316,575 research outputs found
Polyteam semantics
Team semantics is the mathematical framework of modern logics of dependence and independence in which formulae are interpreted by sets of assignments (teams) instead of single assignments as in first-order logic. In order to deepen the fruitful interplay between team semantics and database dependency theory, we define Polyteam Semantics in which formulae are evaluated over a family of teams. We begin by defining a novel polyteam variant of dependence atoms and give a finite axiomatization for the associated implication problem. We relate polyteam semantics to team semantics and investigate in which cases logics over the former can be simulated by logics over the latter. We also characterize the expressive power of poly-dependence logic by properties of polyteams that are downwards closed and definable in existential second-order logic (ESO). The analogous result is shown to hold for poly-independence logic and all ESO-definable properties. We also relate poly-inclusion logic to greatest fixed point logic.Peer reviewe
Counting in Team Semantics
We explore several counting constructs for logics with team semantics. Counting is an important task in numerous applications, but with a somewhat delicate relationship to logic. Team semantics on the other side is the mathematical basis of modern logics of dependence and independence, in which formulae are evaluated not for a single assignment of values to variables, but for a set of such assignments. It is therefore interesting to ask what kind of counting constructs are adequate in this context, and how such constructs influence the expressive power, and the model-theoretic and algorithmic properties of logics with team semantics. Due to the second-order features of team semantics there is a rich variety of potential counting constructs. Here we study variations of two main ideas: forking atoms and counting quantifiers.
Forking counts how many different values for a tuple w occur in assignments with coinciding values for v. We call this the forking degree of bar v with respect to bar w. Forking is powerful enough to capture many of the previously studied atomic dependency properties. In particular we exhibit logics with forking atoms that have, respectively, precisely the power of dependence logic and independence logic.
Our second approach uses counting quantifiers E^{geq mu} of a similar kind as used in logics with Tarski semantics. The difference is that these quantifiers are now applied to teams of assignments that may give different values to mu. We show that, on finite structures, there is an intimate connection between inclusion logic with counting quantifiers and FPC, fixed-point logic with counting, which is a logic of fundamental importance for descriptive complexity theory. For sentences, the two logics have the same expressive power. Our analysis is based on a new variant of model-checking games, called threshold safety games, on a trap condition for such games, and on game interpretations
Intuitionistic Fixed Point Logic
We study the system IFP of intuitionistic fixed point logic, an extension of
intuitionistic first-order logic by strictly positive inductive and coinductive
definitions. We define a realizability interpretation of IFP and use it to
extract computational content from proofs about abstract structures specified
by arbitrary classically true disjunction free formulas. The interpretation is
shown to be sound with respect to a domain-theoretic denotational semantics and
a corresponding lazy operational semantics of a functional language for
extracted programs. We also show how extracted programs can be translated into
Haskell. As an application we extract a program converting the signed digit
representation of real numbers to infinite Gray-code from a proof of inclusion
of the corresponding coinductive predicates.Comment: 65 page
Computing with Infinite Objects: the Gray Code Case
Infinite Gray code has been introduced by Tsuiki as a redundancy-free
representation of the reals. In applications the signed digit representation is
mostly used which has maximal redundancy. Tsuiki presented a functional program
converting signed digit code into infinite Gray code. Moreover, he showed that
infinite Gray code can effectively be converted into signed digit code, but the
program needs to have some non-deterministic features (see also H. Tsuiki, K.
Sugihara, "Streams with a bottom in functional languages"). Berger and Tsuiki
reproved the result in a system of formal first-order intuitionistic logic
extended by inductive and co-inductive definitions, as well as some new logical
connectives capturing concurrent behaviour. The programs extracted from the
proofs are exactly the ones given by Tsuiki. In order to do so, co-inductive
predicates \bS and \bG are defined and the inclusion \bS \subseteq \bG is
derived. For the converse inclusion the new logical connectives are used to
introduce a concurrent version of and \bG \subseteq \bS_{2} is
shown. What one is looking for, however, is an equivalence proof of the
involved concepts. One of the main aims of the present paper is to close the
gap. A concurrent version \bG^{*} of \bG and a modification \bS^{*} of
\bS_{2} are presented such that \bS^{*} = \bG^{*}. A crucial tool in U.
Berger, H. Tsuiki, "Intuitionistic fixed point logic" is a formulation of the
Archimedean property of the real numbers as an induction principle. We
introduce a concurrent version of this principle which allows us to prove that
\bS^{*} and \bG^{*} coincide. A further central contribution is the
extension of the above results to the hyperspace of non-empty compact subsets
of the reals
Towards a Systematic Account of Different Semantics for Logic Programs
In [Hitzler and Wendt 2002, 2005], a new methodology has been proposed which
allows to derive uniform characterizations of different declarative semantics
for logic programs with negation. One result from this work is that the
well-founded semantics can formally be understood as a stratified version of
the Fitting (or Kripke-Kleene) semantics. The constructions leading to this
result, however, show a certain asymmetry which is not readily understood. We
will study this situation here with the result that we will obtain a coherent
picture of relations between different semantics for normal logic programs.Comment: 20 page
Guarded Teams: The Horizontally Guarded Case
Team semantics admits reasoning about large sets of data, modelled by sets of assignments (called teams), with first-order syntax. This leads to high expressive power and complexity, particularly in the presence of atomic dependency properties for such data sets. It is therefore interesting to explore fragments and variants of logic with team semantics that permit model-theoretic tools and algorithmic methods to control this explosion in expressive power and complexity.
We combine here the study of team semantics with the notion of guarded logics, which are well-understood in the case of classical Tarski semantics, and known to strike a good balance between expressive power and algorithmic manageability. In fact there are two strains of guardedness for teams. Horizontal guardedness requires the individual assignments of the team to be guarded in the usual sense of guarded logics. Vertical guardedness, on the other hand, posits an additional (or definable) hypergraph structure on relational structures in order to interpret a constraint on the component-wise variability of assignments within teams.
In this paper we investigate the horizontally guarded case. We study horizontally guarded logics for teams and appropriate notions of guarded team bisimulation. In particular, we establish characterisation theorems that relate invariance under guarded team bisimulation with guarded team logics, but also with logics under classical Tarski semantics
Decidability of predicate logics with team semantics
We study the complexity of predicate logics based on team semantics. We show
that the satisfiability problems of two-variable independence logic and
inclusion logic are both NEXPTIME-complete. Furthermore, we show that the
validity problem of two-variable dependence logic is undecidable, thereby
solving an open problem from the team semantics literature. We also briefly
analyse the complexity of the Bernays-Sch\"onfinkel-Ramsey prefix classes of
dependence logic.Comment: Extended version of a MFCS 2016 article. Changes on the earlier arXiv
version: title changed, added the result on validity of two-variable
dependence logic, restructurin
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