Computability and Turing Machines

Computability and Turing Machines

• Computability refers to the ability of a function to be computed by a Turing machine or other models of computation.
• The Church-Turing thesis states that a function is computable if it can be computed by a Turing machine or Lambda calculus.

Models of Computability

• Turing machines:
  • Consist of a storage tape, read and write head, and a control unit.
  • The tape is divided into cells, each containing a symbol from a finite alphabet.
  • The machine can read and write symbols on the tape and move the head left or right.
  • The control unit contains a program that defines the machine's behavior.
• Lambda calculus:
  • A formal system for expressing functions and performing computations.
  • Developed by Alonzo Church and Stephen Kleene.
  • Can be used to describe mathematical functions and prove their computability.

The Turing Machine

• A Turing machine is a simple computational model that can perform computations by reading and writing symbols on a tape.
• The machine's behavior is defined by a program that specifies its actions based on the current state and input symbol.
• The machine can move the tape left or right, read and write symbols, and change its state.
• The machine halts when it reaches a designated halting state.

Universality of Turing Machines

• A universal Turing machine is a machine that can simulate the behavior of any other Turing machine.
• This is achieved by encoding the description of the machine to be simulated as a string of symbols on the tape.
• The universal machine can then simulate the behavior of the encoded machine.

Non-Deterministic Turing Machines

• A non-deterministic Turing machine is one that has multiple possible next states for a given input and current state.
• This is in contrast to deterministic machines, which have a unique next state for a given input and current state.
• Non-deterministic machines are important in complexity theory and are used to study the complexity of computations.

Other Models of Computability

• Loop programs:
  • A simple programming language that uses loops to iterate over a sequence of statements.
  • Can be used to compute functions that are total (i.e., defined for all inputs).
  • Not Turing-complete, as they cannot generate infinite loops.
• While programs:
  • Similar to loop programs, but use while loops instead of for loops.
  • Also not Turing-complete, as they cannot generate infinite loops.

Church-Turing Thesis

• The Church-Turing thesis states that a function is computable if and only if it can be computed by a Turing machine or Lambda calculus.
• This thesis has far-reaching implications for the study of computability and the characterization of functions as computable or non-computable.### While Programs
• A while program consists of constants, variables, and statements of the form x := a, x := succ a, or x := pred a, which set the value of variable x to a, succ a, or pred a.
• Statements can also be of the form P;Q, which represent the sequential execution of two while programs P and Q.
• WHILE B DO P END statements execute partial program P as long as condition B is true.

Computability and Turing Completeness

• The language of while programs is Turing-complete, meaning it can compute any function that can be computed by a Turing machine.
• This implies that every loop-computable function is while-computable, but not the reverse.
• While programs are Turing-complete, and therefore as powerful as Turing machines.

Other Models of Computation

• Goto programs are similar to loop programs and while programs, but support jump instructions of the form IF B GOTO M.
• Lambda calculus, μ-recursive functions, and one-instruction-set computers (OISC) are all Turing-complete models of computation.
• These models are all equivalent in terms of computability, but differ in their sets of instructions.

Primitive-Recursive and μ-Recursive Functions

• Primitive-recursive functions use a simple form of recursion, equivalent to a for loop.
• μ-recursive functions add a μ-operator, which finds the smallest argument for which a function f is zero, equivalent to a while loop.
• The primitive-recursive functions are exactly the loop-computable functions, and the μ-recursive functions are exactly the Turing-computable functions.

Decidability and the Halting Problem

• Decision problems ask whether an algorithm exists that can make a decision for every possible input.
• The halting problem is the question of whether a given program will terminate for a particular input.
• The halting problem is undecidable, meaning there is no general algorithm to decide whether a program will terminate.
• The halting problem is semi-decidable, meaning we can list all terminating programs, but not all non-terminating programs.

Recursively Enumerable Sets

• A set is recursively enumerable if there is a computable function that lists its elements.
• A set is countable if its elements can be assigned to natural numbers.
• A set is recursively enumerable if and only if it is semi-decidable.

Rice's Theorem and Busy Beavers

• Rice's theorem states that any non-trivial, functional property of a Turing machine is undecidable.
• The busy beaver game is a well-known example of a non-computable function, aiming to find a terminating program of a given size that produces the most output possible.
• The busy beaver function grows faster than any computable function and is not computable itself.

