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It is well known that the resolution method (for propositional logic) is complete. However, completeness proofs found in the literature use an argument by contradiction showing that if a set of clauses is unsatisfiable, then it must have a…
Complemented lattices and uniquely complemented lattices are very important, not only in mathematics, but also in physics, biology, and even in social sciences. They have been investigated for a long time, especially by Huntington,…
The theory of bounded, distributive lattices provides the appropriate language for describing directionality and asymptotics in dynamical systems. For bounded, distributive lattices the general notion of `set-difference' taking values in a…
In this article we introduce theory and algorithms for learning discrete representations that take on a lattice that is embedded in an Euclidean space. Lattice representations possess an interesting combination of properties: a) they can be…
We propose a geometric setting of the axiomatic mathematical formalism of quantum theory. Guided by the idea that understanding the mathematical structures of these axioms is of similar importance as was historically the process of…
Stalnaker and Thomason famously proved that the conditional logic \textsf{C2} with first-order quantifiers is complete with respect to a selection function semantics. However, the selection functions used in this completeness result take…
Necessary and sufficient conditions are presented for the (first-order) theory of a universal class of algebraic structures (algebras) to admit a model completion, extending a characterization provided by Wheeler. For varieties of algebras…
We describe the basic lattice structures of attractors and repellers in dynamical systems. The structure of distributive lattices allows for an algebraic treatment of gradient-like dynamics in general dynamical systems, both invertible and…
Literature involving preferences of artificial agents or human beings often assume their preferences can be represented using a complete transitive binary relation. Much has been written however on different models of preferences. We review…
Substructural logics naturally support a quantitative interpretation of formulas, as they are seen as consumable resources. Distances are the quantitative counterpart of equivalence relations: they measure how much two objects are similar,…
It is shown that the universal theory of the free pseudocomplemented distributive lattice is decidable and a recursive axiomatization is presented. This contrasts with the case of the full elementary theory of the finitely generated free…
We prove that the existence of finite combinatorial objects such as affine planes, mutually orthogonal Latin squares, and resolvable balanced incomplete block designs can be reformulated as the existence of certain algorithmic reductions…
Many different systems with explicit substitutions have been proposed to implement a large class of higher-order languages. Motivations and challenges that guided the development of such calculi in functional frameworks are surveyed in the…
Functional Distributional Semantics provides a computationally tractable framework for learning truth-conditional semantics from a corpus. Previous work in this framework has provided a probabilistic version of first-order logic, recasting…
For many-particle systems defined on lattices we investigate the global structure of effective Hamiltonians and observables obtained by means of a suitable basis transformation. We study transformations which lead to effective Hamiltonians…
This is a survey of characterizations and relationships between some properties of lattices, particularly the modular, Arguesian, linear, and distributive properties, but also some other related properties. The survey emphasizes finite and…
The overarching goal of this thesis is to demonstrate that complementarity is at the heart of quantum information theory, that it allows us to make (some) sense of just what information "quantum information" refers to, and that it is useful…
We discuss the problems of incompleteness and inexpressibility. We introduce almost self-referential formulas, use them to extend set theory, and relate their expressive power to that of infinitary logic. We discuss the nature of proper…
We extend the notion of exact completion on a weakly lex category to elementary doctrines. We show how any such doctrine admits an elementary quotient completion, which freely adds effective quotients and extensional equality. We note that…
Various concepts of grammatical compositionality arise in many theories of both natural and artificial languages, and often play a key role in accounts of the syntax-semantics interface. We propose that many instances of compositionality…