Until recently the method has not extended to security-style properties, principally because classical refinement semantics is inadequate in security contexts . View via Publisher. Alternate Sources. Save to Library. Create Alert. Share This Paper. Topics from this paper.
Semantics computer science Non-interference security Natural language Formal methods. Level of detail Pseudocode Bridging networking Personally identifiable information. Citations Publications citing this paper. An algebraic framework for reasoning about security Solofomampionona Fortunat Rajaona. Common patterns of recursion can be abstracted away using higher-order functions, with catamorphisms and anamorphisms or "folds" and "unfolds" being the most obvious examples. Such recursion schemes play a role analogous to built-in control structures such as loops in imperative languages.
Most general purpose functional programming languages allow unrestricted recursion and are Turing complete , which makes the halting problem undecidable , can cause unsoundness of equational reasoning , and generally requires the introduction of inconsistency into the logic expressed by the language's type system. Some special purpose languages such as Coq allow only well-founded recursion and are strongly normalizing nonterminating computations can be expressed only with infinite streams of values called codata.
As a consequence, these languages fail to be Turing complete and expressing certain functions in them is impossible, but they can still express a wide class of interesting computations while avoiding the problems introduced by unrestricted recursion.
Functional programming limited to well-founded recursion with a few other constraints is called total functional programming. Functional languages can be categorized by whether they use strict eager or non-strict lazy evaluation, concepts that refer to how function arguments are processed when an expression is being evaluated.
The technical difference is in the denotational semantics of expressions containing failing or divergent computations. Under strict evaluation, the evaluation of any term containing a failing subterm fails. For example, the expression:. Under lazy evaluation, the length function returns the value 4 i. In brief, strict evaluation always fully evaluates function arguments before invoking the function. Lazy evaluation does not evaluate function arguments unless their values are required to evaluate the function call itself. The usual implementation strategy for lazy evaluation in functional languages is graph reduction.
Hughes argues for lazy evaluation as a mechanism for improving program modularity through separation of concerns , by easing independent implementation of producers and consumers of data streams. Especially since the development of Hindley—Milner type inference in the s, functional programming languages have tended to use typed lambda calculus , rejecting all invalid programs at compilation time and risking false positive errors , as opposed to the untyped lambda calculus , that accepts all valid programs at compilation time and risks false negative errors , used in Lisp and its variants such as Scheme , though they reject all invalid programs at runtime, when the information is enough to not reject valid programs.
The use of algebraic datatypes makes manipulation of complex data structures convenient; the presence of strong compile-time type checking makes programs more reliable in absence of other reliability techniques like test-driven development , while type inference frees the programmer from the need to manually declare types to the compiler in most cases. Some research-oriented functional languages such as Coq , Agda , Cayenne , and Epigram are based on intuitionistic type theory , which lets types depend on terms.
Such types are called dependent types. These type systems do not have decidable type inference and are difficult to understand and program with.
Some examples include: Trying several different phone numbers to reach a person, perhaps depending on time of day or day of week. The courses will use the new development environment described in the next section. In the Python community, we receive many such contributions including complete foreign-language translations of key documents completely unsolicited! Bondy and U. Preserving information flow properties under refinement Heiko Mantel.
Through the Curry—Howard isomorphism , then, well-typed programs in these languages become a means of writing formal mathematical proofs from which a compiler can generate certified code. While these languages are mainly of interest in academic research including in formalized mathematics , they have begun to be used in engineering as well.
Compcert is a compiler for a subset of the C programming language that is written in Coq and formally verified. A limited form of dependent types called generalized algebraic data types GADT's can be implemented in a way that provides some of the benefits of dependently typed programming while avoiding most of its inconvenience. Functional programs do not have assignment statements, that is, the value of a variable in a functional program never changes once defined. This eliminates any chances of side effects because any variable can be replaced with its actual value at any point of execution.
So, functional programs are referentially transparent. Let us say that the initial value of x was 1 , then two consecutive evaluations of the variable x yields 10 and respectively. In fact, assignment statements are never referentially transparent. Functional programs exclusively use this type of function and are therefore referentially transparent.
In C , anonymous classes are not necessary, because closures and lambdas are fully supported.
Libraries and language extensions for immutable data structures are being developed to aid programming in the functional style in C. Many object-oriented design patterns are expressible in functional programming terms: for example, the strategy pattern simply dictates use of a higher-order function, and the visitor pattern roughly corresponds to a catamorphism , or fold.
Similarly, the idea of immutable data from functional programming is often included in imperative programming languages,  for example the tuple in Python, which is an immutable array. Purely functional data structures are often represented in a different way than their imperative counterparts. Arrays can be replaced by maps or random access lists, which admit purely functional implementation, but have logarithmic access and update times.
Purely functional data structures have persistence , a property of keeping previous versions of the data structure unmodified. In Clojure, persistent data structures are used as functional alternatives to their imperative counterparts. Persistent vectors, for example, use trees for partial updating. Calling the insert method will result in some but not all nodes being created.
Functional programming is very different from imperative programming.
Pure functional programming completely prevents side-effects and provides referential transparency. Higher-order functions are rarely used in older imperative programming. A traditional imperative program might use a loop to traverse and modify a list. There are tasks for example, maintaining a bank account balance that often seem most naturally implemented with state.
The pure functional programming language Haskell implements them using monads , derived from category theory. While existing monads may be easy to apply in a program, given appropriate templates and examples, many students find them difficult to understand conceptually, e. Functional languages also simulate states by passing around immutable states.
This can be done by making a function accept the state as one of its parameters, and return a new state together with the result, leaving the old state unchanged. Impure functional languages usually include a more direct method of managing mutable state. Clojure , for example, uses managed references that can be updated by applying pure functions to the current state. This kind of approach enables mutability while still promoting the use of pure functions as the preferred way to express computations.
Alternative methods such as Hoare logic and uniqueness have been developed to track side effects in programs. Some modern research languages use effect systems to make the presence of side effects explicit. Functional programming languages are typically less efficient in their use of CPU and memory than imperative languages such as C and Pascal.
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Flat arrays may be accessed very efficiently with deeply pipelined CPUs, prefetched efficiently through caches with no complex pointer chasing , or handled with SIMD instructions. It is also not easy to create their equally efficient general-purpose immutable counterparts. For purely functional languages, the worst-case slowdown is logarithmic in the number of memory cells used, because mutable memory can be represented by a purely functional data structure with logarithmic access time such as a balanced tree.
For programs that perform intensive numerical computations, functional languages such as OCaml and Clean are only slightly slower than C according to The Computer Language Benchmarks Game.
Immutability of data can in many cases lead to execution efficiency by allowing the compiler to make assumptions that are unsafe in an imperative language, thus increasing opportunities for inline expansion. Lazy evaluation may also speed up the program, even asymptotically, whereas it may slow it down at most by a constant factor however, it may introduce memory leaks if used improperly.
Launchbury  discusses theoretical issues related to memory leaks from lazy evaluation, and O'Sullivan et al.
However, the most general implementations of lazy evaluation making extensive use of dereferenced code and data perform poorly on modern processors with deep pipelines and multi-level caches where a cache miss may cost hundreds of cycles [ citation needed ]. Imperative programs have the environment and a sequence of steps manipulating the environment.
Functional programs have an expression that is successively substituted until it reaches normal form. An example illustrates this with different solutions to the same programming goal calculating Fibonacci numbers. Printing a list with first 10 Fibonacci numbers, with generators.
Printing first 10 Fibonacci numbers,functional expression style .