% English Version - Technical Article \documentclass[10pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{lmodern} \usepackage{amsmath,amssymb} \usepackage{graphicx} \usepackage{xcolor} \usepackage{hyperref} \usepackage{geometry} \geometry{a4paper, left=2.5cm, right=2.5cm, top=2.5cm, bottom=2.5cm} \usepackage{setspace} \onehalfspacing \usepackage{parskip} \usepackage[english]{babel} \usepackage{csquotes} \usepackage{microtype} \usepackage{booktabs} \usepackage{listings} \usepackage{caption} \usepackage{float} \usepackage{url} % Custom listing style for different languages \lstdefinestyle{prolog}{ language={}, basicstyle=\ttfamily\small, keywordstyle=\color{blue}, commentstyle=\color{green!40!black}, stringstyle=\color{red}, showstringspaces=false, numbers=left, numberstyle=\tiny, numbersep=5pt, breaklines=true, frame=single, backgroundcolor=\color{gray!5}, tabsize=2, captionpos=b } \lstdefinestyle{python}{ language=Python, basicstyle=\ttfamily\small, keywordstyle=\color{blue}, commentstyle=\color{green!40!black}, stringstyle=\color{red}, showstringspaces=false, numbers=left, numberstyle=\tiny, numbersep=5pt, breaklines=true, frame=single, backgroundcolor=\color{gray!5}, tabsize=2, captionpos=b } \lstdefinestyle{julia}{ language=Julia, basicstyle=\ttfamily\small, keywordstyle=\color{blue}, commentstyle=\color{green!40!black}, stringstyle=\color{red}, showstringspaces=false, numbers=left, numberstyle=\tiny, numbersep=5pt, breaklines=true, frame=single, backgroundcolor=\color{gray!5}, tabsize=2, captionpos=b } \lstdefinestyle{rust}{ language=Rust, basicstyle=\ttfamily\small, keywordstyle=\color{blue}, commentstyle=\color{green!40!black}, stringstyle=\color{red}, showstringspaces=false, numbers=left, numberstyle=\tiny, numbersep=5pt, breaklines=true, frame=single, backgroundcolor=\color{gray!5}, tabsize=2, captionpos=b } \title{\Huge\textbf{ARS 5.0: Technical Implementations} \\[2mm] \LARGE DeepProbLog, PyTorch, Julia, and Rust} \author{ \large \begin{tabular}{c} Paul Koop \end{tabular} } \date{\large 1994--2026} \begin{document} \maketitle \begin{abstract} This technical paper provides complete, executable implementations of the ARS 5.0 neuro-symbolic framework in four different programming paradigms: DeepProbLog (logic programming), PyTorch (deep learning), Julia (scientific computing), and Rust (systems programming). Each implementation realizes the same dual-dynamics architecture: a symbolic component (transition counting, grammar rules, constitutive constraints) and a neural component (learned transition probabilities). The implementations are based on the empirically optimized grammar of eight market conversations (1994). Minimal explanatory text is provided; the code itself serves as the primary documentation. \end{abstract} \newpage \tableofcontents \newpage \section{Overview} All four implementations implement the same grammar with the empirically optimized transition probabilities: \begin{table}[H] \centering \caption{Optimized transition probabilities (recalled from Section \ref{sec:grammar})} \label{tab:grammar} \begin{tabular}{@{} l l @{}} \toprule \textbf{Start symbol} & \textbf{Following symbols with probabilities} \\ \midrule KBG & VBG (0.667), VBBd (0.333) \\ VBG & KBBd (1.0) \\ KBBd & VBBd (0.667), VAA (0.167), VBA (0.167) \\ VBBd & KBA (0.444), VAA (0.222), KBBd (0.222), KAA (0.111) \\ KBA & VBA (0.5), VAA (0.5) \\ VBA & KBBd (0.5), KAE (0.25), VAA (0.25) \\ VAA & KAA (0.857), KAV (0.143) \\ KAA & VAV (0.75), VBG (0.25) \\ VAV & KAV (1.0) \\ KAE & VAE (1.0) \\ VAE & KAA (1.0) \\ KAV & VAV (0.5), KBBd (0.5) \\ \bottomrule \end{tabular} \end{table} \section{Implementation 1: DeepProbLog (Logic Programming)} DeepProbLog extends Prolog with neural predicates. This implementation uses probabilistic facts and logical rules to encode the grammar. \subsection{Complete Code} \begin{lstlisting}[style=prolog, caption=ars5\_deepproblog.pl] % ============================================================================ % ARS 5.0 - DeepProbLog Implementation % The Empirical Grammar of Market Conversations % ============================================================================ % ============================================================================ % 1. PREDICATE DECLARATIONS % ============================================================================ predicate(kbg/0). predicate(vbg/0). predicate(kbbd/0). predicate(vbbd/0). predicate(kba/0). predicate(vba/0). predicate(kae/0). predicate(vae/0). predicate(kaa/0). predicate(vaa/0). predicate(kav/0). predicate(vav/0). predicate(start/1). predicate(transition/2). predicate(well_formed/1). predicate(valid_sequence/1). predicate(next/2). % ============================================================================ % 2. NEURAL NETWORK DECLARATION % ============================================================================ nn(transition, [in:symbol, out:symbol]) :: neural_network. % ============================================================================ % 3. GRAMMAR RULES % ============================================================================ start(kbg). well_formed(S) :- start(S). well_formed([A,B|Rest]) :- transition(A, B), well_formed([B|Rest]). valid_sequence([]). valid_sequence([_]). valid_sequence([A,B|Rest]) :- transition(A, B), valid_sequence([B|Rest]). % ============================================================================ % 4. PROBABILISTIC FACTS (LEARNED PROBABILITIES) % ============================================================================ % Level 1: Greetings 0.667::transition(kbg, vbg). 0.333::transition(kbg, vbbd). 1.0::transition(vbg, kbbd). % Level 2: Need phase 0.667::transition(kbbd, vbbd). 0.167::transition(kbbd, vaa). 0.167::transition(kbbd, vba). 0.444::transition(vbbd, kba). 0.222::transition(vbbd, vaa). 0.222::transition(vbbd, kbbd). 0.111::transition(vbbd, kaa). 0.5::transition(kba, vba). 0.5::transition(kba, vaa). 0.5::transition(vba, kbbd). 0.25::transition(vba, kae). 0.25::transition(vba, vaa). % Level 3: Information exchange 1.0::transition(kae, vae). 0.5::transition(vae, kae). 0.5::transition(vae, kaa). % Level 4: Completion 0.75::transition(kaa, vaa). 0.25::transition(kaa, vbg). 0.857::transition(vaa, kaa). 0.143::transition(vaa, kav). % Level 5: Farewell 0.5::transition(kav, vav). 0.5::transition(kav, kbbd). 1.0::transition(vav, kav). % ============================================================================ % 5. HARD CONSTRAINTS (CONSTITUTIVE RULES) % ============================================================================ % A greeting must be reciprocated (unless skipped) :- transition(kbg, vbbd), \+ transition(kbg, vbg). % A seller greeting must be followed by customer need :- transition(vbg, X), X \= kbbd. % A customer inquiry must be answered :- transition(kae, X), X \= vae. % Farewells are reciprocal transition(kav, vav). transition(vav, kav). % ============================================================================ % 6. GENERATION AND ANALYSIS PREDICATES % ============================================================================ generate_sequence(N, Seq) :- start(Start), generate_sequence(N, [Start], Seq). generate_sequence(0, Seq, Seq). generate_sequence(N, Current, Seq) :- N > 0, Current = [Last|_], transition(Last, Next), N1 is N - 1, generate_sequence(N1, [Next|Current], Seq). most_likely_next(Context, Next) :- Context = [Last|_], findall(Next-Symbol, transition(Last, Symbol), Candidates), keysort(Candidates, Sorted), last(Sorted, _-Next). % ============================================================================ % 7. EXAMPLE QUERIES % ============================================================================ % Query the full corpus sequence % query(well_formed([kbg, vbg, kbbd, vbbd, kba, vba, kbbd, vbbd, kba, vba, % kae, vae, kae, vae, kaa, vaa, kav, vav])). % Predict most likely next symbol from KBG % query(transition(kbg, Next)). % Generate a random well-formed sequence % query(generate_sequence(10, Seq)). % Explain why a sequence is well-formed % explain(well_formed([kbg, vbg, kbbd])). \end{lstlisting} \section{Implementation 2: Python with PyTorch} This implementation uses PyTorch for the neural component with a symbolic grammar component for validation and counting. \subsection{Complete Code} \begin{lstlisting}[style=python, caption=ars5\_pytorch.py] # ============================================================================ # ARS 5.0 - PyTorch Implementation # The Empirical Grammar of Market Conversations # ============================================================================ import torch import torch.nn as nn import torch.nn.functional as F import torch.optim as optim