Haken proved that every resolution refutation of the pigeonhole formula has at least exponential size. Groote and Zantema proved that a particular OBDD computation of the pigeonhole formula has an exponential size. Here we show that any arbitrary OBDD refutation of the pigeonhole formula has an exponential size, too: we prove that the size of one of the intermediate OBDDs is Ω(1.025 n ).

We present GDPLL, a generalization of the DPLL procedure. It solves the satisfiability problem for decidable fragments of quantifier-free first-order logic. Sufficient conditions are identified for proving soundness, termination and completeness of GDPLL. We show how the original DPLL procedure is an instance. Subsequently the GDPLL instances for equality logic, and the logic of equality over infinite ground term algebras are presented. Based on this, we implemented a decision procedure for inductive datatypes. We provide some new benchmarks, in order to compare variant

Abstract. In this paper we present a formalization of the theory of hybrid automata and algorithms for building trajectory trees using module types and functors in the Coq proof assistant. PreliminariesHybrid systems are systems in which there is a significant interaction between the continuous and discrete parts. Many of the applications of hybrid systems are safety critical and require the guarantee of a safe operation. The problem of safety verification seeks an answer to the reachability problem: is there a potentially unsafe state reachable from an initial state?The notion of a hybrid automaton was introduced in order to extend verification methods towards the systems with continuous and discrete dynamics [1]. A Hybrid automaton can be defined as a tuple H = (DS, n, S 0 , I, φ, G, R) with the following components: DS is a finite set of discrete locations; n ≥ 0 is the dimension of H. The state space of H is S := DS × R n . Each state has thus the form (l, x), where l ∈ DS and x ∈ R n . S 0 ⊆ S is a set of initial states. I : DS → P(R n ) assigns to each location l an invariant set I(l) ⊆ R n . φ : (DS × R × R ≥0 ) n → R n defines the flow of a system in a discrete location with an initial condition φ(l, x 0 , 0) = x 0 . φ is a vector of n functionsn describes a guard condition. R : DS × DS × R n → R n is a reset function. The semantics of a hybrid automaton is given by the transition system [2].Our method is based on decomposing the continuous state space according to an n-dimensional rectangular grid. We denote by χ an abstract state, by S a the set of all abstract states, and by S a 0 the set of initial abstract states. Definition 1 (Strict Abstract Transition System -SAT S). A hybrid automaton H = (DS, n, S 0 , I, φ, G, R) and an abstract state space S a generate the strict abstract transition system T sats = {S a , ; c , ; d , S a 0 } with ⋆

Groote and Zantema proved that a particular OBDD computation of the pigeonhole formula has exponential size, and that limited OBDD derivations cannot simulate resolution polynomially. Here we show that an arbitrary OBDD refutation of the pigeonhole formula has exponential size: we prove that for any order of computation at least one intermediate OBDD in the proof has size Ω(1.14 n). We also present a family of CNFs that show an exponential blow-up for all OBDD refutations compared to unrestricted resolution refutations.

Modularity and decontextualisation are core principles of a service-oriented architecture. However, the principles are often lost when it comes to an implementation of services, as a result of a rigidly defined service interface. The interface, which defines a data format, is typically specific to a particular context and its change entails significant redevelopment costs. This paper focuses on a two-fold problem. On the one hand, the interface description language must be flexible enough for maintaining service compatibility in a variety of different contexts without modification of the service itself. On the other hand, the composition of interfaces in a distributed environment must be provably consistent. The existing approaches for checking compatibility of service choreographies are either inflexible (WS-CDL and WSCI) or require behaviour specification associated with each service, which is often impossible to provide in practice. We present a novel approach for automatic interface configuration in distributed stream-connected components operating as closed-source services (i.e. the behavioural protocol is unknown). We introduce a Message Definition Language (MDL), which can extend the existing interfaces description languages, such as WSDL, with support of subtyping, inheritance and polymorphism. The MDL supports configuration variables that link input and output interfaces of a service and propagate requirements over an application graph. We present an algorithm that solves the interface reconciliation problem using constraint satisfaction that relies on Boolean satisfiability as a subproblem.