First-order Quasi-canonical Proof Systems Speaker: Yotam Dvir Yotam - - PowerPoint PPT Presentation

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First-order Quasi-canonical Proof Systems

Speaker: Yotam Dvir Yotam Dvir Arnon Avron

Tel-Aviv University, Israel

2019-09-04

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Existential Information Processing A Motivating Example

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Background

◮ Belnap [1977]:

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Background

◮ Belnap [1977]:

• 1. Framework for processing information from a set of sources

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Background

◮ Belnap [1977]:

• 1. Framework for processing information from a set of sources
• 2. Truth value are X ⊆ {0, 1}

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Background

◮ Belnap [1977]:

• 1. Framework for processing information from a set of sources
• 2. Truth value are X ⊆ {0, 1}
• 3. Reasoning is evidence based:

◮ 1 ∈ X - there is evidence supporting ◮ 0 ∈ X - there is evidence opposing

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Background

◮ Belnap [1977]:

• 1. Framework for processing information from a set of sources
• 2. Truth value are X ⊆ {0, 1}
• 3. Reasoning is evidence based:

◮ 1 ∈ X - there is evidence supporting ◮ 0 ∈ X - there is evidence opposing ◮ ⊥ = {} f = {0} t = {1} ⊤ = {0, 1}

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Background

◮ Belnap [1977]:

• 1. Framework for processing information from a set of sources
• 2. Truth value are X ⊆ {0, 1}
• 3. Reasoning is evidence based:

◮ 1 ∈ X - there is evidence supporting ◮ 0 ∈ X - there is evidence opposing ◮ ⊥ = {} f = {0} t = {1} ⊤ = {0, 1}

• 4. Sources provide information on atomic formulas

(good for databases)

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Background

◮ Belnap [1977]:

• 1. Framework for processing information from a set of sources
• 2. Truth value are X ⊆ {0, 1}
• 3. Reasoning is evidence based:

◮ 1 ∈ X - there is evidence supporting ◮ 0 ∈ X - there is evidence opposing ◮ ⊥ = {} f = {0} t = {1} ⊤ = {0, 1}

• 4. Sources provide information on atomic formulas

(good for databases)

◮ Avron and Konikowska [2012]:

4’ Sources provide information on arbitrary formulas (good for more general knowledge bases)

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Background

◮ Belnap [1977]:

• 1. Framework for processing information from a set of sources
• 2. Truth value are X ⊆ {0, 1}
• 3. Reasoning is evidence based:

◮ 1 ∈ X - there is evidence supporting ◮ 0 ∈ X - there is evidence opposing ◮ ⊥ = {} f = {0} t = {1} ⊤ = {0, 1}

• 4. Sources provide information on atomic formulas

(good for databases)

◮ Avron and Konikowska [2012]:

4’ Sources provide information on arbitrary formulas (good for more general knowledge bases)

◮ Problem: both only work with propositional logic, yet real queries involve predicates (“Is there a person with blonde hair and brown eyes?”)

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Sources of Information – Intro

A source provides information on formulas:

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Sources of Information – Intro

A source provides information on formulas: ◮ May have gaps in knowledge Truth values: U 1

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Sources of Information – Intro

A source provides information on formulas: ◮ May have gaps in knowledge Truth values: U 1 ◮ May possess disjunctive information

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Sources of Information – Intro

A source provides information on formulas: ◮ May have gaps in knowledge Truth values: U 1 ◮ May possess disjunctive information, e.g.

◮ ψ ∨ θ

Alice

− − − → 1

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Sources of Information – Intro

A source provides information on formulas: ◮ May have gaps in knowledge Truth values: U 1 ◮ May possess disjunctive information, e.g.

◮ ψ ∨ θ

Alice

− − − → 1 even though ψ, θ

Alice

− − − → U

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Sources of Information – Intro

A source provides information on formulas: ◮ May have gaps in knowledge Truth values: U 1 ◮ May possess disjunctive information, e.g.

◮ ψ ∨ θ

Alice

− − − → 1 even though ψ, θ

Alice

− − − → U ◮ ∃xϕ(x)

Alice

− − − → 1

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Sources of Information – Intro

A source provides information on formulas: ◮ May have gaps in knowledge Truth values: U 1 ◮ May possess disjunctive information, e.g.

◮ ψ ∨ θ

Alice

− − − → 1 even though ψ, θ

Alice

− − − → U ◮ ∃xϕ(x)

Alice

− − − → 1 even though ϕ(a)

Alice

− − − → 1 for no a ∈ Dom

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Sources of Information – Intro

A source provides information on formulas: ◮ May have gaps in knowledge Truth values: U 1 ◮ May possess disjunctive information, e.g.

◮ ψ ∨ θ

Alice

− − − → 1 even though ψ, θ

Alice

− − − → U ◮ ∃xϕ(x)

Alice

− − − → 1 even though ϕ(a)

Alice

− − − → 1 for no a ∈ Dom

◮ Truth tables are consistent with classical logic, e.g.

