3.4 Examples of recursively enumerable sets
5 min read•august 20, 2024
Recursively enumerable sets are a key concept in computability theory. They're sets whose elements can be listed by a , even if the process doesn't always halt. This property makes them crucial for understanding the limits of computation.
These sets highlight the difference between enumeration and decision problems. While some recursively enumerable sets are also recursive, others, like the , reveal fundamental limitations in what computers can solve.
Definition of recursively enumerable sets
- Recursively enumerable sets are sets for which there exists a Turing machine that can list out all the elements of the set, given enough time
- These sets may be infinite, but as long as a Turing machine can generate their elements one by one, they are considered recursively enumerable
- The elements of a can be effectively listed or enumerated by a Turing machine, although the process may not necessarily halt
Recursively enumerable sets vs recursive sets
- Recursive sets are a subset of recursively enumerable sets with the additional property that their complement is also recursively enumerable
- For recursive sets, there exists a Turing machine that can decide whether a given element belongs to the set or not, and this decision process always halts
- In contrast, for recursively enumerable sets, there may not exist a Turing machine that can decide membership in the set; the Turing machine may only be able to enumerate the elements without guaranteeing a halt on non-members
Turing machines for recursively enumerable sets
Turing machine that accepts a set
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- A Turing machine that accepts a recursively enumerable set is one that halts on input strings that belong to the set
- For input strings that are not in the set, the Turing machine may either halt and reject or run forever
- The accepting Turing machine provides a way to verify membership in the set, but it may not be able to decide non-membership
Turing machine that enumerates a set
- A Turing machine that enumerates a recursively enumerable set is one that generates all the elements of the set, one by one, given enough time
- The enumerating Turing machine may produce the elements in any order and may generate the same element multiple times
- The key property is that every element of the set will eventually be generated by the Turing machine, although the process may not halt
Examples of recursively enumerable sets
The halting set
- The halting set is the set of all Turing machine encodings paired with input strings for which the Turing machine halts
- This set is recursively enumerable because a Turing machine can simulate each Turing machine-input pair and enumerate those that halt
- However, the halting set is not recursive because there is no Turing machine that can decide whether a given Turing machine-input pair will halt or not (halting problem)
The diagonal set
- The is the set of all Turing machine encodings that do not halt when given their own encoding as input
- This set is recursively enumerable because a Turing machine can enumerate all the encodings that satisfy this property
- The diagonal set is not recursive because deciding membership in this set would require solving
Set of provably true statements
- In a formal system like first-order logic, the set of all provably true statements (theorems) is recursively enumerable
- A Turing machine can enumerate all the valid proofs in the system and output the theorems they prove
- However, this set is not necessarily recursive, as there may not exist a Turing machine that can decide whether a given statement is provable or not within the system (Gödel's incompleteness theorems)
Properties of recursively enumerable sets
Closure under union
- The union of two recursively enumerable sets is also recursively enumerable
- Given Turing machines that enumerate the elements of two sets, a new Turing machine can be constructed that alternates between the two machines, effectively enumerating the elements of their union
Closure under intersection
- The intersection of a recursively enumerable set and a recursive set is always recursively enumerable
- A Turing machine can enumerate the elements of the recursively enumerable set and use the deciding Turing machine for the recursive set to filter out the elements that belong to both sets
Closure under concatenation
- The concatenation of two recursively enumerable sets (forming strings by concatenating elements from both sets) is also recursively enumerable
- A Turing machine can systematically generate all possible concatenations by enumerating elements from both sets and combining them
Recursively enumerable sets and computability
Church-Turing thesis
- The states that any effectively calculable function can be computed by a Turing machine
- This thesis establishes a connection between the informal notion of computability and the formal concept of Turing machines and recursively enumerable sets
- Functions whose domains are recursively enumerable sets are considered effectively calculable according to the Church-Turing thesis
Recursively enumerable sets as domains of partial computable functions
- Partial computable functions are functions that can be computed by a Turing machine, but may not be defined for all inputs
- The domain of a partial computable function (the set of inputs for which the function is defined) is always a recursively enumerable set
- This property highlights the connection between recursively enumerable sets and the notion of computability
Limitations of recursively enumerable sets
Recursively enumerable sets that are not recursive
- There exist recursively enumerable sets that are not recursive, such as the halting set and the diagonal set
- For these sets, there is no Turing machine that can decide membership in the set for all inputs
- The existence of such sets demonstrates the limitations of computability and the difference between enumeration and decision problems
Impossibility of solving the halting problem
- The halting problem, which asks whether a given Turing machine will halt on a given input, is an example of a problem that is recursively enumerable but not recursive
- proved that there cannot exist a Turing machine that solves the halting problem for all inputs
- This impossibility result showcases the boundaries of what can be computed and the inherent limitations of recursively enumerable sets and Turing machines
Key Terms to Review (20)
Alan Turing: Alan Turing was a British mathematician and logician, widely regarded as one of the fathers of computer science. His pioneering work laid the foundations for modern computing, particularly through his concepts of algorithms, computation, and the development of the Turing machine, which provides a formal framework for understanding computability and recursion.
Algorithm Analysis: Algorithm analysis is the process of evaluating the performance and efficiency of an algorithm in terms of time and space complexity. It helps in understanding how algorithms behave as the size of input data grows, allowing developers to make informed choices when selecting or designing algorithms for specific tasks. This analysis is crucial in determining the practicality and feasibility of algorithms used to solve problems, especially in the realm of recursively enumerable sets.
Church-Turing Thesis: The Church-Turing Thesis posits that any function that can be effectively computed can be computed by a Turing machine, suggesting a deep equivalence between different models of computation. This idea connects to various computational systems, asserting that if a problem can be solved algorithmically, then it can be tackled by either primitive recursive functions or Turing machines, providing insights into the limits of computability and the nature of algorithms.
