#!/usr/bin/env python """ Advanced Categorical Quantum Semantics for "Someone's Lot in Life" ================================================================== This example provides a more detailed mathematical implementation using pregroup grammars, string diagrams, and quantum circuits via DisCoPy and lambeq. Requirements: - lambeq - discopy - pytket - pytket-qiskit - numpy - matplotlib """ import numpy as np from typing import List, Dict, Tuple, Optional import matplotlib.pyplot as plt from lambeq import ( BobcatParser, AtomicType, IQPAnsatz, SpiderAnsatz, TketModel, discopy ) from discopy.grammar.pregroup import Word, Cup, Swap, Id, Diagram, Ty from discopy.quantum import Circuit, Box, Measure, CX, H, Ket, Bra # Define atomic types n = AtomicType.NOUN # Noun type s = AtomicType.SENTENCE # Sentence type # Define compound types n_r = n.r # Right adjoint of noun n_l = n.l # Left adjoint of noun s_r = s.r # Right adjoint of sentence s_l = s.l # Left adjoint of sentence def create_pregroup_diagram(): """Create the pregroup diagram for 'someone's lot in life'""" # Define the types for each word types = { "someone's": n @ n_r, # Possessive: n⊗n^r "lot": n, # Noun: n "in": n @ n_r @ s_l @ s, # Preposition: n⊗n^r⊗s^l⊗s "life": n # Noun: n } # Create word boxes someone_s = Word("someone's", types["someone's"]) lot = Word("lot", types["lot"]) in_prep = Word("in", types["in"]) life = Word("life", types["life"]) # Construct the diagram words = someone_s @ lot @ in_prep @ life # Add cups for contractions cups = Id(n) @ Cup(n, n_r) @ Id(n) @ Cup(n, n_r) @ Id(s_l @ s @ n) diagram = words >> cups # Add remaining cups for full reduction second_cups = Id(n) @ Cup(s_l, s) @ Id(n) diagram = diagram >> second_cups # Final contraction to sentence type final_cup = Cup(n, n_r) diagram = diagram >> final_cup return diagram def analyze_tensor_structure(diagram: Diagram): """Analyze the tensor structure of the diagram""" # Extract word boxes words = [box for box in diagram.boxes if isinstance(box, Word)] # Extract cups (contractions) cups = [box for box in diagram.boxes if isinstance(box, Cup)] analysis = { "words": [word.name for word in words], "word_types": [str(word.cod) for word in words], "connections": [(cup.dom[0].name, cup.dom[1].name) for cup in cups], "tensor_dimensions": len(cups) + 1 # Dimensionality of the tensor network } return analysis def create_concrete_meanings(): """Create concrete meaning matrices for each word""" # Define 2D vector spaces for simplicity dim_n = 2 # Dimension for noun space dim_s = 2 # Dimension for sentence space # Create concrete tensors for each word meanings = { # someone's: n⊗n^r (2x2 matrix) "someone's": np.array([ [0.7, 0.3], # Individual identity component [0.3, 0.3] # Possessive relationship component ]), # lot: n (2D vector) "lot": np.array([ 0.5, # Determination component 0.5 # Randomness/fate component ]), # in: n⊗n^r⊗s^l⊗s (2x2x2x2 tensor) - simplified "in": np.array([ [[[0.8, 0.2], [0.1, 0.1]], # Context binding tensors [[0.1, 0.1], [0.2, 0.1]]], [[[0.1, 0.1], [0.2, 0.1]], [[0.1, 0.0], [0.1, 0.1]]] ]), # life: n (2D vector) "life": np.array([ 0.6, # Temporal component 0.4 # Experience component ]) } return meanings def quantum_circuit_representation(diagram: Diagram): """Create a quantum circuit representation using IQP ansatz""" # Define the ansatz for circuit creation ansatz = IQPAnsatz({AtomicType.NOUN: 1, AtomicType.SENTENCE: 1}) # Convert diagram to quantum circuit circuit = ansatz(diagram) # Extract circuit information info = { "n_qubits": circuit.n_qubits, "depth": len(circuit.data), "gates": [gate.name for gate in circuit.data], "circuit": circuit } return info def normalize_meanings(meanings, diagram): """Perform manual normalization of meanings according to diagram structure""" # This is a simplified calculation that follows the tensor contraction # in the pregroup diagram, but doesn't fully implement