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Uri Andrews

Publications and source records attributed to Uri Andrews.

At least 19 recordsLinked to original sources
arXiv2026

Infinite Belligerent Jump Inversion and Computable Scott Analysis

Scott analysis provides two fundamental tools for studying countable structures: Scott sentences, which characterize structures up to isomorphism, and back-and-forth relations, which measure structural similarity. A recurring phenomenon in computable structure theory is that many notions naturally associated with level $\alpha$ of Scott analysis have effective complexity at approximately $2\alpha$ jumps. This discrepancy appears both in the complexity of the back-and-forth relations and in the passage from arbitrary infinitary formulas to computable infinitary formulas. We develop two new coding tools, the Belligerent Pairs Theorem and Belligerent Jump Inversion Theorem, which allow information at complexity level $2\alpha$ to be reflected in computable structures whose distinguishing features already appear at level $\alpha$. These results extend Harrison-Trainor's finite unfriendly jump inversion uniformly throughout the computable ordinals. As applications, we determine the optimal interaction between syntactic complexity and oracle complexity for computable Scott sentences and for formulas distinguishing computable structures. For every computable infinite ordinal $\alpha$, we determine the oracle needed to compute a $\Pi_\alpha$ Scott sentence for a computable structure which has a $\Pi_\alpha$ Scott sentence. Any computable structure with a $\Pi_\alpha$ Scott sentence has a computable $\Pi_{2\alpha}$ Scott sentence. We show that both of these bounds are sharp. We prove analogous optimal results for formulas witnessing failure of the $\alpha$-back-and-forth relation. We also obtain further applications, including a resolution of a question of Chen, Gonzalez, and Harrison-Trainor concerning the complexity of back-and-forth classes.

math.LO↗
arXiv2026

Tenability and Weak Semantics: Modeling Non-uniform Defense -- Extended Version

In Dung-style abstract argumentation, various semantics capture notions of acceptability of arguments. The admissibility semantics capture the notion that an argument can be consistently defended from any potential counterargument. Weak semantics often relax the demands of admissibility by restricting which counterarguments must be taken seriously (e.g., discounting self-defeating or otherwise incoherent attacks). Many prominent proposals for weak semantics remain extension-based in a stronger sense. While these semantics discount attacks from arguments which are considered unreasonable, they still require a uniform defense against all reasonable arguments, even if they are collectively inconsistent. This uniformity can be too demanding when defensibility is inherently strategic, and thus the appropriate reply depends on the opponent's line of attack. We introduce tenability, a family of dialogue-based semantics that formalize when a designated argument (or a set of arguments) can be maintained in debate by a proponent against any conflict-free attack which the opponent may present. The approach is motivated by three natural benchmark patterns: self-defeating attack, floating assignment, and disjunctive reinstatement, on which tenability behaves differently from all weak semantics previously considered in the literature. We define three variants -- static tenability, tenability, and strong tenability -- via monotone commitment games over finite conflict-free moves, differing in the obligations imposed on the disputants. We establish the relative strength of these notions, prove implications and separations with previously studied weak semantics, and we analyze computational complexity on finite frameworks: deciding static tenability is $\Pi^P_2$-complete, while deciding tenability and strong tenability is PSPACE-complete.

cs.AI↗
arXiv2025

On the Complexity of the Grounded Semantics for Infinite Argumentation Frameworks

Argumentation frameworks, consisting of arguments and an attack relation representing conflicts, are fundamental for formally studying reasoning under conflicting information. We use methods from mathematical logic, specifically computability and set theory, to analyze the grounded extension, a widely-used model of maximally skeptical reasoning, defined as the least fixed-point of a natural defense operator. Without additional constraints, finding this fixed-point requires transfinite iterations. We identify the exact ordinal number corresponding to the length of this iterative process and determine the complexity of deciding grounded acceptance, showing it to be maximally complex. This shows a marked distinction from the finite case where the grounded extension is polynomial-time computable, thus simpler than other reasoning problems explored in formal argumentation.

cs.AI↗
arXiv2025

Complexity in finitary argumentation (extended version)