Closure Properties: Closure properties refer to the ability of a class of sets or functions to remain within that class when certain operations are applied. This concept is crucial in understanding how different classes of sets, like recursively enumerable sets, interact under operations such as union, intersection, and complementation. Exploring closure properties helps clarify the relationships between recursive and recursively enumerable sets, as well as identifying examples and non-examples within these classifications.
Co-recursively enumerable set: A co-recursively enumerable set is a type of set in computability theory where its complement is recursively enumerable. This means that while there exists a Turing machine that can list the elements of the set, there is also a Turing machine that can recognize when an element is not in the set. This concept is key in understanding how certain sets relate to recursively enumerable sets and non-recursively enumerable sets, especially in their applications and implications in theoretical computer science.
Compiler Design: Compiler design refers to the process of creating a compiler, which is a program that translates source code written in a high-level programming language into machine code or an intermediate representation. This process involves various stages including lexical analysis, syntax analysis, semantic analysis, optimization, and code generation. Each of these stages ensures that the resulting machine code accurately reflects the intended functionality of the original source code and is efficient for execution.
Context-free language: A context-free language is a type of formal language that is generated by a context-free grammar, where the production rules allow for the replacement of a single non-terminal symbol with a string of symbols. This kind of language is essential in the field of computer science, particularly in programming language design and parsing. Context-free languages can describe structures that are hierarchical and recursive, making them powerful for defining syntax in various computing applications.
Decidable Set: A decidable set is a collection of elements for which there exists an algorithm that can determine, in a finite amount of time, whether any given element belongs to the set. This concept connects closely to the broader themes of computability and algorithmic solvability, as it highlights the distinction between problems that can be solved by a mechanical process and those that cannot. Understanding decidable sets helps clarify the boundaries of what can be computed and sheds light on related topics such as recursively enumerable sets, non-recursively enumerable sets, and levels of the arithmetical hierarchy.
Diagonal Set: A diagonal set is a specific type of set that is constructed using a diagonalization argument, often associated with Cantor's work on the uncountability of real numbers. This concept illustrates the idea of a set that includes elements which differ from elements in a given countable sequence by at least one characteristic, thus proving that not all sets can be enumerated or counted. Diagonal sets are important in understanding the limitations of recursive enumerability and the nature of certain uncountable sets.
Emil Post: Emil Post was a prominent mathematician and logician known for his work on recursive functions and the foundations of computability theory. His contributions laid the groundwork for many concepts in theoretical computer science, including the relationship between partial and total recursive functions, which is essential for understanding the limits of computation.
Halting Set: A halting set is a specific collection of inputs for which a given computational process will eventually terminate, or 'halt'. Understanding halting sets is crucial because they exemplify the boundaries of computability, particularly in relation to recursively enumerable sets. Halting sets provide insight into which problems can be solved algorithmically and help distinguish between those that can be effectively computed and those that cannot.
Recursive vs. Recursively Enumerable: Recursive sets are those for which there exists a total algorithm that can determine membership for any given element, while recursively enumerable (r.e.) sets are those for which there is a partial algorithm that will list all members but may not halt for non-members. This distinction is important as it highlights the boundaries of computability and the nature of decision problems in mathematical logic and computer science.
Recursively Enumerable Set: A recursively enumerable set is a collection of natural numbers for which there exists a Turing machine that will list its members, possibly without halting if an element is not in the set. This concept connects to various aspects of computability theory, such as the relationship between recursive and recursively enumerable sets, the enumeration theorem, and examples illustrating these sets. Additionally, understanding non-recursively enumerable sets provides insight into the limitations of computation.
Reducibility: Reducibility refers to the ability to transform one problem into another in such a way that a solution to the second problem can be used to solve the first. This concept is crucial for understanding the relationships between different computational problems, especially in determining the computational complexity and decidability of those problems.
Rice's Theorem: Rice's Theorem states that for any non-trivial property of partial recursive functions, it is undecidable whether a given partial recursive function has that property. This highlights the limits of what can be computed and connects deeply with the concepts of recursion and computability.
Set of All Turing Machines That Halt: The set of all Turing machines that halt is the collection of all Turing machines that, given a specific input, eventually reach a halting state after a finite number of steps. This set is crucial in understanding the concept of decidability and computability, as it represents the boundaries of what can be computed within a finite timeframe. The importance of this set lies in its relationship to recursively enumerable sets, as it serves as a key example in discussions about problems that can or cannot be solved by algorithms.
Set of provably true statements: A set of provably true statements refers to a collection of assertions within a formal system that can be demonstrated as true using the axioms and rules of inference of that system. This concept is crucial in understanding the limits of what can be proven within a given logical framework and connects closely with ideas like consistency, completeness, and decidability in recursive function theory.
Set of valid C programs: A set of valid C programs is the collection of all C code snippets that conform to the syntax and semantics defined by the C programming language specification. This set includes programs that will compile successfully and run without errors, encompassing everything from simple statements to complex applications. Understanding this set is crucial for grasping how programming languages classify various types of computable functions and the concept of recursively enumerable sets.
The halting problem: The halting problem is a fundamental question in computer science that asks whether a given program will finish running or continue to run indefinitely for a specific input. This problem reveals the limits of computation, demonstrating that there is no general algorithm that can determine for every possible program-input pair whether the program halts or runs forever. The implications of this problem are crucial for understanding recursively enumerable sets and the classification of problems based on their computational properties.
Turing machine: A Turing machine is a theoretical computational model that consists of an infinite tape, a tape head for reading and writing symbols, and a set of rules that dictate its operations. It serves as a foundational concept in computer science for understanding the limits of what can be computed and is instrumental in studying computable functions and problems.