Minor Project(Automata Simulators):

This project aims to be a combination of simulators for multiple Formal Language and Automata Theory automatons, such as:

• Determinitic Finite Automaton
• Push Down Automata
• Turing Machines
• Cellular automata
• et cetra

Currently, work is complete for Turing Machine and Cellular automata simulators- a preview is available in the releases section. This project is aimed to be the minor project submission for:

• Nikhil Narayanan (RA2111003011390)

• Arnish Arvind Mishra (RA2111003011391)

  
  

Cellular Automata:

Cellular automata (CA) is a mathematical model for simulating complex systems, typically represented as a grid of cells that can be in one of a finite number of states. The cells change state based on a set of rules that determine how their state is influenced by the state of surrounding cells. This creates patterns and structures that evolve over time, often giving rise to complex and seemingly intelligent behavior. CA is used in a variety of fields, including physics, biology, and computer science, to study phenomena ranging from pattern formation to self-organization and emergent behavior.

Connway's Game of Life:

Conway's Game of Life is a cellular automaton that was first proposed by mathematician John Horton Conway in 1970. It is a simulation of simple rules that can generate complex and interesting patterns.

The simulation takes place on a two-dimensional grid of cells, where each cell is either "alive" or "dead". At each step, the state of each cell is updated based on the state of its eight neighboring cells according to the following rules:

If a cell is alive and has two or three live neighbors, it remains alive. If a cell is dead and has exactly three live neighbors, it comes to life. In all other cases, a cell dies or remains dead. These simple rules can generate patterns ranging from simple oscillators to complex shapes that evolve over time. The Game of Life is often used as an example of cellular automata and is widely studied by mathematicians, computer scientists, and other researchers interested in complex systems and emergent behavior. Thus, for Connway's Game of Life, the state transition diagram would be as follows: Ruleset-

• Adjacent live cells <2, Cell death(a)
• Adjacent live cells>3, Cell death(b)
• Exactly 3 cells live in adjacent cells, dead cell comes back to life(c)

Symbols={a,b,c} 
States={Alive, Dead}
Initial State=Alive
Accepted State(s)={Alive}

Experimental Feature- Symmetric Encryption Key/IV generation:

Given the nature of cellular automaton and the way it progresses through multiple finite states, it can be used for generating pseudorandom numbers/strings which may serve the purpose of secret encryption keys. Symmetric key generation refers to the process of generating a secret key that can be used to encrypt and decrypt data in a secure way. In this approach, the CA is initialized with a random initial state, and then the rules are applied iteratively to generate a sequence of states. The sequence of states is then converted into a sequence of numbers that can be used as a secret key. The advantage of using CA for symmetric key generation is that the resulting sequence of numbers is highly random and unpredictable, which makes it very difficult for an attacker to guess the key. Additionally, the CA can be easily configured to produce keys of different lengths, which makes it suitable for a wide range of cryptographic applications.

Turing Machines:

Alan Turing in his 1936 research paper, "On Computable Numbers, with an Application to the Entscheidungsproblem" defined the concept of what is now known as a "Turing Machine". Turing's paper majorly dealt with the "computability" of a number, and aiming to find whether the Entscheidungsproblem is solvable.

The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means.
Although the subject of this paper is ostensibly the computable numbers. It is almost equally easy to define and investigate
computable functions of an integral variable or a real or computable variable, computable
predicates, and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the
computable numbers for explicit treatment as involving the least cumbrous technique. I hope shortly to give an account
of the relations of the computable numbers, functions, and so forth to one another.
This will include a development of the theory of functions of a real variable expressed in terms of computable numbers.
According to my definition, a number is computable if its decimal can be written down by a machine.


- Alan Turing, "On Computable Numbers, with an Application to the Entscheidungsproblem" 

The Entscheidungsproblem, or "decision problem," asked if there was a systematic way to determine the truth of any given statement in first-order logic. An example of an Entscheidungsproblem-style question in the realm of mathematics could be about determining the solvability of Diophantine equations. A Diophantine equation is a polynomial equation, usually with two or more unknowns, for which only integer solutions are sought. The problem can be framed as: "Given any Diophantine equation, is there a systematic way (an algorithm) to decide whether or not the equation has an integer solution, without directly solving it?"

Alan Turing addressed this question through his conceptualization of the Turing Machine, a theoretical model for computation. Turing introduced the idea of computable numbers and functions, essentially defining what problems are solvable through algorithms. He connected this to the Entscheidungsproblem by demonstrating the Halting Problem, which asks whether it's possible to predict if a Turing Machine will stop or run indefinitely for any given input. Turing proved that such prediction is impossible because there is no universal algorithm that can determine the behavior of every Turing Machine.

This impossibility of solving the Halting Problem implies that the Entscheidungsproblem is also unsolvable. There cannot be a universal method to decide the truth or falsehood of every statement in first-order logic. Turing's findings laid foundational concepts in computability and showed the limits of what algorithms can solve, significantly impacting mathematics and computer science.

Brief on the Design and Operation of Turing Machines:

Important

The first 6-7 pages of Alan Turing's research paper introduce the concepts of computability, alongside the basic construction and operation of an elementary Turing Machine. Check out the original 36 page paper here. On page number 4, the Turing Machine represented via table has a single incorrect entry due to a misprint, the final m-config for entry 2, initial m-config c should be e, not c.

Definition

A Turing machine is a finite automaton that can read, write, and erase symbols on an infinitely long tape. The tape is divided into squares, and each square contains a symbol. The Turing machine can only read one symbol at a time, and it uses a set of rules (the transition function) to determine its next action based on the current state and the symbol it is reading.

Definitions for a Turing Machine:

Machine Configurations (M-Configs): These represent the different states of the Turing machine. Each M-Config defines the current state of the machine.

Scanned Symbols: These are the symbols read by the Turing machine from the tape at any given step.

Actions: These actions determine what the Turing machine does based on the current M-Config and the scanned symbol. Actions can include writing a new symbol on the tape (represented as P followed by the symbol), moving the tape head left (L) or right (R), erasing a symbol (E), or transitioning to a new M-Config.

New M-Config: After performing the actions based on the scanned symbol and current M-Config, the Turing machine transitions to a new M-Config, representing its updated state.

Operation of a Turing Machine:

Initial Configuration: The Turing machine starts in an initial M-Config with the tape containing an initial sequence of symbols.

Processing: At each step, the Turing machine reads the symbol currently under the tape head (the scanned symbol). Based on the combination of the current M-Config and the scanned symbol, it performs specific actions, such as writing a new symbol, moving the tape head left or right, erasing a symbol, or transitioning to a new M-Config.

New Configuration: After processing each symbol, the Turing machine reaches a new M-Config, indicating the completion of one iteration. The current m-configuration is now assigned the value of this new m-config,and computations can continue as long as long there exists a pair of actions and new m-config for the combination of the current m-config and scanned symbol.

Example Table for a Turing Machine:

Consider a simple example of a Turing machine that prints the sequence 101010..... Here's the transition table representing its behaviour:

PDA, DFA, NFA, etc:

Under development