Decidable Satisfiability Research Articles

A Tractable Logic for Molecular Biology

We introduce a logic for knowledge representation and reasoning on protein-protein interactions. Modulo a theory, formulas describe protein structures and dynamic changes. They can be composed in order to add or remove static and dynamic observations. A second-order circumscription operator then enables nonmonotonic reasoning on the changes implied by a formula. We introduce deduction rules that produce formulas which are, up to equivalence, in a first-order fragment with decidable satisfiability and validity. Importantly, the rules can produce circumscribed formulas.

Decidability and complexity of the fragments of the modal logic of Allen's relations over the rationals

Interval temporal logics provide a natural framework for temporal reasoning about interval structures over linearly ordered domains, where intervals are taken as first-class citizens. Their expressive power and computational behavior mainly depend on two parameters: the set of modalities they feature and the linear orders over which they are interpreted. In this paper, we consider all fragments of Halpern and Shoham's interval temporal logic HS with a decidable satisfiability problem over the rationals, and we provide a complete classification of them in terms of their expressiveness and computational complexity by solving the last few open problems.

Satisfiability of Xpath on data trees

This is a survey on the satisfiability problem for XPath on data trees. Data trees are finite trees whose every node carries a label from a finite alphabet and a data value from an infinite domain. XPath is an expressive node selecting language for XML documents, which can be abstracted as data trees. Its satisfiability problem is in general undecidable. However, various fragments of XPath have decidable satisfiability problem, these are fragments defined in terms of the navigational axes which are allowed. We explore the state of the art in terms of decidability and discuss briefly some of the algorithmic techniques involved.

A tool for deciding the satisfiability of continuous-time metric temporal logic

Constraint LTL over clocks is a variant of CLTL, an extension of linear-time temporal logic allowing atomic assertions in a concrete constraint system. Satisfiability of CLTL over clocks is here shown to be decidable by means of a reduction to a decidable Satisfiability Modulo Theories (SMT) problem. The result is a complete Bounded Satisfiability Checking procedure, which has been implemented by using standard SMT solvers. The importance of this technique derives from the possibility of translating various continuous-time metric temporal logics, such as MITL and QTL, into CLTL over clocks itself. Although standard decision procedures of these logics do exist, they have never been realized in practice. Suitable translations into CLTL over clocks have instead allowed us the development of the first prototype tool for deciding MITL and QTL. The paper also reports preliminary, but encouraging, experiments on some significant examples of MITL and QTL formulae.

Interval temporal logics over strongly discrete linear orders: Expressiveness and complexity

Interval temporal logics provide a natural framework for temporal reasoning about interval structures over linearly ordered domains, where intervals are taken as the primitive ontological entities. Their computational behavior mainly depends on two parameters: the set of modalities they feature and the linear orders over which they are interpreted. In this paper, we identify all fragments of Halpern and Shoham's interval temporal logic HS with a decidable satisfiability problem over the class of strongly discrete linear orders as well as over its relevant subclasses (the class of finite linear orders, Z, N, and Z−). We classify them in terms of both their relative expressive power and their complexity, which ranges from NP-completeness to non-primitive recursiveness.

Satisfiability vs. Finite Satisfiability in Elementary Modal Logics

We study elementary modal logics, i.e. modal logic considered over first-order definable classes of frames. The classical semantics of modal logic allows infinite structures, but often practical applications require to restrict our attention to finite structures. Many decidability and undecidability results for the elementary modal logics were proved separately for general satisfiability and for finite satisfiability [11, 12, 16, 17]. In this paper, we show that there is a reason why we must deal with both kinds of satisfiability separately -- we prove that there is a universal first-order formula that defines an elementary modal logic with decidable (global) satisfiability problem, but undecidable finite satisfiability problem, and, the other way round, that there is a universal formula that defines an elementary modal logic with decidable finite satisfiability problem, but undecidable general satisfiability problem.

