# Termination Of Nested And Mutually Recursive Algorithms

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### Computable ﬁxpoints in well-structured symbolic model checking

ranging from parameterized distributed algorithms [4] and channel systems [7] to hybrid systems [23] and list-manipulating programs [24]. In regular model checking, one works with regular sets of states and handles them via ﬁnite descriptions, e.g., ﬁnite-state automata or regular expressions. Models amenable to

### Challenges in Decomposing Encodings of Veriﬁcation Problems

In this formula, recursive procedures produce cyclic dependencies of their Summary f predicates. If abstractions are used to lazily solve the formula, the invariant Inv f and the termination argument RR f be-come interdependent. Similarly, invariants of nested loops are dependent on each other. Rewriting nested loops into a single loop with

### A Verifying Core for a Cryptographic Language Compiler

cally prove termination of µCryptol programs, including pro-grams with mutually-recursive cliques of streams. Our ver-ifying compiler has proved the correctness of its core trans-formations for multiple algorithms, including TEA, RC6, and AES. Finally, we describe an ACL2 book of primitive operations for the general speciﬁcation and

### Review of Algorithms for Data Mining

referred to as frequency set). Its core is based on a two-stage frequency set recursive algorithm thought. The basic idea of the algorithm is: first find all the frequent sets, these items appear frequently sets of at least a predefined minimum support same. Then set the frequency generated by the strong

### Anatomy of Alternating Quanti er Satis ability (Work in progress)

i, that are mutually inductive. The main idea is that F iV M iholds for the initialization step for QT, and F i):(ˇx i:(C0jM i)) also holds for each QT iteration; therefore, we strengthen M i in the end of QT and preserve the invariant at the same time. The if branches of QT return hC; M iiwhen M i cannot be strengthened any more. Moreover, QT

### Challenges in Decomposing Encodings of Veriﬁcation Problems

In this formula, recursive procedures produce cyclic dependencies of their Summaryf predicates. If abstractions are used to lazily solve the formula, the invariant Invf and the termination argument RRf be-come interdependent. Similarly, invariants of nested loops are dependent on each other. Rewriting nested

### Combining Induction and Saturation-Based Theorem Proving

allowing for example nested or mutually recursive proofs and arbitrary quantiﬁer al- ternation. However, it does not apply to saturation-based procedures which lack the

### The ACL2 Sedan Theorem Proving System

An interesting termination problem. beginners often write non-terminating programs, and then want to know why termination analysis failed. This lead us to develop an algorithm that generates a simpliﬁed version of the user s program that highlights the reason for the failure. We call the simpliﬁed program that is generated a termination

### Practical programming with total functions

recursive call is made on xs, which is known by the use of deconstruc-tor syntax to be a subcomponent of (x : xs). Conceptually, requiring structural recursion allows the compiler to infer a proof by induction of the recursive function s termination. As such, the second parameter of mapmay be described as an induction variable.

### Compositional Veriﬁcation with Abstraction, Learning, and SAT

nested calls and that use at most biterations of any given loop. We use the term bounded safety to refer to the problem of checking safety for a given 1In general, a procedure can call other procedures, perhaps in a mutually recursive way, in which case bounding the call-stack leads to a more systematic approach. See Chapter 2 for details. 2

### Embedding Cryptol in Higher Order Logic - Gilith

of the nested deﬁnitions tea sums, tea ys and tea zs. These are deﬁned using the technique of Section 3.2 by mutual recursion and equivalence proof. Naturally, since the function tea depends on these mutually recursive sequences, they must be deﬁned ﬁrst. 4 Summary This report has surveyed existing approaches to embedding Cryptol

### A Rude Contract Language for Web Services

have been veriﬁed. This expedient ensures the termination of the algorithm. Deﬁnition 3.2 Let ≤ be the least pre-order on capabilities containing io <:i and io <:o and let first(T)= S↓L(S ;S ) L The structural subschema relation <:A is the smallest relation closed under commutativity of unions and under the rules in Table 2.

