Performance Representatives (PRs)

• Performance Representatives (PRs) are formal constructs that capture key performance attributes in diverse domains such as process verification, electoral systems, and hardware benchmarking.
• They are employed to verify behavioral equivalence in formal models, ensure proportional representation in voting, and optimize benchmarking by reducing sample sizes while preserving accuracy.
• PRs involve trade-offs between quality and quantity, using algorithmic frameworks to balance efficiency, fairness, and computational tractability.

Performance Representatives (PRs) are a concept that appears in theoretical computer science (notably, in process modeling and verification), electoral theory (proportional representation algorithms), and systems benchmarking (statistical and empirical performance profiling). Across diverse fields, PRs serve as either abstract entities or concrete data points that efficiently and accurately capture or allocate essential performance characteristics—be it system behavior, electoral fairness, or hardware/algorithmic execution time.

1. Definition and Contexts of Performance Representatives

In the context of formal verification and Process Rewrite Systems (PRS), a Performance Representative (PR) often refers to a specification process—an abstract object or transition system that encodes the intended, observable behavior of a system, excluding implementation artifacts such as internal or silent (τ) actions. PRs function as reference points for behavioral equivalence checking: an implementation is correct if it is bisimilar to its PR, modulo the chosen equivalence relation (e.g., branching bisimilarity) (Yin et al., 2014).

In computational social choice and electoral systems, PRs typically denote the individuals or seats that are allocated in proportion to voter preferences, under multiwinner approval voting methods or party-list seat allocation algorithms (Pereira, 2016, Rojas, 2023). Here, being "performance representative" means accurately reflecting the support or demands of voter blocs according to proportional representation criteria.

In benchmarking and AI hardware evaluation, PRs are selected measurement points, indicative of critical or step-wise execution thresholds for workloads such as neural network layers; these efficiently summarize the heavy-tailed parameter space in hardware benchmarking, supporting accurate statistical performance prediction without exhaustive sampling (Jung et al., 12 Jun 2024).

2. PRs in Formal Modeling: Branching Bisimilarity and PRS Hierarchy

Within infinite-state process models, PRs manifest as formal specifications written without silent (τ) moves. The main task is verifying whether a given process conforms to its performance representative under behavioral equivalence.

• Branching bisimilarity is the primary equivalence relation considered in this setting. It refines weak bisimilarity by ensuring that all τ-moves in the simulated process are "state preserving," except when necessary to match visible actions. This distinction is critical because, in practice, most specifications (the PRs) are written omitting silent actions; thus, equivalence checking must "ignore" silent moves only if they maintain state (Yin et al., 2014).
• Decision problem boundaries: For the normed process models (nBPA, nBPP), branching bisimilarity is decidable, enabling automated conformance checking of implementations against PRs. However, for more expressive models such as normed one-counter nets (nOCN) and normed process algebras (nPA), branching bisimilarity becomes undecidable. This leads to strict boundaries: only systems expressible in the simpler PRS fragments admit algorithmic behavioral equivalence checking with performance representatives.

3. PRs in Voting Theory: Proportional Representation Algorithms

In voting and representation, PRs are directly associated with committee members or parliamentary seats allocated in accordance with proportional representation principles.

• Methods ensuring PR: Algorithms such as the Proportional Approval Method using Squared loads, Approval removal, and Coin-flip approval transformation (PAMSAC) are designed to guarantee that elected representatives closely match the proportional allocation implied by voter support, including satisfying strong forms of PR (e.g., strong PR, monotonicity, and positive support) (Pereira, 2016).
• Divisor and multiplicative methods: Parliamentary allocation algorithms, such as Hare–Niemeyer, d’Hondt-Jefferson, and Sainte-Laguë, formalize how PRs (seats) are allocated as integer-valued representatives of fractional voter support. The core mathematical formulations ensure that, modulo rounding and tie-breaking, the set of PRs yielded corresponds to quota- or divisor-based apportionment (Rojas, 2023).
• Algorithmic and axiomatic characterizations: Recent research provides formal characterizations of committees that satisfy Proportionality of Solid Coalitions (PSC), demonstrating that the set of all PSC-compliant committees is exactly those generated by sequentially resolving group demands (the Minimal Demand rule). Each selected PR (committee member) can be interpreted as the response to a distinct coalition’s justified demand for representation (Aziz et al., 2020).

4. PRs in Hardware Benchmarking: Efficient Performance Modeling

In empirical performance modeling—particularly of AI accelerators—PRs are defined as measurement configurations selected for their ability to capture key characteristics of step-wise or block-wise hardware execution.

