Computable Model Theory Research Articles

Generically and coarsely computable isomorphisms

Inspired by the study of generic and coarse computability in computability theory, we extend such investigation to the context of computable model theory. In this paper, we continue our study initiated in the previous paper (Journal of Logic and Computation 32 (2022) 581–607) , where we introduced and studied the notions of generically and coarsely computable structures and their generalizations. In this paper, we introduce the notions of generically and coarsely computable isomorphisms, and their weaker variants. We sometimes also require that the isomorphisms preserve the density structure. For example, for any coarsely computable structure A, there is a density preserving coarsely computable isomorphism from A to a computable structure. We demonstrate that each notion of generically and coarsely computable isomorphisms, density preserving or not, gives interesting insights into the structures we consider, focusing on various equivalence structures and injection structures.

On Decidable Categoricity and Almost Prime Models

The complexity of isomorphisms for computable and decidable structures plays an important role in computable model theory. Goncharov [26] defined the degree of decidable categoricity of a decidable model $$\mathcal {M} $$ to be the least Turing degree, if it exists, which is capable of computing isomorphisms between arbitrary decidable copies of $$\mathcal {M} $$ . If this degree is $$\mathbf {0} $$ , we say that the structure $$\mathcal {M} $$ is decidably categorical. Goncharov established that every computably enumerable degree is the degree of categoricity of a prime model, and Bazhenov showed that there is a prime model with no degree of categoricity. Here we investigate the degrees of categoricity of various prime models with added constants, also called almost prime models. We relate the degree of decidable categoricity of an almost prime model $$\mathcal {M} $$ to the Turing degree of the set $$C(\mathcal {M}) $$ of complete formulas. We also investigate uniform decidable categoricity, characterizing it by primality of $$\mathcal {M} $$ and Turing reducibility of $$C(\mathcal {M}) $$ to the theory of $$\mathcal {M} $$ .

Turing degrees and automorphism groups of substructure lattices

The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of investigation in computable model theory. In this paper, we focus on the lattice structure of computably enumerable substructures of a given canonical computable structure. In particular, for a Turing degree $\mathbf{d}$, we investigate the groups of $\mathbf{d}$ -computable automorphisms of the lattice of $\mathbf{d}$-computably enumerable vector spaces, of the interval Boolean algebra $\mathcal{B}_{\eta }$ of the ordered set of rationals, and of the lattice of $\mathbf{d}$ -computably enumerable subalgebras of $\mathcal{B}_{\eta }$. For these groups we show that Turing reducibility can be used to substitute the group-theoretic embedding. We also prove that the Turing degree of the isomorphism types for these groups is the second Turing jump of $\mathbf{d}$ , $\mathbf{d^{\prime \prime }}$.

Turing Degrees and Automorphism Groups of Substructure Lattices

The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of investigation in computable model theory. Here we focus on the lattice structure of computably enumerable substructures of a given canonical computable structure. In particular, for a Turing degree d, we investigate the groups of d-computable automorphisms of the lattice of d-computably enumerable vector spaces, of the interval Boolean algebra Bη of the ordered set of rationals, and of the lattice of d-computably enumerable subalgebras of Bη. For these groups, we show that Turing reducibility can be used to substitute the group-theoretic embedding. We also prove that the Turing degree of the isomorphism types for these groups is the second Turing jump d′′ of d.

Strongly minimal theories with recursive models

We give effectiveness conditions on a strongly minimal theory T guaranteeing that all countable models have computable copies. In particular, we show that if T is strongly minimal and for all n\geq 1 , T\cap\exists_{n+2} is \Delta^0_n , uniformly in n , then every countable model has a computable copy. A longstanding question of computable model theory asked whether for a strongly minimal theory with one computable model, every countable model has an arithmetical copy. Relativizing our main result, we get the fact that if there is one computable model, then every countable model has a \Delta^0_4 copy.

