Turing Networks: Complexity Theory for Network? Message-Passing - PowerPoint PPT Presentation

Turing Networks Jack Romo Introduction So What’s a Turing Turing Networks: Complexity Theory for Network? Message-Passing Parallelism Lower Bounds Upper Bounds Simulation Conclusions Jack Romo University of York jr1161@york.ac.uk April 2019

Turing Networks Overview Jack Romo Introduction So What’s a Turing 1 Introduction Network? Lower Bounds 2 So What’s a Turing Network? Upper Bounds Simulation Conclusions 3 Lower Bounds 4 Upper Bounds 5 Simulation 6 Conclusions

Turing Networks Complexity Theory Jack Romo Introduction So What’s a Turing • The study of resource requirements to solve problems Network? • eg. Time, memory, ... Lower Bounds • Alternatively, the study of complexity classes , ie. sets of Upper Bounds problems with the same computational overhead Simulation • How are they related? Conclusions • eg. P ⊆ NP • Usually studied in context of Turing Machines • Large amount of theory developed here over the last century • Would be ideal to reuse it in other problems • Parallel complexity theory?

Turing Networks Parallel Complexity Theory Jack Romo Introduction So What’s a Turing Network? • Complexity of parallel computations are extremely Lower Bounds nontrivial Upper Bounds • Turing Machines to parallelism: Parallel Turing Machines Simulation • Same, but can create new identical read-write heads at any Conclusions step

Turing Networks Parallel Complexity Theory Jack Romo Introduction So What’s a Turing Network? • Complexity of parallel computations are extremely Lower Bounds nontrivial Upper Bounds • Turing Machines to parallelism: Parallel Turing Machines Simulation • Same, but can create new identical read-write heads at any Conclusions step • However, this models shared memory only, not message passing • Other models can emulate message passing parallelism, but do not connect to classical complexity theory

Turing Networks Parallel Complexity Theory Jack Romo Introduction So What’s a Turing Network? • Complexity of parallel computations are extremely Lower Bounds nontrivial Upper Bounds • Turing Machines to parallelism: Parallel Turing Machines Simulation • Same, but can create new identical read-write heads at any Conclusions step • However, this models shared memory only, not message passing • Other models can emulate message passing parallelism, but do not connect to classical complexity theory • This is a problem!

Turing Networks Modelling Message-Passing Jack Romo Parallelism Introduction So What’s a Turing Network? Lower Bounds Upper Bounds We would like a model that... Simulation 1 Intuitively emulates a network of communicating Conclusions processors. 2 Allows for substantial complexity analysis. 3 Relates classical complexity classes to its own parallel ones. 4 Can simulate other models of parallelism with good complexity.

Turing Networks Network topologies Jack Romo Introduction So What’s a Turing Network? • Take a simple undirected graph G = � V , E � of constant Lower Bounds degree, 1 ∈ V ⊆ N , E ⊆ V × V symmetric relation Upper Bounds • Choose a vertex, index its neighbors from 1 to n ; do this Simulation for every vertex Conclusions • Call this the graph’s orientation , φ : V × N � → V • Call a pair G = � G , φ � a network topology

Turing Networks Network topologies Jack Romo Introduction So What’s a Turing Network? • Take a simple undirected graph G = � V , E � of constant Lower Bounds degree, 1 ∈ V ⊆ N , E ⊆ V × V symmetric relation Upper Bounds • Choose a vertex, index its neighbors from 1 to n ; do this Simulation for every vertex Conclusions • Call this the graph’s orientation , φ : V × N � → V • Call a pair G = � G , φ � a network topology • eg. Linked List • V = N • E = { ( v , v + 1 ) | v ∈ V } ’s symmetric closure • φ ( 1 , 1 ) = 1 • φ ( n + 1 , 1 ) = n , φ ( n + 1 , 2 ) = n + 2 , n ∈ N

Turing Networks Communicative Turing Machines Jack Romo Introduction So What’s a Turing Network? Lower Bounds • Each vertex is a Turing Machine Upper Bounds • However, need some capacity to communicate Simulation Conclusions • Add ’special transitions’ to send/receive a character • Index neighbors to commune with by orientation • Two start states, a ’master’ state for vertex 1 and ’slave’ state for the rest • Slaves must start by waiting for a message

Turing Networks Communicative Turing Machines Jack Romo Introduction So What’s a Definition (Communicative Turing Machine) Turing Network? A Communicative Turing Machine , or CTM, is a 10-tuple Lower Bounds T = � Q , Σ , Γ , q m , q s , h a , h r , δ t , δ s , δ r � Upper Bounds Simulation where Q , Σ are nonempty and finite, Σ ∪ { Λ } ⊂ Γ , Conclusions q m , q s , h a , h r ∈ Q , and δ t : Q × Γ � → Q × Γ × { L , R , S } → N × Γ × Q 2 δ s : Q × Γ � δ r : Q � → N × Q are partial functions, where δ t ( q s , x ) , δ s ( q s , x ) are undefined ∀ x ∈ Γ .

