Concept# Gödel's completeness theorem

Summary

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.
The completeness theorem applies to any first-order theory: If T is such a theory, and φ is a sentence (in the same language) and every model of T is a model of φ, then there is a (first-order) proof of φ using the statements of T as axioms. One sometimes says this as "anything universally true is provable". This does not contradict Gödel's incompleteness theorem, which shows that some formula φu is unprovable although true in the natural numbers, which are a particular model of a first-order theory describing them — φu is just false in some other model of the first-order theory being considered (such as a non-standard model of arithmetic for Peano arithmetic). This kind of failure of consistency between a standard and non-standard model is also called Omega Inconsistency.
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Nicolaus Correll, Alcherio Martinoli

For the multi-robot coverage problem determin- istic deliberative as well as probabilistic approaches have been proposed. Whereas deterministic approaches usually provide provable completeness and promise good performance under perfect conditions, probabilistic approaches are more robust to sensor and actuator noise, but completion cannot be guaranteed and performance is sub-optimal in terms of time to completion. In reality, however, almost all deterministic algorithms for robot coordination can be considered probabilistic when considering the unpredictability of real world factors. This paper investigates experimentally and analytically how probabilistic and deterministic algorithms can be combined for maintaining the robustness of probabilistic approaches, and explicitly model the reliability of a robotic platform. Using realistic simulation and data from real robot experiments, we study system performance of a swarm-robotic inspection system at different levels of noise (wheel-slip). The prediction error of a purely deterministic model increases when the assumption of perfect sensors and actuators is violated, whereas a combination of probabilistic and deterministic models provides a better match with experimental data.

2007MLsub extends traditional Hindley-Milner type inference with subtyping while preserving compact principal types, an exciting new development. However, its specification in terms of biunification is difficult to understand, relying on the new concepts of bisubstitution and polar types, and making use of advanced notions from abstract algebra. In this paper, we show that these are in fact not essential to understanding the mechanisms at play in MLsub. We propose an alternative algorithm called Simple-sub, which can be implemented efficiently in under 500 lines of code (including parsing, simplification, and pretty-printing), looks more familiar, and is easier to understand. We present an experimental evaluation of Simple-sub against MLsub on a million randomly-generated well-scoped expressions, showing that the two systems agree. The mutable automaton-based implementation of MLsub is quite far from its algebraic specification, leaving a lot of space for errors; in fact, our evaluation uncovered several bugs in it. We sketch more straightforward soundness and completeness arguments for Simple-sub, based on a syntactic specification of the type system. This paper is meant to be light in formalism, rich in insights, and easy to consume for prospective designers of new type systems and programming languages. In particular, no abstract algebra is inflicted on readers.

2020Walid Amanhoud, Aude Billard, Mohamed Bouri, Jacob Hernandez Sanchez

Foot devices have been ubiquitously used in surgery to control surgical equipment. Most common applications are foot switches for electro-surgery, endoscope positioning and tele-robotic consoles. Switches fall short of providing continuous control as required for precise use of instruments. We developed a haptic foot interface to provide continuous assistance in surgical procedures. This paper concerns the foot control of simultaneous five degrees of freedom (DoF) of a surgical laparoscopic gripper. We assess systematically precision at controlling position and orientation at the target and closing of the forceps. Our controller provides position:position mapping between the foot and the robotic tool, as well as haptic feedback, compensating for gravity of the lower limb of the operator so as to alleviate fatigue. A dynamic model compensation and closed loop force feedback is used to achieve high transparency and backdrivability. The assistance is based on a novel type of haptic fixtures combining spring-damper with selective dynamic compensation in the direction aligned with the task of grasping, so as to simplify control of certain poses, made difficult due to the coupling between human lower limbs’ DoF’s. We experimentally evaluated the control strategy with six users on a position control surgical task in simulation. Results show the proposed assistance greatly eases the foot grasping task leading to higher completeness, efficiency, and lower mental and physical load.

2021