FO = FO 3 for Linear Orders with Monotone Binary Relations Marie - - PowerPoint PPT Presentation

FO = FO3 for Linear Orders with Monotone Binary Relations

Marie Fortin

University of Liverpool

YR-OWLS, June 16, 2020

1 / 18

The k-variable property

How many variables are needed in first-order logic ?

2 / 18

The k-variable property

How many variables are needed in first-order logic ?

◮ Some properties require unboundedly many variables

∃x1.∃x2.∃x3.∃x4.

1≤i<j≤4 xi = xj

x1 x2 x3 x4

2 / 18

The k-variable property

How many variables are needed in first-order logic ?

◮ Some properties require unboundedly many variables

∃x1.∃x2.∃x3.∃x4.

1≤i<j≤4 xi = xj

x1 x2 x3 x4

◮ ... but not in every class of models:

2 / 18

The k-variable property

How many variables are needed in first-order logic ?

◮ Some properties require unboundedly many variables

∃x1.∃x2.∃x3.∃x4.

1≤i<j≤4 xi = xj

x1 x2 x3 x4

◮ ... but not in every class of models:

∃x.∃y.

• x < y ∧ ∃x. (y < x ∧ ∃y. x < y)
• 2 / 18

The k-variable property

How many variables are needed in first-order logic ?

◮ Some properties require unboundedly many variables

∃x1.∃x2.∃x3.∃x4.

1≤i<j≤4 xi = xj

x1 x2 x3 x4

◮ ... but not in every class of models:

∃x.∃y.

• x < y ∧ ∃x. (y < x ∧ ∃y. x < y)
• x

2 / 18

The k-variable property

How many variables are needed in first-order logic ?

◮ Some properties require unboundedly many variables

∃x1.∃x2.∃x3.∃x4.

1≤i<j≤4 xi = xj

x1 x2 x3 x4

◮ ... but not in every class of models:

∃x.∃y.

• x < y ∧ ∃x. (y < x ∧ ∃y. x < y)
• x

y <

2 / 18

The k-variable property

How many variables are needed in first-order logic ?

◮ Some properties require unboundedly many variables

∃x1.∃x2.∃x3.∃x4.

1≤i<j≤4 xi = xj

x1 x2 x3 x4

◮ ... but not in every class of models:

∃x.∃y.

• x < y ∧ ∃x. (y < x ∧ ∃y. x < y)
• y

x <

2 / 18

The k-variable property

How many variables are needed in first-order logic ?

◮ Some properties require unboundedly many variables

∃x1.∃x2.∃x3.∃x4.

1≤i<j≤4 xi = xj

x1 x2 x3 x4

◮ ... but not in every class of models:

∃x.∃y.

• x < y ∧ ∃x. (y < x ∧ ∃y. x < y)
• x

y <

2 / 18

The k-variable property

How many variables are needed in first-order logic ?

◮ Some properties require unboundedly many variables

∃x1.∃x2.∃x3.∃x4.

1≤i<j≤4 xi = xj

x1 x2 x3 x4

◮ ... but not in every class of models:

∃x.∃y.

• x < y ∧ ∃x. (y < x ∧ ∃y. x < y)
• x

y < Over linear orders, FO = FO3.

2 / 18

Bounded variable logics

Why do we care about the number of variables?

3 / 18

Bounded variable logics

Why do we care about the number of variables?

◮ (Descriptive) complexity

3 / 18

Bounded variable logics

Why do we care about the number of variables?

◮ (Descriptive) complexity ◮ Temporal logics

3 / 18

Bounded variable logics

Why do we care about the number of variables?

◮ (Descriptive) complexity ◮ Temporal logics

[Gabbay 1981] In any class of time flows, TFAE:

◮ There exists an expressively complete finite set of

FO-definable (multi-dimensional) temporal connectives

◮ There exists k such that every first-order sentence is

equivalent to one with at most k variables

3 / 18

Example

Over linear orders, FO = FO3.

