Direct Combinational Synthesis of AI-Inspired Decision Functions Using Contextual Partially Defined Boolean Functions

1. Introduction

AI inference hardware is commonly organized around the computational structure of a model: multiply-accumulate operations, threshold evaluations, table lookups, instruction sequences, or pipelined dataflow. For a fixed quantized decision function, however, the externally required object is ultimately a mapping from a finite set of admissible input encodings to output decisions. This raises a different synthesis question: must the hardware preserve the computational path used to describe or train the decision function, or can the required relation be synthesized directly as Boolean logic?

This paper studies the second possibility. The key observation is contextual. Quantized features are often encoded using constrained code spaces, and only a subset of the nominal Boolean input space is meaningful or reachable. The required function is therefore naturally a partially defined Boolean function (PDBF): output values are mandatory on the admissible feature combinations, while the remaining combinations are outside the contextual care domain. Completion Optimization may exploit this freedom when selecting a realizable completion and synthesizing the resulting circuit.

The experimental contribution is a controlled 28-benchmark suite of synthetic, reproducible AI-inspired decision functions. These benchmarks are not trained neural networks and no claim about prediction accuracy is made. They are deliberately constructed surrogates that isolate the logic-synthesis question: given a quantized contextual decision relation, how compactly and shallowly can it be realized as combinational logic?

Two complementary results emerge. First, GT obtains a lower best gate count and lower best depth than the evaluated direct-PDBF ABC flows on all 28 benchmarks; for every benchmark at least one actual GT implementation simultaneously dominates both evaluated ABC implementations in gate count and logic depth. Second, when each PDBF is first converted to a particular fully specified RTL completion by assigning zero outside the care domain, the resulting RTL→AIG networks are much larger: 22,813 AIG AND nodes in aggregate versus 2,013 for the best evaluated direct-PDBF ABC results. Direct-PDBF ABC is smaller and shallower in all 28 cases. Together, these experiments distinguish synthesis-engine effects from the separate structural cost of committing to one complete off-care behavior before optimization.

2. Relationship to Prior Work

LogicNets demonstrated that sparse quantized neural networks can be mapped to networks of truth-table-based hardware building blocks, achieving very high throughput and low latency on FPGAs [1]. That work establishes an important precedent for moving away from conventional arithmetic-centric neural-network implementation toward logic-centric realization.

Miyasaka et al. studied synthesis and LUT mapping of random-looking dense functions and explicitly investigated don’t-care optimization in a DNN-related setting [2]. Their results show that don’t-cares can provide additional implementation reduction for machine-learning-oriented Boolean functions.

The present work differs in emphasis. It treats the complete fixed decision relation over its admissible quantized feature domain as a contextual PDBF and asks whether the decision block itself can be synthesized directly as combinational logic. The experimental suite is synthetic and controlled; it is intended to establish the synthesis phenomenon before extending the method to trained-model-derived decision functions.

This work also extends the contextual-PDBF viewpoint developed in Paper II [3], the care-domain compatibility formulation developed in Paper IV [4], and the implementation-size perspective of Paper V [5]. Paper IV emphasizes that a PDBF denotes a family of legal completions whose implementation costs may differ; Paper V separates support complexity from realization complexity. Paper VI applies these ideas to AI-inspired quantized decision functions and experimentally compares direct care-domain synthesis with a fixed zero-filled completion synthesized from RTL.

3. Contextual PDBF Formulation

Let q=(q1,…,qr) denote r quantized features. Each feature qi takes values from a finite alphabet and is represented by a Boolean code word. Concatenating the feature codes produces an n-bit Boolean vector x. The nominal Boolean space therefore contains 2^n vectors, but only a subset C⊆{0,1}^n corresponds to admissible feature encodings and contextual combinations.

Let f:C→{0,1}^m specify the required decision outputs on the care domain C. The contextual PDBF is F=(C,f). A combinational implementation N is valid when N(x)=f(x) for every x∈C. For x∉C, the output is unrestricted unless additional application constraints are imposed.

