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Many programmers use dependently typed languages such as Coq to machine-verify high-assurance software. However, existing compilers for these languages provide no guarantees after compiling, nor when linking after compilation. Type-preserving compilers preserve guarantees encoded in types and then use type checking to verify compiled code and ensure safe linking with external code. Unfortunately, standard compiler passes do not preserve the dependent typing of commonly used (intensional) type theories. This is because assumptions valid in simpler type systems no longer hold, and intensional dependent type systems are highly sensitive to syntactic changes, including compilation. We develop an A-normal form (ANF) translation with join-point optimization—a standard translation for making control flow explicit in functional languages—from the Extended Calculus of Constructions (ECC) with dependent elimination of booleans and natural numbers (a representative subset of Coq). Our dependently typed target language has equality reflection, allowing the type system to encode semantic equality of terms. This is key to proving type preservation and correctness of separate compilation for this translation. This is the first ANF translation for dependent types. Unlike related translations, it supports the universe hierarchy, and does not rely on parametricity or impredicativity.
Throughout the COVID-19 pandemic, flexible working has become the norm for many workers. This volume offers an original examination of flexible working using data from 30 European countries and drawing on studies conducted in Australia, the US and India. Rather than providing a better work-life balance, the book reveals how flexible working can lead to exploitation, which manifests differently for women and men, such as more care responsibilities or increased working hours.
Introduction to first-order Predicate Logic. Syntax and semantics, models and quantification. Implementation of “Term,” “Formula,” and “Model” objects for Predicate Logic in Python.
Gödel numbering and Gödel’s Incompleteness Theorem. Undecidability of the halting problem and how it can be used to prove Gödel’s Incompleteness Theorem.
An introduction to formal Logic and to the contrast and connections between syntax and semantics. The unique pedagogical approach of the book of capturing logic in a computer program and of proving by programming.
In this chapter we fix a specific axiomatic system and provide several examples of its formal use. These include logical inferences such as syllogisms, proofs of mathematical theorems in structures such as groups and fields, and foundations of mathematics, specifically Peano Arithmetic and Set Theory.
The semantics of propositional formulas: models and truth tables, tautologies and contradictions. Disjunctive Normal Form and the DNF Theorem. Optional: Conjunctive Normal Form and the CNF Theorem, computational search problems, satisfiability, P and NP.
The coordinated motion between the press and the feeding mechanism directly determines the production efficiency of the high-speed stamping line. In order to generate the high-performance trajectory of the feeding mechanism, this paper investigates the optimization of the trajectory with the local variable period. Based on the quintic B-spline curve and normal distribution, the smooth interpolation method of variable-time interval is proposed to generate the collision-free and energy-jerk-minimal trajectory with variable-time intervals. The advantage of the proposed method is that it can make the feeding mechanism transition smoothly between districts of variable-time and fixed-time intervals. It is beneficial to avoid re-performing the entire process of trajectory planning. ADAMS and actual experiments are used to validate the effectiveness of the proposed method. Results show that the proposed method can maintain the high performance of the initial trajectory, and there is no sharp point in the displacement-time and velocity-time curves. The investigation provides a new direction for the direct generation of local variable-period trajectories in the multi-machine coordination of the stamping line.