59 research outputs found

    Neural Networks Reduction via Lumping

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    The increasing size of recently proposed Neural Networks makes it hard to implement them on embedded devices, where memory, battery and computational power are a non-trivial bottleneck. For this reason during the last years network compression literature has been thriving and a large number of solutions has been published to reduce both the number of operations and the parameters involved with the models. Unfortunately, most of these reducing techniques are actually heuristic methods and usually require at least one re-training step to recover the accuracy. The need of procedures for model reduction is well-known also in the fields of Verification and Performances Evaluation, where large efforts have been devoted to the definition of quotients that preserve the observable underlying behaviour. In this paper we try to bridge the gap between the most popular and very effective network reduction strategies and formal notions, such as lumpability, introduced for verification and evaluation of Markov Chains. Elaborating on lumpability we propose a pruning approach that reduces the number of neurons in a network without using any data or fine-tuning, while completely preserving the exact behaviour. Relaxing the constraints on the exact definition of the quotienting method we can give a formal explanation of some of the most common reduction techniques

    Calculi for higher order communicating systems

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    This thesis develops two Calculi for Higher Order Communicating Systems. Both calculi consider sending and receiving processes to be as fundamental as nondeterminism and parallel composition. The first calculus called CHOCS is an extension of Milner's CCS in the sense that all the constructions of CCS are included or may be derived from more fundamental constructs. Most of the mathematical framework of CCS carries over almost unchanged. The operational semantics of CHOCS is given as a labelled transition system and it is a direct extension of the semantics of CCS with value passing. A set of algebraic laws satisfied by the calculus is presented. These are similar to the CCS laws only introducing obvious extra laws for sending and receiving processes. The power of process passing is underlined by a result showing that the recursion operator is unnecessary in the sense that recursion can be simulated by means of process passing and communication. The CHOCS language is also studied by means of a denotational semantics. A major result is the full abstractness of this semantics with respect to the operational semantics. The denotational semantics is used to provide an easy proof of the simulation of recursion. Introducing processes as first class objects yields a powerful metalanguage. It is shown that it is possible to simulate various reduction strategies of the untyped λ-Calculus in CHOCS. As pointed out by Milner, CCS has its limitations when one wants to describe unboundedly expanding systems, e.g. an unbounded number of procedure invocations in an imperative concurrent programming language P with recursive procedures. CHOCS may neatly describe both call-by-value and call-by-reference parameter mechanisms for P. We also consider call-by-name and lazy parameter mechanisms for P. The second calculus is called Plain CHOCS. Essential to the new calculus is the treatment of restriction as a static binding operator on port names. This calculus is given an operational semantics using labelled transition systems which combines ideas from the applicative transition systems described by Abramsky and the transition systems used for CHOCS. This calculus enjoys algebraic properties which are similar to those of CHOCS only needing obvious extra laws for the static nature of the restriction operator. Processes as first class objects enable description of networks with changing interconnection structure and there is a close connection between the Plain CHOCS calculus and the π-Calculus described by Milner, Parrow and Walker: the two calculi can simulate one another. Recently object oriented programming has grown into a major discipline in computational practice as well as in computer science. From a theoretical point of view object oriented programming presents a challenge to any metalanguage since most object oriented languages have no formal semantics. We show how Plain CHOCS may be used to give a semantics to a prototype object oriented language called 0.Open Acess

    Fifty years of similarity relations: a survey of foundations and applications

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    On the occasion of the 50th anniversary of the publication of Zadeh's significant paper Similarity Relations and Fuzzy Orderings, an account of the development of similarity relations during this time will be given. Moreover, the main topics related to these fuzzy relations will be reviewed.Peer ReviewedPostprint (author's final draft

    Decidability Results for the Boundedness Problem

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    We prove decidability of the boundedness problem for monadic least fixed-point recursion based on positive monadic second-order (MSO) formulae over trees. Given an MSO-formula phi(X,x) that is positive in X, it is decidable whether the fixed-point recursion based on phi is spurious over the class of all trees in the sense that there is some uniform finite bound for the number of iterations phi takes to reach its least fixed point, uniformly across all trees. We also identify the exact complexity of this problem. The proof uses automata-theoretic techniques. This key result extends, by means of model-theoretic interpretations, to show decidability of the boundedness problem for MSO and guarded second-order logic (GSO) over the classes of structures of fixed finite tree-width. Further model-theoretic transfer arguments allow us to derive major known decidability results for boundedness for fragments of first-order logic as well as new ones