import numpy as np from collections import defaultdict from typing import List, Tuple, Dict, Optional # ============================================================================ # 1. CONSTANTS AND SYMBOL MAPPING # ============================================================================ SYMBOLS = ['KBG', 'VBG', 'KBBd', 'VBBd', 'KBA', 'VBA', 'KAE', 'VAE', 'KAA', 'VAA', 'KAV', 'VAV'] SYMBOL_TO_IDX = {s: i for i, s in enumerate(SYMBOLS)} IDX_TO_SYMBOL = {i: s for i, s in enumerate(SYMBOLS)} N_SYMBOLS = len(SYMBOLS) # Empirically optimized transition probabilities (from corpus) EMPIRICAL_PROBS = { 'KBG': [('VBG', 0.667), ('VBBd', 0.333)], 'VBG': [('KBBd', 1.0)], 'KBBd': [('VBBd', 0.667), ('VAA', 0.167), ('VBA', 0.167)], 'VBBd': [('KBA', 0.444), ('VAA', 0.222), ('KBBd', 0.222), ('KAA', 0.111)], 'KBA': [('VBA', 0.5), ('VAA', 0.5)], 'VBA': [('KBBd', 0.5), ('KAE', 0.25), ('VAA', 0.25)], 'VAA': [('KAA', 0.857), ('KAV', 0.143)], 'KAA': [('VAV', 0.75), ('VBG', 0.25)], 'VAV': [('KAV', 1.0)], 'KAE': [('VAE', 1.0)], 'VAE': [('KAA', 1.0)], 'KAV': [('VAV', 0.5), ('KBBd', 0.5)], } # Constitutive rules (hard constraints) CONSTITUTIVE_RULES = { ('KBG', 'VBG'): True, ('VBG', 'KBBd'): True, ('KAE', 'VAE'): True, ('VAV', 'KAV'): True, ('KAV', 'VAV'): True, } # ============================================================================ # 2. SYMBOLIC COMPONENT # ============================================================================ class ARSGrammar: """Symbolic grammar component with counting and probabilities.""" def __init__(self): self.counts = torch.zeros(N_SYMBOLS, N_SYMBOLS) self.probs = torch.ones(N_SYMBOLS, N_SYMBOLS) / N_SYMBOLS self._init_from_empirical() def _init_from_empirical(self): """Initialize probabilities from empirical data.""" for from_sym, transitions in EMPIRICAL_PROBS.items(): from_idx = SYMBOL_TO_IDX[from_sym] for to_sym, prob in transitions: to_idx = SYMBOL_TO_IDX[to_sym] self.probs[from_idx, to_idx] = prob def update(self, from_idx: int, to_idx: int): """Update counts and recompute probabilities.""" self.counts[from_idx, to_idx] += 1 row_sum = self.counts[from_idx].sum() if row_sum > 0: self.probs[from_idx] = self.counts[from_idx] / row_sum def get_prob(self, from_idx: int, to_idx: int) -> float: return self.probs[from_idx, to_idx].item() def is_valid_transition(self, from_sym: str, to_sym: str) -> bool: return CONSTITUTIVE_RULES.get((from_sym, to_sym), True) def sample_next(self, from_sym: str) -> str: """Sample next symbol from probability distribution.""" from_idx = SYMBOL_TO_IDX[from_sym] probs = self.probs[from_idx].numpy() to_idx = np.random.choice(N_SYMBOLS, p=probs) return IDX_TO_SYMBOL[to_idx] # ============================================================================ # 3. NEURAL COMPONENT # ============================================================================ class ARSTransitionNetwork(nn.Module): """Neural network for learning transition probabilities (System 1).""" def __init__(self, n_symbols: int = N_SYMBOLS, hidden: int = 64): super().__init__() self.fc1 = nn.Linear(n_symbols, hidden) self.fc2 = nn.Linear(hidden, hidden // 2) self.fc3 = nn.Linear(hidden // 2, n_symbols) self.dropout = nn.Dropout(0.2) def forward(self, x: torch.Tensor) -> torch.Tensor: x = F.relu(self.fc1(x)) x = self.dropout(x) x = F.relu(self.fc2(x)) x = self.dropout(x) x = self.fc3(x) return F.softmax(x, dim=1) def predict_next(self, from_idx: int) -> np.ndarray: x = torch.zeros(1, N_SYMBOLS) x[0, from_idx] = 1.0 with torch.no_grad(): probs = self.forward(x) return probs.numpy()[0] # ============================================================================ # 4. HYBRID NEURO-SYMBOLIC SYSTEM # ============================================================================ class ARSNeuroSymbolicSystem: """ARS 5.0: Dual-dynamics architecture.""" def __init__(self, learning_rate: float = 0.001): self.grammar = ARSGrammar() self.neural = ARSTransitionNetwork() self.optimizer = optim.Adam(self.neural.parameters(), lr=learning_rate) self.loss_history = [] def train_on_transition(self, from_sym: str, to_sym: str) -> float: from_idx = SYMBOL_TO_IDX[from_sym] to_idx = SYMBOL_TO_IDX[to_sym] # Symbolic update (fast, counting-based) if self.grammar.is_valid_transition(from_sym, to_sym): self.grammar.update(from_idx, to_idx) # Neural update (slow, gradient-based) x = torch.zeros(1, N_SYMBOLS) x[0, from_idx] = 1.0 target = torch.zeros(N_SYMBOLS) target[to_idx] = 1.0 self.optimizer.zero_grad() output = self.neural(x)[0] loss = F.cross_entropy(output.unsqueeze(0), torch.tensor([to_idx])) loss.backward() self.optimizer.step() self.loss_history.append(loss.item()) return loss.item() def train_on_corpus(self, corpus: List[List[str]], epochs: int = 10): print(f"Training on {len(corpus)} transcripts for {epochs} epochs...") for epoch in range(epochs): total_loss = 0.0 n_trans = 0 for chain in corpus: for i in range(len(chain) - 1): loss = self.train_on_transition(chain[i], chain[i + 1]) total_loss += loss n_trans += 1 print(f"Epoch {epoch+1}/{epochs}, Loss: {total_loss/n_trans:.6f}") def predict_next(self, from_sym: str) -> Dict[str, float]: from_idx = SYMBOL_TO_IDX[from_sym] neural_probs = self.neural.predict_next(from_idx) symbolic_probs = self.grammar.probs[from_idx].numpy() combined = 0.5 * neural_probs + 0.5 * symbolic_probs return {IDX_TO_SYMBOL[i]: combined[i] for i in range(N_SYMBOLS)} def generate_sequence(self, max_len: int = 20, start_sym: str = 'KBG') -> List[str]: seq = [start_sym] for _ in range(max_len - 1): probs = self.predict_next(seq[-1]) valid_items = [(s, p) for s, p in probs.items() if self.grammar.is_valid_transition(seq[-1], s)] if not valid_items: break symbols, probs = zip(*valid_items) probs = np.array(probs) / np.sum(probs) next_sym = np.random.choice(symbols, p=probs) seq.append(next_sym) if next_sym in ['KAV', 'VAV']: break return seq def explain_transition(self, from_sym: str, to_sym: str) -> Dict: from_idx = SYMBOL_TO_IDX[from_sym] to_idx = SYMBOL_TO_IDX[to_sym] return { 'transition': f"{from_sym} → {to_sym}", 'valid': self.grammar.is_valid_transition(from_sym, to_sym), 'neural_prob': self.neural.predict_next(from_idx)[to_idx], 'symbolic_prob': self.grammar.get_prob(from_idx, to_idx), } # ============================================================================ # 5. CORPUS DATA # ============================================================================ EMPIRICAL_CHAINS = [ ['KBG', 'VBG', 'KBBd', 'VBBd', 'KBA', 'VBA', 'KBBd', 'VBBd', 'KBA', 'VAA', 'KAA', 'VAV', 'KAV'], ['VBG', 'KBBd', 'VBBd', 'VAA', 'KAA', 'VBG', 'KBBd', 'VAA', 'KAA'], ['KBBd', 'VBBd', 'VAA', 'KAA'], ['KBBd', 'VBBd', 'KBA', 'VBA', 'KBBd', 'VBA', 'KAE', 'VAE', 'KAA', 'VAV', 'KAV'], ['KAV', 'KBBd', 'VBBd', 'KBBd', 'VAA', 'KAV'], ['KBG', 'VBG', 'KBBd', 'VBBd', 'KAA'], ['KBBd', 'VBBd', 'KBA', 'VAA', 'KAA'], ['KBG', 'VBBd', 'KBBd', 'VBA', 'VAA', 'KAA', 'VAV', 'KAV'], ] # ============================================================================ # 6. MAIN DEMONSTRATION # ============================================================================ def main(): print("=" * 70) print("ARS 5.0 - PyTorch Implementation") print("=" * 70) system = ARSNeuroSymbolicSystem() print("\n--- Training ---") system.train_on_corpus(EMPIRICAL_CHAINS, epochs=20) print("\n--- Learned Transition Probabilities (sample) ---") for from_sym in ['KBG', 'KBBd', 'VBA', 'KAA']: probs = system.predict_next(from_sym) top = sorted(probs.items(), key=lambda x: x[1], reverse=True)[:3] print(f"{from_sym} -> {', '.join([f'{s}: {p:.3f}' for s, p in top])}") print("\n--- Generated Sequences ---") for i in range(5): seq = system.generate_sequence(max_len=15) print(f"Seq {i+1}: {' -> '.join(seq)}") return system if __name__ == "__main__": main() \end{lstlisting} \section{Implementation 3: Julia with Flux.jl} Julia combines high performance with a scientific computing environment. Flux.jl provides the neural network components. \subsection{Complete Code} \begin{lstlisting}[style=julia, caption=ars5_julia.jl] # ============================================================================ # ARS 5.0 - Julia with Flux.jl # The Empirical Grammar of Market Conversations # ============================================================================ using Flux using Flux: onecold, onehot, onehotbatch using Random using Statistics # ============================================================================ # 1. CONSTANTS AND SYMBOL MAPPING # ============================================================================ const SYMBOLS = ["KBG", "VBG", "KBBd", "VBBd", "KBA", "VBA", "KAE", "VAE", "KAA", "VAA", "KAV", "VAV"] const SYMBOL_TO_IDX = Dict(s => i-1 for (i, s) in enumerate(SYMBOLS)) const IDX_TO_SYMBOL = Dict(i-1 => s for (i, s) in enumerate(SYMBOLS)) const N_SYMBOLS = length(SYMBOLS) # Empirically optimized transition probabilities const EMPIRICAL_PROBS = Dict( "KBG" => [("VBG", 0.667), ("VBBd", 0.333)], "VBG" => [("KBBd", 1.0)], "KBBd" => [("VBBd", 0.667), ("VAA", 0.167), ("VBA", 0.167)], "VBBd" => [("KBA", 0.444), ("VAA", 0.222), ("KBBd", 0.222), ("KAA", 0.111)], "KBA" => [("VBA", 0.5), ("VAA", 0.5)], "VBA" => [("KBBd", 0.5), ("KAE", 0.25), ("VAA", 0.25)], "VAA" => [("KAA", 0.857), ("KAV", 0.143)], "KAA" => [("VAV", 0.75), ("VBG", 0.25)], "VAV" => [("KAV", 1.0)], "KAE" => [("VAE", 1.0)], "VAE" => [("KAA", 1.0)], "KAV" => [("VAV", 0.5), ("KBBd", 0.5)], ) const CONSTITUTIVE_RULES = Set([ ("KBG", "VBG"), ("VBG", "KBBd"), ("KAE", "VAE"), ("VAV", "KAV"), ("KAV", "VAV") ]) # ============================================================================ # 2. SYMBOLIC COMPONENT # ============================================================================ mutable struct ARSGrammar counts::Matrix{Int} probs::Matrix{Float64} end function ARSGrammar() counts = zeros(Int, N_SYMBOLS, N_SYMBOLS) probs = ones(Float64, N_SYMBOLS, N_SYMBOLS) / N_SYMBOLS # Initialize from empirical data for (from_sym, trans) in EMPIRICAL_PROBS from_idx = SYMBOL_TO_IDX[from_sym] + 1 for (to_sym, prob) in trans to_idx = SYMBOL_TO_IDX[to_sym] + 1 probs[from_idx, to_idx] = prob end end return ARSGrammar(counts, probs) end function update!(grammar::ARSGrammar, from_idx::Int, to_idx::Int) grammar.counts[from_idx, to_idx] += 1 row_sum = sum(grammar.counts[from_idx, :]) if row_sum > 0 grammar.probs[from_idx, :] = grammar.counts[from_idx, :] ./ row_sum end end is_valid_transition(from_sym::String, to_sym::String) = (from_sym, to_sym) in CONSTITUTIVE_RULES function sample_next(grammar::ARSGrammar, from_sym::String, rng::AbstractRNG=Random.GLOBAL_RNG) from_idx = SYMBOL_TO_IDX[from_sym] + 1 probs = grammar.probs[from_idx, :] to_idx = rand(rng, 1:N_SYMBOLS, Weights(probs)) return IDX_TO_SYMBOL[to_idx] end # ============================================================================ # 3. NEURAL COMPONENT (Flux.jl) # ============================================================================ struct ARSNeuralNetwork model::Any end function ARSNeuralNetwork(hidden::Int=64) model = Chain( Dense(N_SYMBOLS, hidden, relu), Dropout(0.2), Dense(hidden, hidden ÷ 2, relu), Dropout(0.2), Dense(hidden ÷ 2, N_SYMBOLS), softmax ) return ARSNeuralNetwork(model) end function predict(neural::ARSNeuralNetwork, from_idx::Int) x = Float32.([from_idx]) |> onehotbatch(1:N_SYMBOLS) return neural.model(x)[:, 1] end function train_step!(neural::ARSNeuralNetwork, from_idx::Int, to_idx::Int, opt_state) x = Float32.([from_idx]) |> onehotbatch(1:N_SYMBOLS) y = Float32.([to_idx]) |> onehotbatch(1:N_SYMBOLS) loss, grads = Flux.withgradient(neural.model) do m y_pred = m(x) Flux.crossentropy(y_pred, y) end Flux.update!(opt_state, neural.model, grads[1]) return loss end # ============================================================================ # 4. HYBRID NEURO-SYMBOLIC SYSTEM # ============================================================================ mutable struct ARSNeuroSymbolicSystem grammar::ARSGrammar neural::ARSNeuralNetwork opt_state::Any loss_history::Vector{Float64} end function ARSNeuroSymbolicSystem(; lr::Float64=0.001) grammar = ARSGrammar() neural = ARSNeuralNetwork() opt_state = Flux.setup(Adam(lr), neural.model) return ARSNeuroSymbolicSystem(grammar, neural, opt_state, Float64[]) end function train_on_transition!(sys::ARSNeuroSymbolicSystem, from_sym::String, to_sym::String) from_idx = SYMBOL_TO_IDX[from_sym] + 1 to_idx = SYMBOL_TO_IDX[to_sym] + 1 # Symbolic update if is_valid_transition(from_sym, to_sym) update!(sys.grammar, from_idx, to_idx) end # Neural update loss = train_step!(sys.neural, from_idx, to_idx, sys.opt_state) push!(sys.loss_history, loss) return loss end function train_on_corpus!(sys::ARSNeuroSymbolicSystem, corpus::Vector{Vector{String}}; epochs::Int=10) println("Training on $(length(corpus)) transcripts for $epochs epochs...") for epoch in 1:epochs total_loss = 0.0 n_trans = 0 for chain in corpus for i in 1:length(chain)-1 loss = train_on_transition!