Binary Speculation Diagram in Circuit Synthesis

• Binary Speculation Diagram (BSD) is a directed acyclic graph representing Boolean functions via speculative expansion to efficiently model complex circuit logic.
• It employs recursive Boolean expansion combined with Monte Carlo sampling and Boolean distance metrics to refine designs and significantly reduce synthesis time.
• Extensions like State-BSD incorporate processor state modeling to capture inter-instruction dependencies, achieving near-perfect accuracy and substantial performance improvements.

A Binary Speculation Diagram (BSD) is a graph-based data structure for representing and synthesizing the Boolean logic of complex digital systems and has emerged as a central abstraction in automated circuit design, particularly in the context of large-scale hardware synthesis from empirical input–output data. BSD and its extensions, such as the Stateful Binary Speculation Diagram (State-BSD), have demonstrated the ability to learn and generate high-performance CPU designs and to address bottlenecks in modern processor architectures, notably by efficiently managing combinatorial and state-dependent logic. In related mathematical contexts, the acronym BSD also refers to the Binary Signed-Digit representation in coding theory and to Bayesian Spectral Decomposition in neural data analysis. This article focuses on BSD as used in circuit logic synthesis and processor design, with comparative notes on other BSD usages when relevant.

1. Definition and Fundamental Structure

A Binary Speculation Diagram is a directed acyclic graph that represents a Boolean function for a circuit. Each internal node is labeled by an input variable and branches to two children corresponding to binary decisions (often $x_i = 0$ and $x_i = 1$), following expansion rules akin to the Shannon Decomposition or the Boolean Expansion Theorem:

$$\mathcal{F}(x) = (\neg x_i) \cdot \mathcal{F}(x|x_i = 0) + x_i \cdot \mathcal{F}(x|x_i = 1)$$

In contrast to conventional Binary Decision Diagrams (BDDs), which enumerate every logic path exhaustively, BSDs implement a speculative mechanism. Subtrees are replaced with constant outputs (0 or 1) when sufficiently supported by empirical data, allowing the diagram to abstain from full expansion in well-constrained regions of the input space. Each leaf represents either a speculated constant or an unresolved function fragment, subject to possible further expansion as dictated by additional samples or criteria.

2. BSD Generation Methodology in Automated Machine Design

The creation of a BSD for synthesizing a complete CPU involves an iterative, data-driven process that combines recursive Boolean expansion, Monte Carlo-based statistical sampling, and structure-reducing function distance metrics. The approach begins with a trivial constant-function speculation for all outputs and proceeds as follows (Cheng et al., 2023):

1. At each expansion step, input–output pairs are sampled at leaf nodes.
2. If sample outputs at a node are uniform (all 0 or all 1), that node is labeled as a constant output (speculation).
3. If variance is detected, the node is expanded on a candidate variable using the Boolean Expansion Theorem.
4. The process repeats recursively, adaptively refining the diagram only in regions where ambiguity remains per observed data.
5. Diagrams are pruned or merged using Boolean distance: subgraphs are compared via a circuit complexity–based metric, allowing isomorphic or near-isomorphic sub-functions to be collapsed.

Accuracy monotonically improves with each expansion: if $\mathcal{F}_k$ is the function after $k$ refinements, then $$\text{Acc}(\mathcal{F}_k) \leq \text{Acc}(\mathcal{F}_{k+1})$$, where accuracy is measured on a holdout sample set.

This process enables the efficient search of a design space of size $10^{10^{540}}$ (for 1798-input, 1826-output Boolean mappings in a RISC-V-scale CPU), a space several orders of magnitude larger than previously tractable for machine-driven design.

3. Extensions: The Stateful Binary Speculation Diagram (State-BSD)

While the basic BSD is limited to stateless, purely combinational logic, the State-BSD (Cheng et al., 6 May 2025) incorporates explicit processor internal states, addressing the challenge of modeling and exploiting inter-instruction data dependencies—key for superscalar CPU design.

State-BSD comprises two coupled components:

• State-Selector: Learns, via simulated annealing, a compact set of processor state snapshots ("reusable" states) to be buffered. The update step is probabilistic, accepting new candidate buffers $S'_k$ with

$P = \exp\left(-\frac{E(S'_k) - E(S_k)}{T}\right)$

where $E(\cdot)$ is a reusability "energy" function and $T$ is the annealing temperature.

