Combination techniques and decision problems for disunification

Previous work on combination techniques considered the question of how to combine unification algorithms for disjoint equational theories E_{1} ,...,E_{n} in order to obtain a unification algorithm for the union E1 unified ... unified En of the theories. Here we want to show that variants of this method may be used to decide solvability and ground solvability of disunification problems in E_{1}cup...cup E_{n}. Our first result says that solvability of disunification problems in the free algebra of the combined theory E_{1}cup...cup E_{n} is decidable if solvability of disunification problems with linear constant restrictions in the free algebras of the theories E_{i}(i = 1,...,n) is decidable. In order to decide ground solvability (i.e., solvability in the initial algebra) of disunification problems in E_{1}cup...cup E_{n} we have to consider a new kind of subproblem for the particular theories Ei, namely solvability (in the free algebra) of disunification problems with linear constant restriction under the additional constraint that values of variables are not Ei-equivalent to variables. The correspondence between ground solvability and this new kind of solvability holds, (1) if one theory Ei is the free theory with at least one function symbol and one constant, or (2) if the initial algebras of all theories Ei are infinite. Our results can be used to show that the existential fragment of the theory of the (ground) term algebra modulo associativity of a finite number of function symbols is decidable; the same result follows for function symbols which are associative and commutative, or associative, commutative and idempotent

Unification Theory - An Introduction

Aus der Einleitung: „Equational unification is a generalization of syntactic unification in which semantic properties of function symbols are taken into account. For example, assume that the function symbol '+' is known to be commutative. Given the unication problem x + y ≐ a + b (where x and y are variables, and a and b are constants), an algorithm for syntactic unification would return the substitution {x ↦ a; y ↦ b} as the only (and most general) unifier: to make x + y and a + b syntactically equal, one must replace the variable x by a and y by b. However, commutativity of '+' implies that {x ↦ b; y ↦ b} also is a unifier in the sense that the terms obtained by its application, namely b + a and a + b, are equal modulo commutativity of '+'. More generally, equational unification is concerned with the problem of how to make terms equal modulo a given equational theory, which specifies semantic properties of the function symbols that occur in the terms to be unified.

Synthetic pheromone reduces the competitive ability of an invasive ant

Contains fulltext : 149035.pdf (preprint version ) (Open Access

Combination techniques and decision problems for disunification

Previous work on combination techniques considered the question of how to combine unification algorithms for disjoint equational theories E_{1} ,...,E_{n} in order to obtain a unification algorithm for the union E1 unified ... unified En of the theories. Here we want to show that variants of this method may be used to decide solvability and ground solvability of disunification problems in E_{1}cup...cup E_{n}. Our first result says that solvability of disunification problems in the free algebra of the combined theory E_{1}cup...cup E_{n} is decidable if solvability of disunification problems with linear constant restrictions in the free algebras of the theories E_{i}(i = 1,...,n) is decidable. In order to decide ground solvability (i.e., solvability in the initial algebra) of disunification problems in E_{1}cup...cup E_{n} we have to consider a new kind of subproblem for the particular theories Ei, namely solvability (in the free algebra) of disunification problems with linear constant restriction under the additional constraint that values of variables are not Ei-equivalent to variables. The correspondence between ground solvability and this new kind of solvability holds, (1) if one theory Ei is the free theory with at least one function symbol and one constant, or (2) if the initial algebras of all theories Ei are infinite. Our results can be used to show that the existential fragment of the theory of the (ground) term algebra modulo associativity of a finite number of function symbols is decidable; the same result follows for function symbols which are associative and commutative, or associative, commutative and idempotent

