13,642 research outputs found
One-dimensional fragment of first-order logic
We introduce a novel decidable fragment of first-order logic. The fragment is
one-dimensional in the sense that quantification is limited to applications of
blocks of existential (universal) quantifiers such that at most one variable
remains free in the quantified formula. The fragment is closed under Boolean
operations, but additional restrictions (called uniformity conditions) apply to
combinations of atomic formulae with two or more variables. We argue that the
notions of one-dimensionality and uniformity together offer a novel perspective
on the robust decidability of modal logics. We also establish that minor
modifications to the restrictions of the syntax of the one-dimensional fragment
lead to undecidable formalisms. Namely, the two-dimensional and non-uniform
one-dimensional fragments are shown undecidable. Finally, we prove that with
regard to expressivity, the one-dimensional fragment is incomparable with both
the guarded negation fragment and two-variable logic with counting. Our proof
of the decidability of the one-dimensional fragment is based on a technique
involving a direct reduction to the monadic class of first-order logic. The
novel technique is itself of an independent mathematical interest
One-Variable Fragments of First-Order Many-Valued Logics
In this thesis we study one-variable fragments of first-order logics. Such a one-variable fragment consists of those first-order formulas that contain only unary predicates and a single variable. These fragments can be viewed from a modal perspective by replacing the universal and existential quantifier with a box and diamond modality, respectively, and the unary predicates with corresponding propositional variables. Under this correspondence, the one-variable fragment of first-order classical logic famously corresponds to the modal logic S5.
This thesis explores some such correspondences between first-order and modal logics. Firstly, we study first-order intuitionistic logics based on linear intuitionistic Kripke frames. We show that their one-variable fragments correspond to particular modal Gödel logics, defined over many-valued S5-Kripke frames. For a large class of these logics, we prove the validity problem to be decidable, even co-NP-complete. Secondly, we investigate the one-variable fragment of first-order Abelian logic, i.e., the first-order logic based on the ordered additive group of the reals. We provide two completeness results with respect to Hilbert-style axiomatizations: one for the one-variable fragment, and one for the one-variable fragment that does not contain any lattice connectives. Both these fragments are proved to be decidable. Finally, we launch a much broader algebraic investigation into one-variable fragments. We turn to the setting of first-order substructural logics (with the rule of exchange). Inspired by work on, among others, monadic Boolean algebras and monadic Heyting algebras, we define monadic commutative pointed residuated lattices as a first (algebraic) investigation into one-variable fragments of this large class of first-order logics. We prove a number of properties for these newly defined algebras, including a characterization in terms of relatively complete subalgebras as well as a characterization of their congruences
The decision problem of modal product logics with a diagonal, and faulty counter machines
In the propositional modal (and algebraic) treatment of two-variable
first-order logic equality is modelled by a `diagonal' constant, interpreted in
square products of universal frames as the identity (also known as the
`diagonal') relation. Here we study the decision problem of products of two
arbitrary modal logics equipped with such a diagonal. As the presence or
absence of equality in two-variable first-order logic does not influence the
complexity of its satisfiability problem, one might expect that adding a
diagonal to product logics in general is similarly harmless. We show that this
is far from being the case, and there can be quite a big jump in complexity,
even from decidable to the highly undecidable. Our undecidable logics can also
be viewed as new fragments of first- order logic where adding equality changes
a decidable fragment to undecidable. We prove our results by a novel
application of counter machine problems. While our formalism apparently cannot
force reliable counter machine computations directly, the presence of a unique
diagonal in the models makes it possible to encode both lossy and
insertion-error computations, for the same sequence of instructions. We show
that, given such a pair of faulty computations, it is then possible to
reconstruct a reliable run from them
On the Complexity of Team Logic and its Two-Variable Fragment
We study the logic FO(~), the extension of first-order logic with team
semantics by unrestricted Boolean negation. It was recently shown
axiomatizable, but otherwise has not yet received much attention in questions
of computational complexity.
In this paper, we consider its two-variable fragment FO2(~) and prove that
its satisfiability problem is decidable, and in fact complete for the recently
introduced non-elementary class TOWER(poly). Moreover, we classify the
complexity of model checking of FO(~) with respect to the number of variables
and the quantifier rank, and prove a dichotomy between PSPACE- and
ATIME-ALT(exp, poly)-completeness.
