arXiv subjects

Erich Grädel

Publications and source records attributed to Erich Grädel.

At least 19 recordsLinked to original sources

arXiv2026

Preservation Theorems in Semiring Semantics

We study the status of preservation theorems such as the Łoś-Tarski theorem and the homomorphism preservation theorem in the context of semiring semantics. Semiring semantics has its origins in the provenance analysis of database queries. Depending on the underlying semiring, it allows us to track which atomic facts are responsible for the truth of a statement or practical information about the evaluation such as costs or confidence. The systematic development of semiring semantics for first-order logic and other logical systems raises the question to what extent classical model-theoretic results can be generalised to this setting and how such results depend on the underlying semiring. The definitions of semantic properties such as preservation under extensions, substructures, or homomorphisms naturally generalise to the setting of semiring semantics. However, the status of the corresponding preservation theorem strongly depends on the algebraic properties of the particular semirings. We prove that these preservation theorems do indeed hold for all lattice semirings (a quite large class, encompassing practically relevant semirings and in particular all min-max semirings). The proofs combine adaptations of the classical compactness and amalgamation methods with specific reduction methods for logical entailment that have been developed in semiring semantics. On the other side, variants of the existential preservation theorem fail for many other semirings, including the tropical semiring, the Viterbi semiring, the Łukasiewicz semiring, and the natural semiring. Surprisingly, the existential preservation theorem does hold for finite interpretations in a number of semirings, including all lattice semirings. Thus, the situation for these semirings is in sharp contrast to the Boolean case, where the Łoś-Tarski theorem holds in general, but not in the finite.

cs.LO↗

arXiv2025

Symmetric Proofs in the Ideal Proof System

We consider the Ideal Proof System (IPS) introduced by Grochow and Pitassi and pose the question of which tautologies admit symmetric proofs, and of what complexity. The symmetry requirement in proofs is inspired by recent work establishing lower bounds in other symmetric models of computation. We link the existence of symmetric IPS proofs to the expressive power of logics such as fixed-point logic with counting and Choiceless Polynomial Time, specifically regarding the graph isomorphism problem. We identify relationships and tradeoffs between the symmetry of proofs and other parameters of IPS proofs such as size, degree and linearity. We study these on a number of standard families of tautologies from proof complexity and finite model theory such as the pigeonhole principle, the subset sum problem and the Cai-Fürer-Immerman graphs, exhibiting non-trivial upper bounds on the size of symmetric IPS proofs.

cs.LO↗

arXiv2024

Provenance Analysis and Semiring Semantics for First-Order Logic

A provenance analysis for a query evaluation or a model checking computation extracts information on how its result depends on the atomic facts of the model or database. Traditional work on data provenance was, to a large extent, restricted to positive query languages or the negation-free fragment of first-order logic and showed how provenance abstractions can be usefully described as elements of commutative semirings -- most generally as multivariate polynomials with positive integer coefficients. We describe and evaluate here a provenance approach for dealing with negation, based on quotient semirings of polynomials with dual indeterminates. This not only provides a semiring provenance analysis for full first-order logic (and other logics and query languages with negation) but also permits a reverse provenance analysis, i.e., finding models that satisfy various properties under given provenance tracking assumptions. We describe the potential for applications to explaining missing query answers or failures of integrity constraints, and to using these explanations for computing repairs. This approach also is the basis of a systematic study of semiring semantics in a broad logical context.

cs.LO↗

arXiv2024

Locality Theorems in Semiring Semantics

Semiring semantics of first-order logic generalises classical Boolean semantics by permitting truth values from a commutative semiring, which can model information such as costs or access restrictions. This raises the question to what extent classical model theoretic properties still apply, and how this depends on the algebraic properties of the semiring. In this paper, we study this question for the classical locality theorems due to Hanf and Gaifman. We prove that Hanf's Locality Theorem generalises to all semirings with idempotent operations, but fails for many non-idempotent semirings. We then consider Gaifman normal forms and show that for formulae with free variables, Gaifman's Theorem does not generalise beyond the Boolean semiring. Also for sentences, it fails in the natural semiring and the tropical semiring. Our main result, however, is a constructive proof of the existence of Gaifman normal forms for min-max and lattice semirings. The proof implies a stronger version of Gaifman's classical theorem in Boolean semantics: every sentence has a Gaifman normal form which does not add negations.