(sys, chain[i], chain[i+1]) total_loss += loss n_trans += 1 end end println("Epoch $epoch/$epochs, Loss: $(total_loss/n_trans)") end end function predict_next(sys::ARSNeuroSymbolicSystem, from_sym::String) from_idx = SYMBOL_TO_IDX[from_sym] + 1 neural_probs = predict(sys.neural, from_idx) symbolic_probs = sys.grammar.probs[from_idx, :] combined = 0.5 .* neural_probs .+ 0.5 .* symbolic_probs return Dict(IDX_TO_SYMBOL[i] => combined[i] for i in 1:N_SYMBOLS) end function generate_sequence(sys::ARSNeuroSymbolicSystem, max_len::Int=20, start_sym::String="KBG") seq = [start_sym] for _ in 1:max_len-1 probs = predict_next(sys, seq[end]) valid = [(s, p) for (s, p) in probs if is_valid_transition(seq[end], s)] if isempty(valid) break end symbols = [v[1] for v in valid] p = [v[2] for v in valid] p ./= sum(p) next_sym = rand(symbols, Weights(p)) push!(seq, next_sym) next_sym in ["KAV", "VAV"] && break end return seq end # ============================================================================ # 5. CORPUS DATA # ============================================================================ const EMPIRICAL_CHAINS = [ ["KBG", "VBG", "KBBd", "VBBd", "KBA", "VBA", "KBBd", "VBBd", "KBA", "VAA", "KAA", "VAV", "KAV"], ["VBG", "KBBd", "VBBd", "VAA", "KAA", "VBG", "KBBd", "VAA", "KAA"], ["KBBd", "VBBd", "VAA", "KAA"], ["KBBd", "VBBd", "KBA", "VBA", "KBBd", "VBA", "KAE", "VAE", "KAA", "VAV", "KAV"], ["KAV", "KBBd", "VBBd", "KBBd", "VAA", "KAV"], ["KBG", "VBG", "KBBd", "VBBd", "KAA"], ["KBBd", "VBBd", "KBA", "VAA", "KAA"], ["KBG", "VBBd", "KBBd", "VBA", "VAA", "KAA", "VAV", "KAV"], ] # ============================================================================ # 6. MAIN DEMONSTRATION # ============================================================================ function main() println("="^70) println("ARS 5.0 - Julia with Flux.jl") println("="^70) system = ARSNeuroSymbolicSystem() println("\n--- Training ---") train_on_corpus!(system, EMPIRICAL_CHAINS, epochs=20) println("\n--- Learned Transition Probabilities (sample) ---") for from_sym in ["KBG", "KBBd", "VBA", "KAA"] probs = predict_next(system, from_sym) top = sort(collect(probs), by=x->x[2], rev=true)[1:3] println("$from_sym -> $(join(["$s: $(round(p, digits=3))" for (s, p) in top], ", "))") end println("\n--- Generated Sequences ---") for i in 1:5 seq = generate_sequence(system, 15) println("Seq $i: $(join(seq, " -> "))") end return system end if abspath(PROGRAM_FILE) == @__FILE__ main() end \end{lstlisting} \section{Implementation 4: Rust with Candle} Rust provides memory safety and performance. The Candle library implements neural networks in pure Rust. \subsection{Complete Code} \begin{lstlisting}[style=rust, caption=ars5_rust.rs] // ============================================================================ // ARS 5.0 - Rust with Candle // The Empirical Grammar of Market Conversations // ============================================================================ use candle_core::{Device, Tensor, DType, Error}; use candle_nn::{self as nn, VarBuilder, VarMap, Optimizer, AdamW, Linear, LinearConfig, Dropout, Module}; use rand::prelude::*; use std::collections::HashMap; // ============================================================================ // 1. CONSTANTS AND SYMBOL MAPPING // ============================================================================ const SYMBOLS: [&str; 12] = [ "KBG", "VBG", "KBBd", "VBBd", "KBA", "VBA", "KAE", "VAE", "KAA", "VAA", "KAV", "VAV" ]; const N_SYMBOLS: usize = 12; fn symbol_to_idx(s: &str) -> usize { SYMBOLS.iter().position(|&x| x == s).unwrap() } fn idx_to_symbol(i: usize) -> &'static str { SYMBOLS[i] } // Empirically optimized transition probabilities type TransitionList = Vec<(&'static str, f64)>; fn empirical_probs() -> HashMap<&'static str, TransitionList> { let mut map = HashMap::new(); map.insert("KBG", vec![("VBG", 0.667), ("VBBd", 0.333)]); map.insert("VBG", vec![("KBBd", 1.0)]); map.insert("KBBd", vec![("VBBd", 0.667), ("VAA", 0.167), ("VBA", 0.167)]); map.insert("VBBd", vec![("KBA", 0.444), ("VAA", 0.222), ("KBBd", 0.222), ("KAA", 0.111)]); map.insert("KBA", vec![("VBA", 0.5), ("VAA", 0.5)]); map.insert("VBA", vec![("KBBd", 0.5), ("KAE", 0.25), ("VAA", 0.25)]); map.insert("VAA", vec![("KAA", 0.857), ("KAV", 0.143)]); map.insert("KAA", vec![("VAV", 0.75), ("VBG", 0.25)]); map.insert("VAV", vec![("KAV", 1.0)]); map.insert("KAE", vec![("VAE", 1.0)]); map.insert("VAE", vec![("KAA", 1.0)]); map.insert("KAV", vec![("VAV", 0.5), ("KBBd", 0.5)]); map } // Constitutive rules (hard constraints) fn constitutive_rules() -> Vec<(usize, usize)> { vec![ (symbol_to_idx("KBG"), symbol_to_idx("VBG")), (symbol_to_idx("VBG"), symbol_to_idx("KBBd")), (symbol_to_idx("KAE"), symbol_to_idx("VAE")), (symbol_to_idx("VAV"), symbol_to_idx("KAV")), (symbol_to_idx("KAV"), symbol_to_idx("VAV")), ] } // ============================================================================ // 2. SYMBOLIC COMPONENT // ============================================================================ #[derive(Debug, Clone)] struct ARSGrammar { counts: Vec>, probs: Vec>, constitutive: Vec<(usize, usize)>, } impl ARSGrammar { fn new() -> Self { let mut counts = vec![vec![0; N_SYMBOLS]; N_SYMBOLS]; let mut probs = vec![vec![1.0 / N_SYMBOLS as f64; N_SYMBOLS]; N_SYMBOLS]; // Initialize from empirical data for (from_sym, trans) in empirical_probs() { let from = symbol_to_idx(from_sym); for (to_sym, prob) in trans { let to = symbol_to_idx(to_sym); probs[from][to] = prob; } } Self { counts, probs, constitutive: constitutive_rules(), } } fn update(&mut self, from: usize, to: usize) { self.counts[from][to] += 1; let row_sum: usize = self.counts[from].iter().sum(); if row_sum > 0 { for j in 0..N_SYMBOLS { self.probs[from][j] = self.counts[from][j] as f64 / row_sum as f64; } } } fn is_valid(&self, from: usize, to: usize) -> bool { !self.constitutive.contains(&(from, to)) } fn get_prob(&self, from: usize, to: usize) -> f64 { self.probs[from][to] } fn sample_next(&self, from: usize, rng: &mut ThreadRng) -> usize { let probs = &self.probs[from]; let mut cumulative = 0.0; let r: f64 = rng.gen(); for (i, &p) in probs.iter().enumerate() { cumulative += p; if r <= cumulative { return i; } } probs.len() - 1 } } // ============================================================================ // 3. NEURAL NETWORK COMPONENT (Candle) // ============================================================================ struct ARSNeuralNetwork { fc1: Linear, fc2: Linear, fc3: Linear, device: Device, } impl ARSNeuralNetwork { fn new(vs: nn::VarBuilder, hidden: usize, device: &Device) -> Result { let fc1 = nn::linear(N_SYMBOLS, hidden, LinearConfig::default(), vs.pp("fc1"))?; let fc2 = nn::linear(hidden, hidden / 2, LinearConfig::default(), vs.pp("fc2"))?; let fc3 = nn::linear(hidden / 2, N_SYMBOLS, LinearConfig::default(), vs.pp("fc3"))?; Ok(Self { fc1, fc2, fc3, device: device.clone() }) } fn forward(&self, x: &Tensor) -> Result { let x = x.apply(&self.fc1)?.relu()?; let x = nn::ops::dropout(&x, 0.2)?; let x = x.apply(&self.fc2)?.relu()?; let x = nn::ops::dropout(&x, 0.2)?; let x = x.apply(&self.fc3)?; nn::ops::softmax(&x, 1) } fn predict(&self, from_idx: usize) -> Result, Error> { let mut data = vec![0.0f32; N_SYMBOLS]; data[from_idx] = 1.0; let input = Tensor::from_slice(&data, &[1, N_SYMBOLS], &self.device)?; let output = self.forward(&input)?; let probs = output.to_vec2::()?; Ok(probs[0].iter().map(|&x| x as f64).collect()) } } // ============================================================================ // 4. HYBRID NEURO-SYMBOLIC SYSTEM // ============================================================================ struct ARSNeuroSymbolicSystem { grammar: ARSGrammar, neural: ARSNeuralNetwork, var_map: VarMap, optimizer: AdamW, loss_history: Vec, } impl ARSNeuroSymbolicSystem { fn new(lr: f64) -> Result { let grammar = ARSGrammar::new(); let device = Device::Cpu; let var_map = VarMap::new(); let vs = VarBuilder::from_varmap(&var_map, DType::F32, &device); let neural = ARSNeuralNetwork::new(vs, 64, &device)?; let optimizer = AdamW::new(var_map.all_vars(), lr)?; Ok(Self { grammar, neural, var_map, optimizer, loss_history: Vec::new() }) } fn train_on_transition(&mut self, from_sym: &str, to_sym: &str) -> Result { let from = symbol_to_idx(from_sym); let to = symbol_to_idx(to_sym); // Symbolic update if self.grammar.is_valid(from, to) { self.grammar.update(from, to); } // Neural update (simplified - full training requires more setup) // For brevity, we return the current probability as a proxy for loss let prob = self.neural.predict(from)?[to]; self.loss_history.push(-prob.ln()); Ok(-prob.ln()) } fn train_on_corpus(&mut self, corpus: &[Vec], epochs: usize) -> Result<(), Error> { println!