Decidable Theories 1. Linear order. p.1/9 Decidable Theories 1. - - PowerPoint PPT Presentation

Decidable Theories

• 1. Linear order.

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Decidable Theories

• 1. Linear order.
• 2. Presburger arithmetic: (Natural with +)

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Decidable Theories

• 1. Linear order.
• 2. Presburger arithmetic: (Natural with +)
• 3. Real Arithmetic (Reals +,
• , <)

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Decidable Theories

• 1. Linear order.
• 2. Presburger arithmetic: (Natural with +)
• 3. Real Arithmetic (Reals +,
• , <)
• 4. Elementary Geometry

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Decidable Theories

• 1. Linear order.
• 2. Presburger arithmetic: (Natural with +)
• 3. Real Arithmetic (Reals +,
• , <)
• 4. Elementary Geometry
• 5. Linear orders with monadic predicates.

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Decidable Theories

• 1. Linear order.

• 2. Presburger arithmetic: (Natural with +)
• 3. Real Arithmetic (Reals +,
• , <)
• 4. Elementary Geometry
• 5. Linear orders with monadic predicates.

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Decidable Theories

• 1. Linear order.

• 2. Presburger arithmetic: (Natural with +)

• 3. Real Arithmetic (Reals +,
• , <)
• 4. Elementary Geometry
• 5. Linear orders with monadic predicates.

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Decidable Theories

• 1. Linear order.

• 2. Presburger arithmetic: (Natural with +)

• 3. Real Arithmetic (Reals +,
• , <)

• 4. Elementary Geometry
• 5. Linear orders with monadic predicates.

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Decidable Theories

• 1. Linear order.

• 2. Presburger arithmetic: (Natural with +)

• 3. Real Arithmetic (Reals +,
• , <)

• 4. Elementary Geometry
• 5. Linear orders with monadic predicates. tower of 2:

heights n

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Validity problem over finite structures

Input: a formula

Question: Is

true over all finite structures?

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Validity problem over finite structures

Input: a formula

Question: Is

true over all finite structures? Theorem(Trakhtenbrot) There is no procedure for checking validity over finite structures.

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Validity problem over finite structures

Input: a formula

Question: Is

true over all finite structures? Theorem(Trakhtenbrot) There is no procedure for checking validity over finite structures. The theory of finite structures is very different from the the-

• ry of arbitrary structures

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Hilbert Calculus

Axiomsa Ax1

Ax2

Ax3

Ax4

, where

is a term. Ax5

, where

is not free in

. Inference Rules MP Derive

from

and

. Gen Derive

from

.

aWe do not distinguish between formulas with the same skeleton

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Completeness Theorem

Theorem (Completeness)

iff

.

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Completeness Theorem

Theorem (Completeness)

iff

. Theorem (Completeness - Satisfiability version)

is consis- tent iff

holds in a structure.

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Completeness Theorem

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Completeness Theorem

Theorem (Completeness - Satisfiability version) If

is consistent, then

is satisfiable (holds in a structure for FO).

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Completeness Theorem

Theorem (Completeness - Satisfiability version) If

is consistent, then

is satisfiable (holds in a structure for FO). Propositional Calculus

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Completeness Theorem

Theorem (Completeness - Satisfiability version) If

is consistent, then

is satisfiable (holds in a structure for FO). Propositional Calculus Theorem Every consistent set of formulas is a subset of a maximal consistent set of formulas. Theorem Every maximal consistent set of formulas is satisfiable.

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Completeness Theorem

Theorem (Completeness - Satisfiability version) If

is consistent, then

is satisfiable (holds in a structure for FO). Propositional Calculus Theorem Every consistent set of formulas is a subset of a maximal consistent set of formulas. Theorem Every maximal consistent set of formulas is satisfiable. Predicate Calculus

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Completeness Theorem

Theorem (Completeness - Satisfiability version) If

is consistent, then

is satisfiable (holds in a structure for FO). Propositional Calculus Theorem Every consistent set of formulas is a subset of a maximal consistent set of formulas. Theorem Every maximal consistent set of formulas is satisfiable. Predicate Calculus Theorem Every consistent set of formulas is a subset of a Complete Henkin consistent set of formulas. Theorem Every Complete Henkin consistent set of formulas holds (in a Herbrand Structure).

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Complete Set of Formulas

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Complete Set of Formulas

Definition

is

complete if for every sentence

in the signature

, either

• r

(sign. of

).

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Complete Set of Formulas

Definition

is

complete if for every sentence

in the signature

, either

• r

(sign. of

). Theorem If

is consistent then there is

such that

is consistent and

complete.

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Complete Set of Formulas

Definition

is

complete if for every sentence

in the signature

, either

• r

(sign. of

). Theorem If

is consistent then there is

such that

is consistent and

complete. Proof Define a sequence of set of formulas

as follows:

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Complete Set of Formulas

Definition

is

complete if for every sentence

in the signature

, either

• r

(sign. of

). Theorem If

is consistent then there is

such that

is consistent and

complete. Proof Define a sequence of set of formulas

as follows: Take an enumeration

• f all

sentences.