Turing Networks Turing Networks Jack Romo Introduction So What’s a Turing Network? Lower Bounds Upper Bounds Simulation Definition (Turing Network) Conclusions A Turing Network is a pair T = � G , T � , such that G is a network topology and T is a Communicative Turing machine.

Turing Networks Defining Computations Jack Romo Introduction So What’s a Turing Network? Lower Bounds Upper Bounds • Define a configuration of one CTM and a transition Simulation Conclusions • Extend to a configuration/transition of a TN • Derivation sequences • Computations as terminating derivation sequences

Turing Networks Configurations Jack Romo Introduction So What’s a Turing Definition (CTM Configuration) Network? A CTM configuration of a CTM T is a 4-tuple of the form Lower Bounds C ∈ Γ ∗ × Γ × Q × Γ ∗ . We name the set of all CTM Upper Bounds Simulation configurations for the CTM T C ( T ) . Conclusions We say, for CTM configurations C n = � r n , s n , q n , t n � , n ∈ N , a network topology G = � V , E � and v 1 , v 2 ∈ V , C 1 ⊢ C 2 ⇔ C 1 transitions to C 2 as TM configs � C 1 , C 2 � ⊢ v 2 v 1 � C 3 , C 4 � ⇔ v 1 in config C 1 sends a char to v 2 in config C 2 , transitioning to C 3 and C 4 C 1 � v 1 C 2 ⇔ v 1 sends to a nonexistent neighbor

Turing Networks Configurations Jack Romo Introduction Definition (TN Configuration) So What’s a Turing Network? A TN configuration of a TN T is a function of the form Lower Bounds Ω : V → C ( T ) . Upper Bounds We say, for CTM configurations Ω n , n ∈ N of a Turing network Simulation T and v 1 , v 2 ∈ V , Conclusions Ω 1 ⊢ v 1 Ω 2 ⇔ Ω 1 | V \{ v 1 } = Ω 2 | V \{ v 1 } ∧ (Ω 1 ( v 1 ) ⊢ Ω 2 ( v 1 ) ∨ Ω 1 ( v 1 ) � v 1 Ω 2 ( v 1 )) Ω 1 ⊢ v 2 v 1 Ω 2 ⇔ Ω 1 | V \{ v 1 } = Ω 2 | V \{ v 1 } ∧ � Ω 1 ( v 1 ) , Ω 1 ( v 2 ) � ⊢ v 2 v 1 � Ω 2 ( v 1 ) , Ω 2 ( v 2 ) � Ω 1 ⊢ Ω 2 ⇔ ( ∃ v ∈ V • Ω 1 ⊢ v Ω 2 ) ∨ ( ∃ v 1 , v 2 ∈ V • Ω 1 ⊢ v 2 v 1 Ω 2 )

Turing Networks Initial and Final States Jack Romo Introduction So What’s a Definition (Initial State) Turing Network? An initial state of a TN T is a configuration Ω S for some Lower Bounds S ∈ Σ ∗ where Upper Bounds Simulation Ω S ( 1 ) = � λ, Λ , q m , S � Conclusions Ω S ( n + 1 ) = � λ, Λ , q s , λ � ∀ n ∈ N Definition (Final State) A final state of T is a configuration Ω h where Ω h ( 1 ) = � A , b , q , C � where q ∈ { h a , h r } . The output string is AbC with all characters not in Σ deleted. We say Ω h is accepting if q = h a and rejecting otherwise.

Turing Networks Computations Jack Romo Introduction So What’s a Turing Network? Definition (Derivation Sequence) Lower Bounds A derivation sequence Ψ = { Ω n } n ∈ X is a sequence of indexed Upper Bounds configurations of T where X ⊆ N and for any n , m ∈ X , Simulation Ω n ⊢ Ω m if m is the least element of X greater than n. Conclusions Say that, for two derivation sequences of T , Ψ 1 , Ψ 2 , Ψ 1 < Ψ 2 if the former is a prefix of the latter as a sequence. Definition (Computation) We say a derivation sequence Ψ is a computation if it starts with an initial state and ends with a final state.

Turing Networks Acceptance, Rejection and Jack Romo Computing Functions Introduction So What’s a Turing Network? Definition (Acceptance and Rejection) Lower Bounds We say T accepts a string S ∈ Σ ∗ if every derivation sequence Upper Bounds Simulation starting with Ω S is less than an accepting computation and all Conclusions rejecting computations are greater than an accepting computation. We say it rejects if there exists a rejecting computation not greater than some accepting computation. Definition (Computing Functions) Say T computes a function f : Σ ∗ � → Σ ∗ if, for every input string s ∈ dom ( f ) , every computation of T with input string s has a final state with output string f ( s ) .

Turing Networks Our Time Function Jack Romo Introduction So What’s a Turing Network? Lower Bounds • Insufficient to analyze length of computations; things Upper Bounds happen in parallel! Simulation • Define parallel sequences Conclusions • Analyze number of parallel sequences a computation is itself a sequence of • τ : Σ ∗ → N gives longest parallel time of any computation starting with an input string