Related papers: Proof Diagrams for Multiplicative Linear Logic: Sy…
We present a polymorphic linear lambda-calculus as a proof language for second-order intuitionistic linear logic. The calculus includes addition and scalar multiplication, enabling the proof of a linearity result at the syntactic level.
Technically speaking, the transcendental syntax is about designing logics with a computational foundation. It suggests a new framework for proof theory where logic (proofs, formulas, truth, ...) is no more primitive but computation is. All…
Proofs are traditionally syntactic, inductively generated objects. This paper reformulates first-order logic (predicate calculus) with proofs which are graph-theoretic rather than syntactic. It defines a combinatorial proof of a formula…
The search for linguistic patterns, stylometry and forensic linguistics have in the theory of complex networks, their structures and associated mathematical tools, allies with which to model and analyze texts. In this paper we present a new…
Architecture styles characterise families of architectures sharing common characteristics. We have recently proposed configuration logics for architecture style specification. In this paper, we study a graphical notation to enhance…
Coinduction refers to both a technique for the definition of infinite streams, so-called codata, and a technique for proving the equality of coinductively specified codata. This article first reviews coinduction in declarative programming.…
We develop an analogue of universal algebra in which generating symbols are interpreted as relations. We prove a variety theorem for these relational algebraic theories, in which we find that their categories of models are precisely the…
Automated theorem proving in first-order logic is an active research area which is successfully supported by machine learning. While there have been various proposals for encoding logical formulas into numerical vectors -- from simple…
Matrix syntax is a formal model of syntactic relations in language. The purpose of this paper is to explain its mathematical foundations, for an audience with some formal background. We make an axiomatic presentation, motivating each axiom…
Semantics of logic programs has been given by proof theory, model theory and by fixpoint of the immediate-consequence operator. If clausal logic is a programming language, then it should also have a compositional semantics. Compositional…
Models of complex systems are widely used in the physical and social sciences, and the concept of layering, typically building upon graph-theoretic structure, is a common feature. We describe an intuitionistic substructural logic called…
A semantic net can be used to represent a sentence. A sentence in a language contains semantics which are polar in nature, that is, semantics which are positive, neutral and negative. Neutrosophy is a relatively new field of science which…
Recent causal probing literature reveals when language models and syntactic probes use similar representations. Such techniques may yield "false negative" causality results: models may use representations of syntax, but probes may have…
ProbNetKAT is a probabilistic extension of NetKAT with a denotational semantics based on Markov kernels. The language is expressive enough to generate continuous distributions, which raises the question of how to compute effectively in the…
The distributed representations currently used are dense and uninterpretable, leading to interpretations that themselves are relative, overcomplete, and hard to interpret. We propose a method that transforms these word vectors into reduced…
Monoidal algebraic structures consist of operations that can have multiple outputs as well as multiple inputs, which have applications in many areas including categorical algebra, programming language semantics, representation theory,…
In logic there is a clear concept of what constitutes a proof and what not. A proof is essentially defined as a finite sequence of formulae which are either axioms or derived by proof rules from formulae earlier in the sequence.…
Two lines of approaches are adopted for complex reasoning with LLMs. One line of work prompts LLMs with various reasoning structures, while the structural outputs can be naturally regarded as intermediate reasoning steps. Another line of…
In this work we propose a multi-valued extension of logic programs under the stable models semantics where each true atom in a model is associated with a set of justifications. These justifications are expressed in terms of causal graphs…
Recent work exhibited that distributed word representations are good at capturing linguistic regularities in language. This allows vector-oriented reasoning based on simple linear algebra between words. Since many different methods have…