16 research outputs found
Elementary Canonical Formulae: A Survey on Syntactic, Algorithmic, and Modeltheoretic Aspects
In terms of validity in Kripke frames, a modal formula expresses a universal monadic second-order condition. Those modal formulae which are equivalent to first-order conditions are called elementary. Modal formulae which have a certain persistence property which implies their validity in all canonical frames of modal logics axiomatized with them, and therefore their completeness, are called canonical. This is a survey of a recent and ongoing study of the class of elementary and canonical modal formulae. We summarize main ideas and results, and outline further research perspectives
Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA
Modal formulae express monadic second-order properties on Kripke frames, but
in many important cases these have first-order equivalents. Computing such
equivalents is important for both logical and computational reasons. On the
other hand, canonicity of modal formulae is important, too, because it implies
frame-completeness of logics axiomatized with canonical formulae.
Computing a first-order equivalent of a modal formula amounts to elimination
of second-order quantifiers. Two algorithms have been developed for
second-order quantifier elimination: SCAN, based on constraint resolution, and
DLS, based on a logical equivalence established by Ackermann.
In this paper we introduce a new algorithm, SQEMA, for computing first-order
equivalents (using a modal version of Ackermann's lemma) and, moreover, for
proving canonicity of modal formulae. Unlike SCAN and DLS, it works directly on
modal formulae, thus avoiding Skolemization and the subsequent problem of
unskolemization. We present the core algorithm and illustrate it with some
examples. We then prove its correctness and the canonicity of all formulae on
which the algorithm succeeds. We show that it succeeds not only on all
Sahlqvist formulae, but also on the larger class of inductive formulae,
introduced in our earlier papers. Thus, we develop a purely algorithmic
approach to proving canonical completeness in modal logic and, in particular,
establish one of the most general completeness results in modal logic so far.Comment: 26 pages, no figures, to appear in the Logical Methods in Computer
Scienc
Inclusions Between Pseudo-euclidean Modal Logics
We describe properties of simply axiomatized modal logics, which are called pseudo-Euclidean modal logics. We will then give a complete description of the inclusion relationship among these logics by showing inclusion relationships for pairs of their logics with fixed m and n
Algorithmic correspondence and completeness in modal logic
Abstract
This thesis takes an algorithmic perspective on the correspondence between modal and hybrid
logics on the one hand, and first-order logic on the other. The canonicity of formulae, and by
implication the completeness of logics, is simultaneously treated.
Modal formulae define second-order conditions on frames which, in some cases, are equiv-
alently reducible to first-order conditions. Modal formulae for which the latter is possible
are called elementary. As is well known, it is algorithmically undecidable whether a given
modal formula defines a first-order frame condition or not. Hence, any attempt at delineating
the class of elementary modal formulae by means of a decidable criterium can only consti-
tute an approximation of this class. Syntactically specified such approximations include the
classes of Sahlqvist and inductive formulae. The approximations we consider take the form
of algorithms.
We develop an algorithm called SQEMA, which computes first-order frame equivalents for
modal formulae, by first transforming them into pure formulae in a reversive hybrid language.
It is shown that this algorithm subsumes the classes of Sahlqvist and inductive formulae, and
that all formulae on which it succeeds are d-persistent (canonical), and hence axiomatize
complete normal modal logics.
SQEMA is extended to polyadic languages, and it is shown that this extension succeeds
on all polyadic inductive formulae. The canonicity result is also transferred.
SQEMA is next extended to hybrid languages. Persistence results with respect to discrete
general frames are obtained for certain of these extensions. The notion of persistence with
respect to strongly descriptive general frames is investigated, and some syntactic sufficient
conditions for such persistence are obtained. SQEMA is adapted to guarantee the persistence
with respect to strongly descriptive frames of the hybrid formulae on which it succeeds, and
hence the completeness of the hybrid logics axiomatized with these formulae. New syntactic
classes of elementary and canonical hybrid formulae are obtained.
Semantic extensions of SQEMA are obtained by replacing the syntactic criterium of nega-
tive/positive polarity, used to determine the applicability of a certain transformation rule, by
its semantic correlate—monotonicity. In order to guarantee the canonicity of the formulae on
which the thus extended algorithm succeeds, syntactically correct equivalents for monotone
formulae are needed. Different version of Lyndon’s monotonicity theorem, which guarantee
the existence of these equivalents, are proved. Constructive versions of these theorems are
also obtained by means of techniques based on bisimulation quantifiers.
Via the standard second-order translation, the modal elementarity problem can be at-
tacked with any second-order quantifier elimination algorithm. Our treatment of this ap-
proach takes the form of a study of the DLS-algorithm. We partially characterize the for-
mulae on which DLS succeeds in terms of syntactic criteria. It is shown that DLS succeeds
in reducing all Sahlqvist and inductive formulae, and that all modal formulae in a single
propositional variable on which it succeeds are canonical
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
Extensions of modal logic KTB and other topics
This thesis covers four topics. They are the extensions of the modal logic KTB, the use of normal forms in modal logic, automated reasoning in the modal logic S4 and the problem of unavoidable words.
Extensions of KTB: The modal logic KTB is the logic of reflexive and symmetric frames. Dually, KTB-algebras have a unary (normal) operator f that satisfies the identities f (x){u2265}x and {u231D}x{u2264}f ({u231D}f(x)). Extensions of KTB are subvarieties of the algebra KTB. Both of these form a lattice, and we investigate the structure of the bottom of the lattice of subvarieties. The unique atom is known to correspond to the modal logic whose frame is a single reflexive point. Yutaka demonstrated that this atom has a unique cover, corresponding to the frame of the two element chain. We construct covers of this element, and so demonstrate that there are a continuum of such covers.
Normal Forms in Modal Logic: Fine proposed the use of normal forms as an alternative to traditional methods of determining Kripke completeness. We expand on this paper and demonstrate the application of normal forms to a number of traditional modal logics, and define new terms needed to apply normal forms in this situation.
Automated reasoning in 84: History based methods for automated reasoning are well understood and accepted. Pliu{u0161}kevi{u010D}ius & Pliu{u0161}kevi{u010D}ien{u0117} propose a new, potentially revolutionary method of applying marks and indices to sequents. We show that the method is flawed, and empirically compare a different mark/index based method to the traditional methods instead.
Unavoidable words: The unavoidable words problem is concerned with repetition in strings of symbols. There are two main ways to identify a word as unavoidable, one based on generalised pattern matching and one from an algorithm. Both methods are in NP, but do not appear to be in P. We define the simple unavoidable words as a subset of the standard unavoidable words that can be identified by the algorithm in P-time. We define depth separating IX x homomorphisms as an easy way to generate a subset of the unavoidable words using the pattern matching method. We then show that the two simpler problems are equivalent to each other