Interval Temporal Logics over Strongly Discrete Linear Orders: the Complete Picture

Interval temporal logics provide a general framework for temporal reasoning about interval structures over linearly ordered domains, where intervals are taken as the primitive ontological entities. In this paper, we identify all fragments of Halpern and Shoham's interval temporal logic HS with a decidable satisfiability problem over the class of strongly discrete linear orders. We classify them in terms of both their relative expressive power and their complexity. We show that there are exactly 44 expressively different decidable fragments, whose complexity ranges from NP to EXPSPACE. In addition, we identify some new undecidable fragments (all the remaining HS fragments were already known to be undecidable over strongly discrete linear orders). We conclude the paper by an analysis of the specific case of natural numbers, whose behavior slightly differs from that of the whole class of strongly discrete linear orders. The number of decidable fragments over natural numbers raises up to 47: three undecidable fragments become decidable with a non-primitive recursive complexity.

THE EXPRESSIVE POWER OF MEMORY LOGICS

We investigate the expressive power of memory logics. These are modal logics extended with the possibility to store (or remove) the current node of evaluation in (or from) a memory, and to perform membership tests on the current memory. From this perspective, the hybrid logic ℋℒ (↓), for example, can be thought of as a particular case of a memory logic where the memory is an indexed list of elements of the domain.This work focuses in the case where the memory is a set, and we can test whether the current node belongs to the set or not. We prove that, in terms of expressive power, the memory logics we discuss here lie between the basic modal logic ${\cal K}$ and ℋℒ (↓). We show that the satisfiability problem of most of the logics we cover is undecidable. The only logic with a decidable satisfiability problem is obtained by imposing strong constraints on which elements can be memorized.

Two-variable logic on data trees and XML reasoning

Motivated by reasoning tasks for XML languages, the satisfiability problem of logics on data trees is investigated. The nodes of a data tree have a label from a finite set and a data value from a possibly infinite set. It is shown that satisfiability for two-variable first-order logic is decidable if the tree structure can be accessed only through the child and the next sibling predicates and the access to data values is restricted to equality tests. From this main result, decidability of satisfiability and containment for a data-aware fragment of XPath and of the implication problem for unary key and inclusion constraints is concluded.

Decidable metric logics

The common metric temporal logic for continuous time were shown to be insufficient, when it was proved that they cannot express a modality suggested by Pnueli. Moreover no finite temporal logic can express all the natural generalizations of this modality. It followed that if we look for an optimal decidable metric logic we must accept infinitely many modalities, or adopt a different formalism. Here we identify a fragment of the second order monadic logic of order with the “ + 1 ” function, that expresses all the Pnueli modalities and much more. Its main advantage over the temporal logics is that it enables us to say not just that within prescribed time there is a point where some punctual event will occur, but also that within prescribed time some process that starts now (or that started before, or that will start soon) will terminate. We prove that this logic is decidable with respect to satisfiability and validity, over continuous time. The proof depends heavily on the theory of compositionality. In particular every temporal logic that has truth tables in this logic is automatically decidable. We extend this result by proving that any temporal logic, that has all its modalities defined by means more general than truth tables, in a logic stronger than the one just described, has a decidable satisfiability problem. We suggest that this monadic logic can be the framework in which temporal logics can be safely defined, with the guarantee that their satisfiability problem is decidable.

A Logic of Probability with Decidable Model Checking

A predicate logic of probability, close to the logics of probability of Halpern et al., is introduced. Our main result concerns the following model-checking problem: deciding whether a given formula holds on the structure defined by a given finite probabilistic process. We show that this model-checking problem is decidable for a rather large subclass of formulas of a second-order monadic logic of probability. We discuss also the decidability of satisfiability and compare our logic of probability with the probabilistic temporal logic pCTL*.

Vertex-minors, monadic second-order logic, and a conjecture by Seese

We prove that one can express the vertex-minor relation on finite undirected graphs by formulas of monadic second-order logic (with no edge set quantification) extended with a predicate expressing that a set has even cardinality. We obtain a slight weakening of a conjecture by Seese stating that sets of graphs having a decidable satisfiability problem for monadic second-order logic have bounded clique-width. We also obtain a polynomial-time algorithm to check that the rank-width of a graph is at most k for any fixed k. The proofs use isotropic systems.

Monodic Packed Fragment with Equality is Decidable

We prove decidability of satisfiability of sentences of the monodic packed fragment of first-order temporal logic with equality and connectives Until and Since, in models with various flows of time and domains of arbitrary cardinality. We also prove decidability over models with finite domains, over flows of time including the real order.