Emil Leon Post, Polish-American mathematician and logician (d.1954)
Emil Leon Post, born on February 11, 1897, and passing away on April 21, 1954, was a profoundly influential American mathematician and logician whose pioneering contributions laid significant groundwork for what would become the vibrant field of theoretical computer science. Despite his relatively short life, Post's intellectual rigor and innovative thinking left an indelible mark on mathematical logic and the fundamental understanding of computation.
Primarily, Emil Post is celebrated for his groundbreaking work in the area now universally recognized as computability theory. This critical branch of mathematical logic and theoretical computer science delves into the fundamental limits of what can be computed by an algorithm. It explores the class of functions that can be computed by a mechanical procedure and the problems that are inherently unsolvable by any algorithmic process, regardless of computational power.
Pioneering Contributions to Computability Theory
Post's work was instrumental in formalizing the concept of an "effective procedure" or algorithm, a cornerstone for understanding computation. His independent development of a model of computation, often referred to as a "Post-Turing machine," paralleled Alan Turing's own groundbreaking work. This convergence of ideas from independent researchers underscored the timeliness and fundamental importance of these abstract computational models.
Key Concepts and Problems Introduced by Post:
- Post-Turing Machine: An abstract computational model, similar to Turing machines, which rigorously defines what it means for a function to be computable. It consists of an infinite tape, a head that reads and writes symbols, and a finite set of states, demonstrating the theoretical limits of computation.
- Post's Correspondence Problem (PCP): Introduced in 1946, PCP is a classic example of an undecidable decision problem in formal language theory. It involves finding a sequence of "dominoes" (pairs of strings) such that the concatenation of the top strings equals the concatenation of the bottom strings. Its undecidability makes it a powerful tool for proving the undecidability of other problems in various areas of computer science.
- Post's Theorem: This fundamental theorem establishes a crucial link between the arithmetical hierarchy (a classification of sets of natural numbers based on the complexity of their definitions in first-order arithmetic) and the Turing degrees (a classification of sets based on their relative computability). It helps to categorize the complexity of undecidable problems.
- Post's Lattice of Degrees of Unsolvability: Post laid the foundation for understanding the structure of unsolvability, demonstrating that there are infinitely many distinct degrees of uncomputability. This concept, often visualized as a lattice, helps mathematicians classify problems based on their inherent difficulty to be solved algorithmically, even if they are all "unsolvable."
The Broader Impact of Emil Post's Work
Beyond his specific contributions to computability theory, Post's visionary approach significantly influenced the development of mathematical logic and laid essential groundwork for the subsequent explosion of theoretical computer science. His work provided the formal tools necessary to analyze the capabilities and limitations of algorithms, long before the advent of modern electronic computers.
His research helped to solidify the understanding of concepts like recursive functions and decidability, which are now fundamental to algorithm design, complexity theory, and the theoretical underpinnings of programming languages. Post's rigorous definitions and challenging problems continue to be studied and applied in advanced computer science and mathematical logic courses worldwide.
Frequently Asked Questions About Emil Leon Post
- What is Emil Leon Post best known for?
- Emil Leon Post is primarily known for his seminal contributions to computability theory, particularly for his independent development of an abstract computing machine model (the Post-Turing machine) and for introducing foundational concepts like Post's Correspondence Problem and Post's Theorem, which profoundly influenced mathematical logic and theoretical computer science.
- When did Emil Leon Post live and work?
- Emil Leon Post was born on February 11, 1897, and passed away on April 21, 1954. His most impactful work was conducted during the mid-20th century, a crucial period for the development of mathematical logic and the nascent ideas of computation.
- How did Emil Post's work relate to Alan Turing's?
- Interestingly, Emil Post independently developed a formal model of computation, often called a Post-Turing machine, which was structurally and functionally equivalent to Alan Turing's famous Turing machine. This independent discovery by two brilliant minds highlights the natural progression and critical importance of these ideas in understanding the foundations of computation at that time.
- What is Post's Correspondence Problem (PCP)?
- Post's Correspondence Problem (PCP) is a famous undecidable decision problem introduced by Post. It involves finding a sequence of 'dominoes' (pairs of strings) such that the concatenation of the top strings equals the concatenation of the bottom strings. Its undecidability makes it a fundamental tool for proving other problems to be undecidable within theoretical computer science.
- Why is computability theory important today?
- Computability theory, largely shaped by pioneers like Emil Post, is crucial because it defines the fundamental limits of what computers can and cannot do. It provides the theoretical framework for understanding algorithms, determining problem solvability, and forms the bedrock for advanced concepts in computer science such as complexity theory, artificial intelligence, and the design of programming languages.