• State-Speculator: Employs BSD expansion on the current instruction and selected states, constructing a decision structure to predict the dependent data and a validity signal. Training proceeds until precision is empirically 100%, guaranteeing functional correctness of predictions essential in parallel execution contexts.

The composite predictor $G: \{\Psi, S\} \to \{D, C\}$ takes all internal states $\Psi$ and buffered states $S$ to produce data predictions $D$ and their validity $C$.

4. Algorithmic Properties and Efficiency Considerations

BSD-based learning leverages three central mechanisms for computational efficiency and accuracy:

• Speculative Leaf Labeling: By speculating output constants early and only refining when data evidence warrants, unnecessary expansions are avoided, greatly reducing diagram size compared to canonical BDDs.
• Monte Carlo Expansion: Statistical sampling at each leaf controls expansion, yielding near-perfect accuracy ($>$99.99999999999% on held-out sets for CPUs) without exponential blowup.
• Boolean Distance–Based Node Reduction: Nodes in the BSD are clustered or merged if their sub-functions are deemed similar under a function-distance metric $Dist(f, g)$ based on circuit complexity, further compressing the diagram and reducing synthesis time.

These strategies enabled the synthesis of an industrial-scale CPU (RISC-V class) from IO data alone in five hours, compared to weeks or months for conventional manual hardware design.

5. Applications and Impact in Processor and System Design

BSD and State-BSD have demonstrated substantial utility in the automated synthesis and optimization of digital systems:

Model	Input Type	Design Scope	Example Achievement
BSD	IO pairs (stateless)	Combinational logic	RISC-V CPU (comparable to Intel 80486SX, runs Linux) (Cheng et al., 2023)
State-BSD	IO + selected internal state	Superscalar pipelines	QiMeng-CPU-v2 (comparable to ARM Cortex A53; 380× speedup) (Cheng et al., 6 May 2025)

By learning combinational and data-dependent logic for control and arithmetic units, BSD-generated CPUs have matched or outperformed baseline automated designs and reached parity with human designs of mid-2010s vintage. State-BSD’s accurate internal state modeling enables robust inter-instruction parallelism in superscalar architectures.

The BSD approach has broader implications:

• It reduces designer effort and domain-specific coding, democratizing high-performance architecture creation.
• The automatic "rediscovery" of the von Neumann decomposition (emergence of control and execution units from BSD structure) suggests that algorithmic learning of architectural primitives is feasible at scale (Cheng et al., 2023).
• The principle can, in theory, be extended to other hardware modules with strong data dependencies (e.g., branch predictors, multistage accelerators).

6. Relationship to Other BSD Usages and Misconceptions

The acronym BSD may refer to other constructs:

• Binary Signed-Digit (BSD) Representation: In integer encoding and arithmetic, BSD represents numbers as signed binary vectors ($b_j \in \{-1,0,1\}$). Connections between BSD representations and the weight distribution coefficients of Stern polynomials have been established, with the count of $i$-bit BSD representations of $n$ having $\ell$ zero digits given by the $t^\ell$ coefficient of the Stern polynomial $B_{2^i - n}(t)$ (Monroe, 2021).
• Bayesian Spectral Decomposition (BSD): In neuroscience, BSD denotes a Bayesian statistical framework for decomposing neural power spectra, providing parametric modeling, rigorous uncertainty propagation, and principled group comparison via log Bayes factors (Medrano et al., 28 Oct 2024).

These usages are distinct in both formalism and application domain; context must be used to distinguish circuit-logic BSDs from those in numerical or statistical modeling.

7. Broader Implications and Future Directions

The adoption of BSD and its stateful extension marks a shift toward data-driven, AI-powered hardware design. The capacity to automatically learn both circuit logic and architectural patterns from empirical behavior, at a scale previously accessible only to human experts, suggests further research into:

• Hierarchical and hybrid extensions for more heterogeneous architectures.
• Automated adaptation to domain-specific workload and power constraints.
• Potential discovery of novel architectures via unconstrained machine-driven exploration.

As BSD-based methods continue to mature, plausible implications include significant reductions in chip design cycle time, cost, and the emergence of self-optimizing hardware that adapts beyond the bounds of traditional, human-crafted blueprints.