Computability and Turing Machines

• Computability refers to the ability of a function to be computed by a Turing machine or other models of computation.
• The Church-Turing thesis states that a function is computable if it can be computed by a Turing machine or Lambda calculus.

Models of Computability

• Turing machines:
  • Consist of a storage tape, read and write head, and a control unit.
  • The tape is divided into cells, each containing a symbol from a finite alphabet.
  • The machine can read and write symbols on the tape and move the head left or right.
  • The control unit contains a program that defines the machine's behavior.
• Lambda calculus:
  • A formal system for expressing functions and performing computations.
  • Developed by Alonzo Church and Stephen Kleene.
  • Can be used to describe mathematical functions and prove their computability.

The Turing Machine

• A Turing machine is a simple computational model that can perform computations by reading and writing symbols on a tape.
• The machine's behavior is defined by a program that specifies its actions based on the current state and input symbol.
• The machine can move the tape left or right, read and write symbols, and change its state.
• The machine halts when it reaches a designated halting state.

Universality of Turing Machines

• A universal Turing machine is a machine that can simulate the behavior of any other Turing machine.
• This is achieved by encoding the description of the machine to be simulated as a string of symbols on the tape.
• The universal machine can then simulate the behavior of the encoded machine.

Non-Deterministic Turing Machines

• A non-deterministic Turing machine is one that has multiple possible next states for a given input and current state.
• This is in contrast to deterministic machines, which have a unique next state for a given input and current state.
• Non-deterministic machines are important in complexity theory and are used to study the complexity of computations.

Other Models of Computability

• Loop programs:
  • A simple programming language that uses loops to iterate over a sequence of statements.
  • Can be used to compute functions that are total (i.e., defined for all inputs).
  • Not Turing-complete, as they cannot generate infinite loops.
• While programs:
  • Similar to loop programs, but use while loops instead of for loops.
  • Also not Turing-complete, as they cannot generate infinite loops.

Church-Turing Thesis

• The Church-Turing thesis states that a function is computable if and only if it can be computed by a Turing machine or Lambda calculus.
• This thesis has far-reaching implications for the study of computability and the characterization of functions as computable or non-computable.### While Programs
• A while program consists of constants, variables, and statements of the form x := a, x := succ a, or x := pred a, which set the value of variable x to a, succ a, or pred a.
• Statements can also be of the form P;Q, which represent the sequential execution of two while programs P and Q.
• WHILE B DO P END statements execute partial program P as long as condition B is true.

Computability and Turing Completeness

• The language of while programs is Turing-complete, meaning it can compute any function that can be computed by a Turing machine.
• This implies that every loop-computable function is while-computable, but not the reverse.
• While programs are Turing-complete, and therefore as powerful as Turing machines.

Other Models of Computation

• Goto programs are similar to loop programs and while programs, but support jump instructions of the form IF B GOTO M.
• Lambda calculus, μ-recursive functions, and one-instruction-set computers (OISC) are all Turing-complete models of computation.
• These models are all equivalent in terms of computability, but differ in their sets of instructions.

Primitive-Recursive and μ-Recursive Functions

• Primitive-recursive functions use a simple form of recursion, equivalent to a for loop.
• μ-recursive functions add a μ-operator, which finds the smallest argument for which a function f is zero, equivalent to a while loop.
• The primitive-recursive functions are exactly the loop-computable functions, and the μ-recursive functions are exactly the Turing-computable functions.

Decidability and the Halting Problem

• Decision problems ask whether an algorithm exists that can make a decision for every possible input.
• The halting problem is the question of whether a given program will terminate for a particular input.
• The halting problem is undecidable, meaning there is no general algorithm to decide whether a program will terminate.
• The halting problem is semi-decidable, meaning we can list all terminating programs, but not all non-terminating programs.

Recursively Enumerable Sets

• A set is recursively enumerable if there is a computable function that lists its elements.
• A set is countable if its elements can be assigned to natural numbers.
• A set is recursively enumerable if and only if it is semi-decidable.

Rice's Theorem and Busy Beavers

• Rice's theorem states that any non-trivial, functional property of a Turing machine is undecidable.
• The busy beaver game is a well-known example of a non-computable function, aiming to find a terminating program of a given size that produces the most output possible.
• The busy beaver function grows faster than any computable function and is not computable itself.

