Separating Data Examples by Description Logic Concepts with Restricted Signatures

We study the separation of positive and negative data examples in terms of description logic concepts in the presence of an ontology. In contrast to previous work, we add a signature that specifies a subset of the symbols that can be used for separation, and we admit individual names in that signature. We consider weak and strong versions of the resulting problem that differ in how the negative examples are treated and we distinguish between separation with and without helper symbols. Within this framework, we compare the separating power of different languages and investigate the complexity of deciding separability. While weak separability is shown to be closely related to conservative extensions, strongly separating concepts coincide with Craig interpolants, for suitably defined encodings of the data and ontology. This enables us to transfer known results from those fields to separability. Conversely, we obtain original results on separability that can be transferred backward. For example, rather surprisingly, conservative extensions and weak separability in ALCO are both 3ExpTime-complete.

Admissibility in Probabilistic Argumentation

argumentation is a prominent reasoning framework. It comes with a variety of semantics, and has lately been enhanced by probabilities to enable a quantitative treatment of argumentation. While admissibility is a fundamental notion in the classical setting, it has been merely reflected so far in the probabilistic setting. In this paper, we address the quantitative treatment of argumentation based on probabilistic notions of admissibility in a way that they form fully conservative extensions of classical notions. In particular, our building blocks are not the beliefs regarding single arguments. Instead we start from the fairly natural idea that whatever argumentation semantics is to be considered, semantics systematically induces constraints on the joint probability distribution on the sets of arguments. In some cases there might be many such distributions, even infinitely many ones, in other cases there may be one or none. Standard semantic notions are shown to induce such sets of constraints, and so do their probabilistic extensions. This allows them to be tackled by SMT solvers, as we demonstrate by a proof-of-concept implementation. We present a taxonomy of semantic notions, also in relation to published work, together with a running example illustrating our achievements.

Proof systems for Geometric theories (PROGEO)

We plan to study the problem of finding conservative extensions of first order logics. In this project we intend to establish a systematic procedure for adding geometric theories in both intuitionistic and classical logics, as well as to extend this procedure to bipolar axioms, a generalization of the set of geometric axioms. This way, we obtain proof systems for several mathematical theories, such as lattices, algebra and projective geometry, being able to reason about such theories using automated deduction.

Conservative Extensions in Horn Description Logics with Inverse Roles

We investigate the decidability and computational complexity of conservative extensions and the related notions of inseparability and entailment in Horn description logics (DLs) with inverse roles. We consider both query conservative extensions, defined by requiring that the answers to all conjunctive queries are left unchanged, and deductive conservative extensions, which require that the entailed concept inclusions, role inclusions, and functionality assertions do not change. Upper bounds for query conservative extensions are particularly challenging because characterizations in terms of unbounded homomorphisms between universal models, which are the foundation of the standard approach to establishing decidability, fail in the presence of inverse roles. We resort to a characterization that carefully mixes unbounded and bounded homomorphisms and enables a decision procedure that combines tree automata and a mosaic technique. Our main results are that query conservative extensions are 2ExpTime-complete in all DLs between ELI and Horn-ALCHIF and between Horn-ALC and Horn-ALCHIF, and that deductive conservative extensions are 2ExpTime-complete in all DLs between ELI and ELHIF_bot. The same results hold for inseparability and entailment.

Arithmetics based on computability logic

This paper is a brief survey of number theories based on em computability logic (CoL) a game-semantically conceived logic of computational tasks of resources. Such theories, termed em clarithmetics, are conservative extensions of first-order Peano arithmetic. The first section of the paper lays out the conceptual basis of CoL and describes the relevant fragment of its formal language, with so called parallel connectives, choice connectives and quantifiers, and blind quantifiers. Both syntactically and semantically, this is a conservative generalization of the language of classical logic. Clarithmetics, based on the corresponding fragment of CoL in the same sense as Peano arithmetic is based on classical logic, are discussed in the second section. The axioms and inference rules of the system of clarithmetic named ${\bf CLA11}$ are presented, and the main results on this system are stated: constructive soundness, extensional completeness, and intensional completeness. In the final section two potential applications of clarithmetics are addressed: clarithmetics as declarative programming languages in an extreme sense, and as tools for separating computational complexity classes. When clarithmetics or similar CoL-based theories are viewed as programming languages, programming reduces to proof-search, as programs can be mechanically extracted from proofs; such programs also serves as their own formal verifications, thus fully neutralizing the notorious (and generally undecidable) program verification problem. The second application reduces the problem of separating various computational complexity classes to separating the corresponding versions of clarithmetic, the potential benefits of which stem from the belief that separating theories should generally be easier than separating complexity classes directly.

