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Saeed Salehi

Publications and source records attributed to Saeed Salehi.

At least 19 recordsLinked to original sources
arXiv2026

Cantor's Non-Equinumerosity Theorems, Inductively

In the first, pre-college level part, we present a proof by mathematical induction for Cantor's theorem on the uncountability of the infinite binary strings and the real numbers. In the second, undergraduate-level part, we prove Cantor's powerset theorem by using transfinite induction. There, we will need the axiom of choice and the concept of ordinals. Some of these proofs by (mathematical and transfinite) induction seem to be new; comparisons will be made with older results.

math.LO↗
arXiv2026

Formalizing the Binding Problem

Representations of the world, arguably, contain information about features (e.g. something is blue, something is a circle) but also information about which features are part of the same object (e.g. the circle is blue), which we call binding information. Any system with the ability to understand scenes with multiple objects must be able to solve the binding problem: it needs to know which features belong together. However, despite work showing that Vision Transformers (ViTs) know which patches belong together, it is not known whether current deep learning models learn to exhibit binding information, i.e., for features. We may believe that there is not much binding information, after all misattributing features to wrong objects is a common failure of ViT-based architectures, especially in scenes with objects sharing features. Here we formalize the binding problem with an information-theoretic approach, and introduce a probing method to measure binding information in model representations. We perform experiments on ViTs, measuring binding from different components of the architecture, such as the image summary token [CLS] or the spatial tokens. We use datasets with different binding challenges, such as feature sharing, occlusion, and natural features, while comparing the performance of several pre-trained ViTs. Overall, our research demonstrates binding as a key ingredient to strong visual recognition and reasoning.

cs.CV↗
arXiv2026

On Chaitin's Heuristic Principle and Halting Probability

It would be a heavenly reward if there were a method of weighing theories and sentences in such a way that a theory could never prove a heavier sentence (Chaitin's Heuristic Principle). Alas, no satisfactory measure has been found so far, and this dream seemed too good ever to come true. In the first part of this paper, we attempt to revive Chaitin's lost paradise of heuristic principle as much as logic allows. In the second part, which is a joint work with M. Jalilvand and B. Nikzad, we study Chaitin's well-known constant Omega and show that this number is not a probability of halting the randomly chosen input-free programs under any infinite discrete measure. We suggest several methods for defining halting probabilities using various measures.

math.LO↗
arXiv2026

Does Object Binding Naturally Emerge in Large Pretrained Vision Transformers?

Object binding, the brain's ability to bind the many features that collectively represent an object into a coherent whole, is central to human cognition. It groups low-level perceptual features into high-level object representations, stores those objects efficiently and compositionally in memory, and supports human reasoning about individual object instances. While prior work often imposes object-centric attention (e.g., Slot Attention) explicitly to probe these benefits, it remains unclear whether this ability naturally emerges in pre-trained Vision Transformers (ViTs). Intuitively, they could: recognizing which patches belong to the same object should be useful for downstream prediction and thus guide attention. Motivated by the quadratic nature of self-attention, we hypothesize that ViTs represent whether two patches belong to the same object, a property we term IsSameObject. We decode IsSameObject from patch embeddings across ViT layers using a quadratic similarity probe, which reaches over 90% accuracy. Crucially, this object-binding capability emerges reliably in DINO, CLIP, and ImageNet-supervised ViTs, but is markedly weaker in MAE, suggesting that binding is not a trivial architectural artifact, but an ability acquired through specific pretraining objectives. We further discover that IsSameObject is encoded in a low-dimensional subspace on top of object features, and that this signal actively guides attention. Ablating IsSameObject from model activations degrades downstream performance and works against the learning objective, implying that emergent object binding naturally serves the pretraining objective. Our findings challenge the view that ViTs lack object binding and highlight how symbolic knowledge of "which parts belong together" emerges naturally in a connectionist system.

cs.CV↗
arXiv2025

A Non-Constructive Proof of Cantor's Theorem

We offer a new proof (and review some known proofs) of Cantor's Powerset Theorem (1891), which concerns the non-existence of a surjective function from a set onto its powerset.

math.LO↗
arXiv2025

Transfer learning strategies for accelerating reinforcement-learning-based flow control