◮ ψ

Alice

− − − → 1 implies that ψ ∨ θ

Alice

− − − → 1

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(n,k)-quantifiers

If: ◮ Q is an n, k-ary quantifier in L ◮ ψ1, . . . ψn are L-formulas ◮ z1, . . . zk are distinct variables then Q z1 . . . zk (ψ1, . . . ψn) is an L-formula (Q binds z1, . . . zk in the connected ψ1, . . . ψn)

Example

◮ 1, 1-ary: ∃, ∀ ◮ 1, 0-ary: ¬, ◮ 2, 0-ary: ∨, ∧

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Generalized Non-deterministic Matrix

Definition

A generalized non-deterministic matrix (GNmatrix) for L is a triplet V, D, O such that:

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Generalized Non-deterministic Matrix

Definition

A generalized non-deterministic matrix (GNmatrix) for L is a triplet V, D, O such that: ◮ ∅ = D V (V – truth values ; D – designated truth values)

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Generalized Non-deterministic Matrix

Definition

A generalized non-deterministic matrix (GNmatrix) for L is a triplet V, D, O such that: ◮ ∅ = D V (V – truth values ; D – designated truth values) ◮ O associates with every n, k-quantifier Q of L a function ˜ Q : P[Vn] → P+[V] (O – truth tables)

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Generalized Non-deterministic Matrix

Definition

A generalized non-deterministic matrix (GNmatrix) for L is a triplet V, D, O such that: ◮ ∅ = D V (V – truth values ; D – designated truth values) ◮ O associates with every n, k-quantifier Q of L a function ˜ Q : P[Vn] → P+[V] (O – truth tables)

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Generalized Non-deterministic Matrix

Definition

A generalized non-deterministic matrix (GNmatrix) for L is a triplet V, D, O such that: ◮ ∅ = D V (V – truth values ; D – designated truth values) ◮ O associates with every n, k-quantifier Q of L a function ˜ Q : P[Vn] → P+[V] (O – truth tables)

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Generalized Non-deterministic Matrix

Definition

A generalized non-deterministic matrix (GNmatrix) for L is a triplet V, D, O such that: ◮ ∅ = D V (V – truth values ; D – designated truth values) ◮ O associates with every n, k-quantifier Q of L a function ˜ Q : P[Vn] → P+[V] (O – truth tables)

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Sources of Information – GNmatrix

Definition

QM3

r =

• {U, 0, 1} , {1} , QO3

r

• , where QO3

r is given by:

a ˜ ¬a U {U} {1} 1 {0} ˜ ∨ U 1 U {U, 1} {U} {1} {U} {0} {1} 1 {1} {1} {1} X ˜ ∃ [X] {U} {U, 1} {U, 0} {U, 1} {0} {0} else {1} ˜ ∧ and ˜ ∀ are defined dually

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Sources of Information – GNmatrix

Definition

QM3

r =

• {U, 0, 1} , {1} , QO3

r

• , where QO3

r is given by:

a ˜ ¬a U {U} {1} 1 {0} ˜ ∨ U 1 U {U, 1} {U} {1} {U} {0} {1} 1 {1} {1} {1} X ˜ ∃ [X] {U} {U, 1} {U, 0} {U, 1} {0} {0} else {1} ˜ ∧ and ˜ ∀ are defined dually ψ ∨ θ Alice − − − → 1 even though ψ, θ Alice − − − → U

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Sources of Information – GNmatrix

Definition

QM3

r =

• {U, 0, 1} , {1} , QO3

r

• , where QO3

r is given by:

a ˜ ¬a U {U} {1} 1 {0} ˜ ∨ U 1 U {U, 1} {U} {1} {U} {0} {1} 1 {1} {1} {1} X ˜ ∃ [X] {U} {U, 1} {U, 0} {U, 1} {0} {0} else {1} ˜ ∧ and ˜ ∀ are defined dually ∃xϕ(x) Alice − − − → 1 even though ϕ(a) Alice − − − → 1 for no a ∈ Dom

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Sources of Information – GNmatrix

Definition

QM3

r =

• {U, 0, 1} , {1} , QO3

r

• , where QO3

r is given by:

a ˜ ¬a U {U} {1} 1 {0} ˜ ∨ U 1 U {U, 1} {U} {1} {U} {0} {1} 1 {1} {1} {1} X ˜ ∃ [X] {U} {U, 1} {U, 0} {U, 1} {0} {0} else {1} ˜ ∧ and ˜ ∀ are defined dually ψ Alice − − − → 1 implies that ψ ∨ θ Alice − − − → 1

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Algebra Instead of Structure

Definition

An L-algebra A consists of the following:

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Algebra Instead of Structure

Definition

An L-algebra A consists of the following: ◮ A non-empty set Dom A

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Algebra Instead of Structure

Definition

An L-algebra A consists of the following: ◮ A non-empty set Dom A ◮ For every m-ary function symbol f in L a function f A : (Dom A)m → Dom A