Abstract argumentation frameworks (AFs) provide a formal setting to analyze many forms of reasoning with conflicting information. While the expressiveness of general infinite AFs make them a tempting tool for modeling many kinds of reasoning scenarios, the computational intractability of solving infinite AFs limit their use, even in many theoretical applications. We investigate the complexity of computational problems related to infinite but finitary argumentations frameworks, that is, infinite AFs where each argument is attacked by only finitely many others. Our results reveal a surprising scenario. On one hand, we see that the assumption of being finitary does not automatically guarantee a drop in complexity. However, for the admissibility-based semantics, we find a remarkable combinatorial constraint which entails a dramatic decrease in complexity. We conclude that for many forms of reasoning, the finitary infinite AFs provide a natural setting for reasoning which balances well the competing goals of being expressive enough to be applied to many reasoning settings while being computationally tractable enough for the analysis within the framework to be useful.

cs.AI↗
arXiv2025

Comparing Dialectical Systems: Contradiction and Counterexample in Belief Change (Extended Version)

Dialectical systems are a mathematical formalism for modeling an agent updating a knowledge base seeking consistency. Introduced in the 1970s by Roberto Magari, they were originally conceived to capture how a working mathematician or a research community refines beliefs in the pursuit of truth. Dialectical systems also serve as natural models for the belief change of an automated agent, offering a unifying, computable framework for dynamic belief management. The literature distinguishes three main models of dialectical systems: (d-)dialectical systems based on revising beliefs when they are seen to be inconsistent, p-dialectical systems based on revising beliefs based on finding a counterexample, and q-dialectical systems which can do both. We answer an open problem in the literature by proving that q-dialectical systems are strictly more powerful than p-dialectical systems, which are themselves known to be strictly stronger than (d-)dialectical systems. This result highlights the complementary roles of counterexample and contradiction in automated belief revision, and thus also in the reasoning processes of mathematicians and research communities.

cs.AI↗
arXiv2025

SCC-recursiveness in infinite argumentation (extended version)

Argumentation frameworks (AFs) are a foundational tool in artificial intelligence for modeling structured reasoning and conflict. SCC-recursiveness is a well-known design principle in which the evaluation of arguments is decomposed according to the strongly connected components (SCCs) of the attack graph, proceeding recursively from "higher" to "lower" components. While SCC-recursive semantics such as \cft and \stgt have proven effective for finite AFs, Baumann and Spanring showed the failure of SCC-recursive semantics to generalize reliably to infinite AFs due to issues with well-foundedness. We propose two approaches to extending SCC-recursiveness to the infinite setting. We systematically evaluate these semantics using Baroni and Giacomin's established criteria, showing in particular that directionality fails in general. We then examine these semantics' behavior in finitary frameworks, where we find some of our semantics satisfy directionality. These results advance the theory of infinite argumentation and lay the groundwork for reasoning systems capable of handling unbounded or evolving domains.

cs.AI↗
arXiv2024

Analogues of the countable Borel equivalence relations in the setting of computable reducibility

Coskey, Hamkins, and Miller [CHM12] proposed two possible analogues of the class of countable Borel equivalence relations in the setting of computable reducibility of equivalence relations on the computably enumerable (c.e.) sets. The first is based on effectivizing the Lusin-Novikov theorem while the latter is based on effectivizing the Feldman-Moore theorem. They asked for an analysis of which degrees under computable reducibility are attained under each of these notions. We investigate these two notions, in particular showing that the latter notion has a strict dichotomy theorem: Every such equivalence relation is either equivalent to the relation of equality ($=^{ce}$) or almost equality ($E_0^{ce}$) between c.e. sets. For the former notion, we show that this is not true, but rather there are both chains and antichains of such equivalence relations on c.e. sets which are between $=^{ce}$ and $E_0^{ce}$. This gives several strong answers to [CHM12, Question 3.5] showing that in general there is no analogue of the Glimm-Efros dichotomy for equivalence relations on the c.e. sets.

