Type Theory

Batanin and Leinster's work on globular operads has provided one of many potential defnitions of a weak $\omega$-category. Through the language of globular operads they construct a monad whose algebras encode weak $\omega$-categories. The purpose of this work is to show how to construct a similar monad which will allow us to formulate weak $\omega$-categorifications of any equational algebraic theory. We first review the classical theory of operads and PROs. We then present how Leinster's globular operads can be extended to a theory of globular PROs via categorical enrichment over the category of collections. It is then shown how a process called globularization allows us to construct from a classical PRO P a globular PRO whose algebras are those algebras for P which are internal to the category of strict $\omega$-categories and strict $\omega$-functors. Leinster's notion of a contraction structure on a globular operad is then extended to this setting of globular PROs in...

The mean-field Landau-type theory is used to analyze the polarization properties of epitaxial ferroelectric thin films grown on dissimilar cubic substrates, which induce biaxial compressive stress in the film plane. The intrinsic effect of the film surfaces on the spontaneous polarization is taken into account via the concept of the extrapolation length. The theory simultaneously allows for the influence of the misfit strain imposed on the film lattice by a thick substrate. Numerical calculations are performed for PbTiO3 and BaTiO3 films under an assumption of the polarization reduction in surface layers. The film mean polarization is calculated as a function of film thickness, temperature, and misfit strain. It is shown that the negative intrinsic size effect is reduced in epitaxial films due to the in-plane compression of the film lattice. At room temperature, strong reduction of the mean polarization may take place only in ultrathin films (thickness ∼1 nm). Theoretical prediction...

We point out the importance of the use of the refined dependent types available in the proof assistant PVS for the correct specification of a geometrical approach to solve the problem of air traffic conflict detection and resolution (between two aircrafts in mid air). A correct PVS specification which completely solves this problem giving a triplet which consists of escape trajectory, recovery trajectory, switching time and whose first component allows exclusively changes in a unique of the parameters ground speed, heading and vertical speed is partially extended in such a way that the formulated escape maneuvers may combine changes in two of these parameters: ground speed and heading.

Functional programming languages are particularly well-suited for building automated reasoning systems, since (among other reasons) a logical term is well modeled by an inductive type, traversing a term can be implemented generically as a higher-order combinator, and backtracking search is dramatically simplified by persistent datastructures. However, existing pure functional programming languages all suffer a major limitation in these domains: traversing a term requires time proportional to the tree size of the term as opposed to its graph size. This limitation would be particularly devastating when building automation for interactive theorem provers such as Lean and Coq, for which the exponential blowup of term-tree sizes has proved to be both common and difficult to prevent. All that is needed to recover the optimal scaling is the ability to perform simple operations on the memory addresses of terms, and yet allowing these operations to be used freely would clearly violate the ba...

Consistent Hamiltonian interactions that can be added to an abelian free BF-type class of theories in any n ≥ 4 spacetime dimensions are constructed in the framework of the Hamiltonian BRST deformation based on cohomological techniques. The resulting model is an interacting field theory in higher dimensions with an open algebra of on-shell reducible first-class constraints. We argue that the Hamiltonian couplings are related to a natural structure of Poisson manifold on the target space.

The origin of life, or abiogenesis, requires integrating models across multiple scales, including chemical reactions, topological structures, informational processes, and evolutionary dynamics, which often remain disconnected, hindering a unified theoretical framework. This paper introduces a categorical framework to bridge prebiotic chemistry and early protocellular evolution by defining the encapsulation functor and demonstrating that autocatalytic sets are universal colimits, linking chemical kinetics with network emergence. We address the chicken-and-egg problem of genetic material versus metabolism by formalizing hypercycles as groupoids, and we model functional closure through limits in compartment categories. The framework also tackles open-ended evolution by representing evolutionary pathways as natural transformations that preserve scale commutativity. While applicable to higher-order emergences like multicellularity and cognition, this work focuses on the transition from prebiotic chemistry to functional protocells, providing a mathematical language to map chemical kinetics to protocellular organization. By incorporating relational biology concepts, such as Robert Rosen's (M,R)-systems, and offering rigorous mathematical constructions, this framework lays a foundation for studying emergent phenomena across biological scales, complementing empirical research with a formal approach to hypothesis generation.

