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69-397: The Glasgow Haskell Compiler ( GHC ) is a native or machine code compiler for the functional programming language Haskell . It provides a cross-platform software environment for writing and testing Haskell code and supports many extensions, libraries , and optimisations that streamline the process of generating and executing code. GHC is the most commonly used Haskell compiler. It

138-766: A capture-avoiding manner. This is defined so that: For example, ( λ x . x ) [ y := y ] = λ x . ( x [ y := y ] ) = λ x . x {\displaystyle (\lambda x.x)[y:=y]=\lambda x.(x[y:=y])=\lambda x.x} , and ( ( λ x . y ) x ) [ x := y ] = ( ( λ x . y ) [ x := y ] ) ( x [ x := y ] ) = ( λ x . y ) y {\displaystyle ((\lambda x.y)x)[x:=y]=((\lambda x.y)[x:=y])(x[x:=y])=(\lambda x.y)y} . The freshness condition (requiring that y {\displaystyle y}

207-492: A code obfuscation technique as a measure against disassembly and tampering. The principle is also used in shared code sequences of fat binaries which must run on multiple instruction-set-incompatible processor platforms. This property is also used to find unintended instructions called gadgets in existing code repositories and is used in return-oriented programming as alternative to code injection for exploits such as return-to-libc attacks . In some computers,

276-510: A function . Lambda calculus may be untyped or typed . In typed lambda calculus, functions can be applied only if they are capable of accepting the given input's "type" of data. Typed lambda calculi are weaker than the untyped lambda calculus, which is the primary subject of this article, in the sense that typed lambda calculi can express less than the untyped calculus can. On the other hand, typed lambda calculi allow more things to be proven. For example, in simply typed lambda calculus , it

345-582: A strictness analyzer and language extensions such as monadic I/O , mutable arrays, unboxed data types, concurrent and parallel programming models (such as software transactional memory and data parallelism ) and a profiler . Peyton Jones, and Marlow, later moved to Microsoft Research in Cambridge , where they continued to be primarily responsible for developing GHC. GHC also contains code from more than three hundred other contributors. From 2009 to about 2014, third-party contributions to GHC were funded by

414-486: A machine with a single accumulator , the accumulator is implicitly both the left operand and result of most arithmetic instructions. Some other architectures, such as the x86 architecture, have accumulator versions of common instructions, with the accumulator regarded as one of the general registers by longer instructions. A stack machine has most or all of its operands on an implicit stack. Special purpose instructions also often lack explicit operands; for example, CPUID in

483-433: A one-to-one mapping to machine code. The assembly language decoding method is called disassembly . Machine code may be decoded back to its corresponding high-level language under two conditions: The first condition is to accept an obfuscated reading of the source code. An obfuscated version of source code is displayed if the machine code is sent to a decompiler of the source language. The second condition requires

552-400: A paging based system, if the current page actually holds machine code by an execute bit — pages have multiple such permission bits (readable, writable, etc.) for various housekeeping functionality. E.g. on Unix-like systems memory pages can be toggled to be executable with the mprotect() system call, and on Windows, VirtualProtect() can be used to achieve a similar result. If an attempt

621-626: A postcard: Dear Professor Church, Russell had the iota operator , Hilbert had the epsilon operator . Why did you choose lambda for your operator? According to Scott, Church's entire response consisted of returning the postcard with the following annotation: " eeny, meeny, miny, moe ". Computable functions are a fundamental concept within computer science and mathematics. The lambda calculus provides simple semantics for computation which are useful for formally studying properties of computation. The lambda calculus incorporates two simplifications that make its semantics simple. The first simplification

690-524: A set of simpler local transformations such as constant folding and beta reduction . The back end of the compiler transforms Core code into an internal representation of C--, via an intermediate language STG (short for "Spineless Tagless G-machine"). The C-- code can then take one of three routes: it is either printed as C code for compilation with GCC , converted directly into native machine code (the traditional " code generation " phase), or converted to LLVM IR for compilation with LLVM. In all three cases,

759-426: A single input. For example, can be reworked into This method, known as currying , transforms a function that takes multiple arguments into a chain of functions each with a single argument. Function application of the s q u a r e _ s u m {\textstyle \operatorname {square\_sum} } function to the arguments (5, 2), yields at once whereas evaluation of

