Concepts inOn the proof complexity of deep inference
Deep inference
Deep inference names a general idea in structural proof theory that breaks with the classical sequent calculus by generalising the notion of structure to permit inference to occur in contexts of high structural complexity.
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Proof complexity
In computer science, proof complexity is a measure of efficiency of automated theorem proving methods that is based on the size of the proofs they produce. The methods for proving contradiction in propositional logic are the most analyzed. The two main issues considered in proof complexity are whether a proof method can produce a polynomial proof of every inconsistent formula, and whether the proofs produced by one method are always of size similar to those produced by another method.
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Gerhard Gentzen
Gerhard Karl Erich Gentzen was a German mathematician and logician. He had his major contributions in the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died in 1945 after the Second World War, because he was deprived of food after being arrested in Prague.
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Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on the philosophy of language and mathematics.
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Proof calculus
In mathematical logic, a proof calculus corresponds to a family of formal systems that use a common style of formal inference for its inference rules. The specific inference rules of a member of such a family characterize the theory of a logic. Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determining and can be used for radically different logics.
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Integration by substitution
In calculus, integration by substitution is a method for finding integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool for mathematicians. It is the counterpart to the chain rule of differentiation. Let be an interval and be a continuously differentiable function. Suppose that is a continuous function.
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Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions.
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