To achieve the lower bounds, we propose a translation from modal team logic
MTL to FO2(~) that extends the well-known standard translation from modal logic
ML to FO2. For the upper bounds, we translate to a fragment of second-order
logic
Undecidability of first-order modal and intuitionistic logics with two variables and one monadic predicate letter
We prove that the positive fragment of first-order intuitionistic logic in
the language with two variables and a single monadic predicate letter, without
constants and equality, is undecidable. This holds true regardless of whether
we consider semantics with expanding or constant domains. We then generalise
this result to intervals [QBL, QKC] and [QBL, QFL], where QKC is the logic of
the weak law of the excluded middle and QBL and QFL are first-order
counterparts of Visser's basic and formal logics, respectively. We also show
that, for most "natural" first-order modal logics, the two-variable fragment
with a single monadic predicate letter, without constants and equality, is
undecidable, regardless of whether we consider semantics with expanding or
constant domains. These include all sublogics of QKTB, QGL, and QGrz -- among
them, QK, QT, QKB, QD, QK4, and QS4.Comment: Pre-final version of the paper published in Studia
Logica,doi:10.1007/s11225-018-9815-
Modal Logics of Topological Relations
Logical formalisms for reasoning about relations between spatial regions play
a fundamental role in geographical information systems, spatial and constraint
databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's
modal logic of time intervals based on the Allen relations, we introduce a
family of modal logics equipped with eight modal operators that are interpreted
by the Egenhofer-Franzosa (or RCC8) relations between regions in topological
spaces such as the real plane. We investigate the expressive power and
computational complexity of logics obtained in this way. It turns out that our
modal logics have the same expressive power as the two-variable fragment of
first-order logic, but are exponentially less succinct. The complexity ranges
from (undecidable and) recursively enumerable to highly undecidable, where the
recursively enumerable logics are obtained by considering substructures of
structures induced by topological spaces. As our undecidability results also
capture logics based on the real line, they improve upon undecidability results
for interval temporal logics by Halpern and Shoham. We also analyze modal
logics based on the five RCC5 relations, with similar results regarding the
expressive power, but weaker results regarding the complexity
Bundled Fragments of First-Order Modal Logic: (Un)Decidability
Quantified modal logic is notorious for being undecidable, with very few known decidable fragments such as the monodic ones. For instance, even the two-variable fragment over unary predicates is undecidable. In this paper, we study a particular fragment, namely the bundled fragment, where a first-order quantifier is always followed by a modality when occurring in the formula, inspired by the proposal of [Yanjing Wang, 2017] in the context of non-standard epistemic logics of know-what, know-how, know-why, and so on.
As always with quantified modal logics, it makes a significant difference whether the domain stays the same across possible worlds. In particular, we show that the predicate logic with the bundle "forall Box" alone is undecidable over constant domain interpretations, even with only monadic predicates, whereas having the "exists Box" bundle instead gives us a decidable logic. On the other hand, over increasing domain interpretations, we get decidability with both "forall Box" and "exists Box" bundles with unrestricted predicates, where we obtain tableau based procedures that run in PSPACE. We further show that the "exists Box" bundle cannot distinguish between constant domain and variable domain interpretations
Lindstrom theorems for fragments of first-order logic
Lindstr\"om theorems characterize logics in terms of model-theoretic
conditions such as Compactness and the L\"owenheim-Skolem property. Most
existing characterizations of this kind concern extensions of first-order
logic. But on the other hand, many logics relevant to computer science are
fragments or extensions of fragments of first-order logic, e.g., k-variable
logics and various modal logics. Finding Lindstr\"om theorems for these
languages can be challenging, as most known techniques rely on coding arguments
that seem to require the full expressive power of first-order logic. In this
paper, we provide Lindstr\"om theorems for several fragments of first-order
logic, including the k-variable fragments for k>2, Tarski's relation algebra,
graded modal logic, and the binary guarded fragment. We use two different proof
techniques. One is a modification of the original Lindstr\"om proof. The other
involves the modal concepts of bisimulation, tree unraveling, and finite depth.
Our results also imply semantic preservation theorems.Comment: Appears in Logical Methods in Computer Science (LMCS
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