### Automatic Generation of Distribution System One-line Diagrams

2. All branches are laid out in only two mutually perpendicular directions. 3. All branches of a lateral are in a straight line 4. The laterals meet one another only at the base nodes i.e no branches intersect. The proposed graph drawing approach consists of two major stages. The ﬁrst stage consists of identiﬁcation of laterals, de-

### A Brief Overview of HOL4 - TS Data61

Datatypes can be mutually and nested recursive and may also use record no-tation. At the term level, support is provided for deﬁning inductively speciﬁed predicates and relations; mutual recursion and inﬁnitary premises are allowed. Total recursive functions speciﬁed as recursion equations, possibly using ML-

CON of conditions. These classes are mutually recursive, i.e. each of themis defined recursively with respect to the other. Formally we should use here a set of BNF equations but for the sake of clarity we restrict ourselves to amore intuitive definition. EXP is the least syntactical class with the following properties: 1) IDE ^ EXP

### THE COMPLETION OF LOCALLY REFINED SIMPLICIAL PARTITIONS

recursive bisections to T is called a level descendant of T. One may verify that the reﬁnement edge of a tagged simplex of type 0 will not be further cut until the creation of level n+1 descendants. Generally, this is not true for a tagged simplex of type unequal to 0. Yet, an edge will never be cut on two consecutive levels.

### Anatomy of Alternating Quantiﬁer Satisﬁability (Work in progress)

Can formulas with nested quantiﬁers be checked effectively for satisﬁability? Several algorithms exist in the context of Quantiﬁed Boolean Formulas that handle alternation of quantiﬁers [12, 1]. They are specialized for eliminating variables over Booleans. An algorithm for alternating quantiﬁer satisﬁability

### Proceedings of the 16th Nordic Workshop on Programming Theory

jects inside cycles and/or recursive functions for Java smart cards. The byte code verier [4], for instance, does not verify properties related to memory usage. There are, undoubtedly, many ways of writing algorithms for detecting syntactic loops, mutually recursive methods and over-approximating the dynamic memory used for an assembler-like

### An Algorithm for Automated Generation of Invariants for Loops

Step.1: We obtain two nested-loop subsystems, each with one outer-loop and two inner loops. Step 2.1: We proceed with simulating the execution of nested loops by assigning the counter j to the main loop, j 1 to the ﬁrst inner loop and j 2 to the second inner loop. Step 2.2: For each nested loop system, rewrite the recursive

### Porting IgorII from MAUDE to HASKELL

termination by construction, handling arbitrary user-deﬁned data types, utilisation of arbitrary background knowledge, automatic invention of auxiliary functions as sub pro-grams, learning complex calling relationships (tree- and nested recursion), allowing for variables in the example equations, simultaneous induction of mutually recursive

### Chapter 1 The Braga Method: Extracting Certi ed Algorithms

terminating schemes of partial algorithms, nested or mutually recursive schemes, etc. The method is based on two main concepts linked together: an inductive description of the computational graph of an algorithm and an inductive characterization of its domain. The computational graph mimics the structure of recursive calls of the algorithm and

### staticif declaration

Handling termination conditions and degenerate cases, most often in recursive and mutually recursive templates. This necessity is cur-rently handled by using template specialization (such as in the classic compile-time factorial example, std::tuple, safe printfusing vari-adics, and many others). We consider this solution undesirable for

### Acomprehensivesurveyofimagesegmentation: clusteringmethods

The next condition implies that all the formed clusters will be mutually exclusive, i.e. a pixel will not be assigned to two clusters. The last condition states that the data values assigned to all the clusters will represent the complete image. In literature, there are number of clustering algorithms for image segmentation. How-

### Performance Analysis of Optimization Techniques for SQL Multi

operators by algorithms, and the costs of diverse sub-plans are estimated. For the given query, the number of plans to be considered is greater than two, because the relations may be retrieved by using either full scan or index scan, and the join has several implementations, such as nested-loop, sort-merge, or index join. One possible

