Formalizing Turing Machines: Advances And Open Problems
Defining Turing Machines
A Turing machine is a mathematical model of computation consisting of states, transitions between states, a tape for input and working storage, and a read-write head to access and modify symbols on the tape. Turing machines provide a formal definition and framework for analyzing the computations realizable by mechanical means.
The Turing machine consists formally of a finite set of states Q, a finite set of tape symbols Γ, a transition function δ, an initial state q0 ∈ Q, and a blank symbol B ∈ Γ. The Turing machine transitions between states based on its current state and the symbol scanned by its tape head, printing a symbol to the tape and moving the tape head left or right one position at each step. The initial state q0 represents the configuration of the Turing machine before computation begins. The blank symbol B allows encoding of infinite tapes.
As a simple example, the following definitions encode a 3-state Turing machine capable of doubling an input of 1s:
- Q = {q0, q1, q2}, the set of states
- Γ = {0, 1, B}, the tape alphabet
- δ: transition function mapping Q × Γ → Q × Γ × {L, R}
- q0, the initial state
- B, the blank symbol
The transition rules encoded in δ for this machine are:
- If in state q0 scanning 1, overwrite with 1, move right, and go to state q1
- If in state q1 scanning 1, overwrite with 1, move right, and stay in state q1
- If in state q1 scanning B, write 1, move left, and go to state q2
- In state q2 scanning any symbol, halt by looping in place
This simple Turing machine thus reads an input of 1s, doubles each 1, writes the doubled input until hitting the blank end marker B, writes one additional 1 past the previous input, and halts.
Simulating and Verifying Turing Machines
In order to analyze and reason about Turing machine computations, techniques to simulate executing Turing machines for given inputs are important. Simulation tools allow tracking full configurations of a Turing machine through each step of transitioning between states based on the transition rules.
Several methods of formally verifying Turing machine simulations have been developed. These verification techniques allow proving through mathematical logic that the simulation correctly encodes the steps a Turing machine takes based on its formal transition function definition. Interactive theorem provers and proof assistants can encode and validate mechanized proofs that simulations correctly capture all possible behaviors of a Turing machine model.
As a simple example, the following Python code simulates the 3-state doubling Turing machine defined earlier on an input tape encoded as a string with the blank symbol represented by space characters:
“`python
tape = “111”
state = “q0”
position = 0
while state != “q2”:
if state == “q0” and tape[position] == “1”:
tape = tape[:position] + “1” + tape[position+1:]
position += 1
state = “q1”
elif state == “q1” and tape[position] == “1”:
tape = tape[:position] + “1” + tape[position+1:]
position += 1
elif state == “q1″ and tape[position] == ” “:
tape = tape[:position] + “1” + tape[position+1:]
position -= 1
state = “q2”
print(tape) # Prints “11111”
“`
This Python simulation could be formally verified against the transition rules of the machine through an interactive theorem prover to confirm correctness.
Turing Machines in Complexity Theory
Turing machines provide a standard model for analyzing computational complexity. The time complexity of a Turing machine is measured by the number of state transition steps executed as a function of the length of the input tape. Space complexity is measured by the maximum distance the tape head travels from the beginning marker during execution.
Many important complexity classes are defined based on bounds on Turing machine resource usage:
- P – polynomial time Turing machines
- PSPACE – polynomial space Turing machines
- EXPTIME – exponential time Turing machines
Reductions are mappings from languages recognized by Turing machines in one complexity class to the languages recognized by Turing machines in another complexity class. Reductions formalize when problems are no harder than other problems. Chains of reductions connect the difficulty of problems across complexity classes.
Major open problems remain around complexity classes defined through Turing machines, including the relationships between classes like P and NP, or NP and PSPACE. Resolving these open problems would significantly advance our understanding of the foundations of computational complexity.
Turing Machines and Computability
Turing machines can be used to recognize or decide formal languages. A language is Turing-recognizable if some Turing machine exists that halts and accepts all valid strings in the language, while possibly looping on inputs not in the language. A language is Turing-decidable if there exists a Turing machine that halts and correctly determines membership for all possible inputs.
Turing machines recognize all possible recursively enumerable languages. However there exist non-recursively enumerable languages that cannot be recognized by a Turing machine. The computable vs non-computable languages distinguishable by Turing machines have implications for the foundations of mathematical logic and computability theory.
The classic example of a non-computable language is the halting language, consisting of encodings of Turing machines paired with inputs such that the Turing machine halts on that input. No single Turing machine can correctly decide membership for all possible pairs, essentially because there is no general mechanical procedure to determine if arbitrary Turing machine code halts.
Beyond Classical Turing Machines
Many extensions of the standard Turing machine model have been studied, adding additional theoretical power. Quantum Turing machines use quantum operations and entanglement in superpositions of states and tapes. Probabilistic Turing machines introduce randomness into state transitions. Other recent Turing machine extensions allow multiple tapes or two-dimensional tapes.
These extended Turing machine models may recognize languages not recognizable by standard Turing machines. However, the Church–Turing thesis argues that standard Turing machines robustly capture the fundamental notion of effective computability. Practical computing devices can simulate standard Turing machines while physical limits constrain actually building useful quantum or probabilistic computing devices.
Applications and Connections
The theory of Turing machine computability and complexity impacts many fields beyond pure computer science. Turing machines provide an executable model used heavily in compiler optimization and code generation algorithms. Modified Turing machines that interact with external environments have inspired research into neural networks and machine learning.
Biologists have mapped DNA copying operations onto Turing machine transition rules to study replication mechanisms. Philosophers use Turing machines to probe the nature of human versus artificial intelligence. The Church-Turing thesis originated from Alonzo Church and Alan Turing’s shared insight that Turing machines formalize the meaning of effective computation.
As both a conceptual breakthrough and formal framework, Turing machines influence theory and applications across computing and other disciplines to this day.
