Enumeration Reducibility in Closure Spaces with Applications to Logic and Algebra

In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of theorems in many finitely axiomatisable theories is nonrecursive, but the set of theorems for any finitely axiomatisable complete theory is recursive. Finitely presented groups might have an nonrecursive word problem, but finitely presented simple groups have a recursive word problem. In this article we introduce a topological framework based on closure spaces to show that many of these proofs can be obtained in a similar setting. We will show in particular that these statements can be generalized to cover arbitrary structures, with no finite or recursive presentation/axiomatization. This generalizes in particular work by Kuznetsov and others. Examples from first order logic and symbolic dynamics will be discussed at length

Adding an Implication to Logics of Perfect Paradefinite Algebras

Perfect paradefinite algebras are De Morgan algebras expanded with a perfection (or classicality) operation. They form a variety that is term-equivalent to the variety of involutive Stone algebras. Their associated multiple-conclusion (Set-Set) and single-conclusion (Set-Fmla) order-preserving logics are non-algebraizable self-extensional logics of formal inconsistency and undeterminedness determined by a six-valued matrix, studied in depth by Gomes et al. (2022) from both the algebraic and the proof-theoretical perspectives. We continue hereby that study by investigating directions for conservatively expanding these logics with an implication connective (essentially, one that admits the deduction-detachment theorem). We first consider logics given by very simple and manageable non-deterministic semantics whose implication (in isolation) is classical. These, nevertheless, fail to be self-extensional. We then consider the implication realized by the relative pseudo-complement over the six-valued perfect paradefinite algebra. Our strategy is to expand such algebra with this connective and study the (self-extensional) Set-Set and Set-Fmla order-preserving logics, as well as the T-assertional logics of the variety induced by the new algebra. We provide axiomatizations for such new variety and for such logics, drawing parallels with the class of symmetric Heyting algebras and with Moisil's `symmetric modal logic'. For the Set-Set logic, in particular, the axiomatization we obtain is analytic. We close by studying interpolation properties for these logics and concluding that the new variety has the Maehara amalgamation property

Equational Formulas and Pattern Operations in Initial Order-Sorted Algebras

A pattern, i.e., a term possibly with variables, denotes the set (language) of all its ground instances. In an untyped setting, symbolic operations on finite sets of patterns can represent Boolean operations on languages. But for the more expressive patterns needed in declarative languages supporting rich type disciplines such as subtype polymorphism untyped pattern operations and algorithms break down. We show how they can be properly defined by means of a signature transformation that enriches the types of the original signature. We also show that this transformation allows a systematic reduction of the first-order logic properties of an initial order-sorted algebra supporting subtype-polymorphic functions to equivalent properties of an initial many-sorted (i.e., simply typed) algebra. This yields a new, simple proof of the known decidability of the first-order theory of an initial order-sorted algebra.Partially supported by NSF Grant CNS 13-19109.Ope

An algebra and a logic for NC1

Presented here are an algebra and a logic characterizing the complexity class NC1, which consists of functions computed by uniform families of polynomial size, log depth circuits. In both characterizations, NC1 functions are regarded as functions from one class of finite relational structures to another. In the algebraic characterization a recursion scheme called upward tree recursion is applied to a class of simple functions. In the logical characterization, first-order logic is augmented by an operator for defining relations by primitive recursion where it is assumed that every structure has an underlying relation BIT giving the binary representations of integers.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28501/1/0000298.pd

Logic and operator algebras

The most recent wave of applications of logic to operator algebras is a young and rapidly developing field. This is a snapshot of the current state of the art.Comment: A minor chang