Optimal Tableau Algorithms for Coalgebraic Logics

ACM Trans. Comput. Logic

2022

Self Cite

We establish a generic upper bound ExpTime for reasoning with global assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier results of this kind, our bound does not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that potentially avoids building the entire exponential-sized space of candidate states, and thus offers a basis for practical reasoning. This algorithm still involves frequent fixpoint computations; we show how these can be handled efficiently in a concrete algorithm modelled on Liu and Smolka’s linear-time fixpoint algorithm. Finally, we show that the upper complexity bound is preserved under adding nominals to the logic, i.e., in coalgebraic hybrid logic.

Section: Introductionmentioning

confidence: 74%

Coalgebraic Reasoning with Global Assumptions in Arithmetic Modal Logics

Kupke

Pattinson

ACM Trans. Comput. Logic

2022

Self Cite

“…Related Work. Our algorithms use a semantic method, and as such complement earlier results on global caching in coalgebraic description logics that rely on tractable sets of tableau rules [21], which are not currently available for our leading examples. (In fact, [28] gives tableau-style axiomatizations of various logics of linear inequalities over the reals and over the integers; however, over the integers the rules appear to be incomplete: if ♯p denotes the integer weight of successors satisfying p, then the formula 2♯⊤ < 1 ∨ 2♯⊤ > 1 is clearly valid, but cannot be derived.…”

Section: Introductionmentioning

confidence: 74%

“…We now develop the type elimination algorithm from the preceding section into a global caching algorithm. Existing global caching algorithms work with systems of tableau rules (satisfiability is guaranteed if every applicable rule has at least one satisfiable conclusion) [21]. The fact that we work with a semantics-based decision procedure impacts on the design of the algorithm in two ways:…”

Section: Global Cachingmentioning

confidence: 99%

See 1 more Smart Citation

Reasoning with Global Assumptions in Arithmetic Modal Logics

Kupke

Pattinson

Fundamentals of Computation Theory

2015

Self Cite

Abstract. We establish a generic upper bound ExpTime for reasoning with global assumptions in coalgebraic modal logics. Unlike earlier results of this kind, we do not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that offers potential for practical reasoning.

“…Almost all instances of coalgebraic logics, including conditional logics, alternating-time/coalition logics, and probabilistic logics, have one-step satisfiability problems in P [19], so that the EXPTIME result applies in these cases. We have little doubt that using global caching, one can show an EXPTIME bound for socalled EXPTIME-tractable instances [9], which would cover essentially all cases of interest. For graded logics (whose one-step satisfiability problem is NP-complete), we have recent results proving an EXPTIME bound even at depth 2 [10].…”

Section: Lemma 13 (Truth)mentioning

confidence: 99%

Narcissists Are Easy, Stepmothers Are Hard

Gorín¹,

Foundations of Software Science and Computational Structures

2012

Abstract. Modal languages are well-known for their robust decidability and relatively low complexity. However, as soon as one adds a selfreferencing construct, like hybrid logic's down-arrow binder, to the basic modal language, decidability is lost, even if one restricts binding to a single variable. Here, we concentrate on the latter case and investigate the logics obtained by restricting the nesting depth of modalities between binding and use. In particular, for distances strictly below 3 we obtain well-behaved logics with a relatively high descriptive power. We investigate the fragment with distance 1 in the framework of coalgebraic modal logic, for which we provide very general decidability and complexity results. For the fragment with distance 2 we focus on the case of Kripke semantics and obtain optimum complexity bounds (no harder than the base logic). We show that this fragment is expressive enough to accommodate the guarded fragment over the correspondence language.