4 / 18

Example

Over linear orders, FO = FO3. Two classical techniques to prove FO = FOk (over a class C)

4 / 18

Example

Over linear orders, FO = FO3. Two classical techniques to prove FO = FOk (over a class C)

• 1. Corollary of expressive completeness of a temporal logic

4 / 18

Example

Over linear orders, FO = FO3. Two classical techniques to prove FO = FOk (over a class C)

• 1. Corollary of expressive completeness of a temporal logic

Example: Over complete linear orders, FO3 ⊆ FO = LTL ⊆ FO3 [Kamp 1968]

4 / 18

Example

Over linear orders, FO = FO3. Two classical techniques to prove FO = FOk (over a class C)

• 1. Corollary of expressive completeness of a temporal logic

Example: Over complete linear orders, FO3 ⊆ FO = LTL ⊆ FO3 [Kamp 1968] Over (arbitrary) linear orders, FO3 ⊆ FO = LTL with Stavi connectives ⊆ FO3 [Gabbay, Hodkinson, Reynolds 1993]

4 / 18

Example

Over linear orders, FO = FO3. Two classical techniques to prove FO = FOk (over a class C)

• 1. Corollary of expressive completeness of a temporal logic
• 2. Ehrenfeucht-Fra¨

ıss´ e games with k pebbles

4 / 18

Example

Over linear orders, FO = FO3. Two classical techniques to prove FO = FOk (over a class C)

• 1. Corollary of expressive completeness of a temporal logic
• 2. Ehrenfeucht-Fra¨

ıss´ e games with k pebbles

Example: Over complete linear orders, FO = FO3 [Immerman, Kozen 1989]

4 / 18

Example

Over linear orders, FO = FO3. Two classical techniques to prove FO = FOk (over a class C)

• 1. Corollary of expressive completeness of a temporal logic

0 or 1 free variables

• 2. Ehrenfeucht-Fra¨

ıss´ e games with k pebbles up to k free variables

4 / 18

Known results (non-exhaustive)

Over linear orders, FO = FO3

[Immerman-Kozen’89]

✓

5 / 18

Known results (non-exhaustive)

Over linear orders, FO = FO3

[Immerman-Kozen’89]

✓

What happens if we have additional binary relations?

5 / 18

Known results (non-exhaustive)

Over linear orders, FO = FO3

[Immerman-Kozen’89]

✓

What happens if we have additional binary relations?

Over ordered graphs, ∀k, FO = FOk

[Rossman’08]

✗

5 / 18

Known results (non-exhaustive)

Over linear orders, FO = FO3

[Immerman-Kozen’89]

✓

What happens if we have additional binary relations?

Over ordered graphs, ∀k, FO = FOk

[Rossman’08]

✗ Over (R, <, +1), FO = FO3

[AHRW’15]

✓

5 / 18

Known results (non-exhaustive)

Over linear orders, FO = FO3

[Immerman-Kozen’89]

✓

What happens if we have additional binary relations?

Over ordered graphs, ∀k, FO = FOk

[Rossman’08]

✗ Over (R, <, +1), FO = FO3

[AHRW’15]

✓ Over Mazurkiewicz traces, FO = FO3

[Gastin-Mukund’02]

✓ Over MSCs, FO = FO3

[Bollig-F.-Gastin’18]

✓

5 / 18

Known results (non-exhaustive)

Over linear orders, FO = FO3

[Immerman-Kozen’89]

✓

What happens if we have additional binary relations?

Over ordered graphs, ∀k, FO = FOk

[Rossman’08]

✗ Over (R, <, +1), FO = FO3

[AHRW’15]

✓ Over Mazurkiewicz traces, FO = FO3

[Gastin-Mukund’02]

✓ Over MSCs, FO = FO3

[Bollig-F.-Gastin’18]

✓

What do these 4 positive results have in common?

