arXiv subjects

Tiago M. L. de Veras

Publications and source records attributed to Tiago M. L. de Veras.

8 recordsLinked to original sources

arXiv2026

Topological Semantics for Scoped Computational Paths

Computational paths record the steps of an equality derivation. We give them a topological semantics that distinguishes derivable rewrites from arbitrary homotopies. Coherent representatives pair traces with paths homotopic to their realizations. We compare a topology retaining the entire trace with one observing only endpoints, length, and paths. Quotienting by the declared rewrites gives a groupoid. Multiplication is continuous when composable pairs carry the quotient topology inherited from composable representatives. This topology can differ from the usual subspace topology on pairs of quotient arrows. We characterize when they agree, give compact-Hausdorff and discrete sufficient conditions, and use the Hawaiian earring to exhibit a failure of agreement. The comparison with geometric homotopy classes is injective exactly when the presentation is geometrically complete. Normal-form certificates give a criterion for completeness. In the universal presentation, all paths are primitive steps and all endpoint-fixed homotopies are allowed rewrites; its quotient recovers the quotient-topologized fundamental groupoid. Circle and torus examples recover the classical based-loop classifications by $\mathbb Z$ and $\mathbb Z^2$. A Lean development supports the construction. A focused Lean 4.32.0 result registered in Palomar covers the topology comparison, additive circle and torus classifications, and a conditional Hawaiian-earring obstruction transfer. We distinguish that result from the earlier Lean 4.24.0 development and from the mathematical exposition.

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arXiv2025

The Seifert-van Kampen Theorem via Computational Paths: A Formalized Approach to Computing Fundamental Groups

The Seifert-van Kampen theorem computes the fundamental group of a space from the fundamental groups of its constituents. We develop a modular SVK framework within the setting of computational paths - an approach to equality where witnesses are explicit sequences of rewrites governed by the LNDEQ-TRS. Our contributions are: (i) pushouts as higher-inductive types with modular typeclass assumptions for computation rules; (ii) free products and amalgamated free products as quotients of word representations; (iii) an SVK equivalence schema parametric in user-supplied encode/decode structure; and (iv) instantiations for classical spaces - figure-eight (pi_1(S^1 v S^1) = Z * Z), 2-sphere (pi_1(S^2) = 1), and 3-sphere (pi_1(S^3) = 1) with Hopf fibration context. Recent extensions include higher homotopy groups pi_n via weak infinity-groupoid structure (with pi_2 abelian via Eckmann-Hilton), and pi_1 >= 1 in the 1-groupoid truncated setting; truncation levels connecting the framework to HoTT; automated path simplification tactics; basic covering space theory with pi_1-actions on fibers; fibration theory with long exact sequences; and Eilenberg-MacLane space characterization (S^1 = K(Z,1)). The development is formalized in Lean 4 with 41,130 lines across 107 modules, using 36 kernel axioms for HIT type-constructor declarations.

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arXiv2025

Computational Paths Form a Weak ω-Groupoid

Lumsdaine (2010) and van den Berg-Garner (2011) proved that types in Martin-Löf type theory carry the structure of weak ω-groupoids. Their proofs, while foundational, rely on abstract properties of the identity type without providing explicit computational content for coherence witnesses. We establish an analogous result for computational paths -- an alternative formulation of equality where witnesses are explicit sequences of rewrites from the LNDEQ-TRS term rewriting system. Our main result is that computational paths on any type form a weak ω-groupoid with fully explicit coherence data. The groupoid operations -- identity, composition, and inverse -- are defined at every dimension, and the coherence laws (associativity, unit laws, inverse laws) are witnessed by concrete rewrite derivations rather than abstract existence proofs. The construction provides: (i) a proper tower of n-cells for all dimensions, with 2-cells as derivations between paths and higher cells mediating between lower-dimensional witnesses; (ii) explicit pentagon and triangle coherences built from the rewrite rules; and (iii) contractibility at dimensions $\geq 3$, ensuring all parallel higher cells are connected. The contractibility property is derived from the normalization algorithm of the rewrite system, grounding the higher-dimensional structure in concrete computational content. The entire construction has been formalized in Lean 4, providing machine-checked verification of the weak ω-groupoid structure.