The word 'actually' often refers to what is in fact the case, but it also often points to what would have been the case in a possible situation that is being envisaged. To capture such nuances, the formal languages discussed in the paper add subscripts to modal operators; in the model theory the subscripts allow an actuality operator to turn the evaluation of a formula to a world introduced by a preceding possibility or necessity operator having the same subscript. The paper covers both propositional and predicate logic and proves the completeness of axiomatizations that extend standard modal systems beginning with K.

In arXiv: quant-ph/0604122v1 (17 April 2006), Stephen Adler published a critique of our paper "The Free Will Theorem" (arXiv: quant-ph/0604079, of which a slightly updated version [1] appears in "Foundations of Physics"). More recently we have discussed our argument with Professor Adler, and are happy to say that he now agrees with us, after certain clarifications, (see Adler ). Adler's main concern is that spin measurements are not instantaneous, and also that the time they take varies with direction. The information available to the particle will vary with this time. This is true, but causes no problem. Particle a's response function θ a may depend on all the information that arrives up to the time of its response. We can in fact allow its domain to include information up to a time T that maximizes this response time over the 40 triples used in our argument. Of course θ a for a particular triple will not in fact vary with information that arrives later than the response time for that triple, but that does not affect its functional nature (a function does not have to vary with all its arguments). A similar argument applies to θ b . (In fact our argument took this into account, though only implicitly, because the information α ′ of which θ a is a function was defined to be independent of x, y, z. However, we are glad to provide the extra clarification.)

In arXiv: quant-ph/0604122v1 (17 April 2006), Stephen Adler published a critique of our paper "The Free Will Theorem" (arXiv: quant-ph/0604079, of which a slightly updated version [1] appears in "Foundations of Physics"). More recently we have discussed our argument with Professor Adler, and are happy to say that he now agrees with us, after certain clarifications, (see Adler ). Adler's main concern is that spin measurements are not instantaneous, and also that the time they take varies with direction. The information available to the particle will vary with this time. This is true, but causes no problem. Particle a's response function θ a may depend on all the information that arrives up to the time of its response. We can in fact allow its domain to include information up to a time T that maximizes this response time over the 40 triples used in our argument. Of course θ a for a particular triple will not in fact vary with information that arrives later than the response time for that triple, but that does not affect its functional nature (a function does not have to vary with all its arguments). A similar argument applies to θ b . (In fact our argument took this into account, though only implicitly, because the information α ′ of which θ a is a function was defined to be independent of x, y, z. However, we are glad to provide the extra clarification.)

On the basis of three physical axioms, we prove that if the choice of a particular type of spin 1 experiment is not a function of the information accessible to the experimenters, then its outcome is equally not a function of the information accessible to the particles. We show that this result is robust, and deduce that neither hidden variable theories nor mechanisms of the GRW type for wave function collapse can be made relativistic. We also establish the consistency of our axioms and discuss the philosophical implications. † Superscript numbers refer to the corresponding Endnotes. ‡ For simplicity, we have spoken of measuring x, y, z in that order, but nothing in the proof is affected if they are measured simultaneously, as in the "spin-Hamiltonian" experiment of Endnote 1. * Nevertheless, relativistic versions of GRW have been claimed; eg. Tumulka[T]. * * Or perhaps the possible free decison ("prompton") at an earlier time t 0 that prompted this response-see the proof of the theorem.

L'ingenierie des lignes de produits logiciels met l'accent sur la gestion de la variabilite et la reutilisation. Dans cet article, nous completons cette approche avec la recherche de garanties de correction sur les produits issus d'une ligne de produits. Nous proposons une methode permettant de produire des produits corrects par construction a partir d'une ligne de produits. Cette methode s'appuie sur un langage, FFML, inspire de FoCaLiZe et incorporant des mecanismes pour exprimer la variabilite, et deux outils : un compilateur vers FoCaLiZe et un composeur. Le composeur permet de generer automatiquement des produits corrects par construction a partir d'une configuration valide. Les outils de FoCaLiZe permettent de verifier les preuves de correction et de generer du code executable OCaml. La methode est illustree dans cet article sur un exemple simple de ligne de produits

