Expressive Completeness over Nat and Finite orders - - PowerPoint PPT Presentation

Expressive Completeness over Nat and Finite orders

MLO=Automata=regular expressions (over finite

• rders).

– p.1/12

Expressive Completeness over Nat and Finite orders

MLO=Automata=regular expressions (over finite

• rders).

MLO=

• Automata=
• regular expressions (over Nat).

– p.1/12

Expressive Completeness over Nat and Finite orders

MLO=Automata=regular expressions (over finite

• rders).

MLO=

• Automata=
• regular expressions (over Nat).

FOMLO=TL(U,S) (over Dedekind complete orders)

– p.1/12

Expressive Completeness over Nat and Finite orders

MLO=Automata=regular expressions (over finite

• rders).

MLO=

• Automata=
• regular expressions (over Nat).

FOMLO=TL(U,S) (over Dedekind complete orders) FOMLO= star free regular expressions (over finite

• rders)

– p.1/12

Expressive Completeness over Nat and Finite orders

MLO=Automata=regular expressions (over finite

• rders).

MLO=

• Automata=
• regular expressions (over Nat).

FOMLO=TL(U,S) (over Dedekind complete orders) FOMLO= star free regular expressions (over finite

• rders)

FOMLO = Counter-free automata (over finite orders)

– p.1/12

Counter-free automata

• Def. A sequence of states

(for

) in an automaton

is a counter for a string

if

where by convention

.

– p.2/12

Counter-free automata

• Def. A sequence of states

(for

) in an automaton

is a counter for a string

if

where by convention

.

• Def. An automaton is counter-free iff it does not have a

counter.

– p.2/12

Counter-free automata

• Def. A sequence of states

(for

) in an automaton

is a counter for a string

if

where by convention

.

• Def. An automaton is counter-free iff it does not have a

counter.

Theorem (MacNaughton) A language is definable by FOMLO

formula iff it is accepted by a deterministic counter-free au- tomaton iff it is definable by a star free regular expression.

– p.2/12

The complexity of TL(U) over Nat

Theorem The satisfiability problem for TL(U) over Nat is in

PSPACE.

– p.3/12

The complexity of TL(U) over Nat

Theorem The satisfiability problem for TL(U) over Nat is in

PSPACE.

Lemma(Small Model property) If

is satisfiable then it is satisfiable on a quasi-periodic model

with

small (

– p.3/12

The complexity of TL(U) over Nat

Theorem The satisfiability problem for TL(U) over Nat is in

PSPACE.

Lemma(Small Model property) If

is satisfiable then it is satisfiable on a quasi-periodic model

with

small (

Lemma The satisfiability of

• ver small model can be

checked in NPSPACE.

– p.3/12

The complexity of TL(U) over Nat

Theorem The satisfiability problem for TL(U) over Nat is in

PSPACE.

Lemma(Small Model property) If

is satisfiable then it is satisfiable on a quasi-periodic model

with

small (

Lemma The satisfiability of

• ver small model can be

checked in NPSPACE. Homework: Prove PSPACE lower bound for the satifiability problem

– p.3/12

The complexity of TL(U) over Nat

Theorem The satisfiability problem for TL(U) over Nat is in

PSPACE.

Lemma(Small Model property) If

is satisfiable then it is satisfiable on a quasi-periodic model

with

small (

Lemma The satisfiability of

• ver small model can be

checked in NPSPACE. Homework: Prove PSPACE lower bound for the satifiability problem Hint: For every PSPACE TM and a word

construct a formula

which is satisfiable iff accepts

.

– p.3/12

Proof of a small model property

Notations: Sub(

) - the set of subformulas of

– p.4/12

Proof of a small model property

Notations: Sub(

) - the set of subformulas of

Example

The Subformulas of

– p.4/12

Proof of a small model property

Notations: Sub(

) - the set of subformulas of

Example

The Subformulas of

Number of subformulas -

– p.4/12

Proof of a small model property

Notations: Sub(

) - the set of subformulas of

Example

The Subformulas of

Number of subformulas -

Def (Type) Let

be a formula

be a linear order with monadic predicates and

an element of

.

– p.4/12

Proof of a small model property

Assume

A1 A2 A3 a b A1 A3 a

Then

– p.5/12

Proof of a small model property

Assume

A1 A2 A3 a b A1 A3 a

Then

• 1. For every

– p.5/12

Proof of a small model property

Assume

A1 A2 A3 a b A1 A3 a

Then

• 1. For every

• 2. For every

– p.5/12

Proof of a small model property

Additional transformations

Image of a point

A1 A2 A3 a b A2 b A2 b A1 a A3 c c’ c’’

Assume

Then

– p.6/12

Proof of a small model property

Additional transformations

Image of a point

A1 A2 A3 a b A2 b A2 b A1 a A3 c c’ c’’

Assume

Then For every

and its image

– p.6/12

Proof of a small model property

A1 c b1 A2 A2 A2 A1 A2 A3 a b2 b3

Assume that

is an unbounded increasing sequence and

for

– p.7/12

Proof of a small model property

A1 c b1 A2 A2 A2 A1 A2 A3 a b2 b3

Assume that

is an unbounded increasing sequence and

for

For

there is

such that

.

– p.7/12

Proof of a small model property

A1 c b1 A2 A2 A2 A1 A2 A3 a b2 b3

Assume that

is an unbounded increasing sequence and

for

For

there is

such that

.

– p.7/12

Proof of a small model property

A1 c b1 A2 A2 A2 A1 A2 A3 a b2 b3

Assume that

is an unbounded increasing sequence and

for

For

there is

such that

. Then for every

and

iff

• ✙

– p.7/12

Proof of a small model property

Hence if

is satisfiable over a linear structure without a maximal element then it is satisfiable over a structure

• ✡

.

