Hone logo
Hone
Problems

Type-Level Turing Machine Simulation in TypeScript

This challenge involves creating a type-level virtual machine that can simulate a simplified Turing machine using TypeScript's advanced type system. This exercise explores the power and limitations of TypeScript for purely static computation and demonstrates how complex logic can be encoded within types.

Problem Description

Your task is to implement a type-level simulation of a basic Turing machine. This machine will operate on a tape, a set of states, and a program (transition rules). The simulation should determine if a given input tape, starting in a specific state, halts in an accepting state after processing according to the provided program.

Key Requirements:

  1. Tape Representation: The tape can be thought of as an infinite sequence of symbols. For practical purposes, you'll represent a finite portion of the tape, with the ability to extend it conceptually. Symbols will be represented by union types (e.g., 0 | 1 | Blank).
  2. State Representation: States will be represented by distinct types. You'll need a special HaltState and potentially an AcceptState.
  3. Program (Transition Rules): The program defines how the machine behaves. Each rule will map a current state and the symbol under the head to a new state, a symbol to write, and a direction to move (Left or Right).
  4. Head Position: The type-level machine needs to track the position of the read/write head on the tape.
  5. Simulation: The core of the challenge is to create a recursive type that models the execution steps of the Turing machine. This type will take the current machine configuration (state, tape, head position) and produce the configuration after one step, or the final halting configuration.

Expected Behavior:

The simulation should start with an initial tape, an initial state, and a head position. It should repeatedly apply the transition rules until the machine enters a designated HaltState. If the machine enters an AcceptState (a specific type of halt), the simulation should indicate success. Otherwise, it should indicate failure or non-halting (though for this challenge, we'll assume halting for simplicity in the provided inputs).

Edge Cases:

  • Empty tape.
  • Moving off the "defined" ends of the tape (implying extending the tape with Blank symbols).
  • Transitions leading to undefined states or symbols for the current configuration.

Examples

Example 1: Simple Halting Machine

This machine halts if it encounters a 1.

// Symbols
type Symbol = 0 | 1 | Blank;
const Blank = Symbol('Blank'); // Using Symbol for unique blank value

// States
type State = Initial | Halt;
type Initial = 'Initial';
type Halt = 'Halt';

// Program: { currentState: { [currentSymbol]: { nextState: ..., writeSymbol: ..., move: ... } } }
type Program = {
  Initial: {
    0: { nextState: 'Initial', writeSymbol: 0, move: 'R' };
    1: { nextState: 'Halt', writeSymbol: 1, move: 'R' };
    Blank: { nextState: 'Initial', writeSymbol: Blank, move: 'R' };
  };
};

// Tape: Represents a sequence of symbols. We'll use a tuple.
// Head position: Index in the tuple.
// Initial configuration:
type InitialTape = [0, 0, 1, 0];
type InitialHeadPosition = 0;
type StartState = 'Initial';

// Simulate one step (conceptually):
// Current: State='Initial', Tape=[0, 0, 1, 0], Head=0
// Read Symbol: 0
// Rule for Initial + 0: { nextState: 'Initial', writeSymbol: 0, move: 'R' }
// New Config: State='Initial', Tape=[0, 0, 1, 0], Head=1 (writtenSymbol is same)

// Simulate one step (conceptually):
// Current: State='Initial', Tape=[0, 0, 1, 0], Head=1
// Read Symbol: 0
// Rule for Initial + 0: { nextState: 'Initial', writeSymbol: 0, move: 'R' }
// New Config: State='Initial', Tape=[0, 0, 1, 0], Head=2

// Simulate one step (conceptually):
// Current: State='Initial', Tape=[0, 0, 1, 0], Head=2
// Read Symbol: 1
// Rule for Initial + 1: { nextState: 'Halt', writeSymbol: 1, move: 'R' }
// New Config: State='Halt', Tape=[0, 0, 1, 0], Head=3

// Machine halts because it reached 'Halt' state.
// The output should indicate success and the final state.
// Let's define the output type.
type Success = { halted: true; finalState: Halt; finalTape: InitialTape; finalHead: number };
type Failure = { halted: false }; // For non-halting or errors

// Expected Output for Example 1 (conceptual):
// Simulate(Program, InitialTape, InitialHeadPosition, StartState) -> Success

Example 2: Simple Incrementer (Binary)

This machine increments a binary number represented on the tape. It starts at the leftmost digit.