5 / 18

Generalisation [F.’19]

FO = FO3 over structures with

◮ one linear order ≤, ◮ “interval-preserving” binary relations R1, R2, . . ., ◮ arbitrary unary predicates p, q, . . .

p p p q q p, q p, q

6 / 18

Generalisation [F.’19]

FO = FO3 over structures with

◮ one linear order ≤, ◮ “interval-preserving” binary relations R1, R2, . . ., ◮ arbitrary unary predicates p, q, . . .

p p p q q p, q p, q

R is interval-preserving if for all intervals I,

◮ R(I) is an interval of (Im(R), ≤) ◮ R−1(I) is an interval of (dom(R), ≤)

6 / 18

Generalisation [F.’19]

FO = FO3 over structures with

◮ one linear order ≤, ◮ “interval-preserving” binary relations R1, R2, . . ., ◮ arbitrary unary predicates p, q, . . .

I

R is interval-preserving if for all intervals I,

◮ R(I) is an interval of (Im(R), ≤) ◮ R−1(I) is an interval of (dom(R), ≤)

6 / 18

Generalisation [F.’19]

FO = FO3 over structures with

◮ one linear order ≤, ◮ “interval-preserving” binary relations R1, R2, . . ., ◮ arbitrary unary predicates p, q, . . .

I R(I)

R is interval-preserving if for all intervals I,

◮ R(I) is an interval of (Im(R), ≤) ◮ R−1(I) is an interval of (dom(R), ≤)

6 / 18

Generalisation [F.’19]

FO = FO3 over structures with

◮ one linear order ≤, ◮ “interval-preserving” binary relations R1, R2, . . ., ◮ arbitrary unary predicates p, q, . . .

I R(I)

R is interval-preserving if for all intervals I,

◮ R(I) is an interval of (Im(R), ≤) ◮ R−1(I) is an interval of (dom(R), ≤)

6 / 18

Generalisation [F.’19]

FO = FO3 over structures with

◮ one linear order ≤, ◮ “interval-preserving” binary relations R1, R2, . . ., ◮ arbitrary unary predicates p, q, . . .

I R(I) ✗

R is interval-preserving if for all intervals I,

◮ R(I) is an interval of (Im(R), ≤) ◮ R−1(I) is an interval of (dom(R), ≤)

6 / 18

Generalisation [F.’19]

FO = FO3 over structures with

◮ one linear order ≤, ◮ “interval-preserving” binary relations R1, R2, . . ., ◮ arbitrary unary predicates p, q, . . .

I R−1(I)

R is interval-preserving if for all intervals I,

◮ R(I) is an interval of (Im(R), ≤) ◮ R−1(I) is an interval of (dom(R), ≤)

6 / 18

A special case: monotone partial functions

Any relation R corresponding to a monotone partial function is interval-preserving.

7 / 18

A special case: monotone partial functions

Any relation R corresponding to a monotone partial function is interval-preserving.

◮ R(I) is an interval of (Im(R), ≤)

7 / 18

A special case: monotone partial functions

Any relation R corresponding to a monotone partial function is interval-preserving.

◮ R(I) is an interval of (Im(R), ≤)

I

7 / 18

A special case: monotone partial functions

Any relation R corresponding to a monotone partial function is interval-preserving.

◮ R(I) is an interval of (Im(R), ≤)

I

R R

7 / 18

A special case: monotone partial functions

Any relation R corresponding to a monotone partial function is interval-preserving.

◮ R(I) is an interval of (Im(R), ≤)

I

R R Im(R)

7 / 18

A special case: monotone partial functions

Any relation R corresponding to a monotone partial function is interval-preserving.

◮ R(I) is an interval of (Im(R), ≤)

I

R R Im(R) R

7 / 18

A special case: monotone partial functions

Any relation R corresponding to a monotone partial function is interval-preserving.

◮ R(I) is an interval of (Im(R), ≤) ◮ R−1(I) is an interval of (dom(R), ≤)

7 / 18

A special case: monotone partial functions

Any relation R corresponding to a monotone partial function is interval-preserving.

◮ R(I) is an interval of (Im(R), ≤) ◮ R−1(I) is an interval of (dom(R), ≤)

I

7 / 18

A special case: monotone partial functions

Any relation R corresponding to a monotone partial function is interval-preserving.

