Metric Encoding Map (MEM) in Model Checking

• Metric Encoding Map (MEM) is a specialized technique that efficiently encodes metric temporal operators by treating them as first-class entities in bounded model checking.
• It introduces auxiliary propositional variables to compactly represent large time constants and accurately handle looping in finite unrollings of temporal models.
• Empirical evaluations demonstrate that MEM significantly reduces SAT solver workload and formula generation overhead, making it crucial for hard real-time system verification.

A Metric Encoding Map (MEM) is a specialized encoding and data structuring technique designed to efficiently support the evaluation of metric-dependent properties within bounded model checking (BMC) of temporal logic systems. In its canonical formulation, MEM is geared toward properties expressed with metric temporal operators—in particular, those found in hard real-time system specifications—which impose stringent demands on both the size and semantics of the generated Boolean formulae during SAT-based verification.

1. Motivation and Conceptual Overview

Bounded model checking requires translation of both the system model and the specification into a propositional formula suitable for SAT solving. Metric temporal logics, such as those featuring the "exactly" ($\lozenge_{=d}$) or "within" ($\Box_{≤d}$) operators, introduce an acute explosion in formula size when naively "unrolled," since each metric operator over a time constant $d$ is expanded into a chain of $d$-dependent non-metric operators. The MEM approach circumvents this by representing metric operators as first-class entities and encoding their semantics via auxiliary propositional variables, encoding the effect of large time constants compactly. It also provides a mechanism for handling the folding of time—“closing the loop”—when simulating infinite or periodic behaviors in a finite structure.

2. Auxiliary Variables and Metric Operator Encoding

The core technique introduces auxiliary propositional variables to "collapse" the effect of large time constants:

• For future-oriented metric temporal operators such as $\lozenge_{=t}\, \phi$, auxiliary variables $MF{\phi}{j}$, with $0 \le j \le t-1$, encode the value of $\phi$ at offsets beyond the finite unfolding boundary, particularly when time wraps in the periodic representation.
• For past-oriented operators (e.g., $\blacklozenge_{=t}\, \phi$), $MP{\phi}{j}$ are introduced, tracking values preceding the past loop selector.

The encoding for $\lozenge_{=t}\, \phi$ is defined piecewise:

$$|[\lozenge_{=t}\,\phi]|_i \iff \begin{cases} \, |[\phi]|_{i+t} & \textrm{if } i+t \le k, \ \, MF{\phi}{\,t+i-k-1} & \textrm{otherwise.} \end{cases}$$

Similarly, for bounded global operators:

$$|[\Box_{\le t}\,\phi]|_i \iff \Big(\bigwedge_{j=1}^{\min(t,k-i)} |[\phi]|_{i+j}\Big) \land \Big(\bigwedge_{j=k+1-i}^{t} MF{\phi}{\,i+j-k-1}\Big).$$

This representation reduces the number of subformulae from $O(d \cdot k)$ (in unrolled form) to a sum proportional to $d + k$, which substantially decreases both the number of variables/constraints and the computational resources required for complex real-time specifications.

3. Handling Forward and Backward Loops

Bounded Model Checking uses a finite unrolling (from $S_0$ to $S_k$) but simulates infinitely repeating behavior via loop selectors ($l_i$ for forward, $l'_i$ for backward). These selectors identify which transitions "wrap" from the end to a point earlier in the unfolding, thereby both simulating infinite (ultimately periodic) or bi-infinite sequences.

Forward loop encoding enforces constraints like:

$l_i \Rightarrow (S_{i-1} = S_k),$

with "in loop" markers and existence flags. For backward loops, analogous variables and structure are applied.

Metric operators whose time indices reach beyond the finite interval must "jump" into this periodic segment. The MEM encoding ensures, through precise index modulation (e.g., $|[\phi]|_{\operatorname{mod}(j-1,i+1)}$) and equations distinguishing cases (e.g., (enc1), (enc2)), that the truth assignments reflect the intended cyclical behavior without ambiguity.

4. Experimental Benchmarks and Quantitative Results

Empirical evaluation demonstrates the substantial benefit of MEM:

• In a shift-register scenario, the MEM reduced variables and constraints for $\lozenge_{=d}\,\mathit{out}$ from $O(d\cdot k)$ to only two main subformulae and a fixed auxiliary overhead.
• For $k = 400$ and $d$ ranging $10$–$150$, speedups reached up to $250\%$ in SAT-solving and $300\%$ in formula generation.
• In case studies (real-time allocator, Fischer’s protocol, railway crossing, lamp timer), mean improvements were $42.2\%$ (generation) and $62.2\%$ (SAT solving)—a combined improvement of approximately $47.9\%$.
• The cost of the MEM encoding grows with $d + k$, not $d \cdot k$, making it vastly more efficient for specifications with large time constants.

The empirical results decisively show that MEM not only reduces SAT instance sizes but also accelerates model checking, particularly for hard real-time and high-metric systems.

5. Technical Summary and Formal Properties

The technique’s systematic management of auxiliary variables, piecewise encoding for cases where the temporal index crosses boundaries, and loop closure via selector variables together establish the MEM as an optimal strategy for handling metric temporal operators in BMC. The approach:

• Avoids explicit unrolling of next-time operators over metric intervals
• Encodes the semantics of metric operators using $MF{\phi}{j}$ and $MP{\phi}{j}$
• Maintains the integrity of infinite or bi-infinite temporal behaviors via dual loop selectors and precise indexing
• Reduces variable/constraint counts and enhances solver efficiency

6. Applicability and Limitations

MEM is highly effective in settings where metric temporal operators with large numerical constants dominate, such as requirements in hard real-time system verification. It is especially advantageous when bounds are tight and loop simulation is critical. The mechanism is less impactful in scenarios where temporal constants are small or properties are dominantly non-metric.

Potential limitations include the more complex case handling required for loop closures and the necessity to maintain additional auxiliary variable layers, though these overheads are sublinear compared to the avoided unrolling cost.

7. Influence on Bounded and Satisfiability Checking

The introduction of MEM and its associated techniques advances the state-of-the-art in SAT-based model checking by providing a direct and efficient route for encoding metric temporal logic. The approach supplies both a conceptual and practical foundation for further extension—such as multimodal temporal logics, different loop types, or sophisticated metric hierarchies in hybrid systems—enhancing the tractability of automated verification for complex temporal requirements in both research and industrial verification contexts.