math.LO↗
arXiv2024

Two results on complexities of decision problems of groups

We answer two questions on the complexities of decision problems of groups, each related to a classical result. First, C. Miller characterized the complexity of the isomorphism problem for finitely presented groups in 1971. We do the same for the isomorphism problem for recursively presented groups. Second, the fact that every Turing degree appears as the degree of the word problem of a finitely presented group is shown independently by multiple people in the 1960s. We answer the analogous question for degrees of ceers instead of Turing degrees. We show that the set of ceers which are computably equivalent to a finitely presented group is $Σ^0_3$-complete, which is the maximal possible complexity.

math.LO↗
arXiv2024

A jump operator on the Weihrauch degrees

A partial order $(P,\le)$ admits a jump operator if there is a map $j\colon P \to P$ that is strictly increasing and weakly monotone. Despite its name, the jump in the Weihrauch lattice fails to satisfy both of these properties: it is not degree-theoretic and there are functions $f$ such that $f\equiv_{\mathrm{W}} f'$. This raises the question: is there a jump operator in the Weihrauch lattice? We answer this question positively and provide an explicit definition for an operator on partial multi-valued functions that, when lifted to the Weihrauch degrees, induces a jump operator. This new operator, called the totalizing jump, can be characterized in terms of the total continuation, a well-known operator on computational problems. The totalizing jump induces an injective endomorphism of the Weihrauch degrees. We study some algebraic properties of the totalizing jump and characterize its behavior on some pivotal problems in the Weihrauch lattice.

math.LO↗
arXiv2024

The Borel complexity of the class of models of first-order theories

We investigate the descriptive complexity of the set of models of first-order theories. Using classical results of Knight and Solovay, we give a sharp condition for complete theories to have a $\pmb\Pi_\omega^0$-complete set of models. In particular, any sequential theory (a class of foundational theories isolated by Pudl\'ak) has a $\pmb\Pi_\omega^0$-complete set of models. We also give sharp conditions for theories to have a $\pmb\Pi^0_n$-complete set of models.

math.LO↗
arXiv2024

Algorithmically finite, universal, and $*$-universal groups

The study of the word problems of groups dates back to Dehn in 1911, and has been a central topic of study in both group theory and computability theory. As most naturally occurring presentations of groups are recursive, their word problems can be thought of as a computably enumerable equivalence relation (ceer). In this paper, we study the word problem of groups in the framework of ceer degrees, introducing a new metric with which to study word problems. This metric is more refined than the classical context of Turing degrees. Classically, every Turing degree is realized as the word problem of some c.e. group, but this is not true for ceer degrees. This motivates us to look at the classical constructions and show that there is a group whose word problem is not universal, but becomes universal after taking any nontrivial free product, which we call $*$-universal. This shows that existing constructions of the Higman embedding theorem do not preserve ceer degrees. We also study the index set of various classes of groups defined by their properties as a ceer: groups whose word problems are dark (equivalently, algorithmically finite as defined by Miasnikov and Osin), universal, and $*$-universal groups.

math.LO↗
arXiv2022

Investigating the computable Friedman-Stanley jump

We answer several questions about the computable Friedman-Stanley jump on equivalence relations. This jump, introduced by Clemens, Coskey, and Krakoff, deepens the natural connection between the study of computable reduction and its Borel analog studied deeply in descriptive set theory.

math.LO↗
arXiv2022

On the structure of computable reducibility on equivalence relations of natural numbers

We examine the degree structure $\mathbf{ER}$ of equivalence relations on $ω$ under computable reducibility. We examine when pairs of degrees have a join. In particular, we show that sufficiently incomparable pairs of degrees do not have a join but that some incomparable degrees do, and we characterize the degrees which have a join with every finite equivalence relation. We show that the natural classes of finite, light, and dark degrees are definable in $\mathbf{ER}$. We show that every equivalence relation has continuum many self-full strong minimal covers, and that $\mathbf{d}\oplus \mathbf{Id_1}$ needn't be a strong minimal cover of a self-full degree $\mathbf{d}$. Finally, we show that the theory of the degree structure $\mathbf{ER}$ as well as the theories of the substructures of light degrees and of dark degrees are each computably isomorphic with second order arithmetic.

