44 research outputs found
A Simple Logic of Functional Dependence
This paper presents a simple decidable logic of functional dependence LFD,
based on an extension of classical propositional logic with dependence atoms
plus dependence quantifiers treated as modalities, within the setting of
generalized assignment semantics for first order logic. The expressive
strength, complete proof calculus and meta-properties of LFD are explored.
Various language extensions are presented as well, up to undecidable
modal-style logics for independence and dynamic logics of changing dependence
models. Finally, more concrete settings for dependence are discussed:
continuous dependence in topological models, linear dependence in vector
spaces, and temporal dependence in dynamical systems and games.Comment: 56 pages. Journal of Philosophical Logic (2021
On Systems of Equations over Free Partially Commutative Groups
Version 2: Corrected Section 3.3: instead of lexicographical normal forms we
now use a normal form due to V. Diekert and A. Muscholl. Consequent changes
made and some misprints corrected.
Using an analogue of Makanin-Razborov diagrams, we give an effective
description of the solution set of systems of equations over a partially
commutative group (right-angled Artin group) . Equivalently, we give a
parametrisation of , where is a finitely generated group.Comment: 117 pages, 22 figure
Implicit function theorem over free groups
We introduce the notion of a regular quadratic equation and a regular NTQ
system over a free group. We prove the results that can be described as
Implicit function theorems for algebraic varieties corresponding to regular
quadratic and NTQ systems. We will also show that the Implicit function theorem
is true only for these varieties. In algebraic geometry such results would be
described as lifting solutions of equations into generic points. From the model
theoretic view-point we claim the existence of simple Skolem functions for
particular -formulas over free groups. Proving these theorems
we describe in details a new version of the Makanin-Razborov process for
solving equations in free groups. We also prove a weak version of the Implicit
function theorem for NTQ systems which is one of the key results in the
solution of the Tarski's problems about the elementary theory of a free group.Comment: 144 pages, 16 figure
Sequent calculus proof systems for inductive definitions
Inductive definitions are the most natural means by which to represent many families of structures
occurring in mathematics and computer science, and their corresponding induction / recursion
principles provide the fundamental proof techniques by which to reason about such
families. This thesis studies formal proof systems for inductive definitions, as needed, e.g., for
inductive proof support in automated theorem proving tools. The systems are formulated as
sequent calculi for classical first-order logic extended with a framework for (mutual) inductive
definitions.
The default approach to reasoning with inductive definitions is to formulate the induction
principles of the inductively defined relations as suitable inference rules or axioms, which are
incorporated into the reasoning framework of choice. Our first system LKID adopts this direct
approach to inductive proof, with the induction rules formulated as rules for introducing atomic
formulas involving inductively defined predicates on the left of sequents. We show this system
to be sound and cut-free complete with respect to a natural class of Henkin models. As a
corollary, we obtain cut-admissibility for LKID.
The well-known method of infinite descent `a la Fermat, which exploits the fact that there are
no infinite descending chains of elements of well-ordered sets, provides an alternative approach
to reasoning with inductively defined relations. Our second proof system LKIDw formalises
this approach. In this system, the left-introduction rules for formulas involving inductively
defined predicates are not induction rules but simple case distinction rules, and an infinitary,
global soundness condition on proof trees ā formulated in terms of ātracesā on infinite paths
in the tree ā is required to ensure soundness. This condition essentially ensures that, for
every infinite branch in the proof, there is an inductive definition that is unfolded infinitely
often along the branch. By an infinite descent argument based upon the well-foundedness of
inductive definitions, the infinite branches of the proof can thus be disregarded, whence the
remaining portion of proof is well-founded and hence sound. We show this system to be cutfree
complete with respect to standard models, and again infer the admissibility of cut.
The infinitary system LKIDw is unsuitable for formal reasoning. However, it has a natural
restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs.
This restricted ācyclicā proof system, CLKIDw, is suitable for formal reasoning since proofs
have finite representations and the soundness condition on proofs is thus decidable.
We show how the formulation of our systems LKIDw and CLKIDw can be generalised to
obtain soundness conditions for a general class of infinite proof systems and their corresponding
cyclic restrictions. We provide machinery for manipulating and analysing the structure of
proofs in these essentially arbitrary cyclic systems, based primarily on viewing them as generating
regular infinite trees, and we show that any proof can be converted into an equivalent
proof with a restricted cycle structure. For proofs in this ācycle normal formā, a finitary, localised soundness condition exists that is strictly stronger than the general, infinitary soundness
condition, but provides more explicit information about the proof.
Finally, returning to the specific setting of our systems for inductive definitions, we show
that any LKID proof can be transformed into a CLKIDw proof (that, in fact, satisfies the finitary
soundness condition). We conjecture that the two systems are in fact equivalent, i.e. that proof
by induction is equivalent to regular proof by infinite descent
Homogeneity and omega-categoricity of semigroups
In this thesis we study problems in the theory of semigroups which arise from model theoretic notions. Our focus will be on omega-categoricity and homogeneity of semigroups, a common feature of both of these properties being symmetricity. A structure is homogeneous if every local symmetry can be extended to a global symmetry, and as such it will have a rich automorphism group.
On the other hand, the Ryll-Nardzewski Theorem dictates that omega-categorical structures have oligomorphic automorphism groups.
Numerous authors have investigated the homogeneity and omega-categoricity of algebras including groups, rings, and of relational structures such as graphs and posets. The central aim of this thesis is to forge a new path through the model theory of semigroups.
The main body of this thesis is split into two parts. The first is an exploration into omega-categoricity of semigroups. We follow the usual semigroup theoretic method of analysing Green's relations on an omega-categorical semigroup, and prove a finiteness condition on their classes. This work motivates a generalization of characteristic subsemigroups, and subsemigroups of this form are shown to inherit omega-categoricity. We also explore methods for building omega-categorical semigroups from given omega-categorical structures.
In the second part we study the homogeneity of certain classes of semigroups, with an emphasis on completely regular semigroups. A complete description of all homogeneous bands is achieved, which shows them to be regular bands with homogeneous structure semilattices. We also obtain a partial classification of homogeneous inverse semigroups. A complete description can be given in a number of cases, including inverse semigroups with finite maximal subgroups, and periodic commutative inverse semigroups. These results extend the classification of homogeneous semilattices by Droste, Truss, and Kuske. We pose a number of open problems, that we believe will open up a rich subsequent stream of research