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828-444: A tag and a Y field. In addition to transfer (branch) instructions, these machines have skip instruction that conditionally skip one or two words, e.g., Compare Accumulator with Storage (CAS) does a three way compare and conditionally skips to NSI, NSI+1 or NSI+2, depending on the result. The MIPS architecture provides a specific example for a machine code whose instructions are always 32 bits long. The general type of instruction

897-476: A typed intermediate language known as "Core" (based on System F , extended with let and case expressions). Core has been extended to support generalized algebraic datatypes in its type system , and is now based on an extension to System F known as System F C . In the tradition of type-directed compiling, GHC's simplifier, or "middle end", where most of the optimizations implemented in GHC are performed,

966-458: A universal lambda function, such as Iota and Jot , which can create any function behavior by calling it on itself in various combinations. Lambda calculus is Turing complete , that is, it is a universal model of computation that can be used to simulate any Turing machine . Its namesake, the Greek letter lambda (λ), is used in lambda expressions and lambda terms to denote binding a variable in

1035-480: A variable that has not been bound, such as the term λ x . ( x + y ) {\displaystyle \lambda x.(x+y)} (which represents the function definition f ( x ) = x + y {\displaystyle f(x)=x+y} ). In this term, the variable y has not been defined and is considered an unknown. The abstraction λ x . ( x + y ) {\displaystyle \lambda x.(x+y)}

1104-625: Is alpha equivalence . It captures the intuition that the particular choice of a bound variable, in an abstraction, does not (usually) matter. For instance, λ x . x {\displaystyle \lambda x.x} and λ y . y {\displaystyle \lambda y.y} are alpha-equivalent lambda terms, and they both represent the same function (the identity function). The terms x {\displaystyle x} and y {\displaystyle y} are not alpha-equivalent, because they are not bound in an abstraction. In many presentations, it

1173-444: Is computer code consisting of machine language instructions , which are used to control a computer's central processing unit (CPU). For conventional binary computers , machine code is the binary representation of a computer program which is actually read and interpreted by the computer. A program in machine code consists of a sequence of machine instructions (possibly interspersed with data). Each machine code instruction causes

1242-625: Is free and open-source software released under a BSD license . GHC originally begun in 1989 as a prototype, written in Lazy ML (LML) by Kevin Hammond at the University of Glasgow . Later that year, the prototype was completely rewritten in Haskell, except for its parser , by Cordelia Hall, Will Partain, and Simon Peyton Jones. Its first beta release was on 1 April 1991. Later releases added

1311-598: Is M applied to N, where M {\displaystyle M} is the lambda term ( λ x . ( λ x . x ) ) {\displaystyle (\lambda x.(\lambda x.x))} being applied to the input N {\displaystyle N} which is x {\displaystyle x} . Both examples 1 and 2 would evaluate to the identity function λ x . x {\displaystyle \lambda x.x} . In lambda calculus, functions are taken to be ' first class values ', so functions may be used as

1380-611: Is a lambda term. That is, a lambda term is valid if and only if it can be obtained by repeated application of these three rules. For convenience, some parentheses can be omitted when writing a lambda term. For example, the outermost parentheses are usually not written. See § Notation , below, for an explicit description of which parentheses are optional. It is also common to extend the syntax presented here with additional operations, which allows making sense of terms such as λ x . x 2 . {\displaystyle \lambda x.x^{2}.} The focus of this article

1449-595: Is a placeholder in both examples. Here, example 1 defines a function λ x . B {\displaystyle \lambda x.B} , where B {\displaystyle B} is ( λ x . x ) x {\displaystyle (\lambda x.x)x} , an anonymous function ( λ x . x ) {\displaystyle (\lambda x.x)} , with input x {\displaystyle x} ; while example 2, M {\displaystyle M}   N {\displaystyle N} ,

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1518-413: Is a syntactically valid term and represents a function that adds its input to the yet-unknown y . Parentheses may be used and might be needed to disambiguate terms. For example, The examples 1 and 2 denote different terms, differing only in where the parentheses are placed. They have different meanings: example 1 is a function definition, while example 2 is a function application. The lambda variable x

1587-552: Is a theorem that every evaluation strategy terminates for every simply typed lambda-term, whereas evaluation of untyped lambda-terms need not terminate (see below ). One reason there are many different typed lambda calculi has been the desire to do more (of what the untyped calculus can do) without giving up on being able to prove strong theorems about the calculus. Lambda calculus has applications in many different areas in mathematics , philosophy , linguistics , and computer science . Lambda calculus has played an important role in