A COMPUTABLE FUNCTOR FROM GRAPHS TO FIELDS

AbstractFried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure${\cal S}$, there exists a countable field${\cal F}$of arbitrary characteristic with the same essential computable-model-theoretic properties as${\cal S}$. Along the way, we develop a new “computable category theory”, and prove that our functor and its partially defined inverse (restricted to the categories of countable graphs and countable fields) are computable functors.

Intelligence vs. Artificial Intelligence: The King Is Naked

The sociological dimension of Artificial Intelligence (AI), and automation, is placed in the path traced by the man-machine integration process, started in the eighteenth century by the Industrial Revolution that assumes its current connotation after World War II. The use of the term intelligence, which appears in the expression Artificial Intelligence, is shown to be improper and misleading, and the expression itself should be replaced, eventually, by Artificial Simulation of the Savant Syndrome. By adopting a theoretical perspective, namely Endo-Dynamo-Tensive Model [Messori 2012B], it is traced the mapping of human’s neurological (neuro-dynamics) and psychological (psycho-dynamics) dimension, and is provided the coordinates to be followed in the phenomenological qualification and definition of the psychic-mental-cognitive function intelligence. The neuro-dynamics of the Nervous System (NS), is taken into account by adopting the theoretical, interpretative and investigative perspective indicated by the particular line of research developed in the context of Quantum Field Theory (QFT) and QED (Quantum Electrodynamic Field Theory), that describes the water of which all living systems are composed, i.e. biological water, as water in a coherent oscillatory phase or state other than that of common water, named super-coherent oscillatory state. By introducing the possible functional role exerted within brain activity by glial cells, cerebrospinal fluid, intra- and extra-cellular fluid is outlined the overcoming of the classical neuroscience paradigm, based on the vision of brain activity as ruled by networks of neurons interconnected by synapses. The body-mind hard problem is taken into account and a solution is advanced. The psycho-dynamics of the humans mind territory is taken into account according to the four poles of mental functions introduced by C.G. Jung in his Psychological Types (1921), where he introduces a hierarchy of mental functions in two mental bipolar dimensions (dichotomies). These are sensing (attentiveness by means of the sense organs) coupled to intuition (awareness in unconscious way or being aware of unconscious contents) and thinking (function of intellectual cognition; the forming of logical conclusions) coupled to feeling (function of subjective estimation). At the conclusion of this work, it is provided the phenomenological definition of intelligence that does not contemplate the possibility to apply to neuroscience, and to natural and human sciences in general, the paradigm that inspires the research on AI, i.e. computational model and Information Theory.

Computably Isometric Spaces

AbstractWe say that an uncountable metric space is computably categorical if every two computable structures on this space are equivalent up to a computable isometry. We show that Cantor space, the Urysohn space, and every separable Hilbert space are computably categorical, but the space [0, 1] of continuous functions on the unit interval with the supremum metric is not. We also characterize computably categorical subspaces of ℝn, and give a sufficient condition for a space to be computably categorical. Our interest is motivated by classical and recent results in computable (countable) model theory and computable analysis.

Metric structures and probabilistic computation

Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of mathematical objects that can be described in classical first-order logic. The present paper shows that probabilistic computation (sometimes called randomized computation) and continuous logic stand in a similar close relationship. The main result of this paper is an effective completeness theorem, showing that every decidable continuous first-order theory has a probabilistically decidable model. We also show that probabilistically computable structures give rise to a model of ACA 0 in a natural way, and describe a connection with complexity theory.

An introduction to computable model theory on groups and fields

We introduce the standard computable-model-theoretic concepts of a computable group and a computable field, and use them to illustrate the sorts of questions about groups and fields which computability theorists investigate. This article is intended for group theorists with some background in algorithmic questions, such as the undecidability of the word problem and the conjugacy problem for finitely presented groups.