Insulin Glargine in the Intensive Care Unit: A Model-Based Clinical Trial Design

Online 4 Oct 2012Introduction: Current succesful AGC (Accurate Glycemic Control) protocols require extra clinical effort and are impractical in less acute wards where patients are still susceptible to stress-induced hyperglycemia. Long-acting insulin Glargine has the potential to be used in a low effort controller. However, potential variability in efficacy and length of action, prevent direct in-hospital use in an AGC framework for less acute wards. Method: Clinically validated virtual trials based on data from stable ICU patients from the SPRINT cohort who would be transferred to such an approach are used to develop a 24-hour AGC protocol robust to different Glargine potencies (1.0x, 1.5x and 2.0x regular insulin) and initial dose sizes (dose = total insulin over prior 12, 18 and 24 hours). Glycemic control in this period is provided only by varying nutritional inputs. Performance is assessed as %BG in the 4.0-8.0mmol/L band and safety by %BG<4.0mmol/L. Results: The final protocol consisted of Glargine bolus size equal to insulin over the previous 18 hours. Compared to SPRINT there was a 6.9% - 9.5% absolute decrease in mild hypoglycemia (%BG<4.0mmol/L) and up to a 6.2% increase in %BG between 4.0 and 8.0mmol/L. When the efficacy is known (1.5x assumed) there were reductions of: 27% BG measurements, 59% insulin boluses, 67% nutrition changes, and 6.3% absolute in mild hypoglycemia. Conclusion: A robust 24-48 clinical trial has been designed to safely investigate the efficacy and kinetics of Glargine as a first step towards developing a Glargine-based protocol for less acute wards. Ensuring robustness to variability in Glargine efficacy significantly affects the performance and safety that can be obtained

Nominal Unification of Higher Order Expressions with Recursive Let

A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in non-deterministic polynomial time. We also explore specializations like nominal letrec-matching for plain expressions and for DAGs and determine the complexity of corresponding unification problems.Comment: Pre-proceedings paper presented at the 26th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2016), Edinburgh, Scotland UK, 6-8 September 2016 (arXiv:1608.02534

Automated Synthesis of Tableau Calculi

This paper presents a method for synthesising sound and complete tableau calculi. Given a specification of the formal semantics of a logic, the method generates a set of tableau inference rules that can then be used to reason within the logic. The method guarantees that the generated rules form a calculus which is sound and constructively complete. If the logic can be shown to admit finite filtration with respect to a well-defined first-order semantics then adding a general blocking mechanism provides a terminating tableau calculus. The process of generating tableau rules can be completely automated and produces, together with the blocking mechanism, an automated procedure for generating tableau decision procedures. For illustration we show the workability of the approach for a description logic with transitive roles and propositional intuitionistic logic.Comment: 32 page

LNCS

We define the model-measuring problem: given a model M and specification φ, what is the maximal distance ρ such that all models M′ within distance ρ from M satisfy (or violate) φ. The model measuring problem presupposes a distance function on models. We concentrate on automatic distance functions, which are defined by weighted automata. The model-measuring problem subsumes several generalizations of the classical model-checking problem, in particular, quantitative model-checking problems that measure the degree of satisfaction of a specification, and robustness problems that measure how much a model can be perturbed without violating the specification. We show that for automatic distance functions, and ω-regular linear-time and branching-time specifications, the model-measuring problem can be solved. We use automata-theoretic model-checking methods for model measuring, replacing the emptiness question for standard word and tree automata by the optimal-weight question for the weighted versions of these automata. We consider weighted automata that accumulate weights by maximizing, summing, discounting, and limit averaging. We give several examples of using the model-measuring problem to compute various notions of robustness and quantitative satisfaction for temporal specifications

Conceptual Design of a Liquid Helium Vertical Test-Stand for 2m long Superconducting Undulator Coils

Superconducting Undulators (SCUs) can produce higher photon flux and cover a wider photon energy range compared to permanent magnet undulators (PMUs) with the same vacuum gap and period length. To build the know-how to implement superconducting undulators for future upgrades of the European XFEL facility, the test stand SUNDAE1 for the characterization of SCU is being developed. The purpose of SUNDAE1 is the training, tuning and development of new SCU coils by means of precise magnetic field measurements. The experimental setup will allow the characterization of magnets up to 2m in length. These magnets will be immersed in a Helium bath at 4K or 2K temperature. In this article, we describe the experimental setup and highlight its expected performances