### THE COMPLETION OF LOCALLY REFINED SIMPLICIAL PARTITIONS

recursive bisections to T is called a level descendent of T. Every di erent tagging of a simplex creates a di erent set of descendents. Remark 2.1. Thebisectionrulesin[Mau95,Tra97], andforn = 3in[B an91,Kos94], arenot equal to ours. If T has re nement edge fx0;x1g, then with our approachthe

### MATHEMATICS OF COMPUTATION Volume 77, Number 261, January

recursive bisections to T is called a level £ descendant of T. One may verify that the refinement edge of a tagged simplex of type 0 will not be further cut until the creation of level n + 1 descendants. Generally, this is not true for a tagged simplex of type unequal to 0. Yet, an edge will never be cut on two consecutive levels.

### Type Inference and Semi-Unification

even though we would expect, as sequential recursive definitions first of map and then of squarelist would yield, the type of map to be map: V a.VP.(a + /3) x cvlist + Plist If we were to use map in another line of the same mutually recursive definition with an argument type

### What s In a Name? Names, Objects, and Bindings Names, Scope

Other ﬁt algorithms Segregation of objects by size More clever data structures Heap Variants Memory pools: Differently-managed heap areas Stack-based pool: only free whole pool at once Nice for build-once data structures Single-size-object pool: Fit, allocation, etc. much faster Good for object-oriented programs Fragmentation malloc

### Memory consumption analysis of Java smart cards

algorithm which determines potential loops and (mutually) recursive methods. The algorithm operates on the bytecode of an applet and is written as a set of rules associating one or more constraints to each bytecode instruction. The rules are designed so that a certiﬁed analyser could be extracted from their proof of correctness.

### The Generalized Counting Method for Recursive Logic Queries

efficiency and, often, to guarantee the termination, of the forward chaining exe- cution phase that implements the fixpoint computation for the recursive query. A

### Recursion Unrolling for Divide and Conquer Programs

Recursion re-rolling rolls back the recursive part of the pro-cedure to ensure that a large unrolled base case is always executed, regardless of the problem size. 1.1 Divide and Conquer Programs We have applied recursion unrolling to divide and conquer programs [10, 8, 5]. Divide and conquer algorithms solve

### Efﬁcient Modular SAT Solving for IC3

recursive call for each of its children. Alg. 1 is related to several recent algorithms. For example, in the case of exactly two modules, Alg. 1 is equivalent to (the simplest case of) the prooﬂess interpolant computing algorithm introduced in [6], and it also resembles typical all-SAT procedures [12] and some 2QBF solvers [18].

### THE COMPLETION OF LOCALLY REFINED SIMPLICIAL PARTITIONS

been bisected. In this paper, we extend this result to bisection algorithms of n-simplices, with that generalizing the result concerning optimality of the adaptive ﬁnite element method to general space dimensions. 1. Introduction Nowadays, adaptive ﬁnite element methods are a popular tool for the numerical solution of boundary value problems.

### Recursive Functions with Higher Order Domains

recursive algorithms in constructive type theory (by simple we mean non-nested and non-mutually recursive). It was extended by Bove and Capretta [BC01] to treat nested recursion by using Dybjer s simultaneous induction-recursion, and by Bove [Bov02a] to treat mutually recursive algorithms, nested or not. A formal

### Modelling general recursion in type theory

The method was introduced in Bove (2001) to formalise simple general recursive algorithms in constructive type theory (by simple we mean non-nested and non-mutually recursive). It was extended in Bove and Capretta (2001) to treat nested recursion, and in Bove (2002b) to treat mutually recursive algorithms, nested or not. All these papers have

### Professor Shang-Hua Teng

tional Geometry and Applications, 9(6):517-532 (1999) Conference version: Workshop on Algorithms and Data Structures, (WADS), pp 188 199, 1993 (with M. Bern and D. Eppstein). Greedy algorithms for low energy and mutually distant sampling. J. Algorithms, 30: 52 67, January 1999. How good is recursive bisection.