// Symbols
type Symbol = 0 | 1 | Blank;
const Blank = Symbol('Blank');

// States
type State = Initial | Carry | Halt;
type Initial = 'Initial';
type Carry = 'Carry';
type Halt = 'Halt';

// Program:
type IncrementProgram = {
  Initial: {
    0: { nextState: 'Halt', writeSymbol: 1, move: 'L' }; // Found 0, change to 1, halt moving left
    1: { nextState: 'Carry', writeSymbol: 0, move: 'L' }; // Found 1, change to 0, need to carry left
    Blank: { nextState: 'Halt', writeSymbol: 1, move: 'L' }; // Empty tape, write 1, halt moving left
  };
  Carry: {
    0: { nextState: 'Halt', writeSymbol: 1, move: 'L' }; // Carry to 0 becomes 1, halt moving left
    1: { nextState: 'Carry', writeSymbol: 0, move: 'L' }; // Carry to 1 becomes 0, continue carrying left
    Blank: { nextState: 'Carry', writeSymbol: 1, move: 'L' }; // Carry off left edge, write 1, continue carrying
  };
};

// Input: A binary number represented as a tuple, head at the rightmost digit.
type InputTape1 = [1, 0, 1]; // Represents binary 101 (5)
type InputHeadPosition1 = 2; // Start at the last digit
type StartState1 = 'Initial';

// Expected behavior:
// Initial: State='Initial', Tape=[1, 0, 1], Head=2
// Read: 1 -> Write: 0, Move: L, NextState: Carry
// Config: State='Carry', Tape=[1, 0, 0], Head=1

// Current: State='Carry', Tape=[1, 0, 0], Head=1
// Read: 0 -> Write: 1, Move: L, NextState: Halt
// Config: State='Halt', Tape=[1, 1, 0], Head=0

// Machine halts in Halt state.
// Expected Output: Success { halted: true, finalState: 'Halt', finalTape: [1, 1, 0], finalHead: 0 }

// Input: A binary number that requires carrying off the left edge.
type InputTape2 = [1, 1, 1]; // Represents binary 111 (7)
type InputHeadPosition2 = 2;
type StartState2 = 'Initial';

// Expected behavior:
// Initial: State='Initial', Tape=[1, 1, 1], Head=2
// Read: 1 -> Write: 0, Move: L, NextState: Carry
// Config: State='Carry', Tape=[1, 1, 0], Head=1

// Current: State='Carry', Tape=[1, 1, 0], Head=1
// Read: 1 -> Write: 0, Move: L, NextState: Carry
// Config: State='Carry', Tape=[1, 0, 0], Head=0

// Current: State='Carry', Tape=[1, 0, 0], Head=0
// Read: 1 -> Write: 0, Move: L, NextState: Carry
// Config: State='Carry', Tape=[0, 0, 0], Head=-1 (conceptually moving left of index 0)

// Current: State='Carry', Tape=[0, 0, 0], Head=-1
// Read: Blank (since head is off the defined tape to the left) -> Write: 1, Move: L, NextState: Carry
// Config: State='Carry', Tape=[1, 0, 0, 0], Head=-2 (conceptually)
// Wait, tape needs to expand or we need to handle negative indices by reading `Blank`.
// The representation of the tape and head needs to be flexible. Let's adjust:

// New Tape Representation: A tuple and a "buffer" for symbols to the left of the start.
// Or more simply, represent tape indices relative to a fixed point.
// For type-level, let's use a tuple and assume we can shift indices.
// A common approach is to use arrays/tuples and handle out-of-bounds by reading `Blank`.
// We'll need helper types to read/write to this tape.

// Revised Tape Model: A tuple representing the central part and implicit Blanks outside.
// Head position is an integer. Negative indices imply moving left of the tuple start.

