Channel system (computer science)

Formally, a channel system (or perfect channel system) ${\displaystyle S}$ is defined as a tuple ${\displaystyle \langle Q,C,\Sigma ,\Delta \rangle }$ with:

• ${\displaystyle Q=\{q_{1},\dots ,q_{m}\}}$ a finite set of control states,
• ${\displaystyle q_{0}\in Q}$ an initial state,
• ${\displaystyle A}$ a finite alphabet (for the sake of notation simplicity, let ${\displaystyle \epsilon \in A}$),
• ${\displaystyle C=\{c_{1},\dots ,c_{n}\}}$ a finite set of channels,
• ${\displaystyle \Sigma =\{a_{1},\dots ,a_{p}\}}$ a finite alphabet of messages,
• ${\displaystyle \Delta \subseteq Q\times C\times \{?,!\}\times \Sigma ^{*}\times A\times Q}$ a finite set of transition rules with ${\displaystyle \Sigma ^{*}}$ being the set of finite (potentially empty) words over the alphabet ${\displaystyle \Sigma }$.[1]

Depending of the author, a channel system may have no initial state and may have an empty alphabet.[2]

A configuration or global state of the channel system is a ${\displaystyle n+1}$ tuple belonging to ${\displaystyle Q\times \prod _{i=1}^{n}(\Sigma ^{*})}$. Intuitively, a configuration ${\displaystyle \gamma =(q,w_{1},\dots ,w_{m})}$ represents that a run is in state ${\displaystyle q}$ and that its ${\displaystyle i}$-th channel contains the word ${\displaystyle w_{i}}$.

The initial configuration is ${\displaystyle (q_{0},\epsilon ,\dots ,\epsilon )}$, with ${\displaystyle \epsilon }$ the empty word.

Intuitively, a transition ${\displaystyle (q,c_{i},!,u,a,q')}$ means that the system may goes to control state ${\displaystyle q}$ to ${\displaystyle q'}$ by writing an ${\displaystyle u}$ to the end of the channel ${\displaystyle c_{i}}$. Similarly ${\displaystyle (q,c_{i},?,u,a,q')}$ means that the system may goes to control state ${\displaystyle q}$ to ${\displaystyle q'}$ by removing a ${\displaystyle u}$ starting the word ${\displaystyle c_{i}}$.

Formally, given a configuration ${\displaystyle (q,w_{1},\dots ,w_{m})}$, and a transition ${\displaystyle (q,c_{i},!,u,q')}$, there is a perfect step ${\displaystyle (q,w_{1},\dots ,w_{i-1},w_{i},w_{i+1},\dots ,w_{m}){\xrightarrow {a}}_{\mathtt {perf}}(q',w_{1},\dots ,w_{i-1},w_{i}u,w_{i+1},\dots ,w_{m})}$, where the step adds a letter ${\displaystyle u}$ to the end of the ${\displaystyle i}$-th word. Similarly, given a transition ${\displaystyle (q,c_{i},?,u,q')}$, there is a perfect step ${\displaystyle (q,w_{1},\dots ,w_{i-1},uw_{i},w_{i+1},w_{m}){\xrightarrow {a}}_{\mathtt {perf}}(q',w_{1},\dots ,w_{i-1},w_{i},w_{i+1},\dots ,w_{m})}$ where the first letter of the ${\displaystyle i}$-th word is ${\displaystyle u}$ and has been removed during the step.

A perfect run is a sequence of perfect step, of the form ${\displaystyle \gamma _{0}{\xrightarrow {a_{0}}}_{\mathtt {perf}}\gamma _{1}\dots }$. We let ${\displaystyle \gamma {\xrightarrow[{\mathtt {perf}}]{*}}\gamma '}$ denote that there is a perfect run starting at ${\displaystyle \gamma }$ and ending at ${\displaystyle \gamma '}$.

Given a perfect or a lossy channel system ${\displaystyle S}$, multiple languages may be defined.

A word over ${\displaystyle A^{*}}$ is accepted by ${\displaystyle S}$ if it is the concatenation of the labels of a run of ${\displaystyle S}$. The language defined by ${\displaystyle S}$ is the set of words accepted by ${\displaystyle S}$.

