Thus the existence of an infinite branch does not necessarily imply a nonterminating computation.
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In the present case this means that if every execution sequence of terminates, then there are only finitely many execution sequences. So if an output set of is infinite, it must contain [a nonterminating computation].
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The c-base is a system of seven concentric rings that can move in relation to one another. Each ring is considered a single module with a special set of functions. The rings are called (from the centre out) "core", "com", "culture", "creactiv", "cience", "carbon" and "clamp". The inner ring, core, provides a nonterminating supply of energy produced by a Möbius strip generator.
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Problem classes having (possibly nonterminating) algorithms with polynomial time average case running time whose output is always correct are said to be in ZPP. The class of problems for which both YES and NO-instances are allowed to be identified with some error is called BPP. This class acts as the randomized equivalent of P, i.e. BPP represents the class of efficient randomized algorithms.
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This is the simplest formal method, and the most general, applying to sequential, parallel, stand-alone, communicating, terminating, nonterminating, natural-time, real-time, deterministic, and probabilistic programs, and includes time and space bounds. This idea has influenced other computer science researchers, including Tony Hoare. Hehner's other research areas include probabilistic programming, unified algebra, and high-level circuit design. In 1979, Hehner invented a generalization of radix complement called quote notation, which is a representation of the rational numbers that allows easier arithmetic and precludes roundoff error.
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Clinger [1981] explained the domain of Actor computations as follows: :The augmented Actor event diagrams [see Actor model theory] form a partially ordered set < , > from which to construct the power domain (see the section on Denotations below). The augmented diagrams are partial computation histories representing "snapshots" [relative to some frame of reference] of a computation on its way to being completed. For ,∈, means is a stage the computation could go through on its way to . The completed elements of represent computations that have terminated and nonterminating computations that have become infinite.
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In fact the value of δ can even be allowed to decrease linearly with time to accommodate Moore's Law. The Actor event timed diagrams form a partially ordered set . The diagrams are partial computation histories representing "snapshots" (relative to some frame of reference) of a computation on its way to being completed. For d1, d2εTimedDiagrams, d1≤d2 means d1 is a stage the computation could go through on its way to d2 The completed elements of TimedDiagrams represent computations that have terminated and nonterminating computations that have become infinite.
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Predicative programming is the original name of a formal method for program specification and refinement, more recently called a Practical Theory of Programming, invented by Eric Hehner. The central idea is that each specification is a binary (boolean) expression that is true of acceptable computer behaviors and false of unacceptable behaviors. It follows that refinement is just implication. This is the simplest formal method, and the most general, applying to sequential, parallel, stand-alone, communicating, terminating, nonterminating, natural-time, real-time, deterministic, and probabilistic programs, and includes time and space bounds.
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He defines the lexicographical order and an addition operation, noting that 0.999... < 1 simply because 0 < 1 in the ones place, but for any nonterminating x, one has 0.999... + x = 1 + x. So one peculiarity of the decimal numbers is that addition cannot always be cancelled; another is that no decimal number corresponds to . After defining multiplication, the decimal numbers form a positive, totally ordered, commutative semiring.Richman pp. 397–399 In the process of defining multiplication, Richman also defines another system he calls "cut D", which is the set of Dedekind cuts of decimal fractions.
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In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation theorem-proving technique for sentences in propositional logic and first-order logic. In other words, iteratively applying the resolution rule in a suitable way allows for telling whether a propositional formula is satisfiable and for proving that a first-order formula is unsatisfiable. Attempting to prove a satisfiable first-order formula as unsatisfiable may result in a nonterminating computation; this problem doesn't occur in propositional logic. The resolution rule can be traced back to Davis and Putnam (1960); Here: p.
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The completed elements may be characterized abstractly as the maximal elements of [see William Wadge 1979]. Concretely, the completed elements are those having non pending events. Intuitively, is not ω-complete because there exist increasing sequences of finite partial computations ::x_0 \le x_1 \le x_2 \le x_3 \le ... :in which some pending event remains pending forever while the number of realized events grows without bound, contrary to the requirement of finite [arrival] delay. Such a sequence cannot have a limit, because any limit would represent a completed nonterminating computation in which an event is still pending.
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Common patterns of recursion can be abstracted away using higher-order functions, with catamorphisms and anamorphisms (or "folds" and "unfolds") being the most obvious examples. Such recursion schemes play a role analogous to built-in control structures such as loops in imperative languages. Most general purpose functional programming languages allow unrestricted recursion and are Turing complete, which makes the halting problem undecidable, can cause unsoundness of equational reasoning, and generally requires the introduction of inconsistency into the logic expressed by the language's type system. Some special purpose languages such as Coq allow only well-founded recursion and are strongly normalizing (nonterminating computations can be expressed only with infinite streams of values called codata).
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In logic, the procedure for obtaining the conjunctive normal form (CNF) of a formula can be implemented as a rewriting system. The rules of an example of such a system would be: : eg eg A \to A (double negation elimination) : eg(A \land B) \to eg A \lor eg B (De Morgan's laws) : eg(A \lor B) \to eg A \land eg B : (A \land B) \lor C \to (A \lor C) \land (B \lor C) (distributivity) : A \lor (B \land C) \to (A \lor B) \land (A \lor C),This variant of the previous rule is needed since the commutative law A∨B = B∨A cannot be turned into a rewrite rule. A rule like A∨B → B∨A would cause the rewrite system to be nonterminating. where the symbol (\to) indicates that an expression matching the left hand side of the rule can be rewritten to one formed by the right hand side, and the symbols each denote a subexpression.
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