32

  67. Processors’ Gentzen Proof System Definition QG 4 EIP is the quasi-canonical system with the following rules (presented in application form): ...Continued Γ ⇒ ϕ { t / x } , ∆ ( ⇒ ∃ ) Γ ⇒ ∃ x ϕ, ∆ Γ , ¬ ϕ { t / x } ⇒ ∆ Γ ⇒ ¬ ϕ { y / x } , ∆ ( ¬∃ ⇒ ) ( ⇒ ¬∃ ) Γ , ¬∃ x ϕ ⇒ ∆ Γ ⇒ ¬∃ x ϕ, ∆ 25/32

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