algorithm composition

Modular complexity analysis of integer programs remains challenging for non-tail-recursive and general function-call structures, as existing frameworks lack systematic support for recursion. Method: We introduce Recursive Ranking Functions (RRFs), a novel formalism enabling modular automatic resource analysis for arbitrary recursive integer programs. Our approach integrates RRFs with instruction-level complexity analysis and implements an end-to-end automated analyzer within the KoAT tool. Contribution/Results: Our method precisely infers polynomial and exponential time complexities—outperforming prior modular techniques in both precision and scalability. It overcomes a fundamental limitation of existing modular analysis frameworks, which were restricted to non-recursive programs. Experimental evaluation demonstrates significant improvements in analytical accuracy and applicability to realistic recursive code patterns, establishing the first fully modular, automated complexity analyzer supporting general recursion in integer programs.

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 study addresses the optimal vertex elimination problem—a core NP-complete challenge in algorithmic differentiation that has long lacked scalable exact solvers for evaluating heuristic performance. The work models the problem as a vertex elimination sequence on directed acyclic graphs and introduces a novel integer programming formulation capable of solving instances one to two orders of magnitude larger than previous approaches. It establishes the first approximate lower bounds for both minimum fill-in and minimum operation count variants and designs a parameterized approximation algorithm based on minimum s-t cuts. Additionally, the paper provides tight theoretical analyses for both forward and reverse modes of automatic differentiation. A new medium-scale graph benchmark with known optimal solutions is introduced, enabling empirical validation that demonstrates the strong practical performance of state-of-the-art heuristics and significantly advances both the tractable scale and theoretical understanding of the problem.

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 addresses the longstanding challenge of reconciling theoretical correctness with practical efficiency by introducing Algorithmist, a multi-agent autonomous research system built upon GitHub Copilot. Through an iterative research-review cycle, Algorithmist collaboratively performs algorithm design, formal verification, proof-guided code generation, and consistency validation. The system establishes a scalable paradigm for provably correct algorithm synthesis by integrating large language models, structured natural-language proof representations, and formal verification techniques to generate algorithms tailored to specific datasets and deployment scenarios. In applications to privacy-preserving data analysis and clustering tasks, Algorithmist automatically produces novel algorithms that simultaneously offer rigorous theoretical guarantees and strong empirical performance, uncovers previously overlooked proof flaws in existing work, and achieves state-of-the-art results in several settings.

This work proposes a compositional and modular automated reasoning framework for simultaneously deriving upper bounds on runtime and proving termination of integer programs with recursion. The approach employs an alternating modular strategy that iteratively infers upper bounds on both the runtime and variable ranges of subroutines, seamlessly integrating multiple static analysis techniques to handle recursive calls and complex control flow. Experimental evaluation demonstrates that the framework substantially outperforms existing tools on large-scale benchmarks, exhibiting both high efficiency and strong scalability in automated complexity analysis and termination verification.

Existing AI approaches to problem solving either rely on costly model-centric strategies or employ fragmented prompting techniques, lacking a unified, interpretable, and efficient algorithmic reasoning framework. This work proposes MAS-Algorithm, the first systematic application of multi-agent systems to algorithmic programming problem solving. Inspired by competitive programming, it constructs a modular and collaborative workflow that enables structured reasoning and seamless integration with external tools. The method demonstrates strong scalability and generality, achieving average pass rate improvements of 6.48% on a newly curated benchmark and 4.72% on LiveCodeBench-Pro. Notably, individual agents contribute performance gains as high as 27.7%, significantly outperforming baseline approaches such as parameter-efficient fine-tuning.

This work addresses the semantic gap between programming languages and pure mathematics—particularly algebraic structures—by proposing a novel functional language design methodology grounded in algebraic hierarchy decomposition theory. Methodologically, it introduces the first categorical generalization of the Krohn–Rhodes theorem to semigroupoids and systematically applies this extension to the design of concatenative functional languages. By constructing a family of hierarchical languages endowed with explicit semigroupoid semantics, the approach enables structural modeling and controlled compositional reasoning about computation. The contributions include: (i) a rigorous algebraic foundation for program decomposition, synthesis, and verification; and (ii) a class of prototype programming languages equipped with precise, mathematically grounded semantics. This bridges formal methods and practical language design, advancing their integration in both theoretical and applied settings.

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.