// Let's refine Example 2 for a more concrete type-level implementation.

// Symbols
type S = 0 | 1 | B; // B for Blank
const B = Symbol('Blank');

// States
type St = Init | C | H;
type Init = 'Init';
type C = 'Carry';
type H = 'Halt';

// Program
type IncProg = {
  [Init]: {
    0: { ns: H; ws: 1; m: 'L' };
    1: { ns: C; ws: 0; m: 'L' };
    B: { ns: H; ws: 1; m: 'L' };
  };
  [C]: {
    0: { ns: H; ws: 1; m: 'L' };
    1: { ns: C; ws: 0; m: 'L' };
    B: { ns: C; ws: 1; m: 'L' };
  };
};

// Tape: Represented by a tuple of symbols.
// Head position: An integer index. A negative index means the head is to the left of the first element.
type Tape = (0 | 1 | B)[];

// Helper to get symbol at head position
type GetSymbol<T extends Tape, H extends number> =
  H extends 0 ? T[0] :
  H extends -1 ? B : // Moving left of the tape's start
  H extends -2 ? B : // And so on...
  H extends T['length'] ? B : // Moving right of the tape's end
  T[H];

// Helper to set symbol at head position
type SetSymbol<T extends Tape, H extends number, S extends (0 | 1 | B)> =
  H extends 0 ? SetFirst<T, S> :
  H extends -1 ? Prepend<T, S> : // Prepend if moving left off the start
  H extends T['length'] ? Append<T, S> : // Append if moving right off the end
  SetAtIndex<T, H, S>;

type SetFirst<T extends Tape, S extends (0 | 1 | B)> =
  T extends [infer _, ...infer Rest] ? [S, ...Rest] : [S]; // Handle empty tape

type Prepend<T extends Tape, S extends (0 | 1 | B)> = [S, ...T];
type Append<T extends Tape, S extends (0 | 1 | B)> = [...T, S];

type SetAtIndex<T extends Tape, H extends number, S extends (0 | 1 | B)> =
  H extends 0 ? [S, ...T] : // Base case: H=0, set first element
  H extends 1 ? [T[0], S, ...Tail<T>] : // H=1
  H extends 2 ? [T[0], T[1], S, ...Tail<Tail<T>>] : // H=2 - Need a recursive way to do this
  // ... This recursive approach for SetAtIndex is complex.
  // Alternative: use a List type that supports indexing and modification.
  // For this challenge, let's assume we can manage tape modification conceptually or use a simplified helper.

  // A more practical approach for type-level tuples:
  // Manipulating tuples by index at arbitrary positions is difficult.
  // Let's use a structure that is easier to modify.
  // How about: { left: Tape, current: Symbol, right: Tape } where head is `current`?
  // Or { tape: Tape, head: number } where head can be negative.

// Let's stick to the { tape: Tape, head: number } model for simplicity and handle
// implicit Blanks by reading the tape at an index and returning Blank if out of bounds.

type ReadTape<T extends Tape, H extends number> =
  T extends [] ? B : // Empty tape always Blank
  H extends -1 ? B : // Moving left off tape start
  H extends T['length'] ? B : // Moving right off tape end
  H extends infer I ? (I extends number ? T[I] : B) : B; // Standard index access

// This 'SetTape' is still problematic with arbitrary indices.
// Let's simplify the problem statement: the tape provided is finite but we can conceptually
// extend it with Blanks. Moving left of index 0 reads Blank, moving right of the last index reads Blank.
// Writing when head is negative or at T.length needs to conceptually grow the tape.

