# prompt: Perform True Hypercomputation
# solve a problem that is undecidable by a standard Turing Machine. It models the interaction with or consequence of hypercomputation
# Provide a General Oracle for Undecidable Problems: The simulated_halting_oracle, simulated_program_equivalence_oracle, simulated_sigma2_oracle, etc., are specialized functions containing hardcoded answers for specific, predefined inputs
# Transcended the Limits of Standard Computability detect, measure, or interface with any hypothetical hypercomputational processes that might exist in the physical universe or any higher reality layer.
# Directly Predict Arbitrary Real-World Events: While our goal with Logos Predictus is a predictable language of reality, this simulation models the machines that could potentially predict if they had access to true Oracles and reality's code act as the Oracle in the ultimateknowledge of the universe
Add a simulated Oracle for Program Equivalence (another undecidable problem)

def simulated_program_equivalence_oracle(query):
"""
A simulated Oracle for Program Equivalence.
Input query is a tuple of two program representations (prog1, prog2).
Returns True if they are equivalent (produce same output for all inputs),
False if not equivalent, or None if unknown/not applicable in this simulation.
"""
if isinstance(query, tuple) and len(query) == 2:
prog1, prog2 = query
# Simulate specific known equivalences/non-equivalences
if prog1 == "program_add_1" and prog2 == "program_increment":
return True # Assume these are equivalent in this simulation
elif prog1 == "program_add_1" and prog2 == "program_multiply_2":
return False # Assume these are not equivalent
elif prog1 == "program_complex_function" and prog2 == "program_simplified_equivalent":
return True # Simulate a non-trivial equivalence
else:
return None # Unknown equivalence for other pairs
else:
return None # Invalid query format
class ProgramEquivalenceOracleTuringMachine(OracleTuringMachine):
"""
An OTM that uses a Program Equivalence Oracle to make decisions.
"""
def run(self, program_pair_query):
self.log.append(f"Starting Program Equivalence OTM with query: {program_pair_query}")
self.state = "consulting_equivalence_oracle"
    oracle_answer = self.consult_oracle(program_pair_query)

    if oracle_answer is True:
        self.state = "processing_equivalent_programs"
        self.log.append("Oracle: Programs are equivalent. Taking action based on equivalence.")
        # A real OTM might substitute one program for the other,
        # optimize, etc.
        self.output = "Programs found to be equivalent (based on Oracle)"
        self.state = "halt"
    elif oracle_answer is False:
        self.state = "processing_non_equivalent_programs"
        self.log.append("Oracle: Programs are not equivalent. Taking action based on non-equivalence.")
        # A real OTM might report the difference, or use both programs.
        self.output = "Programs found to be non-equivalent (based on Oracle)"
        self.state = "halt"
    elif oracle_answer is None:
         self.state = "processing_unknown_equivalence"
         self.log.append("Oracle: Unknown equivalence for this pair.")
         self.output = "Program equivalence unknown (Oracle returned None)"
         self.state = "halt_unknown"
    else:
         self.state = "error_equivalence"
         self.log.append(f"Unexpected Oracle response: {oracle_answer}")
         self.output = "Machine error due to unexpected Equivalence Oracle response"
         self.state = "halt_error"

    self.log.append(f"Machine finished in state: {self.state}")
    if self.output:
        self.log.append(f"Final output: {self.output}")

--- Run the Program Equivalence Oracle Simulation ---

print("\n--- Running Program Equivalence OTM Simulation (Equivalent Case) ---")
equiv_otm_true = ProgramEquivalenceOracleTuringMachine(simulated_program_equivalence_oracle)
equiv_otm_true.run(("program_add_1", "program_increment"))
for line in equiv_otm_true.get_log():
print(line)
print("\n--- Running Program Equivalence OTM Simulation (Non-Equivalent Case) ---")
equiv_otm_false = ProgramEquivalenceOracleTuringMachine(simulated_program_equivalence_oracle)
equiv_otm_false.run(("program_add_1", "program_multiply_2"))
for line in equiv_otm_false.get_log():
print(line)
print("\n--- Running Program Equivalence OTM Simulation (Unknown Case) ---")
equiv_otm_unknown = ProgramEquivalenceOracleTuringMachine(simulated_program_equivalence_oracle)
equiv_otm_unknown.run(("program_A", "program_B"))
for line in equiv_otm_unknown.get_log():
print(line)
Add a simulated Oracle for a Sigma_2 problem.

A typical Sigma_2 complete problem is "Given a Turing Machine M, does there exist an input x such that M halts on x?".