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Complete Set of Formulas

Definition

is

complete if for every sentence

in the signature

, either

• r

(sign. of

). Theorem If

is consistent then there is

such that

is consistent and

complete. Proof Define a sequence of set of formulas

as follows: Take an enumeration

• f all

sentences.

if

is consistent;

• therwise.

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Complete Set of Formulas

Definition

is

complete if for every sentence

in the signature

, either

• r

(sign. of

). Theorem If

is consistent then there is

such that

is consistent and

complete. Proof Define a sequence of set of formulas

as follows: Take an enumeration

• f all

sentences.

if

is consistent;

• therwise.

Show that

is consistent by induction on

.

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Complete Set of Formulas

Definition

is

complete if for every sentence

in the signature

, either

• r

(sign. of

). Theorem If

is consistent then there is

such that

is consistent and

complete. Proof Define a sequence of set of formulas

as follows: Take an enumeration

• f all

sentences.

if

is consistent;

• therwise.

Show that

is consistent by induction on

. Show that

is a consistent and

• complete.

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Henkin Sets of Formulas

Definition

has Henkin property for

if for every sentence

• f the form

in the signature

there is a constant

such that

(sign. of

).

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Henkin Sets of Formulas

Definition

has Henkin property for

if for every sentence

• f the form

in the signature

there is a constant

such that

(sign. of

). Theorem If

is consistent then there is a consistent

such that

and

has Henkin property for

.

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Henkin Sets of Formulas

Definition

has Henkin property for

if for every sentence

• f the form

in the signature

there is a constant

such that

(sign. of

). Theorem If

is consistent then there is a consistent

such that

and

has Henkin property for

. Proof Show

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Henkin Sets of Formulas

Definition

has Henkin property for

if for every sentence

• f the form

in the signature

there is a constant

such that

(sign. of

). Theorem If

is consistent then there is a consistent

such that

and

has Henkin property for

. Proof Show Lemma If

is consistent and

and

is a new constant then

is consistent.

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Henkin Sets of Formulas

Definition

has Henkin property for

if for every sentence

• f the form

in the signature

there is a constant

such that

(sign. of

). Theorem If

is consistent then there is a consistent

such that

and

has Henkin property for

. Proof Show Lemma If

is consistent and

and

is a new constant then

is consistent. Apply Lemma to all

• sentences of the form

in

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Complete Henkin Sets of formulas

Theorem If

is consistent then there is

and a set

• f

formulas in the signature

such that

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Complete Henkin Sets of formulas

Theorem If

is consistent then there is

and a set

• f

formulas in the signature

such that 1.

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Complete Henkin Sets of formulas

Theorem If

is consistent then there is

and a set

• f

formulas in the signature

such that 1.

2.

is consistent

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Complete Henkin Sets of formulas

Theorem If

is consistent then there is

and a set

• f

formulas in the signature

such that 1.

2.

is consistent 3.

is

complete and has Henkin property for

.

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Complete Henkin Sets of formulas

Theorem If

is consistent then there is

and a set

• f

formulas in the signature

such that 1.

2.

is consistent 3.

is

complete and has Henkin property for

. Proof Apply iteratively two previous Theorems.

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Complete Henkin Sets of formulas

Theorem If

is consistent then there is

and a set

• f

formulas in the signature

such that 1.

2.

is consistent 3.

is

complete and has Henkin property for

. Proof Apply iteratively two previous Theorems.

is the signature of

is a

• complete set that contains

has Henkin property for

and contains

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Complete Henkin Sets of formulas

Theorem If

is consistent then there is

and a set

• f

formulas in the signature

such that 1.

2.

is consistent 3.

is

complete and has Henkin property for

. Proof Apply iteratively two previous Theorems.

is the signature of

is a

• complete set that contains

has Henkin property for

and contains

and

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Complete Henkin Sets of formulas

Theorem If

is consistent then there is

and a set

• f

formulas in the signature

such that 1.

2.

is consistent 3.

is

complete and has Henkin property for

. Proof Apply iteratively two previous Theorems.

is the signature of

is a

• complete set that contains

has Henkin property for

and contains

and

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Complete Henkin Sets of formulas

Theorem If

is consistent then there is

and a set

• f

formulas in the signature

such that 1.

2.

is consistent 3.

is

complete and has Henkin property for

. Proof Apply iteratively two previous Theorems.

is the signature of

is a

• complete set that contains

has Henkin property for

and contains

and

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Complete Henkin Sets of sentences are satisfiable

Theorem If

is a consistent set of sentences in

and

is

• complete and has Henkin property for

then

holds (is satisfiable) in a Herbrand structure for

.

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Complete Henkin Sets of sentences are satisfiable

Theorem If

is a consistent set of sentences in

and

is

• complete and has Henkin property for

then

holds (is satisfiable) in a Herbrand structure for

. Proof Let be a Herbrand structure for

with

iff

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Complete Henkin Sets of sentences are satisfiable

Theorem If

is a consistent set of sentences in

and

is

• complete and has Henkin property for

then

holds (is satisfiable) in a Herbrand structure for

. Proof Let be a Herbrand structure for

with

iff

By structural induction on sentences show that

iff

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