math.LO↗
arXiv2021

Recursive spectra of flat strongly minimal theories

We show that for a model complete strongly minimal theory whose pregeometry is flat, the recursive spectrum (SRM($T$)) is either of the form $[0,α)$ for $α\in ω+2$ or $[0,n]\cup\{ω\}$ for $n\in ω$, or $\{ω\}$, or contained in $\{0,1,2\}$. Combined with previous results, this leaves precisely 4 sets for which it is not yet determined whether each is the spectrum of a model complete strongly minimal theory with a flat pregeometry.

math.LO↗
arXiv2020

The Theory of Ceers Computes True Arithmetic

We show that the theory of the partial order of computably enumerable equivalence relations (ceers) under computable reduction is 1-equivalent to true arithmetic. We show the same result for the structure comprised of the dark ceers and the structure comprised of the light ceers. We also show the same for the structure of $\mathcal{I}$-degrees in the dark, light, or complete structure. In each case, we show that there is an interpretable copy of $(\mathbb{N},+,\cdot)$.

math.LO↗
arXiv2020

Self-full ceers and the uniform join operator

A computably enumerable equivalence relation (ceer) $X$ is called self-full if whenever $f$ is a reduction of $X$ to $X$ then the range of $f$ intersects all $X$-equivalence classes. It is known that the infinite self-full ceers properly contain the dark ceers, i.e. the infinite ceers which do not admit an infinite computably enumerable transversal. Unlike the collection of dark ceers, which are closed under the operation of uniform join, we answer a question from \cite{joinmeet} by showing that there are self-full ceers $X$ and $Y$ so that their uniform join $X\oplus Y$ is non-self-full. We then define and examine the hereditarily self-full ceers, which are the self-full ceers $X$ so that for any self-full $Y$, $X\oplus Y$ is also self-full: we show that they are closed under uniform join, and that every non-universal degree in $\textrm{Ceers}_{/{\mathcal{I}}}$ have infinitely many incomparable hereditarily self-full strong minimal covers. In particular, every non-universal ceer is bounded by a hereditarily self-full ceer. Thus the hereditarily self-full ceers form a properly intermediate class in between the dark ceers and the infinite self-full ceers which is closed under $\oplus$.

math.LO↗
arXiv2019

Effective inseparability, lattices, and pre-ordering relations

We study effectively inseparable (e.i.) pre-lattices (i.e. structures of the form $L=\langle ω, \wedge, \lor, 0, 1, \leq_L\rangle$ where $ω$ denotes the set of natural numbers and the following hold: $\wedge, \lor$ are binary computable operations; $\leq_L$ is a c.e. pre-ordering relation, with $0 \leq_{L} x \leq_{L} 1$ for every $x$; the equivalence relation $\equiv_L$ originated by $\leq_L$ is a congruence on $L$ such that the corresponding quotient structure is a non-trivial bounded lattice; the $\equiv_L$-equivalence classes of $0$ and $1$ form an effectively inseparable pair), and show that if $L$ is an e.i. pre-lattice then $\le_{L}$ is universal with respect to all c.e. pre-ordering relations, i.e. for every c.e. pre-ordering relation $R$ there exists a computable function $f$ such that, for all $x,y$, $x \mathrel{R} y$ if and only if $f(x) \le_{L} f(y)$; in fact $\leq_L$ is locally universal, i.e. for every pair $a<_{L} b$ and every c.e. pre ordering relation $R$ one can find a reducing function $f$ from $R$ to $\le_{L}$ such that the range of $f$ is contained in the interval $\{x: a \leq_{L} x \leq_{L} b\}$. Also $\leq_L$ is uniformly dense, i.e. there exists a computable function $f$ such that for every $a,b$ if $a<_{L} b$ then $a<_{L} f(a,b) <_{L} b$, and if $a\equiv_{L} a'$ and $b \equiv_{L} b'$ then $f(a,b)\equiv_{L} f(a',b')$. Some consequences and applications of these results are discussed: in particular for $n \ge 1$ the c.e. pre-ordering relation on $Σ_{n}$ sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson's $Q$ or $R$ is locally universal and uniformly dense; and the c.e. pre-ordering relation of provable implication of Heyting Arithmetic is locally universal and uniformly dense.

math.LO↗