The set of reachable configuration of ${\displaystyle S}$, denoted ${\displaystyle R(S)}$ is defined as the set of configuration ${\displaystyle \gamma }$ reachable from the initial state. I.e. as the set of configurations ${\displaystyle \gamma '}$ such that ${\displaystyle (q_{0},\epsilon ,\dots ,\epsilon ){\xrightarrow {*}}\gamma '}$.

Given a channel ${\displaystyle c}$, the channel of ${\displaystyle c}$ is the set of tuples ${\displaystyle (w_{1},\dots w_{m})}$ such that ${\displaystyle (c,w_{1},\dots ,w_{m})\in R(S)}$.

A one-channel machine is a channel system using a single channel. The same definition also applies for all variants of channel system.

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 machines. We call such systems counter machines. This same definition applies for all variants of channel system.[4]^:337

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 ${\displaystyle (q,w_{1},\dots ,a\cdot w_{i},\dots ,w_{m})\rightsquigarrow (q,w_{1},\dots ,w_{i},\dots ,w_{m})}$.

A lossy channel system or machine capable of lossiness error is an extension of completely specified protocol in which letters may disappear anywhere.

A lossy channel system admits two kinds of steps. Perfect steps, as defined above, and lossy step. We denote a lossy step, ${\displaystyle (q,w_{1},\dots ,w_{i}\cdot a\cdot w'_{i}\dots ,,w_{m})\rightsquigarrow (q,w_{1},\dots ,w_{i}\cdot w'_{i},w'_{m})}$.

A run in which channel are emptied as soon as messages are sent into them is a valid run according to this definition. For this reason, some fairness conditions may be introduced to those systems.

Given a message a channel ${\displaystyle c}$, a run is said to be channel fair with respect to ${\displaystyle c}$ if, assuming there are infinitely many steps in which a letter is sent to ${\displaystyle c}$ then there are infinitely many steps in which a letter is read from ${\displaystyle c}$. [5]^:88

A computation is said to be channel fair if it is channel fair with respect to each channel ${\displaystyle c}$.

The impartiality condition is a restriction to the channel fairness condition in which both the channel and the letter are considered.

Given a message ${\displaystyle m}$ and a channel ${\displaystyle c}$, a run is said to be impartial with respect to ${\displaystyle c}$ and ${\displaystyle m}$ if, assuming there are infinitely many steps in which ${\displaystyle m}$ is sent to ${\displaystyle c}$ then there are infinitely many steps in which ${\displaystyle m}$ is read from ${\displaystyle c}$. [5]^:83

A computation is said to be impartial with respect to a channel ${\displaystyle c}$ if it is impartial with respect to ${\displaystyle c}$ and a messages ${\displaystyle m}$. It is said to be impartial if it is impartial with respect to every channels ${\displaystyle c}$.

The message fairness property is similar to impartiality, but the condition only have to hold if there is an infinite number of step at which ${\displaystyle m}$ may be read. Formally, a run is said to be message faire with respect to ${\displaystyle c}$ and ${\displaystyle m}$ if, assuming there are infinitely many steps in which ${\displaystyle m}$ is sent to ${\displaystyle c}$, and infinitely many step ${\displaystyle i}$ which occurs in a state ${\displaystyle q}$ such that there exists a transition ${\displaystyle (q,m,q')}$, then there are infinitely many steps in which ${\displaystyle m}$ is read from ${\displaystyle c}$. [5]^:88

The run is said to have bounded lossiness if the number of letter removed between two perfect steps is bounded.[4]^:339

A machine capable of insertion of error is an extension of channel system in which letters may appear anywhere.

A machine capable of insertion of error admits two kinds of steps. Perfect steps, as defined above, and insertion steps. We denote an insertion step by ${\displaystyle (q,w_{1},\dots ,w_{i}\cdot w'_{i}\dots ,,w_{m})\rightsquigarrow (q,w_{1},\dots ,w_{i}\cdot a\cdot w'_{i},w'_{m})}$.[3]^:25

A machine capable of duplication error is an extension of machine capable of insertion of error in which the inserted letter is a copy of the previous letter.

