!!!Properties of Expressions

* __Helpful__: Always sends a negative result when it has finished publishing (0 or more results).
* __Side-effect free__: Does not have any side-effects and all information produced is in the forms of publications.[{orc runnable='false'

e >> stop =~~= stop 
}]An asynchronous side-effect free expression can be replaced with an expression {{{v1 | v2 | ... | vn}}} where {{{vi}}} are all the publications of the expression. A synchronous side-effect free expression is equivalent to: {{{Rwait(t1) >> v1 | Rwait(t2) >> v2 | ... | Rwait(tn) >> vn}}} where each pair {{{(ti, vi)}}} is a publication time and value.
* __Pure__: Its publications (0 or more) are always the same given the same arguments or values for free variables, and it is side-effect free. This is a generalization of pure functions similar to pure non-deterministic functions. 
* __Pure Functional__: Always publishes exactly 1 value and is pure. Equivalent to a total mathematical function.[{orc runnable='false'

e =~~= x <x< e
e >> stop =~~= stop 
}] 
* __Asynchronous__: Ordering of publications of subexpressions does not affect the set of possible results of the expression. (TODO: is this correct?)
* __Silent__: Never publishes a value.[{orc runnable='false'

e =~= e >> stop
}]
* __Deterministic__: Will always produce the same result regardless of order of publication or scheduling. Note that asynchronous expressions can still be non-deterministic. (TODO: is this correct?)
* __Resilient__: The expression will ''not'' terminate if pruned.
* __Atomic__: The expression {{{e}}} is not effected by concurrently running expressions (including other copies of {{{e}}}).  This is probably both atomic and co-atomic (in Kitchen terms).
* __Blocking__: The expression will always eventually respond.
* __Strict__: All free variables in {{{e}}} are needed without delay. All site calls are strict. The charactature for strict on {{{x}}} is: [{orc runnable='false'

e <x< f =~~= e <x<$ f
}] where {{{<x<$}}} performs termination but not late binding. Expressions can also be strict in specific variables and non-strict in others.
* __Idempotent__: Executing an expression more than once has the same effect as executing it once.[{orc runnable='false'

e | e =~~= e
}] This is not be an accurate characture because {{{e}}} may not be atomic so it might conflict with itself. However {{{e >> e =~~= e}}} would not be correct either unless {{{e}}} is asynchronous.
* __Talkative__: Publishes at least once.
* __Terminating__: Will eventually terminate even if it is not pruned.

There are probably a few more that would be useful. This is a work in progress.

Many of these properties have characteristic identities which totally encode their meaning. For instance, an expression e is silent iff the following statement is true.
[{orc runnable='false'

e =~= e >> stop
}]
(where =~= is equivalence of expression in the semantics.) Similar statements can and should be written for other properties. However this may not be not possible for all; what about purity it needs to be expressed in terms of all evaluations producing the same publications given the same input.

!!!Decidability
Some of there properties are decidable based on axioms about sites. However most are not. For the undecidable properties it would still be possible to create a conservative estimate that will detect many useful cases of the property.

* __Side-effect free__ is decided by: [{orc runnable='false'

sideEffectFree (f | g) = sideEffectFree f && sideEffectFree g
sideEffectFree (f >x> g) = sideEffectFree f && (if not silent f then sideEffectFree g else true)
sideEffectFree (f <x< g) = sideEffectFree f && sideEffectFree g
sideEffectFree (f ; g) = sideEffectFree f && (if not talkative f then sideEffectFree g else true)
}] (this has not been proved correct)

This section should be completed with deciders and estimates for other properties.

!!!Related Pages

* [Deparallelization]: Has some interesting properties and identities that may be useful.
* [Identities]: A list of algebraic identities some of which depend on these properties.