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Algebra Instead of Structure

Definition

An L-algebra A consists of the following: ◮ A non-empty set Dom A ◮ For every m-ary function symbol f in L a function f A : (Dom A)m → Dom A

Definition

ϕ A ∼ ϕ′ if they are equal up to replacing: ◮ bound variables ◮ closed terms representing the same member of Dom A

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Legal Valuations & Subtitutions

Definition

An M-legal A-valuation is a function v : clFrm → V s.t.:

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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′]

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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 z1 . . . zk (ψ1, . . . ψn)] ∈ ˜

Q[

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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 z1 . . . zk (ψ1, . . . ψn)] ∈ ˜

Q[ {v [ψ1 {a1/z1, . . . ak/zk}], . . . v [ψn {a1/z1, . . . ak/zk}] |a1, . . . ak ∈ Dom A} ]

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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 z1 . . . zk (ψ1, . . . ψn)] ∈ ˜

Q[ {v [ψ1 {a1/z1, . . . ak/zk}], . . . v [ψn {a1/z1, . . . ak/zk}] |a1, . . . ak ∈ Dom A} ]

Definition

An A-source is a QM3

r -legal A-valuation

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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 z1 . . . zk (ψ1, . . . ψn)] ∈ ˜

Q[ {v [ψ1 {a1/z1, . . . ak/zk}], . . . v [ψn {a1/z1, . . . ak/zk}] |a1, . . . ak ∈ Dom A} ]

Definition

An A-source is a QM3

r -legal A-valuation

Definition

A substitution is a function σ : Var → clTrm

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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 z1 . . . zk (ψ1, . . . ψn)] ∈ ˜

Q[ {v [ψ1 {a1/z1, . . . ak/zk}], . . . v [ψn {a1/z1, . . . ak/zk}] |a1, . . . ak ∈ Dom A} ]

Definition

An A-source is a QM3

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)

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Processing Information from Sources

◮ Different sources may disagree

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Processing Information from Sources

◮ Different sources may disagree, e.g. ψ Alice − − − → 1 ψ

Bob

− − − → 0 ψ Carol − − − → U

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Processing Information from Sources

◮ Different sources may disagree, e.g. ψ Alice − − − → 1 ψ

Bob

− − − → 0 ψ Carol − − − → U ◮ Different sources may possess complementary info

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Processing Information from Sources

◮ Different sources may disagree, e.g. ψ Alice − − − → 1 ψ

Bob

− − − → 0 ψ Carol − − − → U ◮ Different sources may possess complementary info, e.g. ψ Alice − − − → 1 ψ ∧ θ Alice − − − → U θ

Bob

− − − → 1 ψ ∧ θ

Bob

− − − → U

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Processing Information from Sources

◮ Different sources may disagree, e.g. ψ Alice − − − → 1 ψ

Bob

− − − → 0 ψ Carol − − − → U ◮ Different sources may possess complementary info, e.g. ψ Alice − − − → 1 ψ ∧ θ Alice − − − → U θ

Bob

− − − → 1 ψ ∧ θ

Bob

− − − → U Processing information from a set of A-sources S:

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Processing Information from Sources

◮ Different sources may disagree, e.g. ψ Alice − − − → 1 ψ

Bob

− − − → 0 ψ Carol − − − → U ◮ Different sources may possess complementary info, e.g. ψ Alice − − − → 1 ψ ∧ θ Alice − − − → U θ

Bob

− − − → 1 ψ ∧ θ

Bob

− − − → U 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

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Processing Information from Sources

◮ Different sources may disagree, e.g. ψ Alice − − − → 1 ψ

Bob

− − − → 0 ψ Carol − − − → U ◮ Different sources may possess complementary info, e.g. ψ Alice − − − → 1 ψ ∧ θ Alice − − − → U θ

Bob

− − − → 1 ψ ∧ θ

Bob

− − − → U 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

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Processing Information from Sources

◮ Different sources may disagree, e.g. ψ Alice − − − → 1 ψ

Bob

− − − → 0 ψ Carol − − − → U ◮ Different sources may possess complementary info, e.g. ψ Alice − − − → 1 ψ ∧ θ Alice − − − → U θ

Bob

− − − → 1 ψ ∧ θ

Bob

− − − → U 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’

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Processing Information from Sources

◮ Different sources may disagree, e.g. ψ Alice − − − → 1 ψ

Bob

− − − → 0 ψ Carol − − − → U ◮ Different sources may possess complementary info, e.g. ψ Alice − − − → 1 ψ ∧ θ Alice − − − → U θ

Bob

− − − → 1 ψ ∧ θ

Bob

− − − → U 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 [∃xϕ(x)]

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Processing Information from Sources

◮ Different sources may disagree, e.g. ψ Alice − − − → 1 ψ

Bob

− − − → 0 ψ Carol − − − → U ◮ Different sources may possess complementary info, e.g. ψ Alice − − − → 1 ψ ∧ θ Alice − − − → U θ

