Executability Hierarchy of RTMs with Infinite Alphabets Bas Luttik - - PowerPoint PPT Presentation

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Executability Hierarchy of RTMs with Infinite Alphabets

Bas Luttik Fei Yang

November 17, 2016

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Outline

Finite Version of the Theory of Executability Infinitary Reactive Turing Machines RTM with Atoms Executability Hierarchy

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Outline

Finite Version of the Theory of Executability Infinitary Reactive Turing Machines RTM with Atoms Executability Hierarchy

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Reactive Turing Machines

A reactive Turing machine M is defined by (S, Aτ, D, ↑, Move, − →), where:

• 1. A is a finite set of actions, τ is an internal action, and Aτ = A∪{τ};
• 2. S is a finite set of control states;
• 3. D is a finite set of data symbols, is a special blank symbol, and

D = D ∪ ;

• 4. ↑∈ S is an initial state;
• 5. Move = {L, R};
• 6. −

→ ⊆ S × D × Aτ × D × Move × S is a finite transition relation.

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Executability

◮ Labelled transition system semantics of RTMs

We associate with every configuration (control state, tape instance) a state, and associate with every execution step a labelled transition.

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Executability

◮ Labelled transition system semantics of RTMs

We associate with every configuration (control state, tape instance) a state, and associate with every execution step a labelled transition.

◮ Executability

A transition system is called executable if it is behaviourally equivalent to the transition system of an RTM.

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Evaluating Expressiveness

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Evaluating Expressiveness

• 1. Can we specify every executable LTS by the LTS associated with P?

(reactive Turing powerfulness)

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Evaluating Expressiveness

• 1. Can we specify every executable LTS by the LTS associated with P?

(reactive Turing powerfulness)

• 2. Is every LTS associated with the process specifiable by P

executable? (executability)

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Some Theorems

Theorem

• 1. For every finite set Aτ and every boundedly branching computable

Aτ-labelled transition system T, there exists an RTM M such that T ↔

b T (M).

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Some Theorems

Theorem

• 1. For every finite set Aτ and every boundedly branching computable

Aτ-labelled transition system T, there exists an RTM M such that T ↔

b T (M).

• 2. For every finite set Aτ and every effective Aτ-labelled transition

system T there exists an RTM M such that T ↔b T (M).

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Limitation of Finite Sets

Many process calculi use infinite sets of action labels.

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Limitation of Finite Sets

Many process calculi use infinite sets of action labels.

◮ π-calculus ◮ ψ-calculus ◮ Value passing calculus ◮ mCRL2

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Limitation of Finite Sets

Many process calculi use infinite sets of action labels.

◮ π-calculus ◮ ψ-calculus ◮ Value passing calculus ◮ mCRL2

We need a more general notion of executability!

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Outline

Finite Version of the Theory of Executability Infinitary Reactive Turing Machines RTM with Atoms Executability Hierarchy

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Allowing Infinite Sets

An RTM M is defined by (S, Aτ, D, ↑, Move, − →).

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Allowing Infinite Sets

An RTM M is defined by (S, Aτ, D, ↑, Move, − →). An infinite set of action labels Aτ is necessary.

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Allowing Infinite Sets

An RTM M is defined by (S, Aτ, D, ↑, Move, − →). An infinite set of action labels Aτ is necessary. The following lemma shows that we also need infinite sets of control states S and/or data symbols D.

Lemma

There does not exist an RTM with infinitely many actions but finitely many states and data symbols that simulates the π-term P = x(y).¯ y.0 modulo branching bisimilarity.

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Infinitary RTMs

An infinitary reactive Turing machine (RTM∞) (S, Aτ, D, ↑, Move, − →), where:

• 1. A is a countable set of actions, τ is an internal action, and

Aτ = A ∪ {τ};

• 2. S is a countable set of control states;
• 3. D is a countable set of data symbols, is a special blank symbol,

and D = D ∪ ;

• 4. ↑∈ S is an initial state;
• 5. Move = {L, R};
• 6. −

→ ⊆ S × D × Aτ × D × Move × S is a countable transition relation.

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Infinitary RTMs

An infinitary reactive Turing machine (RTM∞) (S, Aτ, D, ↑, Move, − →), where:

• 1. A is a countable set of actions, τ is an internal action, and

Aτ = A ∪ {τ};

• 2. S is a countable set of control states;
• 3. D is a countable set of data symbols, is a special blank symbol,

and D = D ∪ ;

• 4. ↑∈ S is an initial state;
• 5. Move = {L, R};
• 6. −

→ ⊆ S × D × Aτ × D × Move × S is a countable transition relation. A transition system is executable by an RTM∞ if it is behaviourally equivalent to a transition system associated with some RTM∞.

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Theorem for RTM∞

Theorem

For every infinite set Aτ and every countable Aτ-labelled transition system T, there exists an RTM∞ M such that T ↔

b T (M).

