FRAIG-BMC: Verification & Bayesian Modeling

• FRAIG-BMC is a dual-method framework that combines formal verification using functional reduction and nonparametric Bayesian modeling for dynamic systems.
• It leverages FRAIG to reduce SAT instance complexity in bounded model checking by eliminating redundant logic through functional equivalence detection.
• In Bayesian modeling, FRAIG-BMC employs fragmentation-coagulation processes to effectively capture evolving community structures in complex networks.

FRAIG-BMC refers to two distinct but technically significant methodologies in computational verification and Bayesian network modeling. In formal property verification, FRAIG-BMC denotes the integration of functional reduction via functionally reduced and-inverter-graphs (FRAIG) into bounded model checking (BMC), accelerating SAT-based property checking by eliminating redundant unrolled logic. Separately, in network science, FRAIG-BMC is used as an alternative label for the fragmentation coagulation based Mixed-Membership Stochastic Blockmodel (fcMMSB), a nonparametric Bayesian approach to dynamic network analysis. This article addresses both perspectives in rigorous detail, referencing foundational works (Yu et al., 7 Dec 2025, Yu et al., 2020).

1. FRAIG-BMC in Bounded Model Checking: Formal Definition

Bounded Model Checking (BMC) is a method for verifying safety properties of transition systems by unrolling the transition relation to a bounded depth and encoding the resultant sequence into a SAT instance. The standard BMC formulation for a finite-state system with Boolean state variables $\Sigma$ is:

• Initial predicate: $I(s_0)$ over state $s_0$
• Transition relation: $T(s_i, s_{i+1})$ between successive states
• Safety property: $P(s)$

At depth $k$, BMC constructs

$$T_k(s_0, s_1, \dots, s_k) = \bigwedge_{i=0}^{k-1} T(s_i, s_{i+1})$$

and the SAT formula:

$$\Phi_k = I(s_0) \wedge \Bigl(\bigwedge_{i=0}^{k-1} T(s_i, s_{i+1})\Bigr) \wedge \neg P(s_k)$$

A SAT solver is invoked on $\Phi_k$ to search for counterexamples, incrementing $k$ if unsatisfiable. Incremental SAT allows clause reuse across successive bounds (Yu et al., 7 Dec 2025).

2. Functionally Reduced And-Inverter-Graph (FRAIG): Data Structures and Reduction Steps

An And-Inverter-Graph (AIG) is a directed acyclic graph representing logic circuits, with internal nodes for 2-input AND gates and optional edge inversions. Each node $n$ computes a Boolean function $f_n$ over inputs and register outputs.

Functional equivalence is defined as:

$$n \equiv m \Longleftrightarrow \forall x \in \{0, 1\}^r: f_n(x) = f_m(x)$$

FRAIG reduction in BMC unrolling proceeds in three stages:

1. Trivial Logic Simplification: Collapse gates with constant or identical inputs.
2. Structural Hashing: Merge nodes with identical fanin structures.
3. SAT Sweeping (Functional Reduction):
   • Assign random-simulation signatures $\sigma(n)$
   • Form Equivalence Classes (ECs)
   • For $n$ and candidate $q$ in its EC, test $SAT(f_n \oplus f_q)$; merge if UNSAT, otherwise refine with counterexamples
   • Cap EC size to maintain scalability

Unique nodes $V_{unq}$ form the reduced AIG, where redundant subgraphs are aliased to representatives (Yu et al., 7 Dec 2025).

3. FRAIG-BMC Algorithmic Workflow and Complexity

The FRAIG-BMC BMC process is defined by a frame-wise loop:

• At each unroll, apply reduction steps to emerging AND nodes
• For each unsimplified and structurally unique node, perform simulation, EC grouping, and SAT sweep for functional merging
• Link frame-to-frame transitions and assert the property negation
• Resolve SAT to determine counterexample existence, leveraging incremental clause learning

Without FRAIG, clause count grows as $O(kn)$; FRAIG merges repetitive gates and reduces SAT variables to $O(n_{reduced})$. While SAT sweep incurs simulation and solver overhead, it is amortized in designs with repeated modules (Yu et al., 7 Dec 2025).

