Executability Hierarchy of RTMs with Infinite Alphabets Bas Luttik Fei Yang November 17, 2016 Where innovation starts
Outline 2/25 Finite Version of the Theory of Executability Infinitary Reactive Turing Machines RTM with Atoms Executability Hierarchy
Outline 3/25 Finite Version of the Theory of Executability Infinitary Reactive Turing Machines RTM with Atoms Executability Hierarchy
Reactive Turing Machines 4/25 A reactive Turing machine M is defined by ( S , A τ , D � , ↑ , Move , − → ) , where: 1. A is a finite set of actions, τ is an internal action, and A τ = 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 � × A τ × D � × Move × S is a finite transition relation.
Executability 5/25 ◮ 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 5/25 ◮ 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.
Evaluating Expressiveness 6/25
Evaluating Expressiveness 6/25 1. Can we specify every executable LTS by the LTS associated with P ? (reactive Turing powerfulness)
Evaluating Expressiveness 6/25 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)
Some Theorems 7/25 Theorem 1. For every finite set A τ and every boundedly branching computable A τ -labelled transition system T, there exists an RTM M such that T ↔ � b T ( M ) .
Some Theorems 7/25 Theorem 1. For every finite set A τ and every boundedly branching computable A τ -labelled transition system T, there exists an RTM M such that T ↔ � b T ( M ) . 2. For every finite set A τ and every effective A τ -labelled transition system T there exists an RTM M such that T ↔ b T ( M ) .
Limitation of Finite Sets 8/25 Many process calculi use infinite sets of action labels.
Limitation of Finite Sets 8/25 Many process calculi use infinite sets of action labels. ◮ π -calculus ◮ ψ -calculus ◮ Value passing calculus ◮ mCRL2
Limitation of Finite Sets 8/25 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!
Outline 9/25 Finite Version of the Theory of Executability Infinitary Reactive Turing Machines RTM with Atoms Executability Hierarchy
Allowing Infinite Sets 10/25 An RTM M is defined by ( S , A τ , D � , ↑ , Move , − → ) .
Allowing Infinite Sets 10/25 An RTM M is defined by ( S , A τ , D � , ↑ , Move , − → ) . An infinite set of action labels A τ is necessary.
Allowing Infinite Sets 10/25 An RTM M is defined by ( S , A τ , D � , ↑ , Move , − → ) . An infinite set of action labels A τ 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.
Infinitary RTMs 11/25 An infinitary reactive Turing machine (RTM ∞ ) ( S , A τ , D � , ↑ , Move , − → ) , where: 1. A is a countable set of actions, τ is an internal action, and A τ = 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 � × A τ × D � × Move × S is a countable transition relation.
Infinitary RTMs 11/25 An infinitary reactive Turing machine (RTM ∞ ) ( S , A τ , D � , ↑ , Move , − → ) , where: 1. A is a countable set of actions, τ is an internal action, and A τ = 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 � × A τ × 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 ∞ .
Theorem for RTM ∞ 12/25 Theorem For every infinite set A τ and every countable A τ -labelled transition system T, there exists an RTM ∞ M such that T ↔ � b T ( M ) .
Theorem for RTM ∞ 12/25 Theorem For every infinite set A τ and every countable A τ -labelled transition system T, there exists an RTM ∞ M such that T ↔ � b T ( M ) . Proof. T = ( S T , − → T , ↑ T ) φ : S T → N
Theorem for RTM ∞ 12/25 Theorem For every infinite set A τ and every countable A τ -labelled transition system T, there exists an RTM ∞ M such that T ↔ � b T ( M ) . Proof. T = ( S T , − → T , ↑ T ) φ : S T → N ◮ S = { s , t , ↑} . ◮ − → consists of the following transitions: τ [ � /φ( ↑ T ) ] R 1. ↑ s − → τ [ � / � ] L 2. s − → t a [ φ( s 1 )/φ( s 2 ) ] R 3. t − → s a if there is a transition s 1 → s 2 for states s 1 , s 2 ∈ S T . −
Some Corollaries 13/25 We restrict the transition relation − → to be effective or computable and get the following corollaries. Corollary 1. For every infinite set A τ and every effective A τ -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 A τ and every computable A τ -labelled transition system T, there exists an RTM ∞ M with a computable transition relation such that T ↔ � b T ( M ) .
Outline 14/25 Finite Version of the Theory of Executability Infinitary Reactive Turing Machines RTM with Atoms Executability Hierarchy
Sets with Atoms 15/25 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.
Sets with Atoms 15/25 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 , A n , A ∗
Legality 16/25 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
Legality 16/25 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 .
Legality 16/25 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.
Legal sets 17/25 Let A be the set of natural numbers, consider the following sets: 1. { 3 }
Legal sets 17/25 Let A be the set of natural numbers, consider the following sets: 1. { 3 } 2. { 3 , { 3 , 5 }}
Legal sets 17/25 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 }
Legal sets 17/25 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 }
Legal sets 17/25 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 = 2 k , k is a natural number }
Orbit-finiteness 18/25 X : a set with atoms x ∈ X x -orbit: { y | y ∈ X , y = π( x ) for some atom automorphism π }
Orbit-finiteness 18/25 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 orbit-finite set.
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