cs.LO↗

arXiv2024

Semiring Provenance for Büchi Games: Strategy Analysis with Absorptive Polynomials

This paper presents a case study for the application of semiring semantics for fixed-point formulae to the analysis of strategies in Büchi games. Semiring semantics generalizes the classical Boolean semantics by permitting multiple truth values from certain semirings. Evaluating the fixed-point formula that defines the winning region in a given game in an appropriate semiring of polynomials provides not only the Boolean information on who wins, but also tells us how they win and which strategies they might use. This is well-understood for reachability games, where the winning region is definable as a least fixed point. The case of Büchi games is of special interest, not only due to their practical importance, but also because it is the simplest case where the fixed-point definition involves a genuine alternation of a greatest and a least fixed point. We show that, in a precise sense, semiring semantics provide information about all absorption-dominant strategies -- strategies that win with minimal effort, and we discuss how these relate to positional and the more general persistent strategies. This information enables applications such as game synthesis or determining minimal modifications to the game needed to change its outcome. Lastly, we discuss limitations of our approach and present questions that cannot be immediately answered by semiring semantics.

cs.LO↗

arXiv2023

Ehrenfeucht-Fraïssé Games in Semiring Semantics

Ehrenfeucht-Fraïssé games provide a fundamental method for proving elementary equivalence (and equivalence up to a certain quantifier rank) of relational structures. We investigate the soundness and completeness of this method in the more general context of semiring semantics. Motivated originally by provenance analysis of database queries, semiring semantics evaluates logical statements not just by true or false, but by values in some commutative semiring; this can provide much more detailed information, for instance concerning the combinations of atomic facts that imply the truth of a statement, or practical information about evaluation costs, confidence scores, access levels or the number of successful evaluation strategies. There is a wide variety of different semirings that are relevant for provenance analysis, and the applicability of classical logical methods in semiring semantics may strongly depend on the algebraic properties of the underlying semiring. While Ehrenfeucht-Fraïssé games are sound and complete for logical equivalences in classical semantics, and thus on the Boolean semiring, this is in general not the case for other semirings. We provide a detailed analysis of the soundness and completeness of model comparison games on specific semirings, not just for classical Ehrenfeucht-Fraïssé games but also for other variants based on bijections or counting. Finally we propose a new kind of games, called homomorphism games, which are based on the fact that there exist certain rather simple semiring interpretations that are locally very different and can be separated even in a one-move game, but which can be proved to be elementarily equivalent via separating sets of homomorphisms into the Boolean semiring. We prove that these homomorphism games provide a sound and complete method for logical equivalences on finite and infinite lattice semirings.

cs.LO↗

arXiv2022

Limitations of the Invertible-Map Equivalences

This note draws conclusions that arise by combining two recent papers, by Anuj Dawar, Erich Grädel, and Wied Pakusa, published at ICALP 2019 and by Moritz Lichter, published at LICS 2021. In both papers, the main technical results rely on the combinatorial and algebraic analysis of the invertible-map equivalences $\equiv^\text{IM}_{k,Q}$ on certain variants of Cai-Fürer-Immerman (CFI) structures. These $\equiv^\text{IM}_{k,Q}$-equivalences, for a number $k$ and a set of primes~$Q$, refine the well-known Weisfeiler-Leman equivalences used in algorithms for graph isomorphism. The intuition is that two graphs $G\equiv^\text{IM}_{k,Q}H$ cannot be distinguished by iterative refinements of equivalences on $k$-tuples defined via linear operators on vector spaces over fields of characteristic $p \in Q$. In the first paper it has been shown, using considerable algebraic machinery, that for a prime $q \notin Q$, the $\equiv^\text{IM}_{k,Q}$ equivalences do not distinguish between non-isomorphic CFI-structures over the field $\mathbb{F}_q$. In the second paper, a similar but not identical construction for CFI-structures over the rings $\mathbb{Z}_{2^i}$ has, again by rather involved combinatorial and algebraic arguments, been shown to be indistinguishable with respect to $\equiv^\text{IM}_{k,\{2\}}$. Together with earlier work, this second result separates rank logic from polynomial time. We unify the two approaches and prove that CFI-structures over the rings $\mathbb{Z}_{2^i}$ are indistinguishable with respect to $\equiv^\text{IM}_{k,\mathbb{P}}$, for the set $\mathbb{P}$ of all primes. This implies the following two results. 1. There is no fixed $k$ such that the invertible-map equivalence $\equiv^\text{IM}_{k,\mathbb{P}}$ coincides with isomorphism on all finite graphs. 2. No extension of fixed-point logic by linear-algebraic operators over fields can capture polynomial time.