Attempts by sensing and intuitive types to diagnose a malfunctioning industrial system by operating a computer-based data bank were compared in terms of (1) the extent of their causal attributions, and ( ) their use of different information-gathering control operations. As predicted by Jungian type theory, intuitives made significantly more causal attributions to factors lying behind the immediately perceived malfunction (P<0.03). There were no significant differences between the two types' use of control operations. The implications of these findings for attribution theory, the theory of diagnostic error, and for the education of diagnosticians are discussed.

We propose an algorithm for constraint solving over hedges and contexts built over individual, sequence, function, and context vari- ables and ∞exible arity symbols, where the admissible bindings of se- quence variables and context variables can be constrained to languages represented by regular hedge or regular context expressions. We identify su-cient syntactic restrictions that enable to solve such constraints by matching techniques, and describe a solving algorithm that is sound and complete.

This paper proposes a technique to limit the number of possible terms to be considered in quantifier instantiation. One of the major hurdles that SMT solvers face when dealing with quantifiers is that there are simply too many terms to instantiate with. So even if the right set of terms is available to the solver, meaning they appear in the formula, the solver might not have enough resources to come upon the right combination. This motivates the technique presented in this paper, which instantiates only by a certain type of terms, called discriminating terms. The paper introduces a class of formulas, where the proposed technique has a considerable impact.

Sometimes machine code turns out to be a better target for verification than source code. RISC machine code is especially advantaged with respect to source code in this regard because it has only two instructions that access memory. That architecture forms the basis here for an inference system that can prove machine code safe against 'hardware aliasing', an effect that occurs in embedded systems. There are programming memes that ensure code is safe from hardware aliasing, but we want to certify that a given machine code is provably safe.

This paper explores the foundational assumptions of Zermelo-Fraenkel (ZF) set theory, revealing a profound geometric underpinning often overlooked in traditional formalism. While ZF formalizes collections and recursive hierarchies of sets, it implicitly encodes a structure of awareness—nested relations that mirror a deeper geometric reality. Introducing the Principle of Geometric Precession (PGP) within the Arithmetic of Order (AoO) framework, we propose that geometry, specifically the finite projective geometry PG(5,2), is the primal source from which algebraic and logical structures emerge. By deriving key set-theoretic operations—pairing, union, powerset—from the intrinsic relations of PG(5,2), AoO repositions geometry as the foundation of mathematical and physical reality. This perspective suggests that logic is not axiomatic but emergent, and that awareness itself is fundamentally geometric. Ultimately, AoO offers a unified view: the recursive architecture of set theory reflects the universe’s self-computation through nested geometric structures, reconciling mathematics, physics, and cognition within a geometric paradigm.

Typing rhythms are one of the rawest form of data stemming from the interaction between humans and computers. When properly analyzed, they may allow to ascertain personal identity. In this paper we provide experimental evidence that the typing dynamics of free text can be used for user identification and authentication even when typing samples are written in different languages. As a consequence, we argue that keystroke analysis can be useful even when people may use different languages, in those areas where ascertaining personal identity is important or crucial, such as within Computer Security.

We investigate linearized gravity on a single de Sitter brane in the anti-de Sitter (AdS) bulk in the Einstein Gauss-Bonnet (EGB) theory. We find that Einstein gravity is recovered for a high energy brane, i.e., in the limit of a large expansion rate, i.e., for H 1, where H is the de Sitter expansion rate and is the curvature radius of the AdS bulk. We also show that, in the short distance limit, r min{ , H -1 }, Brans-Dicke gravity is obtained, whereas in the large distance limit, r max{ , H -1 }, a Brans-Dicke-type theory is obtained for H = O(1), and Einstein gravity is recovered both for H 1 and H 1. In the limit H → 0, these results smoothly match the results known for the Minkowski brane.

