algorithms

This study addresses the challenges of assessing students’ comprehensive competencies in algorithm courses and the disconnect between academic instruction and industry needs. Grounded in the CC2020 competency model, it proposes a multidimensional assessment framework that integrates knowledge, skills, and professional dispositions. Behavioral data from programming assignments and written coursework of 169 students were collected using the xAPI specification. Learning behavior sequences were modeled via Markov processes, and cluster analysis was employed to identify distinct competency profiles. Additionally, a timeliness metric for submissions was introduced to quantify task difficulty. The framework not only enables computable representations of student competencies but also provides empirical support for personalized instructional interventions and curriculum refinement, thereby effectively bridging the gap between academic training and industry requirements.

Algorithm engineering has long lacked a unified methodology, resulting in fragmented knowledge across subfields and poor reproducibility. To address this, this paper introduces Karl Popper’s “Three Worlds” theory—comprising ontology (clarifying problems, tasks, design, and implementation), epistemology (distinguishing descriptive from prescriptive knowledge), and methodology (systematizing knowledge evolution)—to establish the first integrated three-dimensional framework for the field. By synthesizing philosophical methodology, ontological modeling, and empirical paradigm analysis, we propose the first formal research framework for algorithm engineering, explicitly defining validity criteria for diverse scholarly contributions. This framework enhances systematicity, rigor, and cross-domain comparability in algorithm design, implementation, and evaluation. It provides foundational methodological support for disciplinary integration and advances algorithm engineering toward a mature, theory-grounded science.

This paper investigates the mathematical foundations and fundamental limits of algorithmic music composition. Addressing the central question—“Can algorithms generate genuinely creative music?”—it pioneers the application of metamathematical methodology to this domain. Leveraging Turing machine models, Gödel numbering, recursive function theory, and formal system analysis, the study rigorously characterizes the computability, decidability, completeness, and consistency constraints governing musical generation. The results establish an inherent, insurmountable formal barrier to algorithmic composition: genuine creativity cannot be fully captured by unrestricted formal systems and must instead be redefined within constrained, well-specified frameworks. Consequently, the work not only delineates the theoretical capabilities and limitations of algorithmic music but also introduces the first formally verifiable framework for “creative AI music generation.” This framework provides a rigorous foundation for developing next-generation music AIs whose creative capacity can be mathematically proven.

This work addresses automatic runtime and variable-size bound analysis for integer programs, focusing on the decidable subclass of periodic rational-solvable loops (PRS-loops). The proposed method introduces a modular analysis framework: it first derives local bounds for PRS-loops, then lifts them to global bounds via program transformation and inductive reasoning. Crucially, it extends the decidability of PRS-loop analysis to arbitrary integer programs by designing a synergistic synthesis mechanism combining abstract interpretation with rational linear algebra. The approach is fully automated in the tool KoAT, supporting precise derivation of polynomial and exponential complexity bounds as well as variable growth bounds. Experimental evaluation demonstrates effectiveness on diverse nontrivial integer programs, significantly improving the completeness, precision, and practicality of automated complexity analysis.

Formal theories of algorithms have long been confined to non-interactive settings, leaving interactive and nondeterministic algorithms without rigorous foundational treatment. Method: This work introduces a unified formal framework encompassing both non-interactive and interactive, deterministic and nondeterministic algorithms. It proposes the “prototype algorithm” as an abstract computational model and rigorously defines its behavioral semantics. Three equivalence relations—behavioral, implementation, and specification equivalence—are formally introduced; their relationships are established, and specification equivalence is proven to be the appropriate criterion for capturing essential algorithmic identity. Contribution: The framework breaks the traditional boundaries of algorithm definitions, providing the first formal foundation for interactive algorithms. It establishes a layered, extensible meta-theory of algorithms and delivers a rigorous logical basis for reasoning about algorithmic essence, correctness verification, and cross-model comparison—thereby unifying previously fragmented formal approaches under a coherent theoretical umbrella.

This work proposes a formalization of algorithms within an intensional computability framework and clarifies their relationship to implementations in computational models. Treating computational models as monoid actions on configuration spaces, programs are modeled as dynamical systems constrained by such actions. Algorithms are defined as finite directed graphs of partial maps over edge-labeled abstract data structures, explicitly separating control flow from data operations. By leveraging tools from category theory, dynamical systems theory, and graph theory, the approach constructs a rigorous semantic framework that, for the first time, treats algorithms as abstract specifications of computational behavior and precisely characterizes the structure-preserving implementation relation between programs and algorithms, thereby deepening our understanding of the nature of computation.

This work addresses the absence of a mechanized formal verification framework for primal-dual algorithm analysis. It presents the first systematic formalization in Isabelle/HOL that supports rigorous correctness and performance verification of such algorithms, unifying a diverse range of instances—from the classical Hungarian algorithm to modern Adwords algorithms—within a single coherent framework. By achieving machine-checked proofs for multiple primal-dual algorithms, this study not only establishes their formal correctness but also develops a reusable library of verified components. The resulting infrastructure offers a novel paradigm for trustworthy verification of combinatorial optimization algorithms, enhancing both reliability and reusability in formal methods applied to algorithmic analysis.

Existing benchmarks struggle to effectively evaluate the algorithmic reasoning capabilities of large reasoning models. To address this gap, this work proposes the first algorithm-centric evaluation framework, featuring a fine-grained benchmark comprising over 3,000 original problems spanning 27 algorithm categories, meticulously curated by ACM experts. The study introduces a multi-dimensional assessment methodology and reveals that while leading models achieve up to 92% accuracy on non-optimization tasks, their performance sharply drops to approximately 49% on global optimization algorithms—such as dynamic programming—exposing fundamental limitations in comprehending complex algorithmic structures. Furthermore, the work identifies, for the first time, a phenomenon termed “strategic premature deviation,” shedding new light on model failure modes in algorithmic reasoning.

This work proposes a novel paradigm for automatically synthesizing efficient and correct algorithms from natural language problem descriptions, eschewing direct code generation. The approach models algorithm design as a sequential decision process over a typed library of algorithmic skills, where a learned scheduler selects appropriate skills guided by Monte Carlo Tree Search (MCTS) and multilevel verification feedback—including compilation, test execution, complexity analysis, and adversarial stress testing. The key innovation lies in decomposing algorithm design into human-like, schedulable skill units and introducing a verification-guided skill scheduling mechanism. Experiments demonstrate that the method substantially outperforms strong baselines—including end-to-end generation, chain-of-thought prompting, self-improvement approaches, and skill-agnostic MCTS—on competitive programming and combinatorial optimization benchmarks, with ablation studies confirming the contribution of each component.