cs.LO↗

arXiv2022

A Finite-Model-Theoretic View on Propositional Proof Complexity

We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded-width resolution, and the monomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded-width resolution captures existential least fixed-point logic, and that the polynomial calculus with bounded degree over the rationals solves precisely the problems definable in fixed-point logic with counting. We also study the bounded-degree polynomial calculus. Over the rationals, it captures fixed-point logic with counting if we restrict the bit-complexity of the coefficients. For unrestricted coefficients, we can only say that the bounded-degree polynomial calculus is at most as powerful as bounded variable infinitary counting logic, but a precise logical characterisation of its power remains an open problem. These connections between logics and proof systems allow us to establish finite-model-theoretic tools for proving lower bounds for the polynomial calculus over the rationals and also over finite fields. This is a corrected version of the paper (arXiv:1802.09377) published originally on January 23, 2019.

cs.LO↗

arXiv2022

Zero-One Laws and Almost Sure Valuations of First-Order Logic in Semiring Semantics

Semiring semantics evaluates logical statements by values in some commutative semiring K. Random semiring interpretations, induced by a probability distribution on K, generalise random structures, and we investigate here the question of how classical results on first-order logic on random structures, most importantly the 0-1 laws of Glebskii et al. and Fagin, generalise to semiring semantics. For positive semirings, the classical 0-1 law implies that every first-order sentence is, asymptotically, either almost surely evaluated to 0 by random semiring interpretations, or almost surely takes only values different from 0. However, by means of a more sophisticated analysis, based on appropriate extension properties and on algebraic representations of first-order formulae, we can prove much stronger results. For many semirings K, the first-order sentences can be partitioned into classes F(j) for all semiring values j in K, such that every sentence in F(j) evaluates almost surely to j under random semiring interpretations. Further, for finite or infinite lattice semirings, this partition actually collapses to just three classes F(0), F(1), and F(e), of sentences that, respectively, almost surely evaluate to 0, 1, and to the smallest non-zero value e. The problem of computing the almost sure valuation of a first-order sentence on finite lattice semirings is PSPACE-complete. An important semiring where the analysis is somewhat different is the semiring of natural numbers. Here, both addition and multiplication are increasing with respect to the natural semiring order and the classes F(j), for natural numbers j, no longer cover all FO-sentences, but have to be extended by the class of sentences that almost surely evaluate to unboundedly large values.

cs.LO↗

arXiv2022

Unifying Hidden-Variable Problems from Quantum Mechanics by Logics of Dependence and Independence

We study hidden-variable models from quantum mechanics, and their abstractions in purely probabilistic and relational frameworks, by means of logics of dependence and independence, based on team semantics. We show that common desirable properties of hidden-variable models can be defined in an elegant and concise way in dependence and independence logic. The relationship between different properties, and their simultaneous realisability can thus been formulated and a proved on a purely logical level, as problems of entailment and satisfiability of logical formulae. Connections between probabilistic and relational entailment in dependence and independence logic allow us to simplify proofs. In many cases, we can establish results on both probabilistic and relational hidden-variable models by a single proof, because one case implies the other, depending on purely syntactic criteria. We also discuss the no-go theorems by Bell and Kochen-Specker and provide a purely logical variant of the latter, introducing non-contextual choice as a team-semantical property.

cs.LO↗

arXiv2021

Semiring Provenance for Büchi Games: Strategy Analysis with Absorptive Polynomials

This paper presents a case study for the application of semiring semantics for fixed-point formulae to the analysis of strategies in Büchi games. Semiring semantics generalizes the classical Boolean semantics by permitting multiple truth values from certain semirings. Evaluating the fixed-point formula that defines the winning region in a given game in an appropriate semiring of polynomials provides not only the Boolean information on who wins, but also tells us how they win and which strategies they might use. This is well-understood for reachability games, where the winning region is definable as a least fixed point. The case of Bchi games is of special interest, not only due to their practical importance, but also because it is the simplest case where the fixed-point definition involves a genuine alternation of a greatest and a least fixed point. We show that, in a precise sense, semiring semantics provide information about all absorption-dominant strategies - strategies that win with minimal effort, and we discuss how these relate to positional and the more general persistent strategies. This information enables applications such as game synthesis or determining minimal modifications to the game needed to change its outcome.