Bob

− − − → 1 ψ ∧ θ

Bob

− − − → U 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 [∃xϕ(x)] ◮ 0 ∈

a∈Dom d [ϕ(a)] =

⇒ 0 ∈ d [∃xϕ(x)]

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Processing Information from Sources

◮ Different sources may disagree, e.g. ψ Alice − − − → 1 ψ

Bob

− − − → 0 ψ Carol − − − → U ◮ Different sources may possess complementary info, e.g. ψ Alice − − − → 1 ψ ∧ θ Alice − − − → U θ

Bob

− − − → 1 ψ ∧ θ

Bob

− − − → U 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 [∃xϕ(x)] ◮ 0 ∈

a∈Dom d [ϕ(a)] =

⇒ 0 ∈ d [∃xϕ(x)] ◮ b ∈ d [θ] = ⇒ 1 − b ∈ d [¬θ]

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Processing Information from Sources

◮ Different sources may disagree, e.g. ψ Alice − − − → 1 ψ

Bob

− − − → 0 ψ Carol − − − → U ◮ Different sources may possess complementary info, e.g. ψ Alice − − − → 1 ψ ∧ θ Alice − − − → U θ

Bob

− − − → 1 ψ ∧ θ

Bob

− − − → U 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 [∃xϕ(x)] ◮ 0 ∈

a∈Dom d [ϕ(a)] =

⇒ 0 ∈ d [∃xϕ(x)] ◮ b ∈ d [θ] = ⇒ 1 − b ∈ d [¬θ] ◮ etc.

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Processors are Characterized by a GNmatrix

Recall ⊥ = {} f = {0} t = {1} ⊤ = {0, 1}

Definition

QM4

E =

• {⊥, f, t, ⊤} , {t, ⊤} , QO4

E

• , where QO4

E is given by:

a ˜ ¬a ⊥ {⊥} f {t} t {f} ⊤ {⊤} ˜ ∨ ⊥ f t ⊤ ⊥ {⊥, t} {⊥, t} {t} {t} f {⊥, t} {f, ⊤} {t} {⊤} t {t} {t} {t} {t} ⊤ {t} {⊤} {t} {⊤} X ˜ ∃ [X] {⊥} {⊥, t} {⊥, f} {⊥, t} {f} {f, ⊤} {f, ⊤} {⊤} {⊤} {⊤} else {t} ˜ ∧ and ˜ ∀ are defined dually

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Processors are Characterized by a GNmatrix

Recall ⊥ = {} f = {0} t = {1} ⊤ = {0, 1}

Definition

QM4

E =

• {⊥, f, t, ⊤} , {t, ⊤} , QO4

E

• , where QO4

E is given by:

a ˜ ¬a ⊥ {⊥} f {t} t {f} ⊤ {⊤} ˜ ∨ ⊥ f t ⊤ ⊥ {⊥, t} {⊥, t} {t} {t} f {⊥, t} {f, ⊤} {t} {⊤} t {t} {t} {t} {t} ⊤ {t} {⊤} {t} {⊤} X ˜ ∃ [X] {⊥} {⊥, t} {⊥, f} {⊥, t} {f} {f, ⊤} {f, ⊤} {⊤} {⊤} {⊤} else {t} ˜ ∧ and ˜ ∀ are defined dually

Theorem

The function S → d is onto the set of QM4

E-legal A-valuations

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¬-GNmatrix

Definition

A GNmatix M = V, D, O for L is a ¬-GNmatix if: ◮ V ⊆ {⊥, f, t, ⊤} ◮ D = V ∩ {t, ⊤}

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¬-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}

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¬-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) ◮ QM4

E (The processor GNmatrix)

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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

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Quasi-canonical Gentzen-type Proof Systems

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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 ∆”

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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

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Structural Rules

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Structural Rules

(A) X ⇒ X ′ X α ∼ X ′

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Structural Rules

(A) X ⇒ X ′ X α ∼ X ′ Γ ⇒ ∆ (W) Γ′, Γ ⇒ ∆, ∆′

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Structural Rules

(A) X ⇒ X ′ X α ∼ X ′ Γ ⇒ ∆ (W) Γ′, Γ ⇒ ∆, ∆′ Γ′ ⇒ ∆, X X, Γ ⇒ ∆′ (C) Γ′, Γ ⇒ ∆, ∆′

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Structural Rules

(A) X ⇒ X ′ X α ∼ X ′ Γ ⇒ ∆ (W) Γ′, Γ ⇒ ∆, ∆′ Γ′ ⇒ ∆, X X, Γ ⇒ ∆′ (C) Γ′, Γ ⇒ ∆, ∆′ Γ ⇒ ∆ (S) Γ {t1/x1, . . . tm/xm} ⇒ ∆ {t1/x1, . . . tm/xm}