◮ R(I) is an interval of (Im(R), ≤) ◮ R−1(I) is an interval of (dom(R), ≤)

I

R R dom(R)

7 / 18

A special case: monotone partial functions

Any relation R corresponding to a monotone partial function is interval-preserving.

◮ R(I) is an interval of (Im(R), ≤) ◮ R−1(I) is an interval of (dom(R), ≤)

I

R R dom(R) R

7 / 18

Applications

FO = FO3 over

• 1. Linear orders with partial non-decreasing or non-increasing

functions (new)

8 / 18

Applications

FO = FO3 over

• 1. Linear orders with partial non-decreasing or non-increasing

functions (new)

• 2. Linear orders: finite or infinite words, R, Q, ordinals...

8 / 18

Applications

FO = FO3 over

• 1. Linear orders with partial non-decreasing or non-increasing

functions (new)

• 2. Linear orders: finite or infinite words, R, Q, ordinals...
• 3. (R, ≤, +1), (R, ≤, (+q)q∈Q) . . .

8 / 18

Applications

FO = FO3 over

• 1. Linear orders with partial non-decreasing or non-increasing

functions (new)

• 2. Linear orders: finite or infinite words, R, Q, ordinals...
• 3. (R, ≤, +1), (R, ≤, (+q)q∈Q) . . .
• 4. (R, ≤) + polynomial functions (new)

8 / 18

Applications

• 5. Message sequence charts (MSCs)

9 / 18

Applications

• 5. Message sequence charts (MSCs)

a a c a a a a a a a a a a a a a a a a b b a a c a a a p q r

9 / 18

Applications

• 5. Message sequence charts (MSCs)

a a c a a a a a a a a a a a a a a a a b b a a c a a a p q r

Executions of message-passing systems

9 / 18

Applications

• 5. Message sequence charts (MSCs)

a a c a a a a a a a a a a a a a a a a b b a a c a a a p q r

Executions of message-passing systems

◮ Fixed, finite set of processes 9 / 18

Applications

• 5. Message sequence charts (MSCs)

a a c a a a a a a a a a a a a a a a a b b a a c a a a p q r

Executions of message-passing systems

◮ Fixed, finite set of processes ◮ Process order ≤proc 9 / 18

Applications

• 5. Message sequence charts (MSCs)

a a c a a a a a a a a a a a a a a a a b b a a c a a a p q r

Executions of message-passing systems

◮ Fixed, finite set of processes ◮ Process order ≤proc ◮ Message relations ⊳p,q 9 / 18

Applications

• 5. Message sequence charts (MSCs)

a a c a a a a a a a a a a a a a a a a b b a a c a a a p q r

Executions of message-passing systems

◮ Fixed, finite set of processes ◮ Process order ≤proc ◮ Message relations ⊳p,q 9 / 18

Applications

• 5. Message sequence charts (MSCs)

a a c a a a a a a a a a a a a a a a a b b a a c a a a p q r

Executions of message-passing systems

◮ Fixed, finite set of processes ◮ Process order ≤proc

Extended to a linear order

◮ Message relations ⊳p,q 9 / 18

Applications

• 5. Message sequence charts (MSCs)

a a c a a a a a a a a a a a a a a a a b b a a c a a a p q r

Executions of message-passing systems

◮ Fixed, finite set of processes ◮ Process order ≤proc

Extended to a linear order

◮ Message relations ⊳p,q

FIFO → monotone

9 / 18

Applications

• 5. Message sequence charts (MSCs)

a a c a a a a a a a a a a a a a a a a b b a a c a a a p q r

Executions of message-passing systems

◮ Fixed, finite set of processes ◮ Process order ≤proc

Extended to a linear order

◮ Message relations ⊳p,q

FIFO → monotone

→ Interval-preserving structure

9 / 18

Applications

FO = FO3 over structures with

◮ one linear order ≤, ◮ “interval-preserving” binary relations R1, R2, . . ., ◮ arbitrary unary predicates p, q, . . .