This changes the synthesis objective. Instead of implementing one predetermined fully defined Boolean function over all 2^n inputs, synthesis may choose any completion g:{0,1}^n→{0,1}^m satisfying g(x)=f(x) on C. The design problem becomes: choose a favorable legal completion and realize it with low gate count and low logic depth.

The conceptual flow is therefore: Quantized Features → Contextual Care Domain → PDBF → Completion Optimization → Combinational Decision Circuit. The synthesized block need not preserve a multiplier, MAC array, instruction sequence, state machine, or iterative evaluation structure. This statement applies to the synthesized decision block, not to AI systems in general.

4. Benchmark Suite

The experimental suite contains 28 evaluated synthetic AI-inspired contextual PDBFs. Application labels include autonomous braking, battery management, crop health, data-center cooling, drone obstacle response, ECG arrhythmia, financial fraud, food quality, hand gesture, jet-engine health, keyword spotting, malware behavior, medical triage, motor fault, network intrusion, power-grid fault, predictive maintenance, road scene, robot grasp, sepsis risk, smart-camera events, sonar target, supply-chain risk, voice command, wafer defect, water-treatment anomaly, wildfire risk, and maritime navigation.

The benchmarks use constrained quantized feature encodings, including one-hot and mixed-radix structures. Consequently, the contextual care domains occupy only a fraction of the nominal Boolean input space. The suite spans 16–24 primary inputs, 4–7 outputs, and 729–6,561 care terms. Network Intrusion is the densest case at 7.03125% of the nominal input space; most other cases are below 0.3%, and many are below 0.1%.

The decision rules are deterministic and reproducible. They are benchmark surrogates rather than trained classifiers. This distinction is essential: the experiment measures logic realization of specified contextual PDBFs, not model-learning quality, generalization, or inference accuracy.

5. Experimental Methodology

Experiment A synthesizes each benchmark directly as a contextual PDBF with GT Synthesis and with the evaluated ABC synt and transtoch flows. Results are reported in technology-independent two-input logic-gate count and maximum logic depth. Where GT produced multiple implementations, all observed gate/depth tradeoffs were retained.

Two metric-wise comparisons are used. Best-gate comparison selects the minimum observed GT gate count and compares it with the minimum gate count across the two evaluated ABC results. Best-depth comparison independently selects the minimum observed GT depth and compares it with the minimum depth across the two evaluated ABC results. These minima need not correspond to the same GT implementation; they characterize the best observed value of each metric.

A stricter implementation-level test is also reported. GT is said to simultaneously dominate both evaluated ABC flows when at least one actual GT implementation has lower gate count and lower depth than each of the two ABC implementations. This test does not combine metrics from different GT variants.

Percentage reduction is computed as R_G=1−G_GT/G_ABC for gate count and R_L=1−L_GT/L_ABC for depth, using the metric-wise best values defined above. Arithmetic mean, median, range, and a geometric-mean reduction derived from the GT/ABC ratios are reported. Spearman rank correlation is used descriptively to examine whether care density alone tracks the magnitude of the observed reductions.

Experiment B isolates the effect of committing to one fully defined completion before synthesis. For each of the same 28 PDBFs, synthesizable Verilog RTL was generated so that every specified care-domain output is preserved while every off-care input assignment is explicitly assigned output zero. This construction is a deterministic zero-filled completion of the PDBF; it is not claimed to reproduce an original algorithmic RTL implementation. The RTL modules were synthesized with Yosys 0.68 using ABC mapping to AND/inverter form and written as AIGER. AIG size and depth were then measured with yosys-abc using the reported AND-node count and logic level count [6].

The cleanest comparison in Experiment B is therefore between the RTL-derived AIG and the evaluated direct-PDBF ABC networks, because both are measured in the ABC/AIG representation. GT gate counts are also reported relative to the RTL baseline as a technology-independent implementation-size comparison, but they should not be interpreted as cell-for-cell identical to AIG AND-node counts.