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Theorem for RTM∞

Theorem

For every infinite set Aτ and every countable Aτ-labelled transition system T, there exists an RTM∞ M such that T ↔

b T (M).

Proof.

T = (ST, − →T, ↑T) φ : ST → N

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Theorem for RTM∞

Theorem

For every infinite set Aτ and every countable Aτ-labelled transition system T, there exists an RTM∞ M such that T ↔

b T (M).

Proof.

T = (ST, − →T, ↑T) φ : ST → N

◮ S = {s, t, ↑}. ◮ −

→ consists of the following transitions:

• 1. ↑

τ[/φ(↑T )]R

− → s

• 2. s

τ[/]L

− → t

• 3. t

a[φ(s1)/φ(s2)]R

− → s if there is a transition s1

a

− → s2 for states s1, s2 ∈ ST.

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Some Corollaries

We restrict the transition relation − → to be effective or computable and get the following corollaries.

Corollary

• 1. For every infinite set Aτ and every effective Aτ-labelled transition

system T, there exists an RTM∞ M with an effective transition relation such that T ↔

b T (M).

• 2. For every infinite set Aτ and every computable Aτ-labelled

transition system T, there exists an RTM∞ M with a computable transition relation such that T ↔

b T (M).

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Outline

Finite Version of the Theory of Executability Infinitary Reactive Turing Machines RTM with Atoms Executability Hierarchy

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Sets with Atoms

A: a countably infinite set; we call its elements atoms. An atom automorphism: an bijection (permutation) on A. A set with atoms: a set that can contain atoms or other sets with atoms.

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Sets with Atoms

A: a countably infinite set; we call its elements atoms. An atom automorphism: an bijection (permutation) on A. A set with atoms: a set that can contain atoms or other sets with atoms. Examples of sets with atoms may include:

◮ any set without atoms, ◮ an atom a, or an ordered pair of atoms (a,b), encoded by

{{a}, {a, b}}

◮ A, An, A∗

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Legality

X: a set with atoms π: an atom automorphism π(X): the set obtained by application of π to every atom contained in the elements of X, recursively

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Legality

X: a set with atoms π: an atom automorphism π(X): the set obtained by application of π to every atom contained in the elements of X, recursively S ⊆ A S-automorphism: an atom automorphism π is the identity on S.

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Legality

X: a set with atoms π: an atom automorphism π(X): the set obtained by application of π to every atom contained in the elements of X, recursively S ⊆ A S-automorphism: an atom automorphism π is the identity on S. S supports a set with atoms X if X = π(X) for every S-automorphism π. A set with atoms is called legal if it has a finite support, each of its elements has a finite support, and so on recursively.

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Legal sets

Let A be the set of natural numbers, consider the following sets:

• 1. {3}

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Legal sets

Let A be the set of natural numbers, consider the following sets:

• 1. {3}
• 2. {3, {3, 5}}

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Legal sets

Let A be the set of natural numbers, consider the following sets:

• 1. {3}
• 2. {3, {3, 5}}
• 3. {x | x < 3, x is a natural number}

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Legal sets

Let A be the set of natural numbers, consider the following sets:

• 1. {3}
• 2. {3, {3, 5}}
• 3. {x | x < 3, x is a natural number}
• 4. {x | x > 3, x is a natural number}

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Legal sets

Let A be the set of natural numbers, consider the following sets:

• 1. {3}
• 2. {3, {3, 5}}
• 3. {x | x < 3, x is a natural number}
• 4. {x | x > 3, x is a natural number}
• 5. {x | x = 2k, k is a natural number}

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Orbit-finiteness

X: a set with atoms x ∈ X x-orbit: {y | y ∈ X, y = π(x) for some atom automorphism π}

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Orbit-finiteness

X: a set with atoms x ∈ X x-orbit: {y | y ∈ X, y = π(x) for some atom automorphism π} A set with atoms X is partitioned into disjoint orbits. The elements x and y are in the same orbit if π(x) = y for some atom automorphism π. A set with atoms that is partitioned into finitely many orbits is called an

• rbit-finite set.

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Orbit-finiteness

X: a set with atoms x ∈ X x-orbit: {y | y ∈ X, y = π(x) for some atom automorphism π} A set with atoms X is partitioned into disjoint orbits. The elements x and y are in the same orbit if π(x) = y for some atom automorphism π. A set with atoms that is partitioned into finitely many orbits is called an

• rbit-finite set.

Consider the following sets:

• 1. {1, 2, 3}

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Orbit-finiteness

X: a set with atoms x ∈ X x-orbit: {y | y ∈ X, y = π(x) for some atom automorphism π} A set with atoms X is partitioned into disjoint orbits. The elements x and y are in the same orbit if π(x) = y for some atom automorphism π. A set with atoms that is partitioned into finitely many orbits is called an

• rbit-finite set.

