EE290N - Specification and Modeling of Reactive Real-Time Systems


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This is a collection of informal definitions of terms used in this course, meant to supplement, not replace, standard dictionary or mathematical definitions.

actor A process in a dataflow process network.
antisymmetric relation If x != y then at most one of (x,y) and (y,x) are members of the relation.
behavior A tuple of signals.
bijection A one-to-one and onto map.
closed set (metric space) The complement of an open set.
concurrent A working definition is, "running together." This definition reflects uncertainty about exact ordering and the notion of time. Note that "concurrent" is different from "parallel" in that the latter implies separate computational elements, while the former does not.
connected set A set where there do not exist two disjoint open sets such that the union of these open sets is the whole set.
connection A particular kind of process that requires two signals in a tuple to be identical.
continuous time system A metric time system where the set of tags is a continuum.
continuum A closed connected set.
dataflow process An ordered signal process with a firing signal.
determinate A property of a process that given any input has at most one behavior.
discrete-event system A timed system where for each tuple s of signals that satisfies the system, there is an order-preserving bijection from (a) the integers (for a two-sided DE system) or (b) the natural numbers (for a one sided DE system) to T(s), where T(s) is the set of tags in s.
event A value-tag pair.
firing An event in a firing signal.
firing signal An ordered signal associated with a process such that every event in the signal is comparable with every event in all other signals associated with the process.
function A relation between two sets A and B where for each element of A there is at most one distinct member of the relation.
functional process A process with separate input and output signals that has at most one behavior for each tuple of inputs signals.
functional signal A signal that has at most one distinct event for each tag.
injective one-to-one.
input An externally imposed constraint on the set of behaviors of a process.
interactive systems Systems that interact with their environment at their own speed. Unlike transformational systems, the inputs do not need to be available before the system begins to process them, and the outputs may be valid before the system terminates (the system may not terminate, in fact). These types of systems are the emphasis of research in much of the operating systems community. Goals are typified by "please do this task quickly"-styles of kernels.
map A function.
metric Given a set T, a metric is a function d from TxT to the real line such that a set of four properties are satisfied: (1) d(x,y) = d(y,x), (2) d(x,x) = 0, (3) d(x,y) >= 0, and (4) d(x,y) + d(y,z) >= d(x,z).
metric space A set and a metric defined on that set.
monotonic map An order-preserving map.
natural numbers The positive integers.
one-to-one A function f such that if x is not equal to y, then f(x) is not equal to f(y).
onto A function f from X to Y where f(X)=Y.
ontology A vocabulary and set of agreed upon definitions to describe a subject domain. The general subject area of reactive, real-time systems does not have an ontology. However, the various subfields all have their own ontologies. e.g., this glossary. Note that this definition is a good bit narrower than what a philosopher would likely take to be an ontology.
open neighborhood (metric space) A subset with distance less than some threshold from a given point.
open set (metric space) A subset A of T is open in T if for all x in A there is an open neighborhood in A centered at x.
ordered signal A signal s where T(s) is order isomorphic with the integers (two-sided case) or the natural numbers (one sided case).
ordered signal process (OSP) A process where every signal in any behavior is an ordered signal.
order embedding A function f such that x < y if and only if f(x) < f(y).
ordering relation An irreflexive, antisymmetric, transitive relation.
order isomorphism An order embedding that is onto.
order-preserving map A function f such that x < y implies that f(x) < f(y).
parallel Executing on physically separate computational elements.
partial function A function from set A to B that may not have a member for each element of A.
partial order An ordering relation on SxS, where S is some set.
partially ordered set A set S and a partial order.
poset A partially order set.
prefix Signal a is a prefix of signal b if every event in a is also present in b and every event in a is less than every event in b that is not also in a.
prefix order A partial ordering of signals based on prefixes.
process A set of behaviors.
reactive systems Systems that interact with their environment at the speed of the environment. (e.g., hard real-time systems, sampled-data systems, and real-time control systems). The emphasis of this course will be on reactive systems, not on transformational or interactive ones.
reflexive relation (x,x) is a member of the relation for all x in the set.
relation A subset of the cartesian product of two sets.
signal A set of events.
surjective onto.
synchronous events Having the same tag.
synchronous signals Having the same set of tags.
synchronous systems A system where in every behavior, every signal is synchronous with every other.
tag A member of a set that may be partially ordered or totally ordered that is used to denote ordering relationships between events.
token An event in an ordered signal process network.
transformational systems Systems that take a body of data as input and use it to produce a body of data as output. The output is defined only when the system terminates.
transitive relation If (x,y) and (y,z) are members of the relation, then so is (x,z).
ultra metric A metric that replaces the fourth condition with a stronger one: (4) max[d(x,y),d(y,z)] >= d(x,z).
value A member of a set that denotes the information carried by an event.