6. Experimental Results

Table I. Aggregate results over 28 evaluated benchmarks

The 28/28 simultaneous-dominance result is stronger than the metric-wise minima: it states that, for every benchmark, at least one concrete GT netlist lies below both evaluated ABC netlists in both reported dimensions. Thus the aggregate advantage is not merely an artifact of selecting one GT circuit for area and another for depth.

The reductions are substantial but not uniform. Gate-count reduction ranges from 60.0% to 84.3%, while depth reduction ranges from 11.1% to 61.5%. The weaker and more variable depth improvement is consistent with a multiobjective synthesis setting in which some GT variants trade additional gates for shorter depth.

Care density alone is not a strong predictor of the observed advantage. The Spearman coefficients are approximately 0.205 for gate reduction and 0.072 for depth reduction. These descriptive values do not imply that contextual sparsity is unimportant. Rather, they indicate that the fraction |C|/2^n is too coarse to characterize the structure that determines realizability. The arrangement of care points, output conflicts, essential variables, separator structure, and the chosen completion may all matter. This observation provides a direct bridge to the support-complexity questions developed in Paper V [4].

Table II. Detailed 28-Benchmark Results

Note: Gate and depth reductions are metric-wise best comparisons. The GT variant with minimum gates may differ from the GT variant with minimum depth. “G|L” denotes gate count | logic levels.

7. Cost of Premature Full Specification

The second experiment asks a different question from the GT-versus-ABC comparison: what structural cost is observed when the contextual freedom is resolved before synthesis by choosing one particular complete Boolean function? The tested completion assigns zero to every off-care combination while preserving all required care-domain outputs. It is therefore one legal completion of each benchmark, not a claim about the best possible RTL description or the best possible complete function.

Table III. Fully specified RTL→AIG versus direct contextual-PDBF synthesis

*GT gate counts and AIG AND-node counts arise from different technology-independent network representations. Ratios involving GT are therefore descriptive implementation-size comparisons, not strict cell-for-cell equivalences.

The ABC/AIG comparison is broad rather than outlier-driven. Direct contextual-PDBF ABC uses fewer nodes than the zero-filled RTL→AIG completion in every benchmark. The median per-benchmark RTL/ABC node ratio is 11.00×, the arithmetic mean is 11.59×, and the geometric mean is 10.30×. Per-benchmark node reductions range from 78.33% to 96.77%. Depth follows the same direction in all 28 cases, with a mean reduction of 40.66% and a median reduction of 43.75%.

This result is an experimental instance of the completion-family argument developed in Paper IV: legal completions agree on the care domain but need not have equal implementation complexity. It does not establish that zero is a uniquely bad completion, that every fully specified RTL implementation would have comparable cost, or that conventional synthesis cannot exploit don’t-cares when they are represented explicitly. Rather, it demonstrates that resolving contextual freedom by this reproducible zero-filled completion before synthesis carries a substantial structural cost for the present benchmark family.

8. Discussion

The results support a narrow but important claim: when an AI-inspired fixed decision relation is specified only over a constrained contextual domain, direct PDBF synthesis can produce very small and shallow combinational realizations. The experiment does not establish that every trained AI model should be implemented this way, nor that contextual PDBF synthesis will dominate arithmetic accelerators at all scales.

The strongest interpretation is architectural. Once a fixed quantized decision relation has been obtained, the synthesis target need not inherit the structure used to derive that relation. If the admissible domain is sufficiently constrained and the Boolean relation has exploitable structure, a synthesis engine may discover a compact completion and circuit that bear little resemblance to the original algorithmic description.

The weak association between care density and reduction is scientifically useful. A sparse domain creates freedom, but the geometry of the care set and the pattern of required output distinctions determine how much of that freedom is useful. Future analysis should therefore include effective support, essential-variable structure, minimum care-separating support where tractable, and other measures of care-domain separability.