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Quasi-canonical Rules

Canonical

Γ ⇒ A {x/z} , ∆ (⇒ ∀) Γ ⇒ ∀zA, ∆ Γ, A {t/z} ⇒ ∆ ( ∀ ⇒) Γ, ∀zA ⇒ ∆

(x is not free in the bottom sequent)

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Quasi-canonical Rules

Quasi-Canonical

Γ ⇒ ¬A {x/z} , ∆ (⇒ ¬∃) Γ ⇒ ¬∃zA, ∆ Γ, ¬A {t/z} ⇒ ∆ (¬∃ ⇒) Γ, ¬∃zA ⇒ ∆

(x is not free in the bottom sequent)

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Quasi-canonical Rules

Quasi-Canonical

∃ is 1, 1-ary {⇒ ¬p1 (v1)} / (⇒ ¬∃) {¬p1 (c1) ⇒} / (¬∃ ⇒) Γ ⇒ ¬A {x/z} , ∆ (⇒ ¬∃) Γ ⇒ ¬∃zA, ∆ Γ, ¬A {t/z} ⇒ ∆ (¬∃ ⇒) Γ, ¬∃zA ⇒ ∆

(x is not free in the bottom sequent)

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Quasi-canonical Rules

Quasi-Canonical

∃ is 1, 1-ary {⇒ ¬p1 (v1)} / (⇒ ¬∃) {¬p1 (c1) ⇒} / (¬∃ ⇒) Γ ⇒ ¬A {x/z} , ∆ (⇒ ¬∃) Γ ⇒ ¬∃zA, ∆ Γ, ¬A {t/z} ⇒ ∆ (¬∃ ⇒) Γ, ¬∃zA ⇒ ∆

(x is not free in the bottom sequent)

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Quasi-canonical Rules

Quasi-Canonical

∃ is 1, 1-ary {⇒ ¬p1 (v1)} / (⇒ ¬∃) {¬p1 (c1) ⇒} / (¬∃ ⇒) Γ ⇒ ¬A {x/z} , ∆ (⇒ ¬∃) Γ ⇒ ¬∃zA, ∆ Γ, ¬A {t/z} ⇒ ∆ (¬∃ ⇒) Γ, ¬∃zA ⇒ ∆

(x is not free in the bottom sequent)

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Quasi-canonical Rules

Quasi-Canonical

∃ is 1, 1-ary {⇒ ¬p1 (v1)} / (⇒ ¬∃) {¬p1 (c1) ⇒} / (¬∃ ⇒) Γ ⇒ ¬A {x/z} , ∆ (⇒ ¬∃) Γ ⇒ ¬∃zA, ∆ Γ, ¬A {t/z} ⇒ ∆ (¬∃ ⇒) Γ, ¬∃zA ⇒ ∆

(x is not free in the bottom sequent)

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Conflicting Rules

Definition

A pair of rules of the following form are conflicting: ◮ Λ1/ (Q ⇒) and Λ2/ (⇒ Q) ◮ Λ1/ (¬ Q ⇒) and Λ2/ (⇒ ¬ Q)

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Conflicting Rules

Definition

A pair of rules of the following form are conflicting: ◮ Λ1/ (Q ⇒) and Λ2/ (⇒ Q) ◮ Λ1/ (¬ Q ⇒) and Λ2/ (⇒ ¬ Q)

Example

{⇒ ¬p1 (v1)} / (⇒ ¬∃) and {¬p1 (c1) ⇒} / (¬∃ ⇒)

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Coherence

Definition

A quasi-canonical system is coherent if for every pair of conflicting rules Λ1/T1 and Λ2/T2 the set Λ1 ⋒ Λ2 is inconsistent (i.e. ⇒ is derivable using only (C) and (S))

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Coherence

Definition

A quasi-canonical system is coherent if for every pair of conflicting rules Λ1/T1 and Λ2/T2 the set Λ1 ⋒ Λ2 is inconsistent (i.e. ⇒ is derivable using only (C) and (S))

Example

⇒ ¬p (v1) (S) ⇒ ¬p (c1) ¬p (c1) ⇒ (C) ⇒ ({⇒ ¬p1 (v1)} / (⇒ ¬∃) and {¬p1 (c1) ⇒} / (¬∃ ⇒))

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Coherence

Definition

A quasi-canonical system is coherent if for every pair of conflicting rules Λ1/T1 and Λ2/T2 the set Λ1 ⋒ Λ2 is inconsistent (i.e. ⇒ is derivable using only (C) and (S))

Example

⇒ ¬p (v1) (S) ⇒ ¬p (c1) ¬p (c1) ⇒ (C) ⇒ ({⇒ ¬p1 (v1)} / (⇒ ¬∃) and {¬p1 (c1) ⇒} / (¬∃ ⇒))

Theorem

Coherence is decidable

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Strong Cut-elimination

Example

From assumptions {⇒ ¬p (x) , ¬p (c) ⇒} deriving ⇒ ⇒ ¬p (x) (⇒ ¬∃) ⇒ ¬∃xp (x) ¬p (c) ⇒ (¬∃ ⇒) ¬∃xp (x) ⇒ (C) ⇒