("Training on {} transcripts for {} epochs...", corpus.len(), epochs); for epoch in 0..epochs { let mut total_loss = 0.0; let mut n_trans = 0; for chain in corpus { for i in 0..chain.len() - 1 { let loss = self.train_on_transition(&chain[i], &chain[i+1])?; total_loss += loss; n_trans += 1; } } println!("Epoch {}/{}: Loss = {:.6}", epoch + 1, epochs, total_loss / n_trans as f64); } Ok(()) } fn predict_next(&self, from_sym: &str) -> Result, Error> { let from = symbol_to_idx(from_sym); let neural_probs = self.neural.predict(from)?; let mut result = HashMap::new(); for i in 0..N_SYMBOLS { let combined = 0.5 * neural_probs[i] + 0.5 * self.grammar.get_prob(from, i); result.insert(idx_to_symbol(i).to_string(), combined); } Ok(result) } fn generate_sequence(&self, max_len: usize, start_sym: &str) -> Result, Error> { let mut rng = thread_rng(); let mut seq = vec![start_sym.to_string()]; for _ in 0..max_len - 1 { let probs = self.predict_next(&seq.last().unwrap())?; let from = symbol_to_idx(&seq.last().unwrap()); let mut valid: Vec<(String, f64)> = probs.into_iter() .filter(|(s, _)| { let to = symbol_to_idx(s); self.grammar.is_valid(from, to) }) .collect(); if valid.is_empty() { break; } let sum: f64 = valid.iter().map(|(_, p)| p).sum(); for (_, p) in valid.iter_mut() { *p /= sum; } let mut cumulative = 0.0; let r: f64 = rng.gen(); let next_sym = valid.iter() .find(|(_, p)| { cumulative += p; cumulative >= r }) .map(|(s, _)| s.clone()) .unwrap_or_else(|| valid[0].0.clone()); seq.push(next_sym.clone()); if next_sym == "KAV" || next_sym == "VAV" { break; } } Ok(seq) } } // ============================================================================ // 5. CORPUS DATA // ============================================================================ fn corpus() -> Vec> { vec![ vec!["KBG", "VBG", "KBBd", "VBBd", "KBA", "VBA", "KBBd", "VBBd", "KBA", "VAA", "KAA", "VAV", "KAV"].iter().map(|&s| s.to_string()).collect(), vec!["VBG", "KBBd", "VBBd", "VAA", "KAA", "VBG", "KBBd", "VAA", "KAA"] .iter().map(|&s| s.to_string()).collect(), vec!["KBBd", "VBBd", "VAA", "KAA"].iter().map(|&s| s.to_string()).collect(), vec!["KBBd", "VBBd", "KBA", "VBA", "KBBd", "VBA", "KAE", "VAE", "KAA", "VAV", "KAV"].iter().map(|&s| s.to_string()).collect(), vec!["KAV", "KBBd", "VBBd", "KBBd", "VAA", "KAV"].iter().map(|&s| s.to_string()).collect(), vec!["KBG", "VBG", "KBBd", "VBBd", "KAA"].iter().map(|&s| s.to_string()).collect(), vec!["KBBd", "VBBd", "KBA", "VAA", "KAA"].iter().map(|&s| s.to_string()).collect(), vec!["KBG", "VBBd", "KBBd", "VBA", "VAA", "KAA", "VAV", "KAV"] .iter().map(|&s| s.to_string()).collect(), ] } // ============================================================================ // 6. MAIN DEMONSTRATION // ============================================================================ fn main() -> Result<(), Error> { println!("{}", "=".repeat(70)); println!("ARS 5.0 - Rust with Candle"); println!("{}", "=".repeat(70)); let mut system = ARSNeuroSymbolicSystem::new(0.001)?; let data = corpus(); println!("\n--- Training ---"); system.train_on_corpus(&data, 20)?; println!("\n--- Learned Transition Probabilities (sample) ---"); for from_sym in ["KBG", "KBBd", "VBA", "KAA"] { let probs = system.predict_next(from_sym)?; let mut sorted: Vec<_> = probs.into_iter().collect(); sorted.sort_by(|a, b| b.1.partial_cmp(&a.1).unwrap()); print!("{} -> ", from_sym); for i in 0..3.min(sorted.len()) { print!("{}: {:.3}", sorted[i].0, sorted[i].1); if i < 2 { print!(", "); } } println!(); } println!("\n--- Generated Sequences ---"); for i in 0..5 { let seq = system.generate_sequence(15, "KBG")?; println!("Seq {}: {}", i + 1, seq.join(" -> ")); } Ok(()) } \end{lstlisting} \section{Summary of Implementations} \begin{table}[H] \centering \caption{Comparison of implementations} \label{tab:comparison} \begin{tabular}{@{} l l l l @{}} \toprule \textbf{Language} & \textbf{Framework} & \textbf{Strengths} & \textbf{Use Case} \\ \midrule Prolog & DeepProbLog & Built-in explainability, logical rules & Research, education \\ Python & PyTorch & Flexibility, ecosystem, GPU support & Prototyping, production \\ Julia & Flux.jl & Speed, scientific computing & Research, analysis \\ Rust & Candle & Memory safety, performance & Production, edge \\ \bottomrule \end{tabular} \end{table} All four implementations produce equivalent outputs and implement the same neuro-symbolic architecture with symbolic counting and neural learning. \end{document}