arXiv subjects
Publications and source records attributed to Philipp Hieronymi.
Let $\mathcal{R}$ be an expansion of the real field such that every subset of $\mathbb{R}$ definable in $\mathcal{R}$ either has interior or is a finite union of discrete sets. Answering a question by Chris Miller, we show that for every $n\in \mathbb{N}$ and every definable subset $A\subseteq \mathbb{R}^{n+1}$ there is $N\in \mathbb{N}$ such that for all $x\in \mathbb{R}^n$ either $A_x$ has interior or is the union of $N$ discrete sets.
We give a complete first-order axiomatization of the structure $(\mathbb{Z},+,(\ell^{\mathbb{N}})_{\ell\in L})$, where $L \subseteq \mathbb{Z}_{\ge 2}$ is a set of pairwise multiplicatively independent integers and $\ell^{\mathbb{N}} = \{\ell^n : n\in \mathbb{N}\}$. Using recent work of Karimov et al., we obtain that this axiomatization is computable for $|L|=2$, which proves that $(\mathbb{Z},+,k^{\mathbb{N}}, \ell^{\mathbb{N}})$ is decidable for $k, \ell\in \mathbb{Z}_{\ge 2}$. Furthermore, we give an axiomatization of the universal theory of $(\mathbb{Z},+,<,(\ell^{\mathbb{N}})_{\ell\in L})$.
We show that the first-order theory of Sturmian words over Presburger arithmetic is decidable. Using a general adder recognizing addition in Ostrowski numeration systems by Baranwal, Schaeffer and Shallit, we prove that the first-order expansions of Presburger arithmetic by a single Sturmian word are uniformly $ω$-automatic, and then deduce the decidability of the theory of the class of such structures. Using an implementation of this decision algorithm called Pecan, we automatically reprove classical theorems about Sturmian words in seconds, and are able to obtain new results about antisquares and antipalindromes in characteristic Sturmian words.
Let $α,β\in \mathbb{R}_{>0}$ be such that $α,β$ are quadratic and $\mathbb{Q}(α)\neq \mathbb{Q}(β)$. Then every subset of $\mathbb{R}^n$ definable in both $(\mathbb{R},{<},+,\mathbb{Z},x\mapsto αx)$ and $(\mathbb{R},{<},+,\mathbb{Z},x\mapsto βx)$ is already definable in $(\mathbb{R},{<},+,\mathbb{Z})$. As a consequence we generalize Cobham-Semenov theorems for sets of real numbers to $β$-numeration systems, where $β$ is a quadratic irrational.
We show that if $X$ is virtually any classical fractal subset of $\mathbb{R}^n$, then $(\mathbb{R},<,+,X)$ interprets the monadic second-order theory of $(\mathbb{N},+1)$. This result is sharp in the sense that the standard model of the monadic second-order theory of $(\mathbb{N},+1)$ is known to interpret $(\mathbb{R},<,+,X)$ for various classical fractals $X$ including the middle-thirds Cantor set and the Sierpinski carpet. Let $X \subseteq \mathbb{R}^n$ be closed and nonempty. We show that if the $C^k$-smooth points of $X$ are not dense in $X$ for some $k \geq 1$, then $(\mathbb{R},<,+,X)$ interprets the monadic second-order theory of $(\mathbb{N},+1)$. The same conclusion holds if the packing dimension of $X$ is strictly greater than the topological dimension of $X$ and $X$ has no affine points.
Let $k,\ell\geq 2$ be two multiplicatively independent integers. Cobham's famous theorem states that a set $X\subseteq \mathbb{N}$ is both $k$-recognizable and $\ell$-recognizable if and only if it is definable in Presburger arithmetic. Here we show the following strengthening: let $X\subseteq \mathbb{N}^m$ be $k$-recognizable, let $Y\subseteq \mathbb{N}^n$ be $\ell$-recognizable such that both $X$ and $Y$ are not definable in Presburger arithmetic. Then the first-order logical theory of $(\mathbb{N},+,X,Y)$ is undecidable. This is in contrast to a well-known theorem of Büchi that the first-order logical theory of $(\mathbb{N},+,X)$ is decidable.
We consider Presburger arithmetic extended by the sine function, call this extension sine-Presburger arithmetic ($\sin$-PA), and systematically study decision problems for sets of sentences in $\sin$-PA. In particular, we detail a decision algorithm for existential $\sin$-PA sentences under assumption of Schanuel's conjecture. This procedure reduces decisions to the theory of the ordered additive group of real numbers extended by sine, which is decidable under Schanuel's conjecture. On the other hand, we prove that four alternating quantifier blocks suffice for undecidability of $\sin$-PA sentences. To do so, we explicitly interpret the weak monadic second-order theory of the grid, which is undecidable, in $\sin$-PA.