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Strong Cut-elimination

Example

From assumptions {⇒ ¬p (x) , ¬p (c) ⇒} deriving ⇒ ⇒ ¬p (x) (⇒ ¬∃) ⇒ ¬∃xp (x) ¬p (c) ⇒ (¬∃ ⇒) ¬∃xp (x) ⇒ (C) ⇒

• ⇒ ¬p (x)

(S) ⇒ ¬p (c) ¬p (c) ⇒ (C) ⇒

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Strong Cut-elimination

Example

From assumptions {⇒ ¬p (x) , ¬p (c) ⇒} deriving ⇒ ⇒ ¬p (x) (⇒ ¬∃) ⇒ ¬∃xp (x) ¬p (c) ⇒ (¬∃ ⇒) ¬∃xp (x) ⇒ (C) ⇒

• ⇒ ¬p (x)

(S) ⇒ ¬p (c) ¬p (c) ⇒ (C) ⇒ The cut is eliminated and replaced by a cut

• n a substitution instance
• f a formula from the assumptions

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Strong Cut-elimination

Example

From assumptions {⇒ ¬p (x) , ¬p (c) ⇒} deriving ⇒ ⇒ ¬p (x) (⇒ ¬∃) ⇒ ¬∃xp (x) ¬p (c) ⇒ (¬∃ ⇒) ¬∃xp (x) ⇒ (C) ⇒

• ⇒ ¬p (x)

(S) ⇒ ¬p (c) ¬p (c) ⇒ (C) ⇒ The cut is eliminated and replaced by a cut

• n a substitution instance
• f a formula from the assumptions

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Strong Cut-elimination

Example

From assumptions {⇒ ¬p (x) , ¬p (c) ⇒} deriving ⇒ ⇒ ¬p (x) (⇒ ¬∃) ⇒ ¬∃xp (x) ¬p (c) ⇒ (¬∃ ⇒) ¬∃xp (x) ⇒ (C) ⇒

• ⇒ ¬p (x)

(S) ⇒ ¬p (c) ¬p (c) ⇒ (C) ⇒ The cut is eliminated and replaced by a cut

• n a substitution instance
• f a formula from the assumptions

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Strong Cut-elimination

Example

From assumptions {⇒ ¬p (x) , ¬p (c) ⇒} deriving ⇒ ⇒ ¬p (x) (⇒ ¬∃) ⇒ ¬∃xp (x) ¬p (c) ⇒ (¬∃ ⇒) ¬∃xp (x) ⇒ (C) ⇒

• ⇒ ¬p (x)

(S) ⇒ ¬p (c) ¬p (c) ⇒ (C) ⇒ The cut is eliminated and replaced by a cut

• n a substitution instance
• f a formula from the assumptions

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Strong Cut-elimination

Example

From assumptions {⇒ ¬p (x) , ¬p (c) ⇒} deriving ⇒ ⇒ ¬p (x) (⇒ ¬∃) ⇒ ¬∃xp (x) ¬p (c) ⇒ (¬∃ ⇒) ¬∃xp (x) ⇒ (C) ⇒

• ⇒ ¬p (x)

(S) ⇒ ¬p (c) ¬p (c) ⇒ (C) ⇒ The cut is eliminated and replaced by a cut

• n a substitution instance
• f a formula from the assumptions

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Semantics Extended to Sequents

Definition

◮ A, v, σ C if v [σ [C]] ∈ D

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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

• r there exists B ∈ ∆ such that A, v, σ B

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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

• r there exists B ∈ ∆ such that A, v, σ B

◮ A, v, σ Θ if A, v, σ Γ′ ⇒ ∆′ for every Γ′ ⇒ ∆′ ∈ Θ

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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

• r there exists B ∈ ∆ such that A, v, σ B

◮ A, v, σ Θ if A, v, σ Γ′ ⇒ ∆′ for every Γ′ ⇒ ∆′ ∈ Θ ◮ Θ ⊢M Γ ⇒ ∆ if A′, v′, σ′ Θ implies A′, v′, σ′ Γ ⇒ ∆

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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

• r there exists B ∈ ∆ such that A, v, σ B

◮ A, v, σ Θ if A, v, σ Γ′ ⇒ ∆′ for every Γ′ ⇒ ∆′ ∈ Θ ◮ Θ ⊢M Γ ⇒ ∆ if A′, v′, σ′ Θ implies A′, v′, σ′ 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.