cs.LO↗

arXiv2021

Logics of Dependence and Independence: The Local Variants

Modern logics of dependence and independence are based on team semantics, which means that formulae are evaluated not on a single assignment of values to variables, but on a set of such assignments, called a team. This leads to high expressive power, on the level of existential second-order logic. As an alternative, Baltag and van Benthem have proposed a local variant of dependence logic, called logic of functional dependence (LFD). While its semantics is also based on a team, the formulae are evaluated locally on just one of its assignments, and the team just serves as the supply of the possible assignments that are taken into account in the evaluation process. This logic thus relies on the modal perspective of generalized assignments semantics, and can be seen as a fragment of first-order logic. For the variant of LFD without equality, the satisfiability problem is decidable. We extend the idea of localising logics of dependence and independence in a systematic way, taking into account local variants of standard atomic dependency properties: besides dependence and independence, also inclusion, exclusion, and anonymity. We study model-theoretic and algorithmic questions of the localised logics, and also resolve some of the questions that had been left open by Baltag and van Benthem. In particular, we study decidability issues of the local logics, and prove that satisfiability of LFD with equality is undecidable. Further, we establish characterisation theorems via appropriate notions of bisimulation and study the complexity of model checking problems for these logics.

cs.LO↗

arXiv2021

Elementary equivalence versus isomorphism in semiring semantics

We study the first-order axiomatisability of finite semiring interpretations or, equivalently, the question whether elementary equivalence and isomorphism coincide for valuations of atomic facts over a finite universe into a commutative semiring. Contrary to the classical case of Boolean semantics, where every finite structure can obviously be axiomatised up to isomorphism by a first-order sentence, the situation in semiring semantics is rather different, and strongly depends on the underlying semiring. We prove that for a number of important semirings, including min-max semirings, and the semirings of positive Boolean expressions, there exist finite semiring interpretations that are elementarily equivalent but not isomorphic. The same is true for the polynomial semirings that are universal for the classes of absorptive, idempotent, and fully idempotent semirings, respectively. On the other side, we prove that for other, practically relevant, semirings such as the Viterby semiring, the tropical semiring, the natural semiring and the universal polynomial semiring N[X], all finite semiring interpretations are first-order axiomatisable (and thus elementary equivalence implies isomorphism), although some of the axiomatisations that we exhibit use an infinite set of axioms.

math.LO↗

arXiv2020

Logics with Multiteam Semantics

Team semantics is the mathematical basis of modern logics of dependence and independence. In contrast to classical Tarski semantics, a formula is evaluated not for a single assignment of values to the free variables, but on a set of such assignments, called a team. Team semantics is appropriate for a purely logical understanding of dependency notions, where only the presence or absence of data matters, but based on sets, it does not take into account multiple occurrences of data values. It is therefore insufficient in scenarios where such multiplicities matter, in particular for reasoning about probabilities and statistical independencies. Therefore, an extension from teams to multiteams (i.e. multisets of assignments) has been proposed by several authors. In this paper we aim at a systematic development of logics of dependence and independence based on multiteam semantics. We study atomic dependency properties of finite multiteams and discuss the appropriate meaning of logical operators to extend the atomic dependencies to full-fledged logics for reasoning about dependence properties in a multiteam setting. We explore properties and expressive power of a wide spectrum of different multiteam logics and compare them to logics with team semantics. We also study the relationship of logics with multiteam semantics with existential second-order logic for a specific class of metafinite structures. It turns out that inclusion-exclusion logic can be characterised in a precise sense by the Presburger fragment of this logic, but for capturing independence, we need to go beyond it and add some form of multiplication. Finally we also consider multiteams with weights in the reals and study the expressive power of formulae by means of topological properties.

cs.LO↗

arXiv2020

Generalized Absorptive Polynomials and Provenance Semantics for Fixed-Point Logic