Consider the following sets:

• 1. {1, 2, 3}
• 2. A

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Orbit-finiteness

X: a set with atoms x ∈ X x-orbit: {y | y ∈ X, y = π(x) for some atom automorphism π} A set with atoms X is partitioned into disjoint orbits. The elements x and y are in the same orbit if π(x) = y for some atom automorphism π. A set with atoms that is partitioned into finitely many orbits is called an

• rbit-finite set.

Consider the following sets:

• 1. {1, 2, 3}
• 2. A
• 3. A2

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Orbit-finiteness

X: a set with atoms x ∈ X x-orbit: {y | y ∈ X, y = π(x) for some atom automorphism π} A set with atoms X is partitioned into disjoint orbits. The elements x and y are in the same orbit if π(x) = y for some atom automorphism π. A set with atoms that is partitioned into finitely many orbits is called an

• rbit-finite set.

Consider the following sets:

• 1. {1, 2, 3}
• 2. A
• 3. A2
• 4. A∗

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RTM with Atoms

A reactive Turing machine with atoms (RTMA) M is defined by (S, Aτ, D, ↑, Move, − →), where:

• 1. A is a legal and orbit-finite set of actions, τ is an internal action,

and Aτ = A ∪ {τ};

• 2. S is a legal and orbit-finite set of control states;
• 3. D is a legal and orbit-finite set of data symbols, is a special blank

symbol, and D = D ∪ ;

• 4. ↑∈ S is an initial state;
• 5. Move = {L, R};
• 6. −

→ ⊆ S × D × Aτ × D × Move × S is a legal and orbit-finite transition relation.

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RTM with Atoms

A reactive Turing machine with atoms (RTMA) M is defined by (S, Aτ, D, ↑, Move, − →), where:

• 1. A is a legal and orbit-finite set of actions, τ is an internal action,

and Aτ = A ∪ {τ};

• 2. S is a legal and orbit-finite set of control states;
• 3. D is a legal and orbit-finite set of data symbols, is a special blank

symbol, and D = D ∪ ;

• 4. ↑∈ S is an initial state;
• 5. Move = {L, R};
• 6. −

→ ⊆ S × D × Aτ × D × Move × S is a legal and orbit-finite transition relation. A transition system is executable by an RTMA if it is behaviourally equivalent to a transition system associated with some RTMA.

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Legal LTS with Atoms

Let K ⊂ A. A labelled transition system T = (ST, − →T, ↑T) is K-supported, if it satisfies the following conditions:

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Legal LTS with Atoms

Let K ⊂ A. A labelled transition system T = (ST, − →T, ↑T) is K-supported, if it satisfies the following conditions:

• 1. ST, −

→T are sets with atoms with support K; and

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Legal LTS with Atoms

Let K ⊂ A. A labelled transition system T = (ST, − →T, ↑T) is K-supported, if it satisfies the following conditions:

• 1. ST, −

→T are sets with atoms with support K; and

• 2. for every (s, a, t) ∈ −

→T and for every K-automorphism πK, πK(s, a, t) ∈ − →T, where πK(s, a, t) = (πK(s), πK(a), πK(t)).

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Legal LTS with Atoms

Let K ⊂ A. A labelled transition system T = (ST, − →T, ↑T) is K-supported, if it satisfies the following conditions:

• 1. ST, −

→T are sets with atoms with support K; and

• 2. for every (s, a, t) ∈ −

→T and for every K-automorphism πK, πK(s, a, t) ∈ − →T, where πK(s, a, t) = (πK(s), πK(a), πK(t)). We say that T is a legal transition system with atoms if there exists a finite set K ⊂ A such that T is K-supported.

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Theorem for RTMA

Theorem

A transition system T is executable by an RTMA modulo ↔b iff there exists a legal and orbit-finite set Aτ and an effective legal Aτ-labelled transition system with atoms T ′, and T ↔b T ′.

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Outline

Finite Version of the Theory of Executability Infinitary Reactive Turing Machines RTM with Atoms Executability Hierarchy

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A conclusion of Theorems

• 1. The class of executable transition systems by RTMs modulo ↔

b is

the boundedly branching computable transition system with a finite set of labels.

• 2. The class of executable transition systems by RTMs modulo ↔b is

the effective transition system with a finite set of labels.

• 3. The class of executable transition systems by RTMAs modulo ↔b is

the effective legal transition system with atoms.

• 4. The class of executable transition systems by RTM∞s with a

computable transition relation modulo ↔

b is the computable

transition system.

• 5. The class of executable transition systems by RTM∞s with an

effective transition relation modulo ↔

b is the effective transition

system.

• 6. The class of executable transition systems by RTM∞s modulo ↔

b is

the countable transition system.

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A Hierarchy

RTM ↔

b

RTM ↔b RTMA ↔b Computable RTM∞ ↔

b

Effective RTM∞ ↔

b

RTM∞ ↔

b

Figure: A Hierarchy of Executability

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Thank You!