#!/usr/bin/env python """ Categorical Quantum Semantics Representation of "Someone's Lot in Life" ====================================================================== This example implements a precise mathematical representation of the phrase "someone's lot in life" using the lambeq categorical quantum linguistics approach. Requirements: - lambeq - discopy - pytket - pytket-qiskit - matplotlib - numpy """ import numpy as np import matplotlib.pyplot as plt from typing import List, Dict, Tuple # Import lambeq components for categorical quantum NLP from lambeq import ( BobcatParser, AtomicType, IQPAnsatz, TketModel, SpiderAnsatz, discopy, remove_cups, tokenize ) from discopy.quantum import Circuit, Id, Box, Tensor, Swap, Ket, Bra, sqrt, H, X, Z, CX, Measure def main(): """Main execution function demonstrating categorical quantum linguistics representation of 'someone's lot in life'.""" print("=== Categorical Quantum Semantics: 'Someone's Lot in Life' ===") # 1. Pregroup Grammar Typing print("\n1. Pregroup Grammar Typing") print("-------------------------") sentence = "someone's lot in life" # Initialize the parser - Bobcat is lambeq's CCG parser parser = BobcatParser(verbose='text') # Parse the sentence diagram = parser.sentence2diagram(sentence) # Display the pregroup types print(f"Sentence: '{sentence}'") print("\nTypes assigned by pregroup grammar:") words = sentence.split() types_explanation = { "someone's": "n⊗n^r", # Possessive noun phrase "lot": "n", # Noun "in": "n⊗n^r⊗s^l⊗s", # Preposition (noun to noun/sentence modifier) "life": "n" # Noun } for word, type_explanation in types_explanation.items(): print(f" - '{word}': {type_explanation}") print("\nPregroup reduction:") print("someone's lot in life: (n⊗n^r)⊗n⊗(n⊗n^r⊗s^l⊗s)⊗n → s") print("This reduces to: n⊗n^r⊗n⊗n⊗n^r⊗s^l⊗s⊗n → s") print("With cups and caps, this normalizes to the sentence type 's'") # 2. Tensor Network Representation print("\n2. Tensor Network Representation") print("------------------------------") print("Displaying the tensor network (DisCoPy diagram):") display_diagram = diagram.normal_form() # Note: In a Jupyter notebook, this would render graphically print("Tensor diagram structure (textual representation):") print(display_diagram) # 3. Quantum Circuit Translation print("\n3. Quantum Circuit Translation") print("----------------------------") # Define ansatz for mapping to quantum circuit ansatz = IQPAnsatz({AtomicType.NOUN: 1, AtomicType.SENTENCE: 1}) # Create the circuit circuit = ansatz(diagram) print("Quantum circuit representation:") print(circuit) print("\nCircuit analysis:") print(f"Number of qubits: {circuit.n_qubits}") print(f"Number of operations: {len(circuit.data)}") # 4. Normalization with concrete vectors and matrices print("\n4. Normalization with Concrete Vectors and Matrices") print("------------------------------------------------") # Define the vector spaces for each word # We'll use 2-dimensional vector spaces for simplicity n_dim = 2 # Dimensionality for noun spaces s_dim = 2 # Dimensionality for sentence spaces # Define concrete meanings (density matrices) for each word # These would typically come from training, but we'll define them explicitly word_matrices = { "someone's": np.array([[0.7, 0.2], [0.2, 0.3]]), # Possessive: uncertainty, dependency "lot": np.array([[0.5, 0.5], [0.5, 0.5]]), # Lot: equal possibilities "in": np.array([[0.9, 0.1], [0.1, 0.1]]), # In: strong contextual binding "life": np.array([[0.4, 0.6], [0.6, 0.6]]) # Life: existential space } print("Word representations as density matrices:") for word, matrix in word_matrices.items(): print(f"\n'{word}':") print(matrix) # Demonstrate tensor contraction for normalization print("\nTensor contraction process:") print("1. We assign meaning vectors to each word") print("2. For cups (e.g., n⊗n^r → I), we contract indices") print("3. For composition, we use tensor product followed by contraction") # Simplified normalization calculation print("\nSimplified normalization result:") combined = np.tensordot( np.tensordot(word_matrices["someone's"], word_matrices["lot"], axes=0), np.tensordot(word_matrices["in"], word_matrices["life"], axes=0), axes=0 ) # This is a highly simplified contraction - real implementation would follow the cup/cap structure # Show result shape and a normalized result print(f"Result tensor shape: {combined.shape}") print("Normalized to scalar by contracting along appropriate dimensions") final_result = np.trace(np.sum(combined, axis=(1, 3, 5))) print(f"Final scalar: {final_result}") # Create a concrete TketModel for execution print("\nCreating executable quantum model:") model = TketModel.from_diagrams([circuit]) print("\nModel parameters:") print(f"- Circuit parameters: {model.symbols}") print(f"- Parameter count: {len(model.symbols)}") # Initialize with random parameters params = 2 * np.pi * np.random.rand(len(model.symbols)) print("\nRandom initialization of circuit parameters:") print(params) print("\nExpected steps for execution on quantum hardware:") print("1. Prepare initial state |0⟩^⊗n") print("2. Apply parameterized circuit U(θ)") print("3. Measure final state") print("4. Interpret measurement result as semantic meaning") # Demonstrate normalization as projection print("\nNormalization as projection onto sentence space:") print("Pr(s) = ⟨ψ|Πs|ψ⟩ where |ψ⟩ is the quantum state after circuit execution") # Final analysis print("\n=== Interpretation of 'Someone's Lot in Life' ===") print("This categorical quantum representation models how individual components") print("(someone, possession, lot, context, life) combine through tensor operations") print("to represent the meaning of 'someone's lot in life' as a pathway through") print("a quantum computational process.") print("The final normalized value represents the semantic coherence of this phrase.") if __name__ == "__main__": main()