• 1. Linear orders with partial non-decreasing or non-increasing

functions (new)

• 2. Linear orders: finite or infinite words, R, Q, ordinals...
• 3. (R, ≤, +1), (R, ≤, (+q)q∈Q) . . .
• 4. (R, ≤) + polynomial functions (new)
• 5. MSCs
• 6. Mazurkiewicz traces

10 / 18

How does the interval-preserving assumption help?

11 / 18

How does the interval-preserving assumption help?

ϕ(x1, x2, x3) = ∃y. R1(x1, y) ∧ R2(x2, y) ∧ R3(x3, y) y x1 x2 x3

R1 R2 R3

11 / 18

How does the interval-preserving assumption help?

ϕ(x1, x2, x3) = ∃y. R1(x1, y) ∧ R2(x2, y) ∧ R3(x3, y) y x1 x2 x3

R1 R2 R3

Equivalent FO3 formula?

11 / 18

How does the interval-preserving assumption help?

ϕ(x1, x2, x3) = ∃y. R1(x1, y) ∧ R2(x2, y) ∧ R3(x3, y)

R1(x1) R2(x2) R3(x3)

Equivalent FO3 formula?

11 / 18

How does the interval-preserving assumption help?

ϕ(x1, x2, x3) = ∃y. R1(x1, y) ∧ R2(x2, y) ∧ R3(x3, y)

R1(x1) R2(x2) R3(x3) y

Equivalent FO3 formula?

11 / 18

How does the interval-preserving assumption help?

ϕ(x1, x2, x3) = ∃y. R1(x1, y) ∧ R2(x2, y) ∧ R3(x3, y) ≡

• ∃y. R1(x1, y) ∧ R2(x2, y) ∧
• ∧
• ∃y. R1(x1, y) ∧ R3(x3, y) ∧
• ∧
• ∃y. R2(x2, y) ∧ R2(x3, y) ∧
• R1(x1)

R2(x2) R3(x3) y

Equivalent FO3 formula?

11 / 18

How does the interval-preserving assumption help?

ϕ(x1, x2, x3) = ∃y. R1(x1, y) ∧ R2(x2, y) ∧ R3(x3, y) ≡

• ∃y. R1(x1, y) ∧ R2(x2, y) ∧ ∃x. R3(x, y)
• ∧
• ∃y. R1(x1, y) ∧ R3(x3, y) ∧ ∃x. R2(x, y)
• ∧
• ∃y. R2(x2, y) ∧ R2(x3, y) ∧ ∃x. R1(x, y)
• R1(x1)

R2(x2) R3(x3) y

Equivalent FO3 formula?

11 / 18

How does the interval-preserving assumption help?

ϕ(x1, x2, x3) = ∃y. R1(x1, y) ∧ R2(x2, y) ∧ R3(x3, y) ≡

• ∃x3. R1(x1, x3) ∧ R2(x2, x3) ∧ ∃x1. R3(x1, x3)
• ∧
• ∃x2. R1(x1, x2) ∧ R3(x3, x2) ∧ ∃x1. R2(x1, x2)
• ∧
• ∃x1. R2(x2, x1) ∧ R2(x3, x1) ∧ ∃x2. R1(x2, x1)
• R1(x1)

R2(x2) R3(x3) y

Equivalent FO3 formula?

11 / 18

The proof

FO = FO3 over structures with

◮ one linear order ≤, ◮ “interval-preserving” binary relations R1, R2, . . ., ◮ arbitrary unary predicates p, q, . . .

12 / 18

The proof

FO = FO3 over structures with

◮ one linear order ≤, ◮ “interval-preserving” binary relations R1, R2, . . ., ◮ arbitrary unary predicates p, q, . . .