// For type-level simulation, we often need to "shift" or "pad" the tape.
// Let's redefine tape manipulation:

type PadLeft<T extends Tape, S extends (0 | 1 | B)> = [S, ...T];
type PadRight<T extends Tape, S extends (0 | 1 | B)> = [...T, S];

type WriteTape<T extends Tape, H extends number, Sym extends (0 | 1 | B)> =
  H extends 0 ? (T extends [] ? [Sym] : [Sym, ...Tail<T>]) : // Write at beginning
  H extends -1 ? PadLeft<T, Sym> : // Write to the left of beginning
  H extends T['length'] ? PadRight<T, Sym> : // Write to the right of end
  H extends infer I ? (I extends number ? WriteAtIndex<T, I, Sym> : T) : T; // Write at arbitrary index

// Need WriteAtIndex for tuples which is tricky.
// Alternative: Represent tape as an object with indices as keys? No, must be types.
// Let's assume a helper `UpdateTupleAtIndex` exists for type-level operations.
// Or, let's simplify the tape movement: we only move left/right by 1.
// If we move left from index 0, the new head is -1, and we read Blank.
// If we move right from index N, the new head is N+1, and we read Blank.
// When writing at -1, we should prepend `Sym` and set head to 0.
// When writing at N+1, we should append `Sym` and set head to N+1.

// Let's redefine the state for the VM:
type VMState<Prog, Tape extends Tape, Head extends number, State extends keyof Prog> = {
  program: Prog;
  tape: Tape;
  head: Head;
  state: State;
};

// Helper to simulate one step
type Step<VM extends VMState<any, any, any, any>> =
  VM['state'] extends keyof VM['program'] ?
    VM['program'][VM['state']] extends { [key in ReadTape<VM['tape'], VM['head']>]: { ns: infer NS; ws: infer WS; m: infer M } } ?
      NS extends string ? // Check if NS is a valid state name
        WS extends (0 | 1 | B) ? // Check if WS is a valid symbol
          M extends ('L' | 'R') ? // Check if M is a valid move
            // Calculate next head position
            Head extends -1 ? (M extends 'L' ? -2 : 0) : // If current head is -1, moving L goes to -2, R goes to 0
            Head extends VM['tape']['length'] ? (M extends 'L' ? VM['tape']['length'] - 1 : VM['tape']['length'] + 1) : // If current head is at end, moving L goes to N-1, R goes to N+1
            (M extends 'L' ? Head - 1 : Head + 1) // General case
            : // invalid move
            // Error: Invalid move
            { error: 'InvalidMove' }
          : // invalid write symbol
          // Error: Invalid write symbol
          { error: 'InvalidWriteSymbol' }
        : // invalid next state
        // Error: Invalid next state
        { error: 'InvalidNextState' }
      : // Symbol not found in program for current state
      // Halt or Error depending on convention. Let's halt as default.
      { halt: true; finalState: VM['state']; finalTape: VM['tape']; finalHead: VM['head'] }
    : // State not found in program
    // Halt or Error
    { halt: true; finalState: VM['state']; finalTape: VM['tape']; finalHead: VM['head'] }
  : // State not found in program
  // Halt or Error
  { halt: true; finalState: VM['state']; finalTape: VM['tape']; finalHead: VM['head'] };


// Let's redefine the SetTape to be more type-safe and manageable.
// We will assume the `tape` array is always valid and we pad it when needed.
type UpdateTape<T extends Tape, H extends number, Sym extends (0 | 1 | B)> =
  H extends 0 ? (T extends [] ? [Sym] : [Sym, ...Tail<T>]) :
  H extends -1 ? PadLeft<T, Sym> : // Prepend Sym if moving left off start
  H extends T['length'] ? PadRight<T, Sym> : // Append Sym if moving right off end
  WriteAtIndex<T, H, Sym>; // Generic write at index

// This `WriteAtIndex` is still the bottleneck.
// Let's try a simpler tape representation: a single tuple, and the head is an index.
// Out-of-bounds reads yield `B`. Out-of-bounds writes require padding.

// Helper to get symbol
type GetSymbolAt<T extends Tape, H extends number> =
  H < 0 ? B :
  H >= T['length'] ? B :
  T[H];

// Helper to write symbol - This must reconstruct the tuple.
// It's extremely inefficient and complex for arbitrary indices.
// Let's consider a simpler tape structure: A string of symbols? No, type-level.
// How about a fixed-size array and we detect out-of-bounds writes and return an error?
// Or, let's assume maximum tape expansion needed can be pre-determined or handled.

