Non-primitive recursive decidability of products of modal logics with expanding domains (original) (raw)
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Analyzing completeness of axiomatic functional systems for temporal × modal logics
Mathematical Logic Quarterly, 2010
In previous works, we presented a modification of the usual possible world semantics by introducing an independent temporal structure in each world and using accessibility functions to represent the relation among them. Different properties of the accessibility functions (being injective, surjective, increasing, etc.) have been considered and axiomatic systems (called functional) which define these properties have been given. Only a few of these systems have been proved to be complete. The aim of this paper is to make a progress in the study of completeness for functional systems. For this end, we use indexes as names for temporal flows and give new proofs of completeness. Specifically, we focus our attention on the system which defines injectivity, because the system which defines this property without using indexes was proved to be incomplete in previous works. The only system considered which remains incomplete is the one which defines surjectivity, even if we consider a sequence of natural extensions of the previous one.
Temporal Logics over Transitive States
2005
We investigate the computational behaviour of ‘two-dimensional’ propositional temporal logics over (ℕ, \(\Pi^{\rm 1}_{\rm 1}\) -complete) if the domains of states with those relations are assumed to be constant. Motivated by applications in the areas of temporal description logic and specification & verification of hybrid systems, in this paper we analyse the computational impact of allowing the domains of states to expand. We show that over finite expanding domains (with an arbitrary, tree-like, quasi-order, or linear transitive relation) the logics are recursively enumerable, but undecidable. If these finite domains eventually become constant then the resulting O-free logics are decidable (but not in primitive recursive time); on the other hand, when equipped with O they are not even recursively enumerable. Finally, we show that temporal logics over infinite expanding domains as above are undecidable even for the language with the sole temporal operator ‘eventually.’ The proofs are based on Kruskal’s tree theorem and reductions of reachability problems for lossy channel systems.
Decidability of order-based modal logics
Journal of Computer and System Sciences, 2017
Decidability of the validity problem is established for a family of many-valued modal logics, notably Gödel modal logics, where propositional connectives are evaluated according to the order of values in a complete sublattice of the real unit interval [0, 1], and box and diamond modalities are evaluated as infima and suprema over (many-valued) Kripke frames. If the sublattice is infinite and the language is sufficiently expressive, then the standard semantics for such a logic lacks the finite model property. It is shown here, however, that, given certain regularity conditions, the finite model property holds for a new semantics for the logic, providing a basis for establishing decidability and PSPACE-completeness. Similar results are also established for S5 logics that coincide with one-variable fragments of first-order many-valued logics. In particular, a first proof is given of $ Preliminary results from this work were reported in the proceedings of TACL 2013 (as an extended abstract) and WoLLIC 2013 [8].
Products of ‘transitive’modal logics. Part I:‘negative’results
2004
We solve a major open problem concerning algorithmic properties of products of 'transitive' modal logics by showing that products and commutators of such standard logics as K4, S4, S4.1, K4.3, GL, or Grz are undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4, K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, if C1 and C2 are classes of transitive frames such that their depth cannot be bounded by any fixed n < ω, then the logic of the class {F1 × F2 | F1 ∈ C1, F2 ∈ C2} is undecidable. (On the contrary, the product of, say, K4 and the logic of all transitive Kripke frames of depth ≤ n, for some fixed n < ω, is decidable.) The complexity of these undecidable logics ranges from r.e. to co-r.e. and Π 1 1 -complete. As a consequence, we give the first known examples of Kripke incomplete commutators of Kripke complete logics. £ are interpreted by R h , while ¡ and ¤ are interpreted by R v .)
The Boundary Between Decidability and Undecidability for Transitive-Closure Logics
Lecture Notes in Computer Science, 2004
To reason effectively about programs, it is important to have some version of a transitive-closure operator so that we can describe such notions as the set of nodes reachable from a program's variables. On the other hand, with a few notable exceptions, adding transitive closure to even very tame logics makes them undecidable.
On Modal Logics of Partial Recursive Functions
Studia Logica, 2005
The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above interpretation. The cases of deterministic and non-deterministic functions are considered and for both of them semantically complete modal logics are described and decidability of these logics is established.
On the Products of Linear Modal Logics
Journal of Logic and Computation, 2001
We study two-dimensional Cartesian products of modal logics determined by infinite or arbitrarily long finite linear orders and prove a general theorem showing that in many cases these products are undecidable, in particular, such are the squares of standard linear logics like à ¿, Ë ¿, Ä ¿, ÖÞ ¿, or the logic determined by the Cartesian square of any infinite linear order. This theorem solves a number of open problems of Gabbay and Shehtman . We also prove a sufficient condition for such products to be not recursively enumerable and give a simple axiomatisation for the square à ¿ ¢ à ¿ of the minimal liner logic using non-structural Gabbay-type inference rules.
First-order temporal logics are notorious for their bad computational behaviour. It is known that even the two-variable monadic fragment is highly undecidable over various linear timelines, and over branching time even one-variable fragments might be undecidable. However, there have been several attempts on finding well-behaved fragments of first-order temporal logics and related temporal description logics, mostly either by restricting the available quantifier patterns, or considering sub-Boolean languages. Here we analyse seemingly 'mild' extensions of decidable one-variable fragments with counting capabilities, interpreted in models with constant, decreasing, and expanding first-order domains. We show that over most classes of linear orders these logics are (sometimes highly) undecidable, even without constant and function symbols, and with the sole temporal operator 'eventually'.
Dynamic topological logics over spaces with continuous functions
2006
Dynamic topological logics are combinations of topological and temporal modal logics that are used for reasoning about dynamical systems consisting of a topological space and a continuous function on it. Here we partially solve a major open problem in the field by showing (by reduction of the ω-reachability problem for lossy channel systems) that the dynamic topological logic over arbitrary topological spaces as well as those over R n , for each n ≥ 1, are undecidable. Actually, we prove this result for the natural and expressive fragment of the full dynamic topological language where the topological operators cannot be applied to formulas containing the temporal eventuality. Using Kruskal's tree theorem we also show that the formulas of this fragment that are valid in arbitrary topological spaces with continuous functions are recursively enumerable, which is not the case for spaces with homeomorphisms.
Non-finitely axiomatisable two-dimensional modal logics
The Journal of Symbolic Logic, 2012
We show the first examples of recursively enumerable (even decidable) two-dimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linearly ordered first components must be infinite in two senses: It should contain infinitely many propositional variables, and formulas of arbitrarily large modal nesting-depth.