A machine capable of insertion of error admits two kinds of steps. Perfect steps, as defined above, and duplication steps. We denote an insertion step by ${\displaystyle (q,w_{1},\dots ,w_{i}\cdot a\cdot w'_{i}\dots ,,w_{m})\rightsquigarrow (q,w_{1},\dots ,w_{i}\cdot a\cdot a\cdot w'_{i},w'_{m})}$.[3]^:26

A non-duplicate machine capable of duplication error is a machine which ensures that in each channel, the letters alternate between a special new letter #, and a regular letter from the alphabet of message. If it is not the caes, it means a duplication occurred and the run rejects. This process allow to encode any channel system into a machine capable of duplication error, while forcing it not to have errors. Since channel systems can simulate machines, it follows that machines capable of duplication error can simulate Turing machine.

The termination problem consists in deciding, given a channel system ${\displaystyle S}$ and an initial configuration ${\displaystyle \gamma }$ whether all runs of ${\displaystyle S}$ starting at ${\displaystyle \gamma }$ are finite. This problem is undecidable over perfect channel systems, even when the system is a counter machine[4] or when it is a one-channel machine[3]^:26.

This problem is decidable but nonprimitive recursive over lossy channel system.[2]^:10 This problem is trivially decidable over machine capable of insertion of errors[3]^:26.

The reachability problem consists in deciding, given a channel system ${\displaystyle S}$ and two initial configurations ${\displaystyle \gamma }$ and ${\displaystyle \gamma '}$ whether there is a run of ${\displaystyle S}$ from ${\displaystyle \gamma }$ to ${\displaystyle \gamma '}$. This problem is undecidable over perfect channel systems and decidable but nonprimitive recursive over lossy channel system.[2]^:10 This problem is decidable over machine capable of insertion of errors .[3]^:26

The deadlock problem consists in deciding whether there is a reachable configuration without successor. This problem is decidable over lossy channel system[2]^:10 and trivially decidable over machine capable of insertion of errors[3]^:26. It is also decidable over counter machine.[6]

The model checking problem consists in deciding whether given a system ${\displaystyle S}$ and a CTL**-formula or a LTL-formula ${\displaystyle \phi }$ or a whether the language defined by ${\displaystyle S}$ satisfies ${\displaystyle \phi }$. This problem is undecidable over lossy channel system.[3]^:23[5]

The recurrent state problem consists in deciding, given a channel system ${\displaystyle S}$ and an initial configuration ${\displaystyle \gamma }$ and a state ${\displaystyle s}$ whether there exists a run of ${\displaystyle S}$, starting at ${\displaystyle \gamma }$, going infinitely often through state ${\displaystyle s}$. This problem is undecidable over lossy channel system, even with a single channel.[3]^:23[5]^:80

Given a system ${\displaystyle S}$, there is no algorithm which computes a finite state machine representing ${\displaystyle R(S)}$ for the class of lossy channel system.[3]^:24 This problem is decidable over machine capable of insertion of error .[3]^:26

The boundedness problem consists in deciding whether the set of reachable configuration is finite. I.e. the length of the content of each channel is bounded. This problem is trivially decidable over machine capable of insertion of errors[3]^:26. It is also decidable over counter machine.[6]

The eventuality property, or inevitability property problem consists in deciding, given a channel system ${\displaystyle S}$ and a set ${\displaystyle \Gamma }$ of configurations whether all run of ${\displaystyle S}$ starting at ${\displaystyle \gamma }$ goes through a configuration of ${\displaystyle \Gamma }$. This problem is undecidable for lossy channel system with impartiality[5]^:84 and with the two other fairness constraints.[5]^:87

The safety property problem consists in deciding, given a channel system ${\displaystyle S}$ and a regular set ${\displaystyle R}$ whether

The structural termination problem consists in deciding, given a channel system ${\displaystyle S}$ if the termination problem holds for ${\displaystyle S}$ for every initial configuration. This problem is undecidable even over counter machine.[4]^:342