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{{redirect|Channel system|Earth's sea floor|Abyssal channel}}
{{short description|Finite-state machine with fifo buffers for memory}}
In [[computer science]], a '''channel system''' is a [[finite state machine]] similar to [[communicating finite-state machine]] in which there is a single system communicating with itself instead of many systems communicating with each other. A '''channel system''' is similar to a [[pushdown automaton]] where a queue is used instead of a stack. Those queues are called '''channels'''. Intuitively, each channel represents a sequence a message to be sent, and to be read in the order in which they are sent.▼
▲In [[computer science]], a '''channel system''' is a [[finite
== Definition ==
=== Channel system ===
Formally, a '''channel system''' (or '''perfect channel system''') <math>S</math> is defined as a tuple <math>\langle Q,C,\Sigma,\Delta \rangle</math> with:
* <math>Q=\{q_1,\dots,q_m\}</math> a finite set of '''control states''',
* <math>q_0\in Q</math> an '''initial state''',
* <math>A</math> a finite alphabet (for the sake of notation simplicity, let <math>\epsilon \in A</math>),
* <math>C=\{c_1,\dots,c_n\}</math> a finite set of '''channels''',
* <math>\Sigma=\{a_1,\dots,a_p\}</math> a finite alphabet of '''messages''',
* <math>\Delta\subseteq Q\times C\times\{?,!\}\times\Sigma^*\times
Depending
=== Configuration ===
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== Channel system and Turing machine ==
Most problem related to perfect channel system are undecidable{{r|decidable|p=92}}.<ref name="easier">{{cite journal |last1=Cécé |first1=Gérard|last2=Finkel|first2=Alain |title=Unreliable Channels Are Easier to Verify Than Perfect Channels|journal =Information and Computation |date=10 January 1996 |volume=124 |issue=1 |pages=20–31 |doi=10.1006/inco.1996.0003|doi-access=free }}</ref>{{rp|22}} This is due to the fact that such a machine may simulates the run of a [[Turing machine]]. This simulation is now sketched.
Given a [[Turing machine]] <math>M</math>, there exists a perfect channel system <math>S</math> such that any run of <math>M</math> of length <math>n</math> can be simulated by a run of <math>S</math> of length <math>O(n^2)</math>. Intuitively, this simulation consists simply in having the entire tape of the simulated Turing machine in a channel. The content channel is then entirely read and immediately rewritten in the channel, with one exception, the part of the content representing the head of the Turing machine is changed, to simulate a step of the Turing machine computation.
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=== Counter machine ===
When the alphabet of a channel system contains a single message, then each channel is essentially a counter. It follows that those systems are essentially [[Minsky machine]]s. We call such systems '''counter machines'''. This same definition applies for all variants of channel system.<ref name="Counter">{{cite journal|first1=Richard|last1=Mayr|doi=10.1016/S0304-3975(02)00646-1|title=Undecidable problems in unreliable computations|volume=297|issue=1–3|date = 17 March 2008 |pages=337–354|journal=Theoretical Computer Science|doi-access=free}}</ref>{{rp|337}}
=== Completely specified protocol ===
A '''completely specified protocol''' (CSP) is defined exactly as a channel system. However, the notion of step and of run are defined differently.
A CSP admits two kinds of steps. Perfect steps, as defined above, and a '''message loss transition''' step. We denote a message loss transition step by <math>(q,w_1,\dots,a\cdot w_i,\dots,w_m)\rightsquigarrow(q,w_1,\dots,w_i,\dots,w_m)</math>.
=== Looseness ===
A '''lossy channel system''' or '''machine capable of
A lossy channel system admits two kinds of steps. Perfect steps, as defined above, and lossy step. We denote a lossy step, <math>(q,w_1,\dots,w_i\cdot a\cdot w'_i\dots,,w_m)\rightsquigarrow(q,w_1,\dots,w_i\cdot w'_i,w'_m)</math>.
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=== Model checking problem ===
The '''model checking problem''' consists in deciding whether given a system <math>S</math> and a [[CTL*]]*-formula or a [[Linear temporal logic|LTL]]-formula <math>\phi</math> or a whether the language defined by <math>S</math> satisfies <math>\phi</math>. This problem is undecidable over lossy channel system.{{r|easier|p=23}}<ref name="Undecidable over unreliable channels">{{cite journal |last1=Abdulla |first1=Parosh Aziz|last2=Jonsson|first2=Bengt |title=Undecidable Verification Problems for Programs with Unreliable Channels|journal = Information and Computation |date= 10 October 1996 |volume=130 |issue=1 |pages=71–90 |doi=10.1006/inco.1996.0083|doi-access=free }}</ref>
=== Recurrent state problem ===
The '''recurrent state problem''' consists in deciding, given a channel system <math>S</math> and an initial configuration <math>\gamma</math> and a state <math>s</math> whether there exists a run of <math>S</math>, starting at <math>\gamma</math>, going infinitely often through state <math>s</math>. This problem is undecidable over lossy channel system, even with a single channel.{{r|easier|p=23}}{{r|Undecidable over unreliable channels|p=80}}
=== Equivalent finite
Given a system <math>S</math>, there is no algorithm which computes a [[finite
=== Boundedness problem ===
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=== Structural termination ===
The '''structural
==Communicating Hierarchical State Machine==
Hierarchical state machines are finite
However, it was proved that the coexistence of hierarchy and concurrency intrinsically costs language inclusion, language equivalence, and all of universality.<ref>Alur, Rajeev; Kannan, Sampath; Yannakakis, Mihalis. "Communicating hierarchical state machines," Automata, Languages and Programming. Prague: ICALP, 1999</ref>
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