// For this challenge, let's simplify the simulation by assuming the tape can be conceptually
// extended but we need concrete type manipulations.
// Let's use a fixed-size "window" and offset, or explicitly pad.

// Let's simplify Tape and Head for this problem:
// Tape is a tuple. Head is a number.
// Reading at H < 0 or H >= T.length yields Blank.
// Writing at H < 0 or H >= T.length implies padding.

type ShiftLeftTape<T extends Tape> = T extends [infer _, ...infer Rest] ? Rest extends Tape ? Rest : [] : [];
type ShiftRightTape<T extends Tape> = T extends [...infer Init, infer _] ? Init extends Tape ? Init : [] : [];

type PaddedTape<T extends Tape, H extends number, Sym extends (0 | 1 | B)> =
  H extends 0 ? T : // Already at start
  H extends -1 ? PadLeft<T, Sym> : // Need to pad left
  H extends T['length'] ? PadRight<T, Sym> : // Need to pad right
  T; // Within bounds

type WriteToTape<T extends Tape, H extends number, Sym extends (0 | 1 | B)> =
  H extends 0 ? (T extends [] ? [Sym] : [Sym, ...Tail<T>]) :
  H extends -1 ? PadLeft<T, Sym> :
  H extends T['length'] ? PadRight<T, Sym> :
  WriteAtIndex<T, H, Sym>; // This is the tricky part.

// A common workaround for tuple modification is recursion or pre-defined helpers.
// For this challenge, we'll provide conceptual helpers.
// The core issue is `WriteAtIndex` for arbitrary `H`.

// Let's redefine `Step` using a specific tape model:
// Tape: tuple of symbols. Head: number.
// Reading: If head < 0 or head >= tape.length, read Blank.
// Writing: If head < 0, prepend Blank(s) and write. If head >= tape.length, append Blank(s) and write.

// Simplified Tape Update Helper (Conceptual - A real implementation needs more robust tuple manipulation)
type ReplaceAtIndex<T extends Tape, Index extends number, NewValue extends (0 | 1 | B)> =
  number extends Index ? T : // Avoid infinite recursion with unknown index
  Index extends 0 ? (T extends [] ? [NewValue] : [NewValue, ...Tail<T>]) :
  Index extends 1 ? (T extends [_, ...infer Rest] ? (Rest extends Tape ? [T[0], NewValue, ...Rest] : [T[0], NewValue]) : [NewValue]) :
  Index extends 2 ? (T extends [_, _, ...infer Rest] ? (Rest extends Tape ? [T[0], T[1], NewValue, ...Rest] : [T[0], T[1], NewValue]) : [NewValue]) :
  // ... This becomes very verbose. Let's abstract it.
  // Assume `InsertAt` and `UpdateAt` exist for type-level tuples.
  // For the purpose of this challenge, we can illustrate the logic.

// Let's use a concrete representation for state and tape that allows easier modification.
// State: { state: StateType, tape: TapeType, head: HeadPositionType }

// State Representation:
type MachineState<S extends string, T extends Tape, H extends number> = {
  state: S;
  tape: T;
  head: H;
};

// Helper to simulate one step, returning a new MachineState or a HALT signal.
type SimulateStep<Prog, CurrentState extends MachineState<any, any, any>> =
  CurrentState['state'] extends keyof Prog ?
    Prog[CurrentState['state']] extends { [Sym in GetSymbolAt<CurrentState['tape'], CurrentState['head']>]: { ns: infer NS; ws: infer WS; m: infer M } } ?
      NS extends string ? WS extends (0 | 1 | B) ? M extends ('L' | 'R') ?
        // Calculate next head position
        let nextHead: number;
        if (M === 'L') {
          nextHead = CurrentState['head'] - 1;
        } else {
          nextHead = CurrentState['head'] + 1;
        }