– p.8/12

Proof of a small model property

Hence if

is satisfiable over a linear structure without a maximal element then it is satisfiable over a structure

• ✡

. if

is satisfiable over the discrete time then it is satisfiable over a quasiperiodic structure

.

– p.8/12

Proof of a small model property

Hence if

is satisfiable over a linear structure without a maximal element then it is satisfiable over a structure

• ✡

. if

is satisfiable over the discrete time then it is satisfiable over a quasiperiodic structure

.

– p.8/12

Proof of a small model property

Hence if

is satisfiable over a linear structure without a maximal element then it is satisfiable over a structure

• ✡

. if

is satisfiable over the discrete time then it is satisfiable over a quasiperiodic structure

. What is the length of

?

– p.8/12

Proof of a small model property

Hence if

is satisfiable over a linear structure without a maximal element then it is satisfiable over a structure

• ✡

. if

is satisfiable over the discrete time then it is satisfiable over a quasiperiodic structure

. What is the length of

?

the number of types of

.

– p.8/12

Proof of a small model property

Hence if

is satisfiable over a linear structure without a maximal element then it is satisfiable over a structure

• ✡

. if

is satisfiable over the discrete time then it is satisfiable over a quasiperiodic structure

. What is the length of

?

the number of types of

. What is the length of

?

– p.8/12

Proof of a small model property

Hence if

is satisfiable over a linear structure without a maximal element then it is satisfiable over a structure

• ✡

. if

is satisfiable over the discrete time then it is satisfiable over a quasiperiodic structure

. What is the length of

?

the number of types of

. What is the length of

?

.

– p.8/12

The complexity of satisfiability for TL(U)

The small model property Lemma implies

– p.9/12

The complexity of satisfiability for TL(U)

The small model property Lemma implies

Theorem The satisfiability problem for TL(U) over Nat is in

NEXPTIME.

– p.9/12

The complexity of satisfiability for TL(U)

The small model property Lemma implies

Theorem The satisfiability problem for TL(U) over Nat is in

NEXPTIME. Algorithm If

is satisfiable then there are exponentially small

and

such that

. An Algorithm guesses

and

and checks that the guesses are correct.

– p.9/12

The complexity of satisfiability for TL(U)

The small model property Lemma implies

Theorem The satisfiability problem for TL(U) over Nat is in

NEXPTIME. Algorithm If

is satisfiable then there are exponentially small

and

such that

. An Algorithm guesses

and

and checks that the guesses are correct. The algorithm can be implemented on the fly (without explicit construction of

and

) in PSPACE. Hence

– p.9/12

The complexity of satisfiability for TL(U)

The small model property Lemma implies

Theorem The satisfiability problem for TL(U) over Nat is in

NEXPTIME. Algorithm If

is satisfiable then there are exponentially small

and

such that

. An Algorithm guesses

and

and checks that the guesses are correct. The algorithm can be implemented on the fly (without explicit construction of

and

) in PSPACE. Hence

Theorem The satisfiability problem for TL(U) over Nat is in

PSPACE.

– p.9/12

From TL(U) to Automata

• Theorem. For every

there is a Street automata of size

that is equivalent to

.

– p.10/12

From TL(U) to Automata

• Theorem. For every

there is a Street automata of size

that is equivalent to

.

• Def. a set

• f formulas is boolean consistent iff

1.

iff

and

. 2.

iff

– p.10/12

From TL(U) to Automata

• Theorem. For every

there is a Street automata of size

that is equivalent to

.

• Def. a set

• f formulas is boolean consistent iff

1.

iff

and

. 2.

iff

Observation

is a maximal boolean consistent subset of the subformulas of

.

– p.10/12

From TL(U) to Automata

• States. The maximal Consistent subsets of

.

– p.11/12

From TL(U) to Automata

• States. The maximal Consistent subsets of

.

Alphabet Let

be the atomic propositions in

. The alphabet is the subsets of

.

– p.11/12

From TL(U) to Automata

• States. The maximal Consistent subsets of

.

Alphabet Let

be the atomic propositions in

. The alphabet is the subsets of

.

Transitions Let

be the set of atomic propositions which are true at a state

. From

• nly

transitions are enabled.

• ❊

iff for every

either

• r

and

– p.11/12

From TL(U) to Automata

Notations.

the set of states that contain formula

.

– p.12/12

From TL(U) to Automata

Notations.

the set of states that contain formula

.

The Initial States:

– p.12/12

From TL(U) to Automata

Notations.

the set of states that contain formula

.

The Initial States:

The Street Acceptance conditions: For every

we have the pair

(i.e. if

holds infinitely

• ften then

holds infinitely often,)

– p.12/12

From TL(U) to Automata

Notations.

the set of states that contain formula

.

The Initial States:

The Street Acceptance conditions: For every

we have the pair

(i.e. if

holds infinitely

• ften then

holds infinitely often,)

Theorem Let

be a run of the automaton and let

be the corresponding

• string. Then

is an accepting run if and only if

and

.

– p.12/12

From TL(U) to Automata

Notations.

the set of states that contain formula

.

The Initial States:

The Street Acceptance conditions: For every

we have the pair

(i.e. if

holds infinitely

• ften then

holds infinitely often,)

Theorem Let

be a run of the automaton and let

be the corresponding

• string. Then

is an accepting run if and only if

and

.

• Proof. The if direction is easy. The only if direction: by

structural induction on formula for all

simultaneously show: if

is an accepting run then

iff

.

– p.12/12