This work investigates transfer learning strategies to accelerate deep reinforcement learning (DRL) for multifidelity control of chaotic fluid flows. Progressive neural networks (PNNs), a modular architecture designed to preserve and reuse knowledge across tasks, are employed for the first time in the context of DRL-based flow control. In addition, a comprehensive benchmarking of conventional fine-tuning strategies is conducted, evaluating their performance, convergence behavior, and ability to retain transferred knowledge. The Kuramoto-Sivashinsky (KS) system is employed as a benchmark to examine how knowledge encoded in control policies, trained in low-fidelity environments, can be effectively transferred to high-fidelity settings. Systematic evaluations show that while fine-tuning can accelerate convergence, it is highly sensitive to pretraining duration and prone to catastrophic forgetting. In contrast, PNNs enable stable and efficient transfer by preserving prior knowledge and providing consistent performance gains, and are notably robust to overfitting during the pretraining phase. Layer-wise sensitivity analysis further reveals how PNNs dynamically reuse intermediate representations from the source policy while progressively adapting deeper layers to the target task. Moreover, PNNs remain effective even when the source and target environments differ substantially, such as in cases with mismatched physical regimes or control objectives, where fine-tuning strategies often result in suboptimal adaptation or complete failure of knowledge transfer. The results highlight the potential of novel transfer learning frameworks for robust, scalable, and computationally efficient flow control that can potentially be applied to more complex flow configurations.

cs.LG↗
arXiv2025

Pre-trained Transformer-models using chronic invasive electrophysiology for symptom decoding without patient-individual training

Neural decoding of pathological and physiological states can enable patient-individualized closed-loop neuromodulation therapy. Recent advances in pre-trained large-scale foundation models offer the potential for generalized state estimation without patient-individual training. Here we present a foundation model trained on chronic longitudinal deep brain stimulation recordings spanning over 24 days. Adhering to long time-scale symptom fluctuations, we highlight the extended context window of 30 minutes. We present an optimized pre-training loss function for neural electrophysiological data that corrects for the frequency bias of common masked auto-encoder loss functions due to the 1-over-f power law. We show in a downstream task the decoding of Parkinson's disease symptoms with leave-one-subject-out cross-validation without patient-individual training.

cs.HC↗
arXiv2024

Hard Rock Drilling for Super-hot Enhanced Geothermal System Development: Literature Review and Techno-Economic Analysis

The increasing global demand for electricity and the imperative of achieving sustainable and net-zero energy solutions have underscored the importance of exploring alternative sources. Enhanced Geothermal Systems (EGS) have emerged as a promising avenue for renewable and sustainable energy production. However, the development of EGS faces a significant challenge in drilling through hard rock formations at high temperatures, necessitating specialized drilling equipment and techniques. This study aims to investigate the current state-of-the-art technology for drilling in hard rock formations under elevated temperatures, specifically in the context of super-hot EGS development. It involves a comprehensive review of previous projects and a meticulous analysis of existing drilling technologies and techniques. Furthermore, a techno-economic evaluation will be conducted to assess the feasibility of super-hot EGS development in hard igneous formations, considering key factors such as drilling performance, operational challenges, and material costs. The outcomes of this study will enhance the understanding of the technical challenges associated with super-hot EGS development and facilitate the design of efficient and cost-effective drilling technologies for the geothermal energy industry. By improving the drilling process in EGS development, the full potential of geothermal energy can be harnessed as a viable and sustainable energy source to meet the growing global demand for electricity.

physics.geo-ph↗
arXiv2024

Geothermal Energy in Sedimentary Basins: Assessing Techno-economic Viability for Sustainable Development

Drilling deep geothermal wells has proven to be a challenging endeavor, primarily due to issues such as loss circulation events, material limitations under high temperatures, and the production of corrosive fluids. Furthermore, the substantial upfront costs, coupled with geological and technical obstacles associated with drilling super-hot EGS wells in igneous rocks, hinder the widespread implementation of geothermal systems. Alternatively, geothermal energy development in sedimentary basins presents an opportunity for clean energy production with relatively lower investment costs compared to the development of super-hot EGS in igneous rocks. Sedimentary basins exhibit attractive temperatures for geothermal applications, and their wide distribution enhances the potential for nationwide deployment. Decades of drilling and development experience in oil and gas wells have yielded a wealth of data, knowledge, and expertise. Leveraging this experience and data for geothermal drilling can significantly reduce costs associated with subsurface data gathering, well drilling, and completion. This paper explores the economic viability of geothermal energy production systems in sedimentary basins. The study encompasses an analysis of time-to-hit-temperature (THT) and cost-to-hit-temperature (CHT) parameters, as well as Favorability maps across the United States. These maps are based on factors such as well depth, total drilling time, well cost, and subsurface temperature data. By integrating sedimentary basin maps and underground temperature maps, the THT and CHT maps can facilitate the strategic placement of EGS wells and other geothermal system applications in the most favorable locations across the United States.