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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:

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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 MG that is strongly characteristic for G

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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 MG that is strongly characteristic for G ◮ G admits strong cut-elimination

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Existential Information Processing Gentzen-type Proof System

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Processors’ Gentzen Proof System

Definition

QG4

EIP is the quasi-canonical system with the following rules

(presented in application form):

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Processors’ Gentzen Proof System

Definition

QG4

EIP is the quasi-canonical system with the following rules

(presented in application form): Γ, ϕ ⇒ ∆ Γ, ¬¬ϕ ⇒ ∆ (¬¬ ⇒) Γ ⇒ ϕ, ∆ Γ ⇒ ¬¬ϕ, ∆ (⇒ ¬¬)

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Processors’ Gentzen Proof System

Definition

QG4

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

Processors’ Gentzen Proof System

Definition

QG4

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) ⇒ ∆ (¬∨ ⇒)

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Processors’ Gentzen Proof System

Definition

QG4

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) ⇒ ∆ (¬∨ ⇒) Γ ⇒ ¬ϕ, ∆ Γ ⇒ ¬ψ, ∆ Γ ⇒ ¬ (ϕ ∨ ψ) , ∆ (⇒ ¬∨)

24/32

Processors’ Gentzen Proof System

Definition

QG4

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) ⇒ ∆ (¬∨ ⇒) Γ ⇒ ¬ϕ, ∆ Γ ⇒ ¬ψ, ∆ Γ ⇒ ¬ (ϕ ∨ ψ) , ∆ (⇒ ¬∨) Continued next slide...

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Processors’ Gentzen Proof System

Definition

QG4

EIP is the quasi-canonical system with the following rules

(presented in application form): ...Continued Γ ⇒ ϕ {t/x} , ∆ Γ ⇒ ∃xϕ, ∆ (⇒ ∃) Γ, ¬ϕ {t/x} ⇒ ∆ Γ, ¬∃xϕ ⇒ ∆ (¬∃ ⇒) Γ ⇒ ¬ϕ {y/x} , ∆ Γ ⇒ ¬∃xϕ, ∆ (⇒ ¬∃)

25/32

Processors’ Gentzen Proof System

Definition

QG4

EIP is the quasi-canonical system with the following rules

(presented in application form): ...Continued Γ ⇒ ϕ {t/x} , ∆ Γ ⇒ ∃xϕ, ∆ (⇒ ∃) Γ, ¬ϕ {t/x} ⇒ ∆ Γ, ¬∃xϕ ⇒ ∆ (¬∃ ⇒) Γ ⇒ ¬ϕ {y/x} , ∆ Γ ⇒ ¬∃xϕ, ∆ (⇒ ¬∃) (∧ ⇒) (⇒ ∧) (⇒ ¬∧) (∀ ⇒) (⇒ ∀) (⇒ ¬∀)

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Processors’ Correspondence Theorem

Theorem

◮ QM4

E is strongly characteristic for QG4 EIP

◮ QG4

EIP admits strong cut-elimination

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Processors’ Correspondence Theorem

Theorem

◮ QM4

E is strongly characteristic for QG4 EIP

◮ QG4

EIP admits strong cut-elimination

Proof.

• 1. QG4

EIP is a coherent 4-quasi-canonical system

• 2. QM4

E = MQG4

EIP

• 3. Correspondence Theorem

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Processors’ Correspondence Theorem

Theorem

◮ QM4

E is strongly characteristic for QG4 EIP

◮ QG4

EIP admits strong cut-elimination

Proof.

• 1. QG4

EIP is a coherent 4-quasi-canonical system

• 2. QM4

E = MQG4

EIP

• 3. Correspondence Theorem

Thus anything derivable in QG4

EIP holds for all processors

and vice versa

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Thank you!

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Future Research Possibilities

Quasi-canonical Proof Systems: ◮ Known relaxations of the restriction on (¬ ⇒) and (⇒ ¬) rules yield systems for 3-valued logics in propositional

• logic. Will this work in this setting too? Probably...

◮ Extending canonicity to new realms. ◮ Defining rules with function symbols to get explicit dependencies in quantifiers. EIP framework: ◮ How can the restriction on sources sharing the same algebra be lifted? This would seemingly increase the usefulness of the framework. ◮ Sources can be silly geese: ϕ ∨ (ψ ∧ θ)

Mallory

− − − − → 1 & (ϕ ∨ ψ) ∧ (ϕ ∨ θ)

Mallory

− − − − → 0

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References I

Arnon Avron and Beata Konikowska. Finite-valued logics for information processing. Fund. Inform., 114(1):1–30, 2012. ISSN 0169-2968. Nuel D Belnap. How a computer should think. In G. Ryle, editor, Contemporary Aspects of Philosophy. Oriel Press, 1977.

