A short and constructive proof of Tarski’s fixed-point theorem

AbstractWe start by giving a compact representation schema for λ-terms, and show how this leads to an exceedingly small and elegant self-interpreter. We then define the notion of aself-reducer, and show how this too can be written as a small λ-term. Both the self-interpreter and the self-reducer are proved correct. We finally give a constructive proof for the second fixed point theorem for the representation schema. All the constructions have been implemented on a computer, and experiments verify their correctness. Timings show that the self-interpreter and self-reducer are quite efficient, being about 35 and 50 times slower than direct execution using a call-by-need reductions strategy

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Generalizations of Tarski's fixed point theorem for order varieties of complete meet semilattices

Order ◽

10.1007/bf00353657 ◽

1989 ◽

Vol 5 (4) ◽

pp. 381-392 ◽

Cited By ~ 3

Author(s):

Joel Berman ◽

W. J. Blok

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Point Theorem ◽

Tarski’S Fixed Point Theorem

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Financial System

Risk and Liquidity ◽

10.1093/oso/9780198847069.003.0006 ◽

2019 ◽

pp. 96-115

Author(s):

Hyun Song Shin

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Point Theorem ◽

Systemic Risk ◽

Financial System ◽

Comparative Statics ◽

Regularity Conditions ◽

Balance Sheets ◽

System P ◽

Tarski’S Fixed Point Theorem

In a financial system of interlocking balance sheets, the assets of creditors are the liabilities of debtors. A change in the value of underlying assets can ripple through the financial system through valuation changes on balance sheets. Tarski’s fixed point theorem guarantees the existence of consistent valuations. Under mild regularity conditions, there is a unique fixed point. Comparative statics analysis can be used to show how systemic risk propagates.

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Solving Fixed-Point Problems with Inequality and Equality Constraints via a Non-Interior Point Homotopy Path-Following Method

Mathematical Problems in Engineering ◽

10.1155/2017/3456834 ◽

2017 ◽

Vol 2017 ◽

pp. 1-9

Author(s):

Menglong Su ◽

Yufeng Shang

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Point Theorem ◽

Interior Point ◽

Fixed Point Theorems ◽

Path Following ◽

Fixed Point Problems ◽

Constructive Proof ◽

Nonconvex Sets ◽

Smooth Path

In recent years, fixed-point theorems have attracted increasing attention and have been widely investigated by many authors. Moreover, determining a fixed point has become an interesting topic. In this paper, we provide a constructive proof of the general Brouwer fixed-point theorem and then obtain the existence of a smooth path which connects a given point to the fixed point. We also present a non-interior point homotopy algorithm for solving fixed-point problems on a class of nonconvex sets by numerically tricking this homotopy path.

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