Computability and Turing Machines

• Computability refers to the ability of a function to be computed by a Turing machine or other models of computation.
• The Church-Turing thesis states that a function is computable if it can be computed by a Turing machine or Lambda calculus.

Models of Computability

• Turing machines:
  • Consist of a storage tape, read and write head, and a control unit.
  • The tape is divided into cells, each containing a symbol from a finite alphabet.
  • The machine can read and write symbols on the tape and move the head left or right.
  • The control unit contains a program that defines the machine's behavior.
• Lambda calculus:
  • A formal system for expressing functions and performing computations.
  • Developed by Alonzo Church and Stephen Kleene.
  • Can be used to describe mathematical functions and prove their computability.

The Turing Machine

• A Turing machine is a simple computational model that can perform computations by reading and writing symbols on a tape.
• The machine's behavior is defined by a program that specifies its actions based on the current state and input symbol.
• The machine can move the tape left or right, read and write symbols, and change its state.
• The machine halts when it reaches a designated halting state.

Universality of Turing Machines

• A universal Turing machine is a machine that can simulate the behavior of any other Turing machine.
• This is achieved by encoding the description of the machine to be simulated as a string of symbols on the tape.
• The universal machine can then simulate the behavior of the encoded machine.

Non-Deterministic Turing Machines

• A non-deterministic Turing machine is one that has multiple possible next states for a given input and current state.
• This is in contrast to deterministic machines, which have a unique next state for a given input and current state.
• Non-deterministic machines are important in complexity theory and are used to study the complexity of computations.

Other Models of Computability

• Loop programs:
  • A simple programming language that uses loops to iterate over a sequence of statements.
  • Can be used to compute functions that are total (i.e., defined for all inputs).
  • Not Turing-complete, as they cannot generate infinite loops.
• While programs:
  • Similar to loop programs, but use while loops instead of for loops.
  • Also not Turing-complete, as they cannot generate infinite loops.

Church-Turing Thesis

• The Church-Turing thesis states that a function is computable if and only if it can be computed by a Turing machine or Lambda calculus.
• This thesis has far-reaching implications for the study of computability and the characterization of functions as computable or non-computable.### While Programs
• A while program consists of constants, variables, and statements of the form x := a, x := succ a, or x := pred a, which set the value of variable x to a, succ a, or pred a.
• Statements can also be of the form P;Q, which represent the sequential execution of two while programs P and Q.
• WHILE B DO P END statements execute partial program P as long as condition B is true.

Computability and Turing Completeness

• The language of while programs is Turing-complete, meaning it can compute any function that can be computed by a Turing machine.
• This implies that every loop-computable function is while-computable, but not the reverse.
• While programs are Turing-complete, and therefore as powerful as Turing machines.

Other Models of Computation

• Goto programs are similar to loop programs and while programs, but support jump instructions of the form IF B GOTO M.
• Lambda calculus, μ-recursive functions, and one-instruction-set computers (OISC) are all Turing-complete models of computation.
• These models are all equivalent in terms of computability, but differ in their sets of instructions.

Primitive-Recursive and μ-Recursive Functions

• Primitive-recursive functions use a simple form of recursion, equivalent to a for loop.
• μ-recursive functions add a μ-operator, which finds the smallest argument for which a function f is zero, equivalent to a while loop.
• The primitive-recursive functions are exactly the loop-computable functions, and the μ-recursive functions are exactly the Turing-computable functions.

Decidability and the Halting Problem

• Decision problems ask whether an algorithm exists that can make a decision for every possible input.
• The halting problem is the question of whether a given program will terminate for a particular input.
• The halting problem is undecidable, meaning there is no general algorithm to decide whether a program will terminate.
• The halting problem is semi-decidable, meaning we can list all terminating programs, but not all non-terminating programs.

Recursively Enumerable Sets

• A set is recursively enumerable if there is a computable function that lists its elements.
• A set is countable if its elements can be assigned to natural numbers.
• A set is recursively enumerable if and only if it is semi-decidable.

Rice's Theorem and Busy Beavers

• Rice's theorem states that any non-trivial, functional property of a Turing machine is undecidable.
• The busy beaver game is a well-known example of a non-computable function, aiming to find a terminating program of a given size that produces the most output possible.
• The busy beaver function grows faster than any computable function and is not computable itself.