We consider the problem of reconciling a dependently typed functional language with imperative features such as mutable higher-order state, pointer aliasing, and nontermination. We propose Hoare type theory (HTT), which incorporates Hoare-style specifications into types, making it possible to statically track and enforce correct use of side effects. The main feature of HTT is the Hoare type {P }x:A{Q} specifying computations with precondition P and postcondition Q that return a result of type A. Hoare types can be nested, combined with other types, and abstracted, leading to a smooth integration with higher-order functions and type polymorphism. We further show that in the presence of type polymorphism, it becomes possible to interpret the Hoare types in the "small footprint" manner, as advocated by separation logic, whereby specifications tightly describe the state required by the computation. We establish that HTT is sound and compositional, in the sense that separate verifications of individual program components suffice to ensure the correctness of the composite program.

The FOLD v1 Specification defines a functional, category-inspired language for structural computation, physical modeling, and coherence-driven systems. Built on the principles of recursive generation, morphic depth, and coherence, FOLD unifies declarative expressiveness with constraint-based operational semantics. All computation is formulated as convergence toward structural fixpoints under explicit frames, with a minimal, extensible operator set and dual canonical/sugar syntax. The language features a structural type system, rational and real numeric domains, solver integration, and a standard library supporting advanced operations such as partition, entanglement, and optimization. All constructs are compositional, referentially transparent, and grounded in the formal requirements of Fold Theory. Reserved features include surreal number support and superposition-based modeling for future versions. This specification serves as the authoritative reference for language implementers, theorists, and advanced users seeking principled modeling of emergent and constraint-driven systems.

We show that, contrary to the claim made in arXiv:0911.1299, the extended Hořava gravity model proposed in arXiv:0909.3525 does not suffer from a strong coupling problem. By studying the observational constraints on the model we determine the bounds on the scale of the ultraviolet modification for which the proposal yields a phenomenologically viable, renormalizable and weakly coupled model of quantum gravity.

There are several ways to formally represent families of data, such as lambda terms, in a type theory such as the dependent type theory of Coq. Mathematical representations are very compact ones and usually rely on the use of dependent types, but they tend to be difficult to handle in practice. On the contrary, implementations based on a larger (and simpler) data structure combined with a restriction property are much easier to deal with. In this work, we study several families related to lambda terms, among which Motzkin trees, seen as lambda term skeletons, closable Motzkin trees, corresponding to closed lambda terms, and a parameterized family of open lambda terms. For each of these families, we define two different representations, show that they are isomorphic and provide tools to switch from one representation to another. All these datatypes and their associated transformations are implemented in the Coq proof assistant. Furthermore we implement random generators for each representation, using the QuickChick plugin.

There is growing public concern about personal data collected by both private and public sectors. People have very little control over what kinds of data are stored and how such data is used. Moreover, the ability to infer new knowledge from existing data is increasing rapidly with advances in database and data mining technologies. We describe a solution which allows people to take control by specifying constraints on the ways in which their data can be used. User constraints are represented in formal logic, and organizations that want to use this data provide formal proofs that the software they use to process data meets these constraints. Checking the proof by an independent verifier demonstrates that user constraints are (or are not) respected by this software. Our notion of "privacy correctness" differs from general software correctness in two ways. First, properties of interest are simpler and thus their proofs should be easier to automate. Second, this kind of correctness is stricter; in addition to showing a certain relation between input and output is realized, we must also show that only operations that respect privacy constraints are applied during execution. We have therefore an intensional notion of correctness, rather that the usual extensional one. We discuss how our mechanism can be put into practice, and we present the technical aspects via an example. Our example shows how users can exercise control when their data is to be used as input to a decision tree learning algorithm. We have formalized the example and the proof of preservation of privacy constraints in Coq.

There is growing public concern about personal data collected by both private and public sectors. People have very little control over what kinds of data are stored and how such data is used. Moreover, the ability to infer new knowledge from existing data is increasing rapidly with advances in database and data mining technologies. We describe a solution which allows people to take control by specifying constraints on the ways in which their data can be used. User constraints are represented in formal logic, and organizations that want to use this data provide formal proofs that the software they use to process data meets these constraints. Checking the proof by an independent verifier demonstrates that user constraints are (or are not) respected by this software. Our notion of "privacy correctness" differs from general software correctness in two ways. First, properties of interest are simpler and thus their proofs should be easier to automate. Second, this kind of correctness is stricter; in addition to showing a certain relation between input and output is realized, we must also show that only operations that respect privacy constraints are applied during execution. We have therefore an intensional notion of correctness, rather that the usual extensional one. We discuss how our mechanism can be put into practice, and we present the technical aspects via an example. Our example shows how users can exercise control when their data is to be used as input to a decision tree learning algorithm. We have formalized the example and the proof of preservation of privacy constraints in Coq.