1656-514: Is an abstraction representing the function f {\displaystyle f} defined by f ( x ) = x 2 + 2 , {\displaystyle f(x)=x^{2}+2,} using the term x 2 + 2 {\displaystyle x^{2}+2} for t . The name f {\displaystyle f} is superfluous when using abstraction. The syntax ( λ x . t ) {\displaystyle (\lambda x.t)} binds

1725-477: Is generally different from bytecode (also known as p-code), which is either executed by an interpreter or itself compiled into machine code for faster (direct) execution. An exception is when a processor is designed to use a particular bytecode directly as its machine code, such as is the case with Java processors . Machine code and assembly code are sometimes called native code when referring to platform-dependent parts of language features or libraries. From

1794-449: Is given by the op (operation) field, the highest 6 bits. J-type (jump) and I-type (immediate) instructions are fully specified by op . R-type (register) instructions include an additional field funct to determine the exact operation. The fields used in these types are: rs , rt , and rd indicate register operands; shamt gives a shift amount; and the address or immediate fields contain an operand directly. For example, adding

1863-428: Is made to execute machine code on a non-executable page, an architecture specific fault will typically occur. Treating data as machine code , or finding new ways to use existing machine code, by various techniques, is the basis of some security vulnerabilities. Similarly, in a segment based system, segment descriptors can indicate whether a segment can contain executable code and in what rings that code can run. From

1932-407: Is not available. The majority of programs today are written in a high-level language . A high-level program may be translated into machine code by a compiler . Every processor or processor family has its own instruction set . Instructions are patterns of bits , digits, or characters that correspond to machine commands. Thus, the instruction set is specific to a class of processors using (mostly)

2001-425: Is rarely a problem. Systems may also differ in other details, such as memory arrangement, operating systems, or peripheral devices . Because a program normally relies on such factors, different systems will typically not run the same machine code, even when the same type of processor is used. A processor's instruction set may have fixed-length or variable-length instructions. How the patterns are organized varies with

2070-460: Is simply mapped to itself. The second simplification is that the lambda calculus only uses functions of a single input. An ordinary function that requires two inputs, for instance the s q u a r e _ s u m {\textstyle \operatorname {square\_sum} } function, can be reworked into an equivalent function that accepts a single input, and as output returns another function, that in turn accepts

2139-593: Is structured as a series of source-to-source transformations on Core code. The analyses and transformations performed in this compiler stage include demand analysis (a generalization of strictness analysis ), application of user-defined rewrite rules (including a set of rules included in GHC's standard libraries that performs foldr/build fusion ), unfolding (called " inlining " in more traditional compilers), let-floating , an analysis that determines which function arguments can be unboxed, constructed product result analysis , specialization of overloaded functions, and

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2208-401: Is that the lambda calculus treats functions "anonymously;" it does not give them explicit names. For example, the function can be rewritten in anonymous form as (which is read as "a tuple of x and y is mapped to x 2 + y 2 {\textstyle x^{2}+y^{2}} "). Similarly, the function can be rewritten in anonymous form as where the input

2277-441: Is the pure lambda calculus without extensions, but lambda terms extended with arithmetic operations are used for explanatory purposes. An abstraction λ x . t {\displaystyle \lambda x.t} denotes an § anonymous function that takes a single input x and returns t . For example, λ x . ( x 2 + 2 ) {\displaystyle \lambda x.(x^{2}+2)}

2346-468: Is typically set to a hard coded value when the CPU is first powered on, and will hence execute whatever machine code happens to be at this address. Similarly, the program counter can be set to execute whatever machine code is at some arbitrary address, even if this is not valid machine code. This will typically trigger an architecture specific protection fault. The CPU is oftentimes told, by page permissions in

2415-448: Is usual to identify alpha-equivalent lambda terms. The following definitions are necessary in order to be able to define β-reduction: The free variables of a term are those variables not bound by an abstraction. The set of free variables of an expression is defined inductively: For example, the lambda term representing the identity λ x . x {\displaystyle \lambda x.x} has no free variables, but