physics.geo-ph↗
arXiv2023

On a fallacy concerning I-am-unprovable sentences: what to take home from Goedel's introduction

We demonstrate that, in itself and in the absence of extra premises, the following argument scheme is fallacious: The sentence A says about itself that it has a certain property F, and A does in fact have the property F; therefore A is true. We then examine an argument of this form in the informal introduction of Goedel's classic (1931) and examine some auxiliary premises which might have been at work in that context. Philosophically significant as it may be, that particular informal argument plays no role in Goedel's technical results. Going deeper into the issue and investigating truth conditions of Goedelian sentences (i.e., those sentences which are provably equivalent to their own unprovability) will provide us with insights regarding the philosophical debate on the truth of Goedelian sentences of systems--a debate which is at least as old as Dummett (1963).

math.LO↗
arXiv2022

A Reunion of Godel, Tarski, Carnap, and Rosser

We unify Godel's First Incompleteness Theorem (1931), Tarski's Undefinability Theorem (1933), Godel-Carnap's Diagonal Lemma (1934), and Rosser's (strengthening of Godel's first) Incompleteness Theorem (1936), whose proofs resemble much and use almost the same technique.

math.LO↗
arXiv2022

On Godel's "Much Weaker" Assumption

Godelian sentences of a sufficiently strong and recursively enumerable theory, constructed in Godel's 1931 groundbreaking paper on the incompleteness theorems, are unprovable if the theory is consistent; however, they could be refutable. These sentences are independent when the theory is so-called omega-consistent; a notion introduced by Godel, which is stronger than (simple) consistency, but ``much weaker'' than soundness. Godel goes to great lengths to show in detail that omega-consistency is stronger than consistency, but never shows, or seems to forget to say, why it is much weaker than soundness. In this paper, we study this proof-theoretic notion and compare some of its properties with those of consistency and (variants of) soundness.

math.LO↗
arXiv2020

Tarski's Undefinability Theorem and Diagonal Lemma

We prove the equivalence of the semantic version of Tarski's theorem on the undefinability of truth with a semantic version of the Diagonal Lemma, and also show the equivalence of syntactic Tarski's Undefinability Theorem with a weak syntactic diagonal lemma. We outline two seemingly diagonal-free proofs for these theorems from the literature, and show that syntactic Tarski's theorem can deliver Gödel-Rosser's Incompleteness Theorem.

math.LO↗
arXiv2020

On the Truth of Gödelian and Rosserian Sentences

There is a longstanding debate in the logico-philosophical community as to why the Gödelian sentences of a consistent and sufficiently strong theory are true. The prevalent argument seems to be something like this: since every one of the Gödelian sentences of such a theory is equivalent to the theory's consistency statement, even provably so inside the theory, the truth of those sentences follows from the consistency of the theory in question. So, Gödelian sentences of consistent theories should be true. In this paper, we show that Gödelian sentences of only sound theories are true; and there is a long road from consistency to soundness, indeed a hierarchy of conditions which are satisfied by some theories and falsified by others. We also study the truth of Rosserian sentences and provide necessary and sufficient conditions for the truth of Rosserian (and also Gödelian) sentences of theories.

math.LO↗
arXiv2020

From Intuitionism to Many-Valued Logics through Kripke Models

Intuitionistic Propositional Logic is proved to be an infinitely many valued logic by Kurt Gödel (1932), and it is proved by Stanisław Jaśkowski (1936) to be a countably many valued logic. In this paper, we provide alternative proofs for these theorems by using models of Saul Kripke (1959). Gödel's proof gave rise to an intermediate propositional logic (between intuitionistic and classical), that is known nowadays as Gödel or the Gödel-Dummet Logic, and is studied by fuzzy logicians as well. We also provide some results on the inter-definablility of propositional connectives in this logic.

math.LO↗
arXiv2020

Axiomatic (and Non-Axiomatic) Mathematics

Axiomatizing mathematical structures and theories is an objective of Mathematical Logic. Some axiomatic systems are nowadays mere definitions, such as the axioms of Group Theory; but some systems are much deeper, such as the axioms of Complete Ordered Fields with which Real Analysis starts. Groups abound in mathematical sciences, while by Dedekind's theorem there exists only one complete ordered field, up to isomorphism. Cayley's theorem in Abstract Algebra implies that the axioms of group theory completely axiomatize the class of permutation sets that are closed under composition and inversion. In this article, we survey some old and new results on the first-order axiomatizability of various mathematical structures. We will also review identities over addition, multiplication, and exponentiation that hold in the set of positive real numbers.

math.LO↗