The RTL experiment sharpens the architectural interpretation. A PDBF represents a compatibility class, not one predetermined complete function. Selecting a completion first and then optimizing only that function can exclude structurally favorable alternatives that remain legal under the original semantics. In the present suite, the zero-filled completion produces substantially larger AIGs than direct PDBF synthesis even when ABC participates in both flows. This isolates a specification-level effect from the separate question of which synthesis engine is strongest.

9. Limitations and Threats to Validity

First, the 28 benchmarks are synthetic AI-inspired surrogates rather than functions extracted from trained models. They demonstrate a synthesis mechanism, not end-to-end AI accuracy or model compression.

Second, the comparison is against two evaluated ABC flows, synt and transtoch. The results should not be interpreted as a universal statement about all ABC scripts, all synthesis systems, or all technology-mapping objectives.

Third, the reported gate counts and levels are technology-independent logic metrics. Standard-cell area, FPGA LUT count, routing, timing, power, and physical-design effects remain to be measured.

Fourth, the benchmark family deliberately uses constrained quantized encodings. This is the intended contextual-PDBF setting, but it means the results should not be extrapolated to dense unconstrained Boolean functions without additional evidence.

Finally, the metric-wise gate and depth summaries may select different GT variants. For that reason the paper separately reports the stricter 28/28 implementation-level simultaneous-dominance result.

The RTL→AIG baseline introduces an additional limitation. It evaluates one deterministic zero-filled completion generated from the PDBF, not hand-written application RTL and not an exhaustive search over all possible fully specified completions. Its purpose is to measure the cost of one reproducible act of premature full specification. The 11.33× aggregate RTL/ABC ratio therefore must not be generalized to all RTL descriptions or all conventional synthesis flows. A stronger future study should compare several completion policies, explicit RTL don’t-care formulations where semantics are well defined, and technology-mapped implementations.

10. Future Work: From AI-Inspired Surrogates to Trained Models

The next experimental step is to derive contextual PDBFs from small public trained and quantized models. Candidate workflows include sparse quantized networks whose neuron or block behavior can be enumerated, and application domains already used by logic-centric neural-network research such as network intrusion detection and particle-physics triggers [1].

A trained-model experiment should preserve a clear chain of evidence: trained model → quantization → reachable or admitted feature/code domain → exact PDBF extraction → functional equivalence on the care domain → GT/ABC synthesis → technology mapping. Accuracy should be measured before and after any approximation; if no approximation is introduced on the care domain, equivalence rather than accuracy loss is the relevant correctness criterion.

Future work should also connect these experiments to Paper V by measuring effective support and care-separating support complexity. This may help distinguish when sparse contextual encoding merely reduces the nominal domain from when it genuinely reduces the number of variables and distinctions that a circuit must realize.

11. Conclusion

This paper introduced direct combinational synthesis of AI-inspired decision functions as contextual PDBFs and evaluated the approach on 28 reproducible synthetic benchmarks. In the direct-PDBF comparison, GT achieved lower best gate count and lower best logic depth than the two evaluated ABC flows on every benchmark. Mean metric-wise reductions were 70.9% in gate count and 42.6% in logic depth, and at least one concrete GT implementation simultaneously dominated both ABC results in both metrics for all 28 cases.

A second experiment measured the cost of first resolving the same contextual specifications into a zero-filled fully specified RTL completion. The RTL→AIG flow produced 22,813 AIG AND nodes in aggregate, compared with 2,013 nodes for the best evaluated direct-PDBF ABC results: an 11.33× ratio of aggregate counts and a 91.18% aggregate node reduction. Direct-PDBF ABC was smaller and shallower in all 28 benchmarks, showing that the result is not driven by a small number of outliers. This finding does not imply that RTL or conventional synthesis is intrinsically unable to exploit don’t-cares; it shows that, for this benchmark family, committing to the tested complete off-care behavior before synthesis substantially restricts the implementation opportunity available under care-domain compatibility. Together with the GT results, the experiment strengthens the case for treating contextual incompleteness as an optimization resource. The next steps are trained-model-derived PDBFs, alternative completion policies, and technology-mapped area, timing, and power evaluation.

References

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