Key idea: Go through an intermediate language: Star-free Propositional Dynamic Logic. FO Star-free PDL FO3

12 / 18

Star-free Propositional Dynamic Logic

Examples

13 / 18

Star-free Propositional Dynamic Logic

Examples p p p q q q p, q

R

13 / 18

Star-free Propositional Dynamic Logic

Examples p p p q q q p, q

R

(p ∧ ¬q) ∨ (q ∧ ¬p) ✓ ✓ ✓ ✓ ✓ ✓ ✗ ✗ ✗

13 / 18

Star-free Propositional Dynamic Logic

Examples p p p q q q p, q

R

(p ∧ ¬q) ∨ (q ∧ ¬p) ✓ ✓ ✓ ✓ ✓ ✓ ✗ ✗ ✗ R q ✓ ✓ ✗ ✗ ✗ ✗ ✗ ✗ ✗

13 / 18

Star-free Propositional Dynamic Logic

Examples p p p q q q p, q

R

(p ∧ ¬q) ∨ (q ∧ ¬p) ✓ ✓ ✓ ✓ ✓ ✓ ✗ ✗ ✗ R q ✓ ✓ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ≤ · R−1 q ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✗ ✗

13 / 18

Star-free Propositional Dynamic Logic

Examples p p p q q q p, q

R

(p ∧ ¬q) ∨ (q ∧ ¬p) ✓ ✓ ✓ ✓ ✓ ✓ ✗ ✗ ✗ R q ✓ ✓ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ≤ · R−1 q ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✗ ✗ ≤ · {R q}? · ≤ p ✓ ✓ ✓ ✗ ✗ ✗ ✗ ✗ ✗

13 / 18

Star-free Propositional Dynamic Logic

Examples p p p q q q p, q

R

(p ∧ ¬q) ∨ (q ∧ ¬p) ✓ ✓ ✓ ✓ ✓ ✓ ✗ ✗ ✗ R q ✓ ✓ ✗ ✗ ✗ ✗ ✗ ✗ ✗ ≤ · R−1 q ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✗ ✗ ≤ · {R q}? · ≤ p ✓ ✓ ✓ ✗ ✗ ✗ ✗ ✗ ✗ Rc ∩ ≤ (p ∧ q) ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✗ ✗

13 / 18

Star-free Propositional Dynamic Logic

Examples

Over (R, <, {+q | q ∈ Q+}), ϕ U(q,r) ψ ≡ ψ t t + q t + r ϕ

14 / 18

Star-free Propositional Dynamic Logic

Examples

Over (R, <, {+q | q ∈ Q+}), ϕ U(q,r) ψ ≡

• (+q · <) ∩ (+r · <−1) ∩ (< · {¬ϕ}? · <)c

ψ ψ

+q < +r <

ϕ

14 / 18

Star-free Propositional Dynamic Logic

Syntax

State formulas: ϕ ::= P | ϕ ∨ ϕ | ¬ϕ | π ϕ Path formulas: π ::= ≤ | R | {ϕ}? | π−1 | π · π | π ∪ π | πc PDLsf

15 / 18

Star-free Propositional Dynamic Logic

Syntax

State formulas: ϕ ::= P | ϕ ∨ ϕ | ¬ϕ | π ϕ Path formulas: π ::= ≤ | R | {ϕ}? | π−1 | π · π | π ∪ π | πc PDLsf Combines features from

◮ Propositional Dynamic Logic [Fisher-Ladner 1979] ◮ Star-free regular expressions ◮ The calculus of relations

15 / 18

Star-free Propositional Dynamic Logic

Syntax

State formulas: ϕ ::= P | ϕ ∨ ϕ | ¬ϕ | π ϕ Path formulas: π ::= ≤ | R | {ϕ}? | π−1 | π · π | π ∪ π | πc PDLsf Combines features from

◮ Propositional Dynamic Logic [Fisher-Ladner 1979] ◮ Star-free regular expressions ◮ The calculus of relations

Theorem: [Tarski-Givant 1987 (calculus of relations)] PDLsf and FO3 are expressively equivalent