    Equivalence of infinite-state systems with silent steps

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    This dissertation contributes to analysis methods for infinite-state systems. The dissertation focuses on equivalence testing for two relevant classes of infinite-state systems: commutative context-free processes, and one-counter automata. As for equivalence notions, we investigate the classical bisimulation and simulation equivalences. The important point is that we allow for silent steps in the model, abstracting away from internal, unobservable actions. Very few decidability results have been known so far for bisimulation or simulation equivalence for infinite-state systems with silent steps, as presence of silent steps makes the equivalence problem arguably harder to solve. A standard technique for bisimulation or simulation equivalence testing is to use the hierarchy of approximants. For an effective decision procedure the hierarchy must stabilize (converge) at level omega, the first limit ordinal, which is not the case for the models investigated in this thesis. However, according to a long-standing conjecture, the community believed that the convergence actually takes place at level omega+ omega in the class of commutative context free processes. We disprove the conjecture and provide a lower bound of omega * omega for the convergence level. We also show that all previously known positive decidability results for BPPs can be re-proven uniformly using the improved approximants techniques. Moreover dissertation contains an unsuccesfull attack on one of the main open problems in the area: decidability of weak bisimulation equivalence for commutative context-free processes. Our technical development of this section is not sufficient to solve the problem, but we believe it is a serious step towards a solution. Furtermore, we are able to show decidability of branching (stuttering) bisimulation equivalence, a slightly more discriminating variant of bisimulation equivalence. It is worth emphesizing that, until today, our result is the only known decidability result for bisimulation equivalence in a class of inifinite-state systems with silent steps that is not known to admit convergence of (some variant of) standard approximants at level omega. Finally we consider weak simulation equivalence over one-counter automata without zero tests (allowing zero tests implies undecidability). While weak bisimulation equivalence is known to be undecidable in this class, we prove a surprising result that weak simulation equivalence is actually decidable. Thus we provide a first example going against a trend, widely-believed by the community, that simulation equivalence tends to be computationally harder than bisimulation equivalence. In short words, the dissertation contains three new results, each of them solving a non-trivial open problem about equivalence testing of infinite-state systems with silent steps

    Robustly Complete Temporal Logic Control Synthesis for Nonlinear Systems

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    Modern systems such as spacecrafts and autonomous vehicles are complex yet safety-critical, and therefore the control methods that can deal with different dynamics and constraints while being provably correct are sought after. Formal methods are rigorous techniques originally used for developing and verifying finite-state systems with respect to specifications in formal languages. This thesis is concerned with using formal methods in control synthesis for nonlinear systems, which can guarantee the correctness of the resulting control strategies. For nonlinear continuous-state dynamical systems, formal control synthesis relies on finite abstractions of the original system by discretizing the system state space and over approximating system transitions. Without further assumptions, control synthesis is usually not complete in the way that no control strategies can be found even if there exists one. To deal with this problem, this thesis proposes a formal control synthesis approach that is sound and robustly complete in the sense that correct control strategies can be found whenever the specifications can be realized for the system with additional disturbance. Fundamental to the soundness and robust completeness is a fixed-point characterization of the winning set of the system with respect to a given specification, which is the set of initial conditions that can be controlled to satisfy the specification. Regarding discrete-time systems, such characterizations are first presented by using iterative computation of predecessors for basic linear temporal logic (LTL) specifications, including invariance, reachability and reach-and-stay. A more general class of LTL formulas, which can be translated into deterministic B\"uchi automata (DBA), is also considered, and an algorithm guided by the graph structure of the LTL-equivalent DBA is proposed for characterizing the winning set in this situation. It is then shown that the computational complexity of the algorithm can be reduced by using a pre-processing procedure to the graphs of the DBA. Because of the general nonlinearity, exact computation of winning sets is currently almost impossible. In this work, the conditions for set approximations are derived so that control synthesis is robustly complete. To meet such conditions, the proposed approach adopts interval arithmetic and a subdivision scheme in the approximation of predecessors. Under such a scheme, the system state space is adaptively partitioned with respect to both the given dynamics and specification and set approximation can be made arbitrarily precise to satisfy the robust completeness conditions. The proposed method is also shown applicable to sampled-data systems by computing validated solutions over one sampling period based on high-order Taylor expansion. Applications such as converter voltage regulation, parallel parking, and reactive locomotion planning problems are studied to show the effectiveness and efficiency of the proposed approach

    Syntax-Guided Optimal Synthesis for Chemical Reaction Networks

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    We study the problem of optimal syntax-guided synthesis of stochastic Chemical Reaction Networks (CRNs) that plays a fundamental role in design automation of molecular devices and in the construction of predictive biochemical models. We propose a sketching language for CRNs that concisely captures syntactic constraints on the network topology and allows its under-specification. Given a sketch, a correctness specification, and a cost function defined over the CRN syntax, our goal is to find a CRN that simultaneously meets the constraints, satisfies the specification and minimizes the cost function. To ensure computational feasibility of the synthesis process, we employ the Linear Noise Approximation allowing us to encode the synthesis problem as a satisfiability modulo theories problem over a set of parametric Ordinary Differential Equations (ODEs). We design and implement a novel algorithm for the optimal synthesis of CRNs that employs almost complete refutation procedure for SMT over reals and ODEs, and exploits a meta-sketching abstraction controlling the search strategy. Through relevant case studies we demonstrate that our approach significantly improves the capability of existing methods for synthesis of biochemical systems and paves the way towards their automated and provably-correct design
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