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arXiv2025

Formalizing Computational Paths and Fundamental Groups in Lean

Computational paths treat propositional equality as explicit paths built from labelled deduction steps and rewrite rules. This view originates in work by de Queiroz and collaborators [1] and yields a weak groupoid structure for equality, together with a computational account of homotopy inspired by homotopy type theory. In this paper we present a complete mechanization of this framework in Lean 4 and show how it supports concrete homotopy theoretic computations. Our contributions are threefold. First, we formalize the theory of computational paths in Lean, including path formation, composition, inverses, and a rewrite system that identifies redundant or trivial paths. We prove that equality types with computational paths carry a weak groupoid structure in the sense of the original theory. Second, we organize this material into a reusable Lean library, ComputationalPathsLean, which exposes an interface for paths, rewrites, and loop spaces. This library allows later developments to treat computational paths as a drop-in replacement for propositional equality when reasoning about homotopical structure. Third, we apply the library to six canonical examples in algebraic topology. We give Lean proofs that the fundamental group of the circle is isomorphic to the integers, the cylinder and Mobius band also have fundamental group isomorphic to the integers (via retraction to the circle), the fundamental group of the torus is isomorphic to the product of two copies of the integers, the fundamental group of the Klein bottle is isomorphic to the semidirect product Z cross Z, and the fundamental group of the real projective plane is isomorphic to Z_2. These case studies demonstrate that the computational paths approach scales to nontrivial homotopical computations in a modern proof assistant. All the definitions and proofs described here are available in an open-source Lean 4 repository.

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arXiv2025

Quantum Multiplexer Simplification for State Preparation

The initialization of quantum states or Quantum State Preparation (QSP) is a basic subroutine in quantum algorithms. In the worst case, general QSP algorithms are expensive due to the application of multi-controlled gates required to build the quantum state. Here, we propose an algorithm that detects whether a given quantum state can be factored into substates, increasing the efficiency of compiling the QSP circuit when we initialize states with some level of unentanglement. The simplification is done by eliminating controls of quantum multiplexers, significantly reducing circuit depth and the number of CNOT gates with a better execution and compilation time than the previous QSP algorithms. Considering efficiency in terms of depth and number of CNOT gates, our method is competitive with the methods in the literature. However, when it comes to run-time and compilation efficiency, our result is significantly better, and the experiments show that by increasing the number of qubits, the gap between the temporal efficiency of the methods increases.

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arXiv2021

Double sparse quantum state preparation

Initializing classical data in a quantum device is an essential step in many quantum algorithms. As a consequence of measurement and noisy operations, some algorithms need to reinitialize the prepared state several times during its execution. In this work, we propose a quantum state preparation algorithm called CVO-QRAM with computational cost O(kM), where M is the number of nonzero probability amplitudes and $k$ is the maximum number of bits with value 1 in the patterns to be stored. The proposed algorithm can be an alternative to create sparse states in future NISQ devices.

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arXiv2020

Circuit-based quantum random access memory for classical data with continuous amplitudes

Loading data in a quantum device is required in several quantum computing applications. Without an efficient loading procedure, the cost to initialize the algorithms can dominate the overall computational cost. A circuit-based quantum random access memory named FF-QRAM can load M n-bit patterns with computational cost O(CMn) to load continuous data where C depends on the data distribution. In this work, we propose a strategy to load continuous data without post-selection with computational cost O(Mn). The proposed method is based on the probabilistic quantum memory, a strategy to load binary data in quantum devices, and the FF-QRAM using standard quantum gates, and is suitable for noisy intermediate-scale quantum computers.

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arXiv2019

An alternative approach to the calculation of fundamental groups based on labeled natural deduction

In this work, we use a labelled deduction system based on the concept of computational paths (sequence of rewrites) as equalities between two terms of the same type. We also define a term rewriting system that is used to make computations between these computational paths, establishing equalities between equalities. We use a labelled deduction system based on the concept of computational paths (sequence of rewrites) to obtain some results of algebraic topology and with support of the Seifet-Van Kampen Theorem we will calculate, in a way less complex than the one made in mathematics \cite{Munkres} and the technique of homotopy type theory \cite{hott}, the fundamental group of Klein Blottle $\mathbb{K}^2$, of the Torus $\mathbb{T}^2$ and Two holed Torus $\mathbb{M}_2=\mathbb{T}^2\# \mathbb{T}^2$ (the connected sum two torus).

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