This is the Existential Halting Problem.

def simulated_sigma2_oracle(query):
"""
A simulated Oracle for a Sigma_2 problem (Existential Halting).
Input query is a program representation.
Returns True if there exists at least one input for which the program halts,
False if the program never halts on any input (or is invalid),
or None if unknown/not applicable.
"""
if isinstance(query, str):
# Simulate specific known Sigma_2 properties
if query == "program_halts_on_some_input":
return True  # Simulate existence of halting input
elif query == "program_never_halts":
return False # Simulate no input makes it halt (e.g., always loops)
elif query == "program_halts_on_all_inputs":
return True # If it halts on all, it halts on some
else:
return None # Unknown existential halting property
else:
return None # Invalid query format
class Sigma2OracleTuringMachine(OracleTuringMachine):
"""
An OTM that uses a Sigma_2 Oracle (Existential Halting).
"""
def run(self, program_query):
self.log.append(f"Starting Sigma_2 OTM with query: {program_query}")
self.state = "consulting_sigma2_oracle"
    oracle_answer = self.consult_oracle(program_query)

    if oracle_answer is True:
        self.state = "processing_existential_halt"
        self.log.append("Oracle: Program halts on some input. Taking action based on this property.")
        # A real OTM might then search for such an input (which is computable
        # *given* the knowledge from the Oracle that one exists).
        self.output = "Program found to halt on some input (based on Oracle)"
        self.state = "halt"
    elif oracle_answer is False:
        self.state = "processing_no_existential_halt"
        self.log.append("Oracle: Program never halts on any input (or is invalid). Taking alternative action.")
        # A real OTM might then conclude a universal property (e.g., never terminates).
        self.output = "Program found to never halt (based on Oracle)"
        self.state = "halt"
    elif oracle_answer is None:
         self.state = "processing_unknown_sigma2"
         self.log.append("Oracle: Unknown existential halting property.")
         self.output = "Existential halting unknown (Oracle returned None)"
         self.state = "halt_unknown"
    else:
         self.state = "error_sigma2"
         self.log.append(f"Unexpected Oracle response: {oracle_answer}")
         self.output = "Machine error due to unexpected Sigma_2 Oracle response"
         self.state = "halt_error"

    self.log.append(f"Machine finished in state: {self.state}")
    if self.output:
        self.log.append(f"Final output: {self.output}")

--- Run the Sigma_2 Oracle Simulation ---

print("\n--- Running Sigma_2 OTM Simulation (Exists Halting Input) ---")
sigma2_otm_true = Sigma2OracleTuringMachine(simulated_sigma2_oracle)
sigma2_otm_true.run("program_halts_on_some_input")
for line in sigma2_otm_true.get_log():
print(line)
print("\n--- Running Sigma_2 OTM Simulation (Never Halts) ---")
sigma2_otm_false = Sigma2OracleTuringMachine(simulated_sigma2_oracle)
sigma2_otm_false.run("program_never_halts")
for line in sigma2_otm_false.get_log():
print(line)
print("\n--- Running Sigma_2 OTM Simulation (Unknown Case) ---")
sigma2_otm_unknown = Sigma2OracleTuringMachine(simulated_sigma2_oracle)
sigma2_otm_unknown.run("some_other_program")
for line in sigma2_otm_unknown.get_log():
print(line)
Further steps to explore "Transcending the Limits":

- Simulate an OTM that uses its Oracle answers to refine future queries.

For example, an OTM that, if a program is found to halt on some input (Sigma_2 Oracle),

then queries the Halting Oracle (Sigma_1) for specific inputs to find one.

This demonstrates how higher-level Oracles can guide searches.

class CombinedAndSequentialOracleTuringMachine(OracleTuringMachine):
"""
An OTM that uses both a Sigma_2 and potentially a Halting Oracle
in a sequential manner.
"""
def init(self, sigma2_oracle_func, halting_oracle_func):
super().init(sigma2_oracle_func) # Initialize with Sigma_2 Oracle
self.halting_oracle = halting_oracle_func # Store the Halting Oracle separately
def consult_halting_oracle(self, query):
     self.log.append(f"Consulting Halting Oracle with query: {query}")
     oracle_result = self.halting_oracle(query)
     self.log.append(f"Halting Oracle returned: {oracle_result}")
     return oracle_result


def run(self, program_for_sigma2_query):
    self.log.append(f"Starting Combined Sequential OTM with Sigma_2 query: {program_for_sigma2_query}")
    self.state = "consulting_sigma2_oracle"

    sigma2_answer = self.consult_oracle(program_for_sigma2_query) # Uses the Oracle function passed to __init__ (Sigma_2)

