Related papers: Extensional Taylor Expansion
Higher-order representations of objects such as programs, proofs, formulas and types have become important to many symbolic computation tasks. Systems that support such representations usually depend on the implementation of an intensional…
Taylor's law describes the fluctuation characteristics underlying a system in which the variance of an event within a time span grows by a power law with respect to the mean. Although Taylor's law has been applied in many natural and social…
The relational semantics of linear logic is a powerful framework for defining resource-aware models of the $\lambda$-calculus. However, its quantitative aspects are not reflected in the preorders and equational theories induced by these…
We introduce a linear infinitary $\lambda$-calculus, called $\ell\Lambda_{\infty}$, in which two exponential modalities are available, the first one being the usual, finitary one, the other being the only construct interpreted…
Herbrand's theorem is one of the most fundamental insights in logic. From the syntactic point of view, it suggests a compact representation of proofs in classical first- and higher-order logic by recording the information of which instances…
Addition theorems can be constructed by doing three-dimensional Taylor expansions according to $f (\mathbf{r} + \mathbf{r}') = \exp (\mathbf{r}' \cdot \mathbf{\nabla}) f (\mathbf{r})$. Since, however, one is normally interested in addition…
We present a general and user-extensible equality checking algorithm that is applicable to a large class of type theories. The algorithm has a type-directed phase for applying extensionality rules and a normalization phase based on…
Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given…
Taxonomies play a vital role in structuring and categorizing information across domains. However, many existing taxonomies suffer from limited coverage and outdated or ambiguous nodes, reducing their effectiveness in knowledge retrieval. To…
For any given spacetime the choice of time coordinate is undetermined. A particular choice is the absolute time associated with a preferred vector field. Using the absolute time Hamilton's equations are $- (\delta H_{c})/(\delta…
In reductive proof search, proofs are naturally generalized by solutions, comprising all possibly infinite structures generated by locally correct, bottom-up application of inference rules. We propose an extension of the Curry-Howard…
We introduce the structural resource lambda-calculus, a new formalism in which strongly normalizing terms of the lambda-calculus can naturally be represented, and at the same time any type derivation can be internally rewritten to its…
Hereditary substitution is a form of type-bounded iterated substitution, first made explicit by Watkins et al. and Adams in order to show normalization of proof terms for various constructive logics. This paper is the first to apply…
We give a brief introduction to the clocked lambda calculus, an extension of the classical lambda calculus with a unary symbol tau used to witness the beta-steps. In contrast to the classical lambda calculus, this extension is infinitary…
We explore recursive programming with extensible data types. Row types make the structure of data types first class, and can express a variety of type system features including record subtyping and combination of case branches. Our goal is…
We describe a type system for the linear-algebraic $\lambda$-calculus. The type system accounts for the linear-algebraic aspects of this extension of $\lambda$-calculus: it is able to statically describe the linear combinations of terms…
It is well known that the length of a beta-reduction sequence of a simply typed lambda-term of order k can be huge; it is as large as k-fold exponential in the size of the lambda-term in the worst case. We consider the following relevant…
The Krivine machine is an abstract machine implementing the linear head reduction of lambda-calculus. Ehrhard and Regnier gave a resource sensitive version returning the annotated form of a lambda-term accounting for the resources used by…
Balancing exploration and exploitation is a central goal in reinforcement learning (RL). Despite recent advances in enhancing large language model (LLM) reasoning, most methods lean toward exploitation, and increasingly encounter…
A method for introducing the higher order terms in the potential expansion to study the continuous limits of the Toda hierarchy is proposed in this paper. The method ensures that the higher order terms are differential polynomials of the…