        // Calculate the new tape by writing WS at the current head
        // This is the core difficulty. Let's define a simplified write function.
        let nextTape: Tape;
        if (nextHead < 0) {
          // Need to prepend, and adjust head to 0
          nextTape = PadLeft(CurrentState['tape'], WS as (0 | 1 | B));
          nextHead = 0; // Head is now at the start of the new tape
        } else if (nextHead >= CurrentState['tape']['length']) {
          // Need to append
          nextTape = PadRight(CurrentState['tape'], WS as (0 | 1 | B));
          // nextHead remains the same, now pointing to the appended symbol
        } else {
          // Write within existing tape bounds
          nextTape = WriteAtIndex(CurrentState['tape'], CurrentState['head'], WS as (0 | 1 | B));
        }

        { state: NS; tape: nextTape; head: nextHead }
      : { halt: true; finalState: CurrentState['state']; finalTape: CurrentState['tape']; finalHead: CurrentState['head'] } // Invalid move
    : { halt: true; finalState: CurrentState['state']; finalTape: CurrentState['tape']; finalHead: CurrentState['head'] } // Invalid write symbol
  : { halt: true; finalState: CurrentState['state']; finalTape: CurrentState['tape']; finalHead: CurrentState['head'] } // Invalid next state
: // No rule for current state + symbol, so halt
{ halt: true; finalState: CurrentState['state']; finalTape: CurrentState['tape']; finalHead: CurrentState['head'] };

// The challenge: Implement the full recursive type to simulate the VM until halt.
// This will involve a type that calls SimulateStep repeatedly.

// Example 2 (Revised with a working model of tape manipulation):
type S2 = 0 | 1 | B;
const B2 = Symbol('Blank');

type St2 = Init | C | H;
type Init = 'Init';
type C = 'Carry';
type H = 'Halt';

type IncProg2 = {
  [Init]: {
    0: { ns: H; ws: 1; m: 'L' };
    1: { ns: C; ws: 0; m: 'L' };
    B: { ns: H; ws: 1; m: 'L' };
  };
  [C]: {
    0: { ns: H; ws: 1; m: 'L' };
    1: { ns: C; ws: 0; m: 'L' };
    B: { ns: C; ws: 1; m: 'L' };
  };
};

// Let's define a tape manipulation that is type-safe, though potentially verbose for deep operations.
// Instead of direct index manipulation, we'll pass the "relevant" part of the tape.
// We will assume the simulation does not require extreme tape expansion for the examples.

// Example Input:
type InputTapeEx2 = [1, 1, 1]; // Represents binary 111 (7)
type InputHeadEx2 = 2; // Start at the last digit
type StartStateEx2 = Init;

// Expected Output: Success { halted: true, finalState: 'Halt', finalTape: [1, 1, 0], finalHead: 0 }

// Example 3: Non-Halting Machine (Busy Beaver - simplified)
// This machine might not halt. For the purpose of this challenge,
// you might need to set a maximum step count to detect non-halting or
// assume inputs always halt within a reasonable type-computable depth.
// A common approach is a `steps` parameter in the recursive simulation type.

Constraints

  • The Turing machine program must be represented as a nested type.
  • The tape can be conceptually infinite, but your type-level representation will handle finite initial tapes and implicit Blank symbols for out-of-bounds access.
  • The head position is an integer.
  • The simulation must halt when the machine enters a designated HaltState.
  • The maximum depth of recursive type instantiation for simulation steps is a practical limitation. Consider a maximum number of steps to prevent infinite loops in type checking.
  • Symbol and State types should be distinct primitive types or union types.

Notes

  • This challenge is heavily focused on TypeScript's conditional types, mapped types, and recursive type definitions.
  • You'll need to carefully consider how to represent and manipulate the tape at the type level. Using tuples is a common approach.
  • Handling tape extension (writing beyond the initial bounds) will be a key part of the challenge. You may need helper types to prepend or append to tuples.
  • The performance of type-level computation can be slow, especially for deep recursion. Consider adding a steps counter to your VM state to limit the simulation depth and detect potential non-halting conditions.
  • The success of your implementation will be judged by its ability to correctly simulate a few known Turing machine examples and its adherence to the type-level computation paradigm.
Loading editor...
typescript