15 / 18

A fragment of Star-free PDL

16 / 18

A fragment of Star-free PDL

State formulas: ϕ ::= P | ϕ ∨ ϕ | ¬ϕ | π ϕ Path formulas: π ::= ≤ | R | {ϕ}? | π−1 | π · π | π ∪ π | πc PDLsf π ::= ≤ | R | {ϕ}? | π−1 | π · π | π ∩ π | (≤ · π · ≤)c | (≤ · π · ≥)c | (≥ · π · ≤)c | (≥ · π · ≥)c PDLint

sf

16 / 18

A fragment of Star-free PDL

State formulas: ϕ ::= P | ϕ ∨ ϕ | ¬ϕ | π ϕ Path formulas: π ::= ≤ | R | {ϕ}? | π−1 | π · π | π ∪ π | πc PDLsf π ::= ≤ | R | {ϕ}? | π−1 | π · π | π ∩ π | (≤ · π · ≤)c | (≤ · π · ≥)c | (≥ · π · ≤)c | (≥ · π · ≥)c PDLint

sf

Lemma: ∀π ∈ PDLint

sf , π is interval-preserving

16 / 18

Equivalences over interval-preserving structures

FO PDLint

sf

FO3 PDLsf

17 / 18

Equivalences over interval-preserving structures

FO PDLint

sf

FO3 PDLsf

def. def.

17 / 18

Equivalences over interval-preserving structures

FO PDLint

sf

FO3 PDLsf

def. trivial induction def.

17 / 18

Equivalences over interval-preserving structures

FO PDLint

sf

FO3 PDLsf

def. trivial induction def.

◮ State formula ϕ ∈ PDLsf

• ϕFO(x) ∈ FO

◮ Path formula π ∈ PDLsf

• πFO(x, y) ∈ FO

17 / 18

Equivalences over interval-preserving structures

FO PDLint

sf

FO3 PDLsf

def. trivial induction def.

◮ State formula ϕ ∈ PDLsf

• ϕFO(x) ∈ FO

π ϕ

• ∃y.πFO(x, y) ∧ ϕFO(y)

◮ Path formula π ∈ PDLsf

• πFO(x, y) ∈ FO

π1 · π2

• ∃z.πFO

1 (x, z) ∧ πFO 2 (z, y)

17 / 18

Equivalences over interval-preserving structures

FO PDLint

sf

?

Any FO formula Φ(x1, . . . , xn) is equivalent to a finite positive boolean combination of formulas of the form πFO(xi, xj), where π ∈ PDLint

sf .

17 / 18

Equivalences over interval-preserving structures

FO PDLint

sf

?

Any FO formula Φ(x1, . . . , xn) is equivalent to a finite positive boolean combination of formulas of the form πFO(xi, xj), where π ∈ PDLint

sf .

Proof: by induction on Φ.

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Equivalences over interval-preserving structures

FO PDLint

sf

?

Any FO formula Φ(x1, . . . , xn) is equivalent to a finite positive boolean combination of formulas of the form πFO(xi, xj), where π ∈ PDLint

sf .

Proof: by induction on Φ.

◮ Atomic formulas, disjunction: easy

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Equivalences over interval-preserving structures

FO PDLint

sf

?

Any FO formula Φ(x1, . . . , xn) is equivalent to a finite positive boolean combination of formulas of the form πFO(xi, xj), where π ∈ PDLint

sf .

Proof: by induction on Φ.

◮ Negation: Express πc using

(≤ · π · ≤)c, (≤ · π · ≥)c, (≥ · π · ≤)c, (≥ · π · ≥)c.

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Equivalences over interval-preserving structures

FO PDLint

sf

?

Any FO formula Φ(x1, . . . , xn) is equivalent to a finite positive boolean combination of formulas of the form πFO(xi, xj), where π ∈ PDLint

sf .

Proof: by induction on Φ.

◮ Existential quantification: Similar to the example before.

17 / 18

Equivalences over interval-preserving structures

FO PDLint

sf

?

Any FO formula Φ(x1, . . . , xn) is equivalent to a finite positive boolean combination of formulas of the form πFO(xi, xj), where π ∈ PDLint

sf .