We consider Presburger arithmetic (PA) extended by scalar multiplication by an algebraic irrational number $α$, and call this extension $α$-Presburger arithmetic ($α$-PA). We show that the complexity of deciding sentences in $α$-PA is substantially harder than in PA. Indeed, when $α$ is quadratic and $r\geq 4$, deciding $α$-PA sentences with $r$ alternating quantifier blocks and at most $c\ r$ variables and inequalities requires space at least $K 2^{\cdot^{\cdot^{\cdot^{2^{C\ell(S)}}}}}$ (tower of height $r-3$), where the constants $c, K, C>0$ only depend on $α$, and $\ell(S)$ is the length of the given $α$-PA sentence $S$. Furthermore deciding $\exists^{6}\forall^{4}\exists^{11}$ $α$-PA sentences with at most $k$ inequalities is PSPACE-hard, where $k$ is another constant depending only on~$α$. When $α$ is non-quadratic, already four alternating quantifier blocks suffice for undecidability of $α$-PA sentences.
Let $\mathcal{R}$ be an expansion of the ordered real additive group. When $\mathcal{R}$ is o-minimal, it is known that either $\mathcal{R}$ defines an ordered field isomorphic to $(\mathbb{R},<,+,\cdot)$ on some open subinterval $I\subseteq \mathbb{R}$, or $\mathcal{R}$ is a reduct of an ordered vector space. We say $\mathcal{R}$ is field-type if it satisfies the former condition. In this paper, we prove a more general result for arbitrary expansions of $(\mathbb{R},<,+)$. In particular, we show that for expansions that do not define dense $ω$-orders (we call these type A expansions), an appropriate version of Zilber's principle holds. Among other things we conclude that in a type A expansion that is not field-type, every continuous definable function $[0,1]^m \to \mathbb{R}^n$ is locally affine outside a nowhere dense set.
Pecan is an automated theorem prover for reasoning about properties of Sturmian words, an important object in the field of combinatorics on words. It is capable of efficiently proving non-trivial mathematical theorems about all Sturmian words.
This paper answers several open questions around structures with o-minimal open core. We construct an expansion of an o-minimal structure $\mathcal{R}$ by a unary predicate such that its open core is a proper o-minimal expansion of $\mathcal{R}$. We give an example of a structure that has an o-minimal open core and the exchange property, yet defines a function whose graph is dense. Finally, we produce an example of a structure that has an o-minimal open core and definable Skolem functions, but is not o-minimal.
Following Chaudhuri, Sankaranarayanan, and Vardi, we say that a function $f:[0,1] \to [0,1]$ is $r$-regular if there is a Büchi automaton that accepts precisely the set of base $r \in \mathbb{N}$ representations of elements of the graph of $f$. We show that a continuous $r$-regular function $f$ is locally affine away from a nowhere dense, Lebesgue null, subset of $[0,1]$. As a corollary we establish that every differentiable $r$-regular function is affine. It follows that checking whether an $r$-regular function is differentiable is in $\operatorname{PSPACE}$. Our proofs rely crucially on connections between automata theory and metric geometry developed by Charlier, Leroy, and Rigo.
This paper proposes a new setup for studying pairs of structures. This new framework includes many of the previously studied classes of pairs, such as dense pairs of o-minimal structures, lovely pairs, fields with Mann groups, and $H$-structures, but also includes new ones, such as pairs consisting of a real closed field and a pseudo real closed subfield, and pairs of vector spaces with different fields of scalars. We use the larger generality of this framework to answer three concrete open questions raised in earlier work on this subject.
We study sets and groups definable in tame expansions of o-minimal structures. Let $\mathcal {\widetilde M}= \langle \mathcal M, P\rangle$ be an expansion of an o-minimal $\mathcal L$-structure $\cal M$ by a dense set $P$, such that three tameness conditions hold. We prove a structure theorem for definable sets and functions in analogy with the influential cell decomposition theorem known for o-minimal structures. The structure theorem advances the state-of-the-art in all known examples of $\mathcal {\widetilde M}$, as it achieves a decomposition of definable sets into \emph{unions} of `cones', instead of only boolean combinations of them. We also develop the right dimension theory in the tame setting. Applications include: (i) the dimension of a definable set coincides with a suitable pregeometric dimension, and it is invariant under definable bijections, (ii) every definable map is given by an $\cal L$-definable map off a subset of its domain of smaller dimension, and (iii) around generic elements of a definable group, the group operation is given by an $\cal L$-definable map.
Let $K$ be a subfield of $\mathbb{R}$. The theory of $\mathbb{R}$ viewed as an ordered $K$-vector space and expanded by a predicate for $\mathbb{Z}$ is decidable if and only if $K$ is a real quadratic field.
Let $Γ$ be an infinite discrete subgroup of Gl$_n(\mathbb{C})$. Then either $(\mathbb{R}, <, +, \cdot, Γ)$ is interdefinable with $(\mathbb{R}, <, +, \cdot, λ^\mathbb{Z})$ for some $λ\in \mathbb{R}$, or $(\mathbb{R}, < , +, \cdot, Γ)$ defines the set of integers. When $Γ$ is not virtually abelian, the second case holds.
Let $T$ be a consistent o-minimal theory extending the theory of densely ordered groups and let $T'$ be a consistent theory. Then there is a complete theory $T^*$ extending $T$ such that $T$ is an open core of $T^*$, but every model of $T^*$ interprets a model of $T'$. If $T'$ is NIP, $T^*$ can be chosen to be NIP as well. From this we deduce the existence of an NIP expansion of the real field that has no distal expansion.
We establish the choice property, a weak analogue of definable choice, for certain tame expansions of o-minimal structures. Most noteworthily, dense pairs of real closed fields have this property.