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Register a new userBundled fragments of first-order modal logic: (un)decidability
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Added by
Yanjing Wang
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Quantified modal logic provides a natural logical language for reasoning about modal attitudes even while retaining the richness of quantification for referring to predicates over domains. But then most fragments of the logic are undecidable, over many model classes. Over the years, only a few fragments (such as the monodic) have been shown to be decidable. In this paper, we study fragments that bundle quantifiers and modalities together, inspired by earlier work on epistemic logics of know-how/why/what. As always with quantified modal logics, it makes a significant difference whether the domain stays the same across worlds, or not. In particular, we show that the bundle $forall Box$ is undecidable over constant domain interpretations, even with only monadic predicates, whereas $exists Box$ bundle is decidable. On the other hand, over increasing domain interpretations, we get decidability with both $forall Box$ and $exists Box$ bundles with unrestricted predicates. In these cases, we also obtain tableau based procedures that run in PSPACE. We further show that the $exists Box$ bundle cannot distinguish between constant domain and increasing domain interpretations.
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Weakly Aggregative Modal Logic (WAML) is a collection of disguised polyadic modal logics with n-ary modalities whose arguments are all the same. WAML has some interesting applications on epistemic logic and logic of games, so we study some basic model theoretical aspects of WAML in this paper. Specifically, we give a van Benthem-Rosen characterization theorem of WAML based on an intuitive notion of bisimulation and show that each basic WAML system K_n lacks Craig Interpolation.
We study the decidability of termination for two CHR dialects which, similarly to the Datalog like languages, are defined by using a signature which does not allow function symbols (of arity >0). Both languages allow the use of the = built-in in the body of rules, thus are built on a host language that supports unification. However each imposes one further restriction. The first CHR dialect allows only range-restricted rules, that is, it does not allow the use of variables in the body or in the guard of a rule if they do not appear in the head. We show that the existence of an infinite computation is decidable for this dialect. The second dialect instead limits the number of atoms in the head of rules to one. We prove that in this case, the existence of a terminating computation is decidable. These results show that both dialects are strictly less expressive than Turing Machines. It is worth noting that the language (without function symbols) without these restrictions is as expressive as Turing Machines.
Let 2<nleq l<m< omega. Let L_n denote first order logic restricted to the first n variables. We show that the omitting types theorem fails dramatically for the n--variable fragments of first order logic with respect to clique guarded semantics, and for its packed n--variable fragments. Both are modal fragments of L_n. As a sample, we show that if there exists a finite relation algebra with a so--called strong l--blur, and no m--dimensional relational basis, then there exists a countable, atomic and complete L_n theory T and type Gamma, such that Gamma is realizable in every so--called m--square model of T, but any witness isolating Gamma cannot use less than $l$ variables. An $m$--square model M of T gives a form of clique guarded semantics, where the parameter m, measures how locally well behaved M is. Every ordinary model is k--square for any n<k<omega, but the converse is not true. Any model M is omega--square, and the two notions are equivalent if M is countable. Such relation algebras are shown to exist for certain values of l and m like for nleq l<omega and m=omega, and for l=n and mgeq n+3. The case l=n and m=omega gives that the omitting types theorem fails for L_n with respect to (usual) Tarskian semantics: There is an atomic countable L_n theory T for which the single non--principal type consisting of co--atoms cannot be omitted in any model M of T. For n<omega, positive results on omitting types are obained for L_n by imposing extra conditions on the theories and/or the types omitted. Positive and negative results on omitting types are obtained for infinitary variants and extensions of L_{omega, omega}.
We investigate the decidability and complexity status of model-checking problems on unlabelled reachability graphs of Petri nets by considering first-order and modal languages without labels on transitions or atomic propositions on markings. We consider several parameters to separate decidable problems from undecidable ones. Not only are we able to provide precise borders and a systematic analysis, but we also demonstrate the robustness of our proof techniques.
We consider a specific class of tree structures that can represent basic structures in linguistics and computer science such as XML documents, parse trees, and treebanks, namely, finite node-labeled sibling-ordered trees. We present axiomatizations of the monadic second-order logic (MSO), monadic transitive closure logic (FO(TC1)) and monadic least fixed-point logic (FO(LFP1)) theories of this class of structures. These logics can express important properties such as reachability. Using model-theoretic techniques, we show by a uniform argument that these axiomatizations are complete, i.e., each formula that is valid on all finite trees is provable using our axioms. As a backdrop to our positive results, on arbitrary structures, the logics that we study are known to be non-recursively axiomatizable.
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