Proof: by induction on Φ.

◮ Existential quantification: Similar to the example before.

∃x.

i πFO i (xi, x)

17 / 18

Equivalences over interval-preserving structures

FO PDLint

sf

?

Any FO formula Φ(x1, . . . , xn) is equivalent to a finite positive boolean combination of formulas of the form πFO(xi, xj), where π ∈ PDLint

sf .

Proof: by induction on Φ.

◮ Existential quantification: Similar to the example before.

∃x.

i πFO i (xi, x)

• intersection of n intervals

17 / 18

Equivalences over interval-preserving structures

FO PDLint

sf

?

Any FO formula Φ(x1, . . . , xn) is equivalent to a finite positive boolean combination of formulas of the form πFO(xi, xj), where π ∈ PDLint

sf .

Proof: by induction on Φ.

◮ Existential quantification: Similar to the example before.

∃x.

i πFO i (xi, x)

• intersection of n intervals

xi xj ∃x ϕ πi πj

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Equivalences over interval-preserving structures

FO PDLint

sf

?

Any FO formula Φ(x1, . . . , xn) is equivalent to a finite positive boolean combination of formulas of the form πFO(xi, xj), where π ∈ PDLint

sf .

Proof: by induction on Φ.

◮ Existential quantification: Similar to the example before.

∃x.

i πFO i (xi, x)

• intersection of n intervals

≡

i,j(πi · {ϕ}? · π−1 j )FO(xi, xj)

• pairwise intersections

xi xj ∃x ϕ πi πj

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Conclusion

◮ Over linearly ordered structures with interval-preserving

binary relations, FO = PDLsf = FO3

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Conclusion

◮ Over linearly ordered structures with interval-preserving

binary relations, FO = PDLsf = FO3

◮ Covers many classical classes of structures: linear orders,

real-time signals, MSCs, . . .

18 / 18

Conclusion

◮ Over linearly ordered structures with interval-preserving

binary relations, FO = PDLsf = FO3

◮ Covers many classical classes of structures: linear orders,

real-time signals, MSCs, . . .

◮ Star-free PDL is a useful technical tool, but also an

interesting logic on its own.

18 / 18

Conclusion

◮ Over linearly ordered structures with interval-preserving

binary relations, FO = PDLsf = FO3

◮ Covers many classical classes of structures: linear orders,

real-time signals, MSCs, . . .

◮ Star-free PDL is a useful technical tool, but also an

interesting logic on its own. Further directions:

◮ Generalizations to ther types of orders (trees. . . ),

relations of arity > 2?

18 / 18

Conclusion

◮ Over linearly ordered structures with interval-preserving

binary relations, FO = PDLsf = FO3

◮ Covers many classical classes of structures: linear orders,

real-time signals, MSCs, . . .

◮ Star-free PDL is a useful technical tool, but also an

interesting logic on its own. Further directions:

◮ Generalizations to ther types of orders (trees. . . ),

relations of arity > 2?

◮ Uniform approach for proving completeness of temporal

logics?

18 / 18

Conclusion

◮ Over linearly ordered structures with interval-preserving

binary relations, FO = PDLsf = FO3

◮ Covers many classical classes of structures: linear orders,

real-time signals, MSCs, . . .

◮ Star-free PDL is a useful technical tool, but also an

interesting logic on its own. Further directions:

◮ Generalizations to ther types of orders (trees. . . ),

relations of arity > 2?

◮ Uniform approach for proving completeness of temporal

logics?

18 / 18

Conclusion

◮ Over linearly ordered structures with interval-preserving

binary relations, FO = PDLsf = FO3

◮ Covers many classical classes of structures: linear orders,

real-time signals, MSCs, . . .

◮ Star-free PDL is a useful technical tool, but also an

interesting logic on its own. Further directions:

◮ Generalizations to ther types of orders (trees. . . ),

relations of arity > 2?

◮ Uniform approach for proving completeness of temporal

logics?

Thank you!

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