ON A QUESTION OF KRAJEWSKI’S

In this paper, we study finitely axiomatizable conservative extensions of a theory U in the case where U is recursively enumerable and not finitely axiomatizable. Stanisław Krajewski posed the question whether there are minimal conservative extensions of this sort. We answer this question negatively.Consider a finite expansion of the signature of U that contains at least one predicate symbol of arity ≥ 2. We show that, for any finite extension α of U in the expanded language that is conservative over U, there is a conservative extension β of U in the expanded language, such that $\alpha \vdash \beta$ and $\beta \not \vdash \alpha$. The result is preserved when we consider either extensions or model-conservative extensions of U instead of conservative extensions. Moreover, the result is preserved when we replace $\dashv$ as ordering on the finitely axiomatized extensions in the expanded language by a relevant kind of interpretability, to wit interpretability that identically translates the symbols of the U-language.We show that the result fails when we consider an expansion with only unary predicate symbols for conservative extensions of U ordered by interpretability that preserves the symbols of U.

REXPANSIONS OF NONDETERMINISTIC MATRICES AND THEIR APPLICATIONS IN NONCLASSICAL LOGICS

The operations of expansion and refinement on nondeterministic matrices (Nmatrices) are composed to form a new operation called rexpansion. Properties of this operation are investigated, together with their effects on the induced consequence relations. Using rexpansions, a semantic method for obtaining conservative extensions of (N)matrix-defined logics is introduced and applied to fragments of the classical two-valued matrix, as well as to other many-valued matrices and Nmatrices. The main application of this method is the construction and investigation of truth-preserving ¬-paraconsistent conservative extensions of Gödel fuzzy logic, in which ¬ has several desired properties. This is followed by some results regarding the relations between the constructed logics.

CONSERVATIVITY OF ULTRAFILTERS OVER SUBSYSTEMS OF SECOND ORDER ARITHMETIC

We extend the usual language of second order arithmetic to one in which we can discuss an ultrafilter over of the sets of a given model. The semantics are based on fixing a subclass of the sets in a structure for the basic language that corresponds to the intended ultrafilter. In this language we state axioms that express the notion that the subclass is an ultrafilter and additional ones that say it is idempotent or Ramsey. The axioms for idempotent ultrafilters prove, for example, Hindman’s theorem and its generalizations such as the Galvin--Glazer theorem and iterated versions of these theorems (IHT and IGG). We prove that adding these axioms to IHT produce conservative extensions of ACA0+IHT,${\rm{ACA}}_{\rm{0}}^ +$, ATR0,${\rm{\Pi }}_2^1$-CA0, and${\rm{\Pi }}_2^1$-CA0for all sentences of second order arithmetic and for full Z2for the class of${\rm{\Pi }}_4^1$sentences. We also generalize and strengthen a metamathematical result of Wang (1984) to show, for example, that any${\rm{\Pi }}_2^1$theorem ∀X∃YΘ(X,Y) provable in ACA0or${\rm{ACA}}_{\rm{0}}^ +$there aree,k∈ ℕ such that ACA0or${\rm{ACA}}_{\rm{0}}^ +$proves that ∀X(Θ(X, Φe(J(k)(X))) where Φeis theeth Turing reduction andJ(k)is thekth iterate of the Turing or Arithmetic jump, respectively. (A similar result is derived for${\rm{\Pi }}_3^1$theorems of${\rm{\Pi }}_1^1$-CA0and the hyperjump.)

Computational Coverage of TLG: Nonlinearity

We study nonlinear connectives (exponentials) in the context of Type Logical Grammar(TLG). We devise four conservative extensions of theDisplacement calculus with brackets, \DbC, \DbCM, \DbCb and \DbCbMr which contain the universal and existential exponential modalities of linear logic (\LL). These modalitiesdo not exhibit the same structural properties as in \LL, which in TLG are especially adapted for linguistic purposes. The universal modality \univexpfor TLG allows only the commutative and contraction rules, but not weakening, whereas the existential modality \exstexp allows the so-called (intuitionistic) mingle rule, whichderives a restricted version of weakening called \emph{expansion}. We provide a Curry-Howard labelling for both exponential connectives. As it turns out,controlled contraction by \univexp gives a way to account for the so-called parasitic gaps, and controlled Mingle \exstexp iterability, in particular iteratedcoordination. Finally, the four calculi are proved to be Cut-Free but decidability is only proved for $\DbCb$, whereasfor the rest the question of decidability remains open.