Complexity classifications for different equivalence and audit problems for Boolean circuits

We study Boolean circuits as a representation of Boolean functions and consider different equivalence, audit, and enumeration problems. For a number of restricted sets of gate types (bases) we obtain efficient algorithms, while for all other gate types we show these problems are at least NP-hard.Comment: 25 pages, 1 figur

Quantifier-Free Boolean Algebra with Presburger Arithmetic is NP-Complete

Boolean Algebra with Presburger Arithmetic (BAPA) combines1) Boolean algebras of sets of uninterpreted elements (BA)and 2) Presburger arithmetic operations (PA). BAPA canexpress the relationship between integer variables andcardinalities of unbounded finite sets and can be used toexpress verification conditions in verification of datastructure consistency properties.In this report I consider the Quantifier-Free fragment ofBoolean Algebra with Presburger Arithmetic (QFBAPA).Previous algorithms for QFBAPA had non-deterministicexponential time complexity. In this report I show thatQFBAPA is in NP, and is therefore NP-complete. My resultyields an algorithm for checking satisfiability of QFBAPAformulas by converting them to polynomially sized formulasof quantifier-free Presburger arithmetic. I expect thisalgorithm to substantially extend the range of QFBAPAproblems whose satisfiability can be checked in practice

Knowledge Compilation of Logic Programs Using Approximation Fixpoint Theory

To appear in Theory and Practice of Logic Programming (TPLP), Proceedings of ICLP 2015 Recent advances in knowledge compilation introduced techniques to compile \emph{positive} logic programs into propositional logic, essentially exploiting the constructive nature of the least fixpoint computation. This approach has several advantages over existing approaches: it maintains logical equivalence, does not require (expensive) loop-breaking preprocessing or the introduction of auxiliary variables, and significantly outperforms existing algorithms. Unfortunately, this technique is limited to \emph{negation-free} programs. In this paper, we show how to extend it to general logic programs under the well-founded semantics. We develop our work in approximation fixpoint theory, an algebraical framework that unifies semantics of different logics. As such, our algebraical results are also applicable to autoepistemic logic, default logic and abstract dialectical frameworks

Querying Schemas With Access Restrictions

We study verification of systems whose transitions consist of accesses to a Web-based data-source. An access is a lookup on a relation within a relational database, fixing values for a set of positions in the relation. For example, a transition can represent access to a Web form, where the user is restricted to filling in values for a particular set of fields. We look at verifying properties of a schema describing the possible accesses of such a system. We present a language where one can describe the properties of an access path, and also specify additional restrictions on accesses that are enforced by the schema. Our main property language, AccLTL, is based on a first-order extension of linear-time temporal logic, interpreting access paths as sequences of relational structures. We also present a lower-level automaton model, Aautomata, which AccLTL specifications can compile into. We show that AccLTL and A-automata can express static analysis problems related to "querying with limited access patterns" that have been studied in the database literature in the past, such as whether an access is relevant to answering a query, and whether two queries are equivalent in the accessible data they can return. We prove decidability and complexity results for several restrictions and variants of AccLTL, and explain which properties of paths can be expressed in each restriction.Comment: VLDB201

Satisfiability Games for Branching-Time Logics

The satisfiability problem for branching-time temporal logics like CTL*, CTL and CTL+ has important applications in program specification and verification. Their computational complexities are known: CTL* and CTL+ are complete for doubly exponential time, CTL is complete for single exponential time. Some decision procedures for these logics are known; they use tree automata, tableaux or axiom systems. In this paper we present a uniform game-theoretic framework for the satisfiability problem of these branching-time temporal logics. We define satisfiability games for the full branching-time temporal logic CTL* using a high-level definition of winning condition that captures the essence of well-foundedness of least fixpoint unfoldings. These winning conditions form formal languages of \omega-words. We analyse which kinds of deterministic {\omega}-automata are needed in which case in order to recognise these languages. We then obtain a reduction to the problem of solving parity or B\"uchi games. The worst-case complexity of the obtained algorithms matches the known lower bounds for these logics. This approach provides a uniform, yet complexity-theoretically optimal treatment of satisfiability for branching-time temporal logics. It separates the use of temporal logic machinery from the use of automata thus preserving a syntactical relationship between the input formula and the object that represents satisfiability, i.e. a winning strategy in a parity or B\"uchi game. The games presented here work on a Fischer-Ladner closure of the input formula only. Last but not least, the games presented here come with an attempt at providing tool support for the satisfiability problem of complex branching-time logics like CTL* and CTL+