Showing papers by "Erich Grädel published in 2021"

Semiring Provenance for Guarded Logics

Abstract: Provenance analysis aims at understanding how the result of a computational process with a complex input, consisting of multiple items, depends on the various parts of this input. Here we investigate this for the model checking problem of guarded logics on finite relational structures. Semiring provenance was originally developed for positive database query languages, to understand which combinations of the atomic facts in a database can be used for computing the result of a given query. Based on interpretations of the atomic facts not just by true or false, but by values in an appropriate semiring, one can then answer questions such as the minimal cost of a query evaluation, the confidence one can have that the result is true, or the clearance level that is required for obtaining the output. Semiring provenance was recently extended by Gradel and Tannen to logics with negation, notably first-order logic, dealing with negation by transformation into negation normal form and by semirings of polynomials with a duality on the indeterminates. Here we develop this approach further for the guarded fragment (GF), introduced by Andreka, van Benthem and Nemeti, based on an analysis of the associated model checking games. Guarded quantification permits to control the complexity of the semiring computations since once has to take sums or products only over those tuples of elements that appear in the guards. Finally, we extend our provenance analysis to the more powerful guarded negation fragment of first-order logics.

Semiring Provenance for Fixed-Point Logic

Abstract: Semiring provenance is a successful approach, originating in database theory, to providing detailed information on how atomic facts combine to yield the result of a query In particular, general provenance semirings of polynomials or formal power series provide precise descriptions of the evaluation strategies or "proof trees" for the query By evaluating these descriptions in specific application semirings, one can extract practical information for instance about the confidence of a query or the cost of its evaluation This paper develops semiring provenance for very general logical languages featuring the full interaction between negation and fixed-point inductions or, equivalently, arbitrary interleavings of least and greatest fixed points This also opens the door to provenance analysis applications for modal μ-calculus and temporal logics, as well as for finite and infinite model-checking games Interestingly, the common approach based on Kleene’s Fixed-Point Theorem for ω-continuous semirings is not sufficient for these general languages We show that an adequate framework for the provenance analysis of full fixed-point logics is provided by semirings that are (1) fully continuous, and (2) absorptive 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-points and make sure that provenance values of greatest fixed points are informative We identify semirings of generalized absorptive polynomials S^{∞}[X] and prove universal properties that make them the most general appropriate semirings for our framework These semirings have the further property of being (3) chain-positive, which is responsible for having truth-preserving interpretations that give non-zero values to all true formulae We relate the provenance analysis of fixed-point formulae with provenance values of plays and strategies in the associated model-checking games Specifically, we prove that the provenance value of a fixed point formula gives precise information on the evaluation strategies in these games

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

Abstract: 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.

Elementary equivalence versus isomorphism in semiring semantics

Abstract: 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.

Semiring Provenance for B\"uchi Games: Strategy Analysis with Absorptive Polynomials

Abstract: This paper presents a case study for the application of semiring semantics for fixed-point formulae to the analysis of strategies in Buchi 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 Buchi 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 further 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.

Limitations of the Invertible-Map Equivalences.

Abstract: This note draws conclusions that arise by combining two recent papers, by Anuj Dawar, Erich Gradel, 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-Furer-Immerman (CFI) structures. These $\equiv^\text{IM}_{k,Q}$-equivalences, for a natural 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 that for a prime $q otin Q$, the $\equiv^\text{IM}_{k,Q}$ equivalences are not strong enough to 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 been shown to be indistinguishable with respect to $\equiv^\text{IM}_{k,\{2\}}$. Together with earlier work on rank logic, this second result suffices to separate rank logic from polynomial time. We show here that the two approaches can be unified to 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.

Logics of dependence and independence: The local variants

Abstract: 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.

Logics of Dependence and Independence: The Local Variants

Abstract: 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.