Conway originally conjectured that no pattern can grow indefinitely i.e., that for any initial configuration with a finite number of living cells, the population cannot grow beyond some finite upper limit.

The development ... leads to ... an identification of computability with effective calculability.

Such an automaton is called an infinite tree automaton States Finite states An automaton that contains only a finite number of states.

Infinite states An automaton that may not have a finite number of states, or even a countable number of states.

The grid can be in any finite number of dimensions.

One significant disadvantage of flash memory is the finite number of read write cycles in a specific block.

For example, the quantum finite automaton or topological automaton has uncountable infinity of states.

Direct and iterative methods Direct methods compute the solution to a problem in a finite number of steps.

In 1912 Nernst stated the law thus It is impossible for any procedure to lead to the isotherm in a finite number of steps.

Even using infinite precision arithmetic these methods would not reach the solution within a finite number of steps (in general).

Proof by exhaustion In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately.

The number of boxes is finite, so eventually our choice procedure comes to an end.

If there were an entropy difference at absolute zero, T 0 could be reached in a finite number of steps.

There are only countably many finite sequences, so also there are only countably many finite subsets.

So even though the universe lasts forever, there's only a finite number of things that can possibly happen in the universe.

An automaton contains a finite set of states .

Every finite ordinal (natural number) is initial, but most infinite ordinals are not initial.

189 190)An even stronger assertion is that It is impossible by any procedure to reduce the temperature of a system to zero in a finite number of operations.

If the devil can force a win then it can do so in a finite number of moves.

Chomsky frequently quotes Humboldt's description of language as a system which makes infinite use of finite means , meaning that an infinite number of sentences can be created using a finite number of grammatical rules.

The finite cardinals are just the natural numbers, i.e., a set X is finite if and only if _ X _ _ n _ n for some natural number n .

An example of an accepting state appears in Fig.5 a deterministic finite automaton (DFA) that detects whether the binary input string contains an even number of 0s.

In other words, a formal grammar of a language can explain the ability of a hearer speaker to produce and interpret an infinite number of utterances, including novel ones, with a limited set of grammatical rules and a finite set of terms.

In that case it is equivalent to saying that if we have several (a finite number of) boxes, each containing at least one item, then we can choose exactly one item from each box.

In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.

A theory is satisfiable if it has a model formula_14, i.e.

It is finite harmony is infinite.

This might come in the form of a proof that the number in question is in fact irrational (or rational, as the case may be) or a finite algorithm that could determine whether the number is rational.

There is a finite limit to the electric field strength that may be withstood by any medium.

Σ is a finite set of symbols , called the alphabet of the automaton.

Man's knowledge is finite.

Finite state machines can model a large number of problems, among which are electronic design automation, communication protocol design, language parsing and other engineering applications.

These countries already understand that water is a renewable but finite resource.

This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a continuum.

In mathematics and computer science, a directed acyclic graph (DAG ), is a directed graph with no directed cycles.

Antiphon the Sophist believed that inscribing regular polygons within a circle and doubling the number of sides will eventually fill up the area of the circle, and since a polygon can be squared, it means the circle can be squared.

From the point of view of quantum field theory, such situations are described by quantum states that are eigenstates of the number operator formula_50, which measures the total number of particles present.

The figure on the right illustrates a deterministic finite automaton using a state diagram.

Regular expressions in this sense can express the regular languages, exactly the class of languages accepted by deterministic finite automata.

If you have a finite subset, you can order the elements into a finite sequence.

Consider a finite state automaton where each transition is labelled by a transition from the Petri net.

We list the elements of A effectively, n0, n1, n2, n3, ... From this list we extract an increasing sublist put m0 n0, after finitely many steps we find an nk such that nk m0, put m1 nk.

First we might try to proceed as if X were finite.

This result generalizes to the Cartesian product of any finite collection of countable sets and the proof follows by induction on the number of sets in the collection.

But fortunately there is a finite size that you can see, and that's known as the Schwarzschild radius.