Related papers: Abstractions of Sequences, Functions and Operators
Induction is the process by which we obtain predictive laws or theories or models of the world. We consider the structural aspect of induction. We answer the question as to whether we can find a finite and minmalistic set of operations on…
Recursive relational specifications are commonly used to describe the computational structure of formal systems. Recent research in proof theory has identified two features that facilitate direct, logic-based reasoning about such…
Abstraction is a core tenet of human cognition and communication. When composing natural language instructions, humans naturally evoke abstraction to convey complex procedures in an efficient and concise way. Yet, interpreting and grounding…
Working with causal models at different levels of abstraction is an important feature of science. Existing work has already considered the problem of expressing formally the relation of abstraction between causal models. In this paper, we…
Recent e-graph applications have typically considered concrete semantics of expressions, where the notion of equivalence stems from concrete interpretation of expressions. However, equivalences that hold over one interpretation may not hold…
The Transformer architecture has revolutionized the field of sequence modeling and underpins the recent breakthroughs in large language models (LLMs). However, a comprehensive mathematical theory that explains its structure and operations…
Abstraction is essential for reducing the complexity of systems across diverse fields, yet designing effective abstraction methodology for probabilistic models is inherently challenging due to stochastic behaviors and uncertainties. Current…
Implicit networks are a class of neural networks whose outputs are defined by the fixed point of a parameterized operator. They have enjoyed success in many applications including natural language processing, image processing, and numerous…
A previously established correspondence between definite-parity real functions and inner analytic functions is generalized to real functions without definite parity properties. The set of inner analytic functions that corresponds to the set…
The relationship between abstract interpretation and partial deduction has received considerable attention and (partial) integrations have been proposed starting from both the partial deduction and abstract interpretation perspectives. In…
We consider abstraction-based design of output-feedback controllers for non-linear dynamical systems against specifications over state-based predicates in linear-time temporal logic (LTL). In this context, our contribution is two-fold: (I)…
Calculational abstract interpretation, long advocated by Cousot, is a technique for deriving correct-by-construction abstract interpreters from the formal semantics of programming languages. This paper addresses the problem of deriving…
Value-based static analysis techniques express computed program invariants as logical formula over program variables. Researchers and practitioners use these invariants to aid in software engineering and verification tasks. When selecting…
Finite-state abstractions are widely studied for the automated synthesis of correct-by-construction controllers for stochastic dynamical systems. However, existing abstraction methods often lead to prohibitively large finite-state models.…
The purpose of this article is to propose and investigate a partial order structure weaker than the lattice structure and which have nice properties regarding closure operators. We extend accordingly closed pattern mining and formal concept…
We explore denotational interpreters: denotational semantics that produce coinductive traces of a corresponding small-step operational semantics. By parameterising our denotational interpreter over the semantic domain and then varying it,…
The recently developed theory of extended generating functions of symplectic maps are combined with methods to prove invertibility via high-order Taylor model methods to obtain rigorous lower bounds for the domains of definition of…
Categorical semantics of type theories are often characterized as structure-preserving functors. This is because in category theory both the syntax and the domain of interpretation are uniformly treated as structured categories, so that we…
Abstract interpretation has been shown to be a promising technique for the thread-modular verification of concurrent programs. Central to this is the generation of interferences, in the form of rely-guarantee conditions, conforming to a…
We consider a sequence of successively more restrictive definitions of abstraction for causal models, starting with a notion introduced by Rubenstein et al. (2017) called exact transformation that applies to probabilistic causal models,…