The Higher Arithmetic

Davenport, James H.

doi:10.1017/cbo9780511818097

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“…In particular, the inequality ι 1 ≡ 3x + y ≤ 25 is mapped to ι 1 ≡ tx + uy ≤ 25 and ι 2 ≡ 3x + 5y ≤ 50 to ι 2 ≡ x ≤ 50. The restriction det(T ) ∈ {1, −1} implies that T is unimodular, that is, the transformation maps every integral point in the original system to an integral point in the new coordinate system [26]. In the example above, a suitable matrix is…”

Section: Calculating Cuts Between Two Inequalitiesmentioning

confidence: 99%

“…The idea is to consider the slope of ι 1 as a fraction 12/7 and to calculate approximations to 12/7 using fractions made up of smaller numbers. These approximations provide potential slopes for a cut that originates in 41,26 . Note that an inequality touching 41, 26 with approximated slope a/b ≥ 12/7 touches the next integral vertex at 41+a, 26+b and thus does not cut off integral points with x-coordinates between 41 and 41 + a.…”

Section: Calculating Cuts Between Two Inequalitiesmentioning

confidence: 99%

“…Hence, it suffices to find two consecutive slopes a/b and a /b such that 41 + a ≤ 50 < 41 + a to ensure that no feasible integral point is cut off. In order to find these slopes, observe that every rational number can be represented as a finite continued fraction that takes on the following form: The coefficients a i can be inferred by observing the intermediate results of Euclid's greatest common divisor algorithm when applied to 12 and 7 [26]:…”

Section: Calculating Cuts Between Two Inequalitiesmentioning

confidence: 99%

“…For the example, these seven slopes are shown in Fig. 28 as a displacement to the integral point 41,26 . Note that slopes with odd indices i are not feasible with respect to ι 1 and can therefore be discarded as an end-point for a cut.…”

Section: Calculating Cuts Between Two Inequalitiesmentioning

confidence: 99%

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The two variable per inequality abstract domain

Simon

King

2010

Higher-Order Symb Comput

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Abstract. This article presents the Two Variable Per Inequality abstract domain (TVPI domain for short). This so-called weakly-relational domain is able to express systems of linear inequalities where each inequality has at most two variables. The domain represents a sweet-point in the performance-cost tradeoff between the faster Octagon domain and the more expressive domain of general convex polyhedra. In particular, we detail techniques to closely approximate integral TVPI systems, thereby finessing the problem of excessively growing coefficients, yielding -to our knowledge -the only relational domain that combines linear relations with arbitrary coefficients and strongly polynomial performance. Keywords: polyhedral analysis, integer programming, abstract interpretationStatic analysis methods have evolved from inferring prerequisite invariants for compiler optimisations to tools in their own right that are able to prove the absence of run-time errors of software. The abstract interpretation framework [21] provides a way of constructing and justifying program analyses. The key idea is to simulate each operation in a program with an abstract analog that operates on a description of the program state, rather than the state itself. The descriptions are chosen to trace a property of interest, for instance, an interval that encloses all possible values of a variable would be of value when reasoning about array bounds. These descriptions constitute what is known as an abstract domain. An abstract domain is usually presented as a lattice D, , , and the analysis itself is formulated as a set of recursive equations over D. The recursive equations are solved iteratively, until a fixpoint is reached which is detected using the ordering predicate . Each equation expresses how a program statement transforms the state at one program point to the state at another. Equations may be recursive because of loops in the program. The meet and join † This paper is a revised extract of the first author's PhD thesis [65], which, in turn, extends [72].

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Section: Calculating Cuts Between Two Inequalitiesmentioning

confidence: 99%

Section: Calculating Cuts Between Two Inequalitiesmentioning

confidence: 99%

Section: Calculating Cuts Between Two Inequalitiesmentioning

confidence: 99%

Section: Calculating Cuts Between Two Inequalitiesmentioning

confidence: 99%

See 2 more Smart Citations

The two variable per inequality abstract domain

Simon

King

2010

Higher-Order Symb Comput

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“…A 26). If we accept the view (Davenport 1952(Davenport /1960) that the object of a science is to discover and establish propositions about its subject matter, we can say that science involves heuristics (for discovering) and apodictics (for establishing). Besides the unknown conclusion, we also need known premises-demonstrative logic does not explain how the premises are known to be true.…”

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confidence: 99%

Aristotle's Demonstrative Logic

Corcoran

2009

History and Philosophy of Logic

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Demonstrative logic, the study of demonstration as opposed to persuasion, is the subject of Aristotle's twovolume Analytics. Many examples are geometrical. Demonstration produces knowledge (of the truth of propositions). Persuasion merely produces opinion. Aristotle presented a general truth-and-consequence conception of demonstration meant to apply to all demonstrations. According to him, a demonstration, which normally proves a conclusion not previously known to be true, is an extended argumentation beginning with premises known to be truths and containing a chain of reasoning showing by deductively evident steps that its conclusion is a consequence of its premises. In particular, a demonstration is a deduction whose premises are known to be true. Aristotle's general theory of demonstration required a prior general theory of deduction presented in the Prior Analytics. His general immediate-deductionchaining conception of deduction was meant to apply to all deductions. According to him, any deduction that is not immediately evident is an extended argumentation that involves a chaining of intermediate immediately evident steps that shows its final conclusion to follow logically from its premises. To illustrate his general theory of deduction, he presented an ingeniously simple and mathematically precise special case traditionally known as the categorical syllogistic. IntroductionThis expository paper on Aristotle's demonstrative logic is intended for a broad audience that includes non-specialists. Demonstrative logic is the study of demonstration as opposed to persuasion. It presupposes the Socratic knowledge/ opinion distinction-separating knowledge (beliefs that are known to be true) from opinion (beliefs that are not so known). Demonstrative logic is the subject of Aristotle's two-volume Analytics, as he said in its first sentence. Many of his examples are geometrical. Every non-repetitive demonstration produces or confirms knowledge of (the truth of) its conclusion for every person who comprehends the demonstration. Persuasion merely produces opinion. Aristotle presented a general truth-and-consequence conception of demonstration meant to apply to all demonstrations. According to him, a demonstration is an extended argumentation that begins with premises known to be truths and involves a chain of reasoning showing by deductively evident steps that its conclusion is a consequence of its premises. In short, a demonstration is a deduction whose premises are known to be true. For Aristotle, starting with premises known to be true and a conclusion not known to be true, the knower demonstrates the conclusion by deducing it from the premises-thereby acquiring knowledge of the conclusion.It often happens that a person will 'redemonstrate' a proposition after having previously demonstrated it-perhaps using fewer premises or a simpler chain of reasoning. The new argumentation has a conclusion already known to be true; so knowledge of the truth of the conclusion is not produced. In this case, the new argumentation is s...

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Partial Orders, Lattices, Well-Founded Orderings, Unique Prime Factorization in ℤ and GCDs, Equivalence Relations, Fibonacci and Lucas Numbers, Public Key Cryptography and RSA, Distributive Lattices, Boolean Algebras, Heyting Algebras

Gallier

2011

Discrete Mathematics

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The Higher Arithmetic

Cited by 50 publications

References 0 publications

The two variable per inequality abstract domain

The two variable per inequality abstract domain

Aristotle's Demonstrative Logic

Partial Orders, Lattices, Well-Founded Orderings, Unique Prime Factorization in ℤ and GCDs, Equivalence Relations, Fibonacci and Lucas Numbers, Public Key Cryptography and RSA, Distributive Lattices, Boolean Algebras, Heyting Algebras

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