Semiring provenance is a successful approach to provide detailed information on the combinations of atomic facts that are responsible for the result of a query. In particular, interpretations in general provenance semirings of polynomials or formal power series give precise descriptions of the successful evaluation strategies for the query. While provenance analysis in databases has, for a long time, been largely confined to negation-free query languages, a recent approach extends this to model checking problems for logics with full negation. Algebraically this relies on new quotient semirings of dual-indeterminate polynomials or power series. So far, this approach has been developed mainly for first-order logic and for the positive fragment of least fixed-point logic. What has remained open is an adequate treatment for fixed-point calculi that admit arbitrary interleavings of least and greatest fixed points. We show that an adequate framework for the provenance analysis of full fixed-point logics is provided by semirings that are (1) fully continuous, (2) absorptive, and (3) chain-positive. Full continuity guarantees that provenance values of least and greatest fixed-points are well-defined. Absorptive semirings provide a symmetry between least and greatest fixed-point computations and make sure that provenance values of greatest fixed points are informative. Finally, chain-positivity is responsible for having truth-preserving interpretations, which give non-zero values to all true formulae. We further identify semirings of generalized absorptive polynomials and prove universal properties that make them the most general appropriate semirings for LFP. We illustrate the power of provenance interpretations in these semirings by relating them to provenance values of plays and strategies in the associated model-checking games.

cs.LO↗

arXiv2020

Provenance Analysis for Logic and Games

A model checking computation checks whether a given logical sentence is true in a given finite structure. Provenance analysis abstracts from such a computation mathematical information on how the result depends on the atomic data that describe the structure. In database theory, provenance analysis by interpretations in commutative semirings has been rather succesful for positive query languages (such a unions of conjunctive queries, positive relational algebra, or datalog). However, it did not really offer an adequate treatment of negation or missing information. Here we propose a new approach for the provenance analysis of logics with negation, such as first-order logic and fixed-point logics. It is closely related to a provenance analysis of the associated model-checking games, and based on new semirings of dual-indeterminate polynomials or dual-indeterminate formal power series. These are obtained by taking quotients of traditional provenance semirings by congruences that are generated by products of positive and negative provenance tokens. Beyond the use for model-checking problems in logics, provenance analysis of games is of independent interest. Provenance values in games provide detailed information about the number and properties of the strategies of the players, far beyond the question whether or not a player has a winning strategy from a given position.

cs.LO↗

arXiv2019

Approximations of Isomorphism and Logics with Linear-Algebraic Operators

Invertible map equivalences are approximations of graph isomorphism that refine the well-known Weisfeiler-Leman method. They are parametrised by a number k and a set Q of primes. The intuition is that two graphs G and H which are equivalent with respect to k-Q-IM-equivalence cannot be distinguished by a refinement of k-tuples given by linear operators acting on vector spaces over fields of characteristic p, for any p in Q. These equivalences first appeared in the study of rank logic, but in fact they can be used to delimit the expressive power of any extension of fixed-point logic with linear-algebraic operators. We define an infinitary logic with k variables and all linear-algebraic operators over finite vector spaces of characteristic p in Q and show that the k-Q-IM-equivalence is the natural notion of elementary equivalence for this logic. By means of a new and much deeper algebraic analysis of a generalized variant, for any prime p, of the CFI-structures due to Cai, Fürer, and Immerman, we prove that, as long as Q is not the set of all primes, there is no k such that k-Q-IM-equivalence is the same as isomorphism. It follows that there are polynomial-time properties of graphs which are not definable in the infinitary logic with all Q-linear-algebraic operators and finitely many variables, which implies that no extension of fixed-point logic with linear-algebraic operators can capture PTIME, unless it includes such operators for all prime characteristics. Our analysis requires substantial algebraic machinery, including a homogeneity property of CFI-structures and Maschke's Theorem, an important result from the representation theory of finite groups.

cs.LO↗

arXiv2017

Semiring Provenance for First-Order Model Checking

Given a first-order sentence, a model-checking computation tests whether the sentence holds true in a given finite structure. Data provenance extracts from this computation an abstraction of the manner in which its result depends on the data items that describe the model. Previous work on provenance was, to a large extent, restricted to the negation-free fragment of first-order logic and showed how provenance abstractions can be usefully described as elements of commutative semirings --- most generally as multivariate polynomials with positive integer coefficients. In this paper we introduce a novel approach to dealing with negation and a corresponding commutative semiring of polynomials with dual indeterminates. These polynomials are used to perform reverse provenance analysis, i.e., finding models that satisfy various properties under given provenance tracking assumptions.

cs.LO↗