Effectiveness in RPL, with applications to continuous logic

In this paper, we introduce a foundation for computable model theory of rational Pavelka logic (an extension of Łukasiewicz logic) and continuous logic, and prove effective versions of some related theorems in model theory. We show how to reduce continuous logic to rational Pavelka logic. We also define notions of computability and decidability of a model for logics with computable, but uncountable, set of truth values; we show that provability degree of a formula with respect to a linear theory is computable, and use this to carry out an effective Henkin construction. Therefore, for any effectively given consistent linear theory in continuous logic, we effectively produce its decidable model. This is the best possible, since we show that the computable model theory of continuous logic is an extension of computable model theory of classical logic. We conclude with noting that the unique separable model of a separably categorical and computably axiomatizable theory (such as that of a probability space or an L p Banach lattice) is decidable.

On effective presentations of formal concept lattices

We study the isomorphism types of ordered structures of concepts for computable formal contexts from the point of view of computable model theory. We show that, although these structures could have the cardinality of the continuum or—if they are countable—could have an arbitrary high hyperarithmetical complexity, they are in some sense very close to computable orderings. We give some sufficient conditions for these orderings to have computable presentations. A full description is given of the isomorphism types of discrete concept lattices. We also give some counterexamples.

Prime models of finite computable dimension

AbstractWe study the following open question in computable model theory: does there exist a structure of computable dimension two which is the prime model of its first-order theory? We construct an example of such a structure by coding a certain family of c.e. sets with exactly two one-to-one computable enumerations into a directed graph. We also show that there are examples of such structures in the classes of undirected graphs, partial orders, lattices, and integral domains.

Applications of Kolmogorov complexity to computable model theory

AbstractIn this paper we answer the following well-known open question in computable model theory. Does there exist a computable not ℵ0-categorical saturated structure with a unique computable isomor-phism type? Our answer is affirmative and uses a construction based on Kolmogorov complexity. With a variation of this construction, we also provide an example of an ℵ1-categorical but not ℵ0-categorical saturated -structure with a unique computable isomorphism type. In addition, using the construction we give an example of an ℵ1-categorical but not ℵ0-categorical theory whose only non-computable model is the prime one.

On the computability-theoretic complexity of trivial, strongly minimal models

We show the existence of a trivial, strongly minimal (and thus uncountably categorical) theory for which the prime model is computable and each of the other countable models computes 0. This result shows that the result of Goncharov/Harizanov/Laskowski/Lempp/McCoy (2003) is best possible for trivial strongly minimal theories in terms of computable model theory. We conclude with some remarks about axiomatizability.

Enumerations in computable structure theory

We exploit properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a computable successor ordinal α , we transform a countable directed graph G into a structure G ∗ such that G has a Δ α 0 isomorphic copy if and only if G ∗ has a computable isomorphic copy. A computable structure A is Δ α 0 categorical ( relatively Δ α 0 categorical, respectively) if for all computable ( countable, respectively) isomorphic copies B of A , there is an isomorphism from A onto B , which is Δ α 0 ( Δ α 0 relative to the atomic diagram of B , respectively). We prove that for every computable successor ordinal α , there is a computable, Δ α 0 categorical structure, which is not relatively Δ α 0 categorical. This generalizes the result of Goncharov that there is a computable, computably categorical structure, which is not relatively computably categorical. An additional relation R on the domain of a computable structure A is intrinsically Σ α 0 ( relatively intrinsically Σ α 0 , respectively) on A if in all computable ( countable, respectively) isomorphic copies B of A , the image of R is Σ α 0 ( Σ α 0 relative to the atomic diagram of B , respectively). We prove that for every computable successor ordinal α , there is an intrinsically Σ α 0 relation on a computable structure, which not relatively intrinsically Σ α 0 . This generalizes the result of Manasse that there is an intrinsically computably enumerable relation on a computable structure, which is not relatively intrinsically computably enumerable. The Δ α 0 dimension of a structure A is the number of computable isomorphic copies, up to Δ α 0 isomorphisms. We prove that for every computable successor ordinal α and every n ≥ 1 , there is a computable structure with Δ α 0 dimension n . This generalizes the result of Goncharov that there is a structure of computable dimension n for every n ≥ 1 . Finally, we prove that for every computable successor ordinal α , there is a countable structure with isomorphic copies in just the Turing degrees of sets X such that Δ α 0 relative to X is not Δ α 0 . In particular, for every finite n , there is a structure with isomorphic copies in exactly the non- low n Turing degrees. This generalizes the result obtained by Wehner, and independently by Slaman, that there is a structure A with isomorphic copies in exactly the nonzero Turing degrees.