4. FRAIG-BMC for Bayesian Network Modeling: Fragmentation-Coagulation Mixed Membership Stochastic Blockmodel

In a distinct context, FRAIG-BMC (fcMMSB) designates a nonparametric Bayesian network model with temporal dynamics:

• Entity-based clustering for communities; linkage-based clustering for group assignment
• Communities evolve by fragmentation and coagulation via Discrete Fragmentation Coagulation Process (DFCP)
• Community memberships $z_i^t$, group indicators $g_{i \rightarrow j}^t$, compatibility matrix $B$, adjustments $Q$
• Link probability defined by logistic regression on $y_{ij}^t$ derived from community and group assignments

DFCP progression involves:

• Initialization by Chinese Restaurant Process (CRP) with concentration parameter $\zeta$
• Community fragmentation into subclusters via CRP
• Coagulation merging fragments to communities via global CRP with concentration $\eta$

Polya-Gamma augmentation enables efficient Gibbs sampling for Bernoulli likelihoods and parameter inference. Fragmentation and coagulation flexibly model community birth, death, splits, and merges; $\zeta$ and $\eta$ control process volatility (Yu et al., 2020).

5. Experimental Results and Observed Performance

FRAIG-BMC (in the BMC context) demonstrates substantial speedups:

• Benchmarks: Sequential equivalence checking (SEC), partial retention register detection (PartRet), information flow checking (IFC)
• Platforms: aigbmc with incremental CaDiCaL SAT; baselines include MiniSAT-AIGBMC and ABC-bmc3
• Metrics: Number of merged nodes, bound depths reached (≤3600s), SAT time, peak memory

Table: Application Results (adapted from (Yu et al., 7 Dec 2025)) | Application | Instances | FRAIG-BMC Solved (%) | Baselines Solved (%) | |-------------|-----------|----------------------|----------------------| | SEC | 61 | ~30% more cases | - | | PartRet | 60 | ~60 | MiniSAT: ~50, ABC: ~30 | | IFC | 56 | ~55 | MiniSAT: ~40 |

Typical speedups range 1.5×–4×, with highest gains on designs exhibiting module repetition. Designs lacking such repetition may see neutral or marginally negative impact due to FRAIG overhead (Yu et al., 7 Dec 2025).

6. Practical Implications, Applicability, and Limitations

FRAIG-BMC substantially improves SAT-solving efficiency in BMC for verification tasks involving repeated circuit modules—dual-rail, shadow circuits, retimed clones—by enabling aggressive merging of functionally and structurally equivalent nodes. Learned SAT clauses propagate more broadly, formulas shrink, and deeper bounds become tractable.

Constraint-aware simulation is necessary under tight property or environment constraints to avoid missing valid equivalences. On random logic networks with minimal functional sharing, FRAIG-BMC's overhead may outweigh its reduction benefits, suggesting careful benchmarking and tuning.

In Bayesian network analysis, FRAIG-BMC (fcMMSB) supports accurate modeling of dynamic community structures, accommodating complex birth/death/split/merge phenomena via hyperparametric control within DFCP (Yu et al., 2020).

7. Illustrative Case Study and Conclusion

A circuit containing two identical 32-bit ripple-carry adders, instantiated per timeframe, demonstrates the advantage: standard BMC encodes 64 gates/frame, whereas FRAIG-BMC merges bit-slices and introduces only 32 unique variables per frame, yielding a ~50% reduction in formula size and halving SAT solve time in deep unrolls (Yu et al., 7 Dec 2025).

In summary, FRAIG-BMC in verification employs functional reduction during unrolling to compress Boolean SAT instances, fostering superior scalability in BMC—whereas, in probabilistic modeling, FRAIG-BMC provides a generative and inference framework for temporal network community evolution. Both methodologies exploit redundancy, whether logical or structural, to deliver enhanced computational tractability.