This paper presents a formalization of Nuprl&#39;s metatheory in Coq. It includes a nominal-style de nition of the Nuprl language, its reduction rules, a coinductive computational equivalence, and a Currystyle type system where a type is de ned as a Partial Equivalence Relation (PER) à la Allen. This type system includes Martin-Löf dependent types, a hierarchy of universes, inductive types and partial types. We then prove that the typehood rules of Nuprl are valid w.r.t. this PER semantics and hence reduce Nuprl&#39;s consistency to Coq&#39;s consistency.

Luo's Extended Calculus of donstructions (ECC) is a higher order functional calculus based on Coquand's and Huet's Calculus of Constructions, but has in addition strong sums and a predicative cumulative type hierarchy. In this paper I introduce inductive types on the predicative type levels of ECC. I also show how the o-Set model for ECC can be extended to a model for this augmented calculus.

Este artículo presenta una defensa sistemática y exhaustiva de la teoría del valortrabajo (TVT) de Karl Marx frente a las críticas formuladas por Juan R. Rallo en su obra Anti-Marx. Se argumenta que la refutación de Rallo, si bien articulada, se fundamenta en un error metodológico al tratar el argumento dialéctico-genealógico de Marx como un silogismo de lógica formal. Se analiza cada objeción de Rallo, citándola textualmente, para luego refutarla punto por punto. Se demuestra que los "seis problemas prácticos" que expone son malentendidos de la teoría, límites que el propio Marx reconoció, o problemas técnicos resueltos por la literatura moderna. El análisis se apoya en los textos de Marx, las soluciones teóricas de la Interpretación Temporal de Sistema Único (TSSI), y, en la evidencia empírica que demuestra la fuerte correlación entre valores-trabajo y precios de mercado.

We propose a uniform, category-theoretic account of structural induction for inductively defined data types. The account is based on the understanding of inductively defined data types as initial• algebras for certain kind of endofunctors T: llll-.JB on a bicartesian/distributive category B. Regarding a predicate logic as a fibration p : !Pl-+llll over llll, we consider a logical predicate lifting of T to the total category lF. Then, a predicate is inductive precisely when it carries an algebra structure for such lifted endofunctor. The validity of the induction principle is formulated by requiring that the 'truth' predicate functor T: JBl-.JP' preserve initial algebras. We then show that when the fibration admits a comprehension principle, analogous to the one in set theory, it satisfies the induction principle. We also consider the appropriate extensions of the above formulation to deal with initiality (and induction) in arbitrary contexts, i.e. the 'stability' property of the induction principle.

We consider pseudo-descent in the context of 2-fibrations. A 2-category of descent data is associated to a 3-truncated simplicial object in the base 2-category. A morphism q in the base induces (via comma-objects and pullbacks) an internal category whose truncated nerve allows the definition of the 2-category of descent data for q. When the 2-fibration admits direct images, we provide the analogous of the Beck-Bénabou-Roubaud theorem, identifying the 2-category of descent data with that of pseudo-algebras for the pseudo-monad q * Σq. We introduce a notion of 2-regularity for a 2-category R, so that its basic 2-fibration of internal fibrations cod : Fib(R) → R admits direct images. In this context, we show that essentiallysurjective-on-objects morphisms, defined by a certain lax colimit, are of effective descent by means of a Beck-style pseudo-monadicity theorem.

We consider a search for phenomenological signatures from an hypothetical space-time granularity that respects Lorentz invariance. The model is based on the idea that the metric description of Einstein's gravity corresponds to a hydrodynamic characterization of some deeper underlying structure, and that Einstein's gravity is thus to be seen as emergent. We present the specific phenomenological model in detail and analyze the bounds on its free parameters established by an experiment specifically designed to test this model.