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End Extra Slides

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Processors – Effective Variant

◮ Some of the ‘integrity conditions’ are not semi-decidable with inf. many individuals (|Dom A| = ∞):

◮ 0 ∈

a∈Dom A d [ϕ(a)] =

⇒ 0 ∈ d [∃xϕ(x)] ◮ 1 ∈

a∈Dom A d [ϕ(a)] =

⇒ 1 ∈ d [∀xϕ(x)]

31/32

Processors – Effective Variant

◮ Some of the ‘integrity conditions’ are not semi-decidable with inf. many individuals (|Dom A| = ∞):

◮ 0 ∈

a∈Dom A d [ϕ(a)] =

⇒ 0 ∈ d [∃xϕ(x)] ◮ 1 ∈

a∈Dom A d [ϕ(a)] =

⇒ 1 ∈ d [∀xϕ(x)]

◮ Konikowska suggested removing them trading off precision for computability

31/32

Processors – Effective Variant

◮ Some of the ‘integrity conditions’ are not semi-decidable with inf. many individuals (|Dom A| = ∞):

◮ 0 ∈

a∈Dom A d [ϕ(a)] =

⇒ 0 ∈ d [∃xϕ(x)] ◮ 1 ∈

a∈Dom A d [ϕ(a)] =

⇒ 1 ∈ d [∀xϕ(x)]

◮ Konikowska suggested removing them trading off precision for computability

Definition

◮ obtain QG4sd

EIP from QG4 EIP by removing (⇒ ¬∃), (⇒ ∀)

◮ obtain QM4sd

E

from QM4

E by expanding ˜

∃, ˜ ∀: ... (e.g. in QM4sd

E

if h[X] = {f, ⊤} then ˜ ∃X [h] = {t, ⊤})

31/32

Processors – Effective Variant

◮ Some of the ‘integrity conditions’ are not semi-decidable with inf. many individuals (|Dom A| = ∞):

◮ 0 ∈

a∈Dom A d [ϕ(a)] =

⇒ 0 ∈ d [∃xϕ(x)] ◮ 1 ∈

a∈Dom A d [ϕ(a)] =

⇒ 1 ∈ d [∀xϕ(x)]

◮ Konikowska suggested removing them trading off precision for computability

Definition

◮ obtain QG4sd

EIP from QG4 EIP by removing (⇒ ¬∃), (⇒ ∀)

◮ obtain QM4sd

E

from QM4

E by expanding ˜

∃, ˜ ∀: ... (e.g. in QM4sd

E

if h[X] = {f, ⊤} then ˜ ∃X [h] = {t, ⊤})

Theorem

◮ QG4sd

EIP is strongly characteristic for QM4sd E

◮ QG4sd

EIP admits strong cut-elimination

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The Crux of the Correspondence Theorem

Lemma

If Γ ⇒ ∆ doesn’t mix free and bound variables and has no Θ-cut-free proof from Θ in a Quasi-canonical G, then Θ MG Γ ⇒ ∆

Outline of Proof.

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The Crux of the Correspondence Theorem

Lemma

If Γ ⇒ ∆ doesn’t mix free and bound variables and has no Θ-cut-free proof from Θ in a Quasi-canonical G, then Θ MG Γ ⇒ ∆

Outline of Proof.

◮ Extend Γ, ∆ to maximal Γ∗, ∆∗

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The Crux of the Correspondence Theorem

Lemma

If Γ ⇒ ∆ doesn’t mix free and bound variables and has no Θ-cut-free proof from Θ in a Quasi-canonical G, then Θ MG Γ ⇒ ∆

Outline of Proof.

◮ Extend Γ, ∆ to maximal Γ∗, ∆∗ ◮ Construct an MG-legal valuation v for a Herbrand structure s.t. v (ϕ) depends on the appearances of ϕ and ¬ϕ in Γ∗ and ∆∗

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The Crux of the Correspondence Theorem

Lemma

If Γ ⇒ ∆ doesn’t mix free and bound variables and has no Θ-cut-free proof from Θ in a Quasi-canonical G, then Θ MG Γ ⇒ ∆

Outline of Proof.

◮ Extend Γ, ∆ to maximal Γ∗, ∆∗ ◮ Construct an MG-legal valuation v for a Herbrand structure s.t. v (ϕ) depends on the appearances of ϕ and ¬ϕ in Γ∗ and ∆∗ ◮ Prove that

◮ A ∈ Γ∗ = ⇒ 1 ∈ v [σ∗ [A]] ◮ A ∈ ∆∗ = ⇒ 0 ∈ v [σ∗ [A]]

where σ∗ (x) = x

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The Crux of the Correspondence Theorem

Lemma

If Γ ⇒ ∆ doesn’t mix free and bound variables and has no Θ-cut-free proof from Θ in a Quasi-canonical G, then Θ MG Γ ⇒ ∆

Outline of Proof.

◮ Extend Γ, ∆ to maximal Γ∗, ∆∗ ◮ Construct an MG-legal valuation v for a Herbrand structure s.t. v (ϕ) depends on the appearances of ϕ and ¬ϕ in Γ∗ and ∆∗ ◮ Prove that

◮ A ∈ Γ∗ = ⇒ 1 ∈ v [σ∗ [A]] ◮ A ∈ ∆∗ = ⇒ 0 ∈ v [σ∗ [A]]

where σ∗ (x) = x ◮ This implies v Γ ⇒ ∆, and indirectly also v | = Θ