2484-575: The IA-32 instruction set; and the PowerPC 615 microprocessor, which can natively process both PowerPC and x86 instruction sets. Machine code is a strictly numerical language, and it is the lowest-level interface to the CPU intended for a programmer. Assembly language  provides a direct map between the numerical machine code and a human-readable mnemonic. In assembly, numerical opcodes and operands are replaced with mnemonics and labels. For example,

2553-459: The Kleene–Rosser paradox . Subsequently, in 1936 Church isolated and published just the portion relevant to computation, what is now called the untyped lambda calculus. In 1940, he also introduced a computationally weaker, but logically consistent system, known as the simply typed lambda calculus . Until the 1960s when its relation to programming languages was clarified, the lambda calculus

2622-539: The Kruskal count , sometimes possible through opcode-level programming to deliberately arrange the resulting code so that two code paths share a common fragment of opcode sequences. These are called overlapping instructions , overlapping opcodes , overlapping code , overlapped code , instruction scission , or jump into the middle of an instruction . In the 1970s and 1980s, overlapping instructions were sometimes used to preserve memory space. One example were in

2691-599: The Zilog Z80 processor, the machine code 00000101 , which causes the CPU to decrement the B general-purpose register , would be represented in assembly language as DEC B . The IBM 704, 709, 704x and 709x store one instruction in each instruction word; IBM numbers the bit from the left as S, 1, ..., 35. Most instructions have one of two formats: For all but the IBM 7094 and 7094 II, there are three index registers designated A, B and C; indexing with multiple 1 bits in

2760-405: The runtime system for Haskell, essential to run programs, is written in C and C-- . GHC's front end , incorporating the lexer , parser and typechecker , is designed to preserve as much information about the source language as possible until after type inference is complete, toward the goal of providing clear error messages to users. After type checking, the Haskell code is desugared into

2829-416: The x86 architecture has available the 0x90 opcode; it is represented as NOP in the assembly source code . While it is possible to write programs directly in machine code, managing individual bits and calculating numerical addresses is tedious and error-prone. Therefore, programs are rarely written directly in machine code. However, an existing machine code program may be edited if the assembly source code

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2898-558: The CPU to perform a specific task. Examples of such tasks include: In general, each architecture family (e.g., x86 , ARM ) has its own instruction set architecture (ISA), and hence its own specific machine code language. There are exceptions, such as the VAX architecture, which includes optional support of the PDP-11 instruction set; the IA-64 architecture, which includes optional support of

2967-480: The Industrial Haskell Group. Since early releases the official website has referred to GHC as The Glasgow Haskell Compiler , whereas in the executable version command it is identified as The Glorious Glasgow Haskell Compilation System . This has been reflected in the documentation. Initially, it had the internal name of The Glamorous Glasgow Haskell Compiler . GHC is written in Haskell , but

3036-410: The curried version requires one more step to arrive at the same result. The lambda calculus consists of a language of lambda terms , that are defined by a certain formal syntax, and a set of transformation rules for manipulating the lambda terms. These transformation rules can be viewed as an equational theory or as an operational definition . As described above, having no names, all functions in

3105-459: The development of the theory of programming languages . Functional programming languages implement lambda calculus. Lambda calculus is also a current research topic in category theory . Lambda calculus was introduced by mathematician Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics . The original system was shown to be logically inconsistent in 1935 when Stephen Kleene and J. B. Rosser developed

3174-763: The function λ x . y {\displaystyle \lambda x.y} x {\displaystyle x} has a single free variable, y {\displaystyle y} . Suppose t {\displaystyle t} , s {\displaystyle s} and r {\displaystyle r} are lambda terms, and x {\displaystyle x} and y {\displaystyle y} are variables. The notation t [ x := r ] {\displaystyle t[x:=r]} indicates substitution of r {\displaystyle r} for x {\displaystyle x} in t {\displaystyle t} in

3243-568: The function that always returns y {\displaystyle y} , no matter the input. As an example of a function operating on functions, the function composition can be defined as λ f . λ g . λ x . ( f ( g x ) ) {\displaystyle \lambda f.\lambda g.\lambda x.(f(gx))} . There are several notions of "equivalence" and "reduction" that allow lambda terms to be "reduced" to "equivalent" lambda terms. A basic form of equivalence, definable on lambda terms,

3312-469: The implementation of error tables in Microsoft 's Altair BASIC , where interleaved instructions mutually shared their instruction bytes. The technique is rarely used today, but might still be necessary to resort to in areas where extreme optