General Proof Research Articles

Optimal finite-time heat engines under constrained control

We optimize finite-time stochastic heat engines with a periodically scaled Hamiltonian under experimentally motivated constraints on the bath temperature $T$ and the scaling parameter $\lambda$. We present a general geometric proof that maximum-efficiency protocols for $T$ and $\lambda$ are piecewise constant, alternating between the maximum and minimum allowed values. When $\lambda$ is restricted to a small range and the system is close to equilibrium at the ends of the isotherms, a similar argument shows that this protocol also maximizes output power. These results are valid for arbitrary dynamics. We illustrate them for an overdamped Brownian heat engine, which can experimentally be realized using optical tweezers with stiffness $\lambda$.

A distributed simplex architecture for multi-agent systems

We present the Distributed Simplex Architecture (DSA), a new runtime assurance technique that provides safety guarantees for multi-agent systems (MASs). DSA is inspired by the Simplex control architecture of Sha et al., but with some significant differences. The traditional Simplex approach is limited to single-agent systems or a MAS with a centralized control scheme. DSA addresses this limitation by extending the scope of Simplex to include MASs under distributed control. In DSA, each agent runs a local instance of traditional Simplex such that the preservation of safety in the local instances implies safety for the entire MAS. Control Barrier Functions (CBFs) play a critical role. They are used to define DSA’s core components – the baseline controller and the decision module’s logic for switching between advanced and baseline control – and they provide the basis for the proof of safety. We present a general proof of safety for DSA, provided the CBF-related optimization problem solved by the baseline controller is feasible (has a solution) at each time step for which the baseline controller is in control. We also propose a novel extension to the switching logic designed to avoid states in which this optimization problem is infeasible. Finally, we present experimental results for several case studies, including flocking with collision avoidance, safe navigation of ground rovers through way-points, and safe operation of a microgrid.

Kant’s Die falsche Spitzfindigkeit and Proof-theoretic Semantics

According to Schroeder-Heister 2018, proof-theoretic semantics is ‘an alternative to truth-condition semantics. It is based on the fundamental assumption that the central notion in terms of which meanings are assigned to certain expressions of our language, in particular to logical constants, is that of proof rather than truth. In this sense proof-theoretic semantics is semantics in terms of proof. Proof-theoretic semantics also means the semantics of proofs, i.e. the semantics of entities which describe how we arrive at certain assertions given certain assumptions.' This text advocates that proof-theoretic semantics, as described above, was an approach to semantical issues in logic that appeared in mainstream philosophical literature at least a couple of centuries before the tradition of general proof theory (cf. Prawitz 1971) came into being. This is done by means of an interpretive analysis of Kant’s 1762 essay Die falsche Spitzfindigkeit der vier syllogistischen Figuren, in which two theses are argued: first, that the issue of justifying the logical validity of inferences is approached by Kant in this text, in a strong sense, in terms of proofs; and second, that the purpose of his effort is to establish a point concerning the semantical equivalence between certain distinct valid inferences.

A JURIDICAL REVIEW OF LEGALITY AND BURDEN OF PROOF OF WIRETAPPING EVIDENCE IN CASES OF CORRUPTION CRIMES BASED ON LEGAL REGULATIONS IN INDONESIA

Amid national development efforts in various fields, people's aspirations to eradicate corruption and other irregularities are increasing. It is because, in reality, the existence of criminal acts of corruption has caused enormous state losses, which can impact the emergence of crises in various fields. The number of defendants in corruption cases acquitted by general courts or state courts was based on various reasons. One was the corruption charge against the defendant, but it proved it. The legislators were well aware of the difficulties that law enforcers may face in proving criminal acts of corruption. Thus, besides continuing to refer to the legal aspects of general proof regulated in the Criminal Procedure Code (KUHAP), legislators provided exceptions for proving corruption cases. One of them was using digital/electronic proof, such as email, telegram, wiretapping, teleconference, video conference, or CCTV footage. Based on the previous explanation, this study was aimed at elaborating on the legality of wiretapping at the stage of the investigation process in corruption cases and the evidentiary value of wiretapping communication/telephone devices based on the provisions of Article 183 of the Criminal Procedure Code to optimize the disclosure of corruption in Indonesia. The research method used was a qualitative method through a normative juridical method, namely the method used by mastering the law for a particular problem and how to implement or apply these legal regulations. From the results, the research found that in the Criminal Procedure Code (KUHAP), there was also an article that regulates wiretapping, namely Article 7 paragraph (1) j states that: "Investigators, because of their obligations, have the authority to carry out other actions according to applicable law." This article needs to state that investigators are allowed to wiretap clearly. However, in the elaboration of this article, at least it states that other actions according to the applicable law are already a part of wiretapping that carry out other actions according to the responsible law

The Hrushovski–Lang–Weil estimates

In this work we give a geometric proof of Hrushovski's generalization of the Lang-Weil estimates on the number of points in the intersection of a correspondence with the graph of Frobenius.

Model-guided synthesis of inductive lemmas for FOL with least fixpoints

Recursively defined linked data structures embedded in a pointer-based heap and their properties are naturally expressed in pure first-order logic with least fixpoint definitions (FO+lfp) with background theories. Such logics, unlike pure first-order logic, do not admit even complete procedures. In this paper, we undertake a novel approach for synthesizing inductive hypotheses to prove validity in this logic. The idea is to utilize several kinds of finite first-order models as counterexamples that capture the non-provability and invalidity of formulas to guide the search for inductive hypotheses. We implement our procedures and evaluate them extensively over theorems involving heap data structures that require inductive proofs and demonstrate the effectiveness of our methodology.

Visualizing the sum of squares induction proof

We provide a visual proof for the inductive step of the sum of squares formula proof.

Block-diagonalization of infinite-volume lattice Hamiltonians with unbounded interactions

In this paper we extend the local iterative Lie-Schwinger block-diagonalization method – introduced in [8] for quantum lattice systems with bounded interactions in arbitrary dimension– to systems with unbounded interactions, i.e., systems of bosons. We study Hamiltonians that can be written as the sum of a gapped operator consisting of a sum of on-site terms and a perturbation given by relatively bounded (but unbounded) interaction potentials of short range multiplied by a real coupling constant t. For sufficiently small values of |t|independent of the size of the lattice, we prove that the spectral gap above the ground-state energy of such Hamiltonians remains strictly positive.As in [8], we iteratively construct a sequence of local block-diagonalization steps based on unitary conjugations of the original Hamiltonian and inspired by the Lie-Schwinger procedure. To control the supports of the effective potentials generated in the course of our block-diagonalization steps, we use methods introduced in [8] for Hamiltonians with bounded interactions potentials. However, due to the unboundedness of the interaction potentials, weighted operator norms must be introduced, and some of the steps of the inductive proof by which we control the weighted norms of the effective potentials require special care to cope with matrix elements of unbounded operators.We stress that no “large-field problems” appear in our construction. In this respect our operator methods turn out to be an efficient tool to separate the low-energy spectral region of the Hamiltonian from other spectral regions, where the unbounded nature of the interaction potentials would become manifest.

Rigorous analysis of optical forces in dielectric structures based on the Minkowski-Helmholtz formula

Optical forces in dielectric structures are typically analyzed by utilizing either the Maxwell stress tensor or energy-based methods from which they can be derived by means of the eigenfrequencies and the effective refractive indices involved. While the equivalence of these two methods has been discussed in several studies, it would seem that a general electrodynamic proof of this aspect is still lacking. In this work, we provide a rigorous electrodynamic derivation based on the Minkowski-Helmholtz formula and the electromagnetic variation theorem, from which one can directly conclude that under Hermitian conditions these two approaches are formally equivalent to each other. The results of our study universally apply to any dielectric waveguide or cavity configuration. In addition, this methodology can be employed in graded index systems that do not exhibit sharp interfaces. Importantly, our analysis offers a straightforward route for predicting optical forces in a variety of photonic arrangements, including dielectric scatterers and multielement array configurations.

Gauge-independent description of the Aharonov-Bohm effect

The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region \textit{inaccessible} to the particle. Classically, such a non-local effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle, and the electromagnetic vector potential $\mathbf{A}$, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics $\mathbf{A}$ is in some sense more "fundamental" than $\mathbf {B}$ in spite of the former being gauge dependent, and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB-effect can be fully accounted for by considering only the gauge invariant $\mathbf{B}$ field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge invariant formulation is that we must give up locality -- the AB-phase for the particle will arise from the change in the action for the $\mathbf{B}$ field in the region inaccessible to the particle.

Counterexample-Guided Prophecy for Model Checking Modulo the Theory of Arrays

We develop a framework for model checking infinite-state systems by automatically augmenting them with auxiliary variables, enabling quantifier-free induction proofs for systems that would otherwise require quantified invariants. We combine this mechanism with a counterexample-guided abstraction refinement scheme for the theory of arrays. Our framework can thus, in many cases, reduce inductive reasoning with quantifiers and arrays to quantifier-free and array-free reasoning. We evaluate the approach on a wide set of benchmarks from the literature. The results show that our implementation often outperforms state-of-the-art tools, demonstrating its practical potential.

Practical generic programming over a universe of native datatypes

Datatype-generic programming makes it possible to define a construction once and apply it to a large class of datatypes. It is often used to avoid code duplication in languages that encourage the definition of custom datatypes, in particular state-of-the-art dependently typed languages where one can have many variants of the same datatype with different type-level invariants. In addition to giving access to familiar programming constructions for free, datatype-generic programming in the dependently typed setting also allows for the construction of generic proofs. However, the current interfaces available for this purpose are needlessly hard to use or are limited in the range of datatypes they handle. In this paper, we describe the design of a library for safe and user-friendly datatype-generic programming in the Agda language. Generic constructions in our library are regular Agda functions over a broad universe of datatypes, yet they can be specialized to native Agda datatypes with a simple one-liner. Furthermore, we provide building blocks so that library designers can too define their own datatype-generic constructions.

The arithmetic of modular grids

A modular grid is a pair of sequences ( f m ) m $ (f_m ) _m$ and ( g n ) n $ (g_n ) _n$ of weakly holomorphic modular forms such that for almost all m and n, the coefficient of q n $q^n$ in f m $f_m$ is the negative of the coefficient of q m $q^m$ in g n $g_n$ . Zagier proved this coefficient duality in weights 1/2 and 3/2 in the Kohnen plus space, and such grids have appeared for Poincaré series, for modular forms of integral weight, and in many other situations. We give a general proof of coefficient duality for canonical row-reduced bases of spaces of weakly holomorphic modular forms of integral or half-integral weight for every group Γ ⊆ SL 2 ( R ) $\Gamma \subseteq {\text{\rm SL}}_2(\mathbb {R})$ commensurable with SL 2 ( Z ) ${\text{\rm SL}}_2(\mathbb {Z})$ . We construct bivariate generating functions that encode these modular forms, and study linear operations on the resulting modular grids.

A note on matrix multiplication appearing as element concatenation or coinciding with matrix addition

In this note we demonstrate two instances where matrix multiplication can be easily verified. In the first setting, the matrix product appears as matrix element concatenation, and in the second, the product coincides with matrix addition. General proofs for some results are provided with a more complete description for matrices. Suggested for consolidation of the algorithm rather than as an introduction, we believe the example base that can be created, combined with an analysis for understanding the context, acts as a positive motivator to secure mastery as it needs to be practised.

ON THE PROOF OF THE THEOREMS OF FOUNDATIONS OF GEOMETRY USING ISABELLE/HOL

Isabelle/HOL is a generic proof assistant. Using Isabelle/HOL requires insight into procedures as well as into the concepts involved. In addition, how a computer manages procedures can affect mathematical concepts. Use of Isabelle/HOL can correct a current weakness in mathematical studies. The advantage of the theorem proving support system represented by Isabelle/HOL is that it mechanically guarantees the “correctness” of both human-written programs and mathematical proofs. It can allow us to clearly understand mathematical concepts and can minimize the burden of operation opportunities. However, in order to take advantage of its high versatility and reliability, the problem that all certification procedures must be clearly formalized when creating certification must be overcome. “Foundations of Geometry” is a book on mathematics written by Hilbert in 1899. The book is famous as the most rigorous study of the axiom system of Euclidean geometry by axioms and formalism. When we tried to implement Hilbert’s axioms in Isabelle/HOL, the proofs based on human cognition hindered the implementation. The purpose of this paper is “correctly” reconstruct the proofs as automated theorem proving. We are aiming to implement them “accurately” on Isabelle/ HOL and have done so for many of them. This is the originality of this study.

Engaging, Large-Scale Functional Programming Education in Physical and Virtual Space

Worldwide, computer science departments have experienced a dramatic increase in the number of student enrolments. Moreover, the ongoing COVID-19 pandemic requires institutions to radically replace the traditional way of on-site teaching, moving interaction from physical to virtual space. We report on our strategies and experience tackling these issues as part of a Haskell-based functional programming and verification course, accommodating over 2000 students in the course of two semesters. Among other things, we fostered engagement with weekly programming competitions and creative homework projects, workshops with industry partners, and collaborative pair-programming tutorials. To offer such an extensive programme to hundreds of students, we automated feedback for programming as well as inductive proof exercises. We explain and share our tools and exercises so that they can be reused by other educators.

Topological Drawings Meet Classical Theorems from Convex Geometry

In this article we discuss classic theorems from Convex Geometry in the context of topological drawings and beyond. In a simple topological drawing of the complete graph \(K_n\), any two edges share at most one point: either a common vertex or a point where they cross. Triangles of simple topological drawings can be viewed as convex sets. This gives a link to convex geometry. As our main result, we present a generalization of Kirchberger’s theorem that is of purely combinatorial nature. It turned out that this classic theorem also applies to “generalized signotopes”—a combinatorial generalization of simple topological drawings, which we introduce and investigate in the course of this article. As indicated by their name they are a generalization of signotopes, a structure studied in the context of encodings for arrangements of pseudolines. We also present a family of simple topological drawings with arbitrarily large Helly number, and a new proof of a topological generalization of Carathéodory’s theorem in the plane and discuss further classic theorems from Convex Geometry in the context of simple topological drawings.

Context-independent mapping and free choice are equivalent: a general proof

Free choice (or statistical independence) assumption in a hidden variable model (HVM) means that the settings chosen by experimenters do not depend on the values of the hidden variable. The assumption of context-independent (CI) mapping in an HVM means that the results of a measurement do not depend on settings for other measurements. If the measurements are spacelike separated, this assumption is known as local causality. Both free choice and CI mapping assumptions are considered necessary for derivation of the Bell-type criteria of contextuality/nonlocality. It is known, however, for a variety of special cases, that the two assumptions are not logically independent. We show here, in complete generality, for any system of random variables with or without disturbance/signaling, that an HVM that postulates CI mapping is equivalent to an HVM that postulates free choice. If one denies the possibility that a given empirical scenario can be described by an HVM in which measurements depend on other measurements’ settings, free choice violations should be denied too, and vice versa.

Skip letters for short supersequence of all permutations

A supersequence over a finite set is a sequence that contains as subsequence all permutations of the set. This paper defines an infinite array of methods to create supersequences of decreasing lengths. This yields the shortest known supersequences over larger sets. It also provides the best results asymptotically. It is based on a general proof using a new property called strong completeness. The same technique also can be used to prove existing supersequences which combines the old and new ones into an unified conceptual framework.

Witten index for weak supersymmetric systems: Invariance under deformations

When a 4D supersymmetric theory is placed on [Formula: see text], the supersymmetric algebra is necessarily modified to [Formula: see text] and we are dealing with a weak supersymmetric system. For such systems, the excited states of the Hamiltonian are not all paired. As a result, the Witten index [Formula: see text] is no longer an integer number, but a [Formula: see text]-dependent function. However, this function stays invariant under deformations of the theory that keep the supersymmetry algebra intact. Based on the Hilbert space analysis, we give a simple general proof of this fact. We then show how this invariance works for two simplest weak supersymmetric quantum mechanical systems involving a real or a complex bosonic degree of freedom.