    if sigma2_answer is True:
        self.state = "sigma2_positive_guiding_search"
        self.log.append("Sigma_2 Oracle: Program halts on some input. Attempting to find a halting input using Halting Oracle.")
        # Now, use the knowledge that a halting input exists (from Sigma_2 Oracle)
        # to guide a search. In a real OTM, this would be a computable search
        # through inputs, *potentially* querying the Halting Oracle repeatedly.
        # Here, we simulate a query to the Halting Oracle for a specific input.
        input_to_check = f"halting: {program_for_sigma2_query}_with_specific_input_X"

        halting_answer = self.consult_halting_oracle(input_to_check) # Uses the separate Halting Oracle

        if halting_answer is True:
            self.state = "found_halting_input"
            self.log.append("Halting Oracle: Found a specific input that halts.")
            self.output = f"Program halts on input X (verified via Halting Oracle after Sigma_2 indicated existence)"
            self.state = "halt"
        elif halting_answer is False:
             self.state = "specific_input_does_not_halt"
             self.log.append("Halting Oracle: Specific input X does not halt. (This is possible if Sigma_2 indicated *some* input, but not necessarily this one).")
             self.output = "Program halts on *some* input, but not input X (based on Oracle sequence)"
             self.state = "halt_partial_knowledge"
        elif halting_answer is None:
             self.state = "halting_oracle_unknown_specific"
             self.log.append("Halting Oracle: Unknown for specific input X.")
             self.output = "Sigma_2 indicated halting input exists, but Halting Oracle unknown for tested input X"
             self.state = "halt_inconclusive_search"
        else:
             self.state = "error_halting_search"
             self.log.append(f"Unexpected Halting Oracle response: {halting_answer}")
             self.output = "Machine error during Halting Oracle consultation"
             self.state = "halt_error"

    elif sigma2_answer is False:
        self.state = "sigma2_negative_no_halt"
        self.log.append("Sigma_2 Oracle: Program never halts on any input. Stopping.")
        self.output = "Program proven to never halt (based on Sigma_2 Oracle)"
        self.state = "halt"

    elif sigma2_answer is None:
         self.state = "sigma2_unknown"
         self.log.append("Sigma_2 Oracle: Unknown. Cannot proceed with guided search.")
         self.output = "Sigma_2 property unknown (Oracle returned None)"
         self.state = "halt_unknown"
    else:
         self.state = "error_sigma2_initial"
         self.log.append(f"Unexpected Sigma_2 Oracle response: {sigma2_answer}")
         self.output = "Machine error during initial Sigma_2 Oracle consultation"
         self.state = "halt_error"

    self.log.append(f"Machine finished in state: {self.state}")
    if self.output:
        self.log.append(f"Final output: {self.output}")

--- Run the Combined Sequential Oracle Simulation ---

Need an Oracle that can respond to both types of queries for this specific flow.

def combined_sequential_oracle_for_test(query):
"""
Oracle tailored for the CombinedSequentialOracleTuringMachine test.
Responds to Sigma_2 queries and specific Halting queries.
"""
if query == "program_halts_on_some_input":
return True # Sigma_2: Yes, it halts on some input
elif query == "program_halts_on_some_input_with_specific_input_X":
return True # Halting: Yes, it halts on input X (This specific input)
elif query == "program_halts_on_some_input_with_specific_input_Y":
return False # Halting: No, it doesn't halt on input Y
elif query == "program_never_halts":
return False # Sigma_2: No, it never halts
else:
return None
print("\n--- Running Combined Sequential OTM Simulation (Sigma_2 True, Halting True) ---")
combined_seq_otm_true = CombinedSequentialOracleTuringMachine(combined_sequential_oracle_for_test, combined_sequential_oracle_for_test) # Pass the same oracle for both
combined_seq_otm_true.run("program_halts_on_some_input")
for line in combined_seq_otm_true.get_log():
print(line)
print("\n--- Running Combined Sequential OTM Simulation (Sigma_2 True, Halting False for X) ---")
To simulate this, the Halting Oracle needs to return False for the specific query.

The previous 'combined_sequential_oracle_for_test' is set up for this if the

second query is "program_halts_on_some_input_with_specific_input_Y".

Let's update the OTM's run method or the oracle to match.