• PR selection methodology: By conducting parameter sweeps and knowledge-driven analysis of the accelerator's mapping properties (e.g., kernel sizes, tiling factors), informative measurement points are identified. Each PR is typically the final configuration before a step increase in measured execution time, capturing the maximal efficiency within that step. Algorithmic procedures, such as mapping parameters to multiples of detected step widths, systematize PR extraction (Jung et al., 12 Jun 2024).
• Advantages: Using PRs reduces the number of required benchmarks by orders of magnitude while preserving model accuracy. Statistical estimators (e.g., Random Forests) trained on PRs exhibit much lower mean absolute percentage error (MAPE) than those trained on randomly sampled datasets of equal size (e.g., 0.33% MAPE versus >10% for random sampling in some single-layer UltraTrail accelerator benchmarks). This approach also yields faster convergence and data-efficiency.
• Applications: The PR methodology supports rapid and accurate prediction of layer-wise and full-DNN execution times, benefiting Neural Architecture Search, accelerator selection, and benchmarking scenarios with limited hardware access.

5. PRs as an Operational Principle: Algorithms, Tradeoffs, and Fairness

Across domains, the concept of PRs is operationalized through principled algorithms that seek to balance accuracy, representativity, fairness, and computational tractability.

• Game-theoretic and optimization frameworks: In PRS, the bisimulation game (between Attacker and Defender) structures equivalence checking, while in voting, optimization-based or sequential rules allocate PRs to groups proportionally to their justified demands (Yin et al., 2014, Aziz et al., 2020).
• Quality-quantity tradeoff: Selecting a set of PRs often involves a formal tradeoff. In representative democracy models, increasing group (district) size raises the mean competence of a PR but reduces the number of independent votes in aggregate decision-making. There exists an optimal group size—yielding a linear scaling of PRs (representatives) with population—that maximizes the probability of correct ensemble decisions (Magdon-Ismail et al., 2018).
• Algorithmic efficiency: Many PR selection and allocation routines are polynomial time (e.g., Expanding Approvals Rule in voting (Aziz et al., 2017), or step-wise mapping for hardware PRs (Jung et al., 12 Jun 2024)), enhancing both practical deployability and transparency.

6. Future Directions, Open Problems, and Practical Implications

Current research outlines several implications and open areas related to the use and optimization of PRs:

• Decidability boundaries: In process verification, the established undecidability border on branching bisimilarity for richer PRS models strongly constrains the classes of systems for which fully automated conformance to PRs is feasible. Future work seeks to identify subclasses above this boundary, or alternative equivalences with tractable checking (Yin et al., 2014).
• Axiomatic extensions and algorithmic variants: In voting, ongoing developments include refining PR criteria (e.g., from weak PSC to generalised PSC or PJR), optimizing for computational scalability under large-scale participatory budgets, and better integrating indifferences and monotonicity in committee selection (Aziz et al., 2017).
• Systemic fairness and real-world adoption: Ensuring that PRs—whether as seats, committee members, or pandemic response plans—fairly and proportionally reflect community support or systemic requirements is linked to both sustained trust and practical functionality. Algorithmic clarity, transparent rounding, and formal performance guarantees are critical for widespread adoption (Pereira, 2016, Rojas, 2023).
• Hardware modeling extensions: The PR methodology is being extended to capture not just computational dimensions but also memory behaviors in accelerators. Such multidimensional PRs would further enhance prediction accuracy and benchmarking efficiency (Jung et al., 12 Jun 2024).

7. Summary Table: Domains and Roles of PRs

Domain	Definition / Function of PRs	Key Reference
Process Verification (PRS)	Reference behavior (specification) for equivalence	(Yin et al., 2014)
Multiwinner Voting / Parliament	Committee or seat allocated using PR algorithm	(Pereira, 2016, Rojas, 2023)
Empirical Benchmarking (AI/Hardware)	Representative benchmark configuration	(Jung et al., 12 Jun 2024)
Representative Democracy	Elected decision-maker for a population subset	(Magdon-Ismail et al., 2018)

Conclusion

Performance Representatives (PRs) serve as mathematically and operationally principled stand-ins for system behaviors, voter interests, or workload characteristics, enabling both efficient computation and principled fairness. Their precise definition, method of selection, and role in optimization are context-dependent, but across domains PRs distill the essence of complex systems or populations into manageable, representative, and actionable entities. Theoretical boundaries (such as undecidability in process verification) and practical tradeoffs (between quality and number of PRs) continue to shape the ongoing evolution and application of this fundamental concept.