Well-founded Functions and Extreme Predicates in Dafny: A Tutorial

A recursive function is well defined if its every recursive callcorresponds a decrease in some well-founded order. Such a function issaid to be _terminating_ and is in many applications the standard wayto define a function. A boolean function can also be defined asan extreme solution to a recurrence relation, that is, as a least orgreatest fixpoint of some functor. Such _extreme predicates_ areuseful to encode a set of inductive or coinductive inference rulesand are at the core of many a constructive logic. Theverification-aware programming language Dafny supports bothterminating functions and extreme predicates. This tutorialdescribes the difference in general terms, and then describes novelsyntactic support in Dafny for defining and proving lemmas withextreme predicates. Various examples and considerations are given.Although Dafny's verifier has at its core a first-order SMT solver,Dafny's logical encoding makes it possible to reason about fixpointsin an automated way.

Definition of Flat Poset and Existence Theorems for Recursive Call

Summary This text includes the definition and basic notions of product of posets, chain-complete and flat posets, flattening operation, and the existence theorems of recursive call using the flattening operator. First part of the article, devoted to product and flat posets has a purely mathematical quality. Definition 3 allows to construct a flat poset from arbitrary non-empty set [12] in order to provide formal apparatus which eanbles to work with recursive calls within the Mizar langauge. To achieve this we extensively use technical Mizar functors like BaseFunc or RecFunc. The remaining part builds the background for information engineering approach for lists, namely recursive call for posets [21].We formalized some facts from Chapter 8 of this book as an introduction to the next two sections where we concentrate on binary product of posets rather than on a more general case.

The Calculation of Product Carbon Footprint and Optimization Design

With more and more governments and organizations taking Carbon Footprint as the measure of greenhouse gas emission, the study about the calculation of carbon footprint has become a hot spot. The paper analyzed the carbon footprint in different stages of a product life circle, including manufacturing, transporting, using and disposing and also studied the part contributing the largest carbon emission. Especially in the calculation of carbon emission of manufacturing stage, recursive call algorithm was applied. The optimization design model of carbon footprint was also depicted. All the work this paper had undertaken facilitates to formulate specific carbon emission reduction measures.

Transforming Recursion to Iteration in Programming

The main advantage of a Recursive Algorithm (an algorithm defined in terms of itself) is that it can be easily described and easily implemented in a programming language (van Breughel, 1997). On the other hand, the efficiency of such an algorithm is relatively low because for every recursive call not yet terminated, a number of data should be maintained in a stack, causing time delays and requiring higher memory space (Rohl, 1984). Solving the same problem iteratively instead of recursively can improve time and space efficiency. For example, to solve a problem that involves N recursive procedure calls, it will require stack space linear to N. On the contrary, using iteration, the program will need a constant amount of space, independent of the number of iterations. There are programming languages, such as Prolog, that do not possess built-in iterative structures and so recursion should be used instead. Nevertheless, there are ways to write recursive programs that have similar behaviour with that of the corresponding iterative programs.

Chip Architecture for Data Sorting Using Recursive Algorithm

“This paper suggests a way to implement recursive algorithm on hardware with an example of sorting of numeric data. Every recursive call/return needs a mechanism to store/restore parameters, local variables and return addresses respectively. Also a control sequence is needed to control the flow of execution as in case of recursive call and recursive return. The number of states required for the execution of a recursion in hardware can be reduced compared with software. This paper describes all the details that are required to implement recursive algorithm in hardware. For implementation, all the entities are designed using VHDL and are synthesized, configured on Spartan-2 XC2S200-5PQ208. “