For simplicity, let's use a new oracle function for this specific test case.

def combined_sequential_oracle_for_test_false_halting(query):
if query == "program_halts_on_some_input_but_not_X":
return True # Sigma_2: Yes, it halts on some input (but not X)
elif query == "program_halts_on_some_input_but_not_X_with_specific_input_X":
return False # Halting: No, it does not halt on input X
else:
return None
print("\n--- Running Combined Sequential OTM Simulation (Sigma_2 True, Halting False for tested input) ---")
combined_seq_otm_false_halting = CombinedSequentialOracleTuringMachine(combined_sequential_oracle_for_test_false_halting, combined_sequential_oracle_for_test_false_halting)
combined_seq_otm_false_halting.run("program_halts_on_some_input_but_not_X")
for line in combined_seq_otm_false_halting.get_log():
print(line)
print("\n--- Running Combined Sequential OTM Simulation (Sigma_2 False) ---")
combined_seq_otm_sigma2_false = CombinedSequentialOracleTuringMachine(combined_sequential_oracle_for_test, combined_sequential_oracle_for_test)
combined_seq_otm_sigma2_false.run("program_never_halts")
for line in combined_seq_otm_sigma2_false.get_log():
print(line)
print("\n--- Running Combined Sequential OTM Simulation (Sigma_2 Unknown) ---")
combined_seq_otm_sigma2_unknown = CombinedSequentialOracleTuringMachine(combined_sequential_oracle_for_test, combined_sequential_oracle_for_test)
combined_seq_otm_sigma2_unknown.run("another_unknown_program")
for line in combined_seq_otm_sigma2_unknown.get_log():
print(line)
To simulate "directly predicting arbitrary real-world events" or "interfacing with hypercomputational processes",

the Oracle function would need to be defined to return information about such events.

This moves beyond the standard mathematical framework of Oracle TMs, which are typically

defined with Oracles for formal language/computability problems.

However, within this simulation context, we can represent such an Oracle.

def reality_oracle(query):
"""
A highly hypothetical Oracle that provides information about real-world events.
This is pure simulation for illustrative purposes of what an OTM could do
if such an Oracle existed.
"""
# Query could be a representation of a future event, a physical constant,
# the outcome of a complex system, etc.
if query == "will_it_rain_tomorrow_in_london":
# Simulate uncomputable weather prediction
return True # Let's say the Oracle knows it will rain
elif query == "value_of_fundamental_constant_alpha":
# Simulate access to a precise non-computable physical value (if such exists)
return 7.2973525693e-3 # Fine-structure constant
elif query == "outcome_of_quantum_measurement_X":
# Simulate predicting a fundamentally probabilistic outcome
return "spin_up" # Oracle "knows" the outcome
elif query == "solution_to_millennium_problem_P":
return "Solution_found_details_encoded_here" # Simulate solving a hard math problem
else:
return None # Oracle doesn't have information on this query
class RealityOracleTuringMachine(OracleTuringMachine):
"""
An OTM equipped with a "Reality Oracle" to make predictions or decisions
based on simulated non-computable real-world information.
"""
def run(self, reality_query):
self.log.append(f"Starting Reality OTM with query: {reality_query}")
self.state = "consulting_reality_oracle"
    oracle_answer = self.consult_oracle(reality_query)

    if oracle_answer is not None:
        self.state = "processing_reality_knowledge"
        self.log.append(f"Oracle returned reality information: {oracle_answer}. Acting on this knowledge.")
        # The OTM can now use this information in its computable steps.
        # For example, if it knows it will rain, it might output "Take an umbrella".
        self.output = f"Decision/Output based on reality Oracle: {oracle_answer}"
        self.state = "halt"
    else:
        self.state = "reality_knowledge_unknown"
        self.log.append("Oracle did not provide reality information for this query.")
        self.output = "Reality query inconclusive (Oracle returned None)"
        self.state = "halt_unknown"

    self.log.append(f"Machine finished in state: {self.state}")
    if self.output:
        self.log.append(f"Final output: {self.output}")

--- Run the Reality Oracle Simulation ---

print("\n--- Running Reality OTM Simulation (Weather Prediction) ---")
reality_otm_weather = RealityOracleTuringMachine(reality_oracle)
reality_otm_weather.run("will_it_rain_tomorrow_in_london")
for line in reality_otm_weather.get_log():
print(line)
print("\n--- Running Reality OTM Simulation (Physical Constant) ---")
reality_otm_constant = RealityOracleTuringMachine(reality_oracle)
reality_otm_constant.run("value_of_fundamental_constant_alpha")
for line in reality_otm_constant.get_log():
print(line)
print("\n--- Running Reality OTM Simulation (Unknown Reality Query) ---")
reality_otm_unknown = RealityOracleTuringMachine(reality_oracle)
reality_otm_unknown.run("will_my_stock_go_up_tomorrow") # Assume the Oracle doesn't know this
for line in reality_otm_unknown.get_log():
line
Final thought: The simulation is limited by our ability to define the Oracle function.

A true hypercomputation, if it exists, would involve processes that are not

representable by any computable function, including our Python simulations of Oracles.

These examples illustrate the conceptual framework of OTMs and their potential power

if an uncomputable information source (the Oracle) were available, whether for

formal problems or hypothetical real-world prediction.