PSPACE Bounds for Rank-1 Modal Logics

For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACE-bounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant proof-theoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way

Instantiation of SMT problems modulo Integers

Many decision procedures for SMT problems rely more or less implicitly on an instantiation of the axioms of the theories under consideration, and differ by making use of the additional properties of each theory, in order to increase efficiency. We present a new technique for devising complete instantiation schemes on SMT problems over a combination of linear arithmetic with another theory T. The method consists in first instantiating the arithmetic part of the formula, and then getting rid of the remaining variables in the problem by using an instantiation strategy which is complete for T. We provide examples evidencing that not only is this technique generic (in the sense that it applies to a wide range of theories) but it is also efficient, even compared to state-of-the-art instantiation schemes for specific theories.Comment: Research report, long version of our AISC 2010 pape

Expressiveness of the modal mu-calculus on monotone neighborhood structures

We characterize the expressive power of the modal mu-calculus on monotone neighborhood structures, in the style of the Janin-Walukiewicz theorem for the standard modal mu-calculus. For this purpose we consider a monadic second-order logic for monotone neighborhood structures. Our main result shows that the monotone modal mu-calculus corresponds exactly to the fragment of this second-order language that is invariant for neighborhood bisimulations

Combining Spatial and Temporal Logics: Expressiveness vs. Complexity

In this paper, we construct and investigate a hierarchy of spatio-temporal formalisms that result from various combinations of propositional spatial and temporal logics such as the propositional temporal logic PTL, the spatial logics RCC-8, BRCC-8, S4u and their fragments. The obtained results give a clear picture of the trade-off between expressiveness and computational realisability within the hierarchy. We demonstrate how different combining principles as well as spatial and temporal primitives can produce NP-, PSPACE-, EXPSPACE-, 2EXPSPACE-complete, and even undecidable spatio-temporal logics out of components that are at most NP- or PSPACE-complete

Tropically convex constraint satisfaction

A semilinear relation S is max-closed if it is preserved by taking the componentwise maximum. The constraint satisfaction problem for max-closed semilinear constraints is at least as hard as determining the winner in Mean Payoff Games, a notorious problem of open computational complexity. Mean Payoff Games are known to be in the intersection of NP and co-NP, which is not known for max-closed semilinear constraints. Semilinear relations that are max-closed and additionally closed under translations have been called tropically convex in the literature. One of our main results is a new duality for open tropically convex relations, which puts the CSP for tropically convex semilinaer constraints in general into NP intersected co-NP. This extends the corresponding complexity result for scheduling under and-or precedence constraints, or equivalently the max-atoms problem. To this end, we present a characterization of max-closed semilinear relations in terms of syntactically restricted first-order logic, and another characterization in terms of a finite set of relations L that allow primitive positive definitions of all other relations in the class. We also present a subclass of max-closed constraints where the CSP is in P; this class generalizes the class of max-closed constraints over finite domains, and the feasibility problem for max-closed linear inequalities. Finally, we show that the class of max-closed semilinear constraints is maximal in the sense that as soon as a single relation that is not max-closed is added to L, the CSP becomes NP-hard.Comment: 29 pages, 2 figure

Delta-Complete Decision Procedures for Satisfiability over the Reals

We introduce the notion of "\delta-complete decision procedures" for solving SMT problems over the real numbers, with the aim of handling a wide range of nonlinear functions including transcendental functions and solutions of Lipschitz-continuous ODEs. Given an SMT problem \varphi and a positive rational number \delta, a \delta-complete decision procedure determines either that \varphi is unsatisfiable, or that the "\delta-weakening" of \varphi is satisfiable. Here, the \delta-weakening of \varphi is a variant of \varphi that allows \delta-bounded numerical perturbations on \varphi. We prove the existence of \delta-complete decision procedures for bounded SMT over reals with functions mentioned above. For functions in Type 2 complexity class C, under mild assumptions, the bounded \delta-SMT problem is in NP^C. \delta-Complete decision procedures can exploit scalable numerical methods for handling nonlinearity, and we propose to use this notion as an ideal requirement for numerically-driven decision procedures. As a concrete example, we formally analyze the DPLL framework, which integrates Interval Constraint Propagation (ICP) in DPLL(T), and establish necessary and sufficient conditions for its \delta-completeness. We discuss practical applications of \delta-complete decision procedures for correctness-critical applications including formal verification and theorem proving.Comment: A shorter version appears in IJCAR 201

Global Numerical Constraints on Trees

We introduce a logical foundation to reason on tree structures with constraints on the number of node occurrences. Related formalisms are limited to express occurrence constraints on particular tree regions, as for instance the children of a given node. By contrast, the logic introduced in the present work can concisely express numerical bounds on any region, descendants or ancestors for instance. We prove that the logic is decidable in single exponential time even if the numerical constraints are in binary form. We also illustrate the usage of the logic in the description of numerical constraints on multi-directional path queries on XML documents. Furthermore, numerical restrictions on regular languages (XML schemas) can also be concisely described by the logic. This implies a characterization of decidable counting extensions of XPath queries and XML schemas. Moreover, as the logic is closed under negation, it can thus be used as an optimal reasoning framework for testing emptiness, containment and equivalence