Constructive Zermelo-Fraenkel Set Theory, CZF, has emerged as a standard reference theory that relates to constructive predicative mathematics as ZFC relates to classical Cantorian mathematics. A hallmark of this theory is that it possesses a type-theoretic model. Aczel showed that it has a formulae-as-types interpretation in Martin-Löf's intuitionist theory of types . This paper, though, is concerned with a rather different interpretation. It is shown that Kleene realizability provides a self-validating semantics for CZF, viz. this notion of realizability can be formalized in CZF and demonstrably in CZF it can be verified that every theorem of CZF is realized. This semantics, then, is put to use in establishing several equiconsistency results. Specifically, augmenting CZF by well-known principles germane to Russian constructivism and Brouwer's intuitionism turns out to engender theories of equal proof-theoretic strength with the same stock of provably recursive functions.

In this paper we shall investigate fragments of Kripke-Platek set theory with Infinity which arise from the full theory by restricting Foundation to Π n Foundation, where n ≥ 2. The strength of such fragments will be characterized in terms of the smallest ordinal α such that L α is a model of every Π 2 sentence which is provable in the theory.

This paper is concerned with metamathematical properties of intuitionistic set theories with choice principles. It is proved that the disjunction property, the numerical existence property, Church's rule, and several other metamathematical properties hold true for Constructive Zermelo-Fraenkel Set Theory and full Intuitionistic Zermelo-Fraenkel augmented by any combination of the principles of Countable Choice, Dependent Choices and the Presentation Axiom. Also Markov's principle may be added. Moreover, these properties hold effectively. For instance from a proof of a statement ∀n ∈ ω ∃m ∈ ω ϕ(n, m) one can effectively construct an index e of a recursive function such that ∀n ∈ ω ϕ(n, {e}(n)) is provable. Thus we have an explicit method of witness and program extraction from proofs involving choice principles. As for the proof technique, this paper is a continuation of . [32] introduced a selfvalidating semantics for CZF that combines realizability for extensional set theory and truth.

In order to build the collection of Cauchy reals as a set in constructive set theory, the only Power Set-like principle needed is Exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that Exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, CZF with Subset Collection replaced by Exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class.

Constructive Zermelo-Fraenkel set theory, CZF, can be interpreted in Martin-Löf type theory via the so-called propositions-as-types interpretation. However, this interpretation validates more than what is provable in CZF. We now ask ourselves: is there a reasonably simple axiomatization (by a few axiom schemata say) of the set-theoretic formulae validated in Martin-Löf type theory? The answer is yes for a large collection of statements called the mathematical formulae. The validated mathematical formulae can be axiomatized by suitable forms of the axiom of choice.

The paper investigates inaccessible set axioms and their consistency strength in constructive set theory. In ZFC inaccessible sets are of the form V κ where κ is a strongly inaccessible cardinal and V κ denotes the κ -th level of the von Neumann hierarchy. Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory. The objective of this paper is to show that the consistency strength of inaccessible set axioms heavily depends on the context in which they are embedded. The context here will be the theory CZF -of constructive Zermelo Fraenkel set theory but without ∈ -Induction (foundation). Let INAC be the statement that for every set there is an inaccessible set containing it. CZF -+ INAC is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics and a great deal of category theory. CZF -+ INAC also has a realizability interpretation in type theory which gives its theorems a direct computational meaning. The main result presented here is that the proof theoretic ordinal of CZF -+ INAC is a small ordinal known as the Feferman -Schütte ordinal Γ 0 .

We characterize the proof-theoretic strength of systems of explicit mathematics with a general principle (MID) asserting the existence of least fixed points for monotone inductive definitions, in terms of certain systems of analysis and set theory. In the case of analysis, these are systems which contain the C&axiom of choice and @-comprehension for formulas without set parameters. In the case of set theory, these are systems containing the Kripke-Platek axioms for a recursively inaccessible universe together with the existence of a stable ordinal. In all cases, the exact strength depends on what forms of induction are admitted in the respective systems.