Bounding prime models

Abstract.A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model of T decidable in X. It is easy to see that X = 0′ is prime bounding. Denisov claimed that every X <T 0′ is not prime bounding, but we discovered this to be incorrect. Here we give the correct characterization that the prime bounding sets X ≤τ 0′ are exactly the sets which are not low2. Recall that X is low2 if X″ ≤τ 0″. To prove that a low2 set X is not prime bounding we use a 0′ -computable listing of the array of sets {Y : Y ≤τX } to build a CAD theory T which diagonalizes against all potential X-decidable prime models of T, To prove that any non-low2X is indeed prime bounding. we fix a function f ≤TX that is not dominated by a certain 0′-computable function that picks out generators of principal types. Given a CAD theory T. we use f to eventually find, for every formula φ(x̄) con sistent with T. a principal type which contains it. and hence to build an X-decidable prime model of T. We prove the prime bounding property equivalent to several other combinatorial properties, including some related to the limitwise monotonic functions which have been introduced elsewhere in computable model theory.

Effective completeness theorems for modal logic

We initiate the study of computable model theory of modal logic, by proving effective completeness theorems for a variety of first-order modal logics. We formulate a natural definition of a decidable Kripke model, and show how to construct such a decidable Kripke model of a given decidable theory. Our construction is inspired by the effective Henkin construction for classical logic. The Henkin construction, however, depends in an essential way on the Deduction Theorem. In its usual form the Deduction Theorem fails for modal logic. In our construction, the Deduction Theorem is replaced by a result about objects called finite Kripke diagrams. We argue that this result can be viewed as an analogue of the Deduction Theorem for modal logic.

Effective completeness theorems for modal logic

We initiate the study of computable model theory of modal logic, by proving effective completeness theorems for a variety of first-order modal logics. We formulate a natural definition of a decidable Kripke model, and show how to construct such a decidable Kripke model of a given decidable theory. Our construction is inspired by the effective Henkin construction for classical logic. The Henkin construction, however, depends in an essential way on the Deduction Theorem. In its usual form the Deduction Theorem fails for modal logic. In our construction, the Deduction Theorem is replaced by a result about objects called finite Kripke diagrams. We argue that this result can be viewed as an analogue of the Deduction Theorem for modal logic.

Predicting the Morphologies of Confined Copolymer/Nanoparticle Mixtures

To isolate the factors that control the structure of nanocomposite thin films, we develop a computational model and scaling theory to investigate the behavior of diblock/nanoparticle mixtures that are confined between two hard walls. We find that in such restricted geometries a polymer-induced depletion attraction drives the particles to these walls. If the particles are chemically distinct from the walls, they will effectively modify the chemical nature of these substrates. This change in chemistry, in turn, affects the polymer−wall interactions and consequently the structure of the film. We illustrate this point by considering mixtures of particles and symmetric diblocks and show that the confining walls can be exploited to promote the self-assembly of the system into particle nanowires that extend throughout the films and are separated by nanoscale stripes of polymer domains. Such films constitute vital components in the fabrication of nanoscale devices. Furthermore, the results point to a novel technique for modifying the chemical nature of coatings and films entirely through self-assembly. Since this technique relies on entropic effects, it constitutes a fairly robust method that can be applied more generally than approaches that rely primarily on chemistry-specific enthalpic effects.