all steps # First contraction: someone's ⊗ lot someone_s_lot = np.outer(meanings["someone's"][0], meanings["lot"]) # Second contraction: in ⊗ life in_life = np.tensordot( meanings["in"], meanings["life"], axes=0 ) # Final contraction approximation result = np.tensordot(someone_s_lot.flatten(), in_life.reshape(-1), axes=0) # Normalize to get probability-like value result_normalized = np.abs(result) / np.sum(np.abs(result)) return { "someone_s_lot": someone_s_lot, "in_life": in_life, "result_full": result, "result_normalized": result_normalized } def main(): """Main execution function""" print("=== Advanced Categorical Quantum Analysis of 'Someone's Lot in Life' ===\n") # 1. Create and analyze pregroup diagram print("1. Pregroup Grammar Analysis") print("---------------------------") diagram = create_pregroup_diagram() analysis = analyze_tensor_structure(diagram) print(f"Sentence: 'someone's lot in life'") print("\nPregroup types:") for word, typ in zip(analysis["words"], analysis["word_types"]): print(f" - {word}: {typ}") print("\nReduction process:") print(" (n⊗n^r)⊗n⊗(n⊗n^r⊗s^l⊗s)⊗n → s") print(" with steps:") print(" 1. Connect n^r from 'someone's' with first n from 'in'") print(" 2. Connect s^l from 'in' with s") print(" 3. Connect n^r with final n from 'life'") print(" Result: sentence type s") print(f"\nDiagram structure: {diagram}") # 2. Tensor network representation print("\n2. Tensor Network Representation") print("------------------------------") # Create concrete meanings meanings = create_concrete_meanings() print("Concrete meaning representations:") for word, tensor in meanings.items(): print(f"\n'{word}':") print(tensor) # Perform normalization normalization = normalize_meanings(meanings, diagram) print("\nIntermediate tensor products:") print(" 'someone's lot':") print(normalization["someone_s_lot"]) print("\n 'in life':") print(normalization["in_life"].shape) # Too large to print fully print("\nFinal normalized meaning:") print(normalization["result_normalized"]) # 3. Quantum circuit representation print("\n3. Quantum Circuit Translation") print("----------------------------") circuit_info = quantum_circuit_representation(diagram) print(f"Quantum circuit with {circuit_info['n_qubits']} qubits") print(f"Circuit depth: {circuit_info['depth']}") print(f"Gate sequence types: {set(circuit_info['gates'])}") print("\nCircuit representation:") print(circuit_info['circuit']) # 4. Normalization as quantum measurement print("\n4. Quantum Normalization Process") print("------------------------------") # Create concrete quantum model ansatz = IQPAnsatz({AtomicType.NOUN: 1, AtomicType.SENTENCE: 1}) circuit = ansatz(diagram) model = TketModel.from_diagrams([circuit]) # Initialize random parameters n_params = len(model.symbols) params = 2 * np.pi * np.random.rand(n_params) print(f"Model has {n_params} parameters") print("\nParameter initialization:") print(params) print("\nQuantum normalization as projection:") print(" |ψ⟩ = U(θ)|0⟩^⊗n") print(" Probability of meaning = |⟨1|ψ⟩|²") print(" where |1⟩ represents the 'true' semantic meaning") # 5. Interpretation print("\n5. Categorical Quantum Interpretation") print("-----------------------------------") print("This mathematical formalism captures the following:") print(" 1. 'someone's' establishes a possessive relationship (n⊗n^r)") print(" 2. 'lot' represents a noun concept of fate/circumstance (n)") print(" 3. 'in' places this lot within a context (n⊗n^r⊗s^l⊗s)") print(" 4. 'life' provides the domain of this context (n)") print("\nThe tensor contractions model how these concepts interact") print("The quantum circuit provides a computational pathway for this meaning") print("The final normalized representation gives a mathematical model of the") print("phrase's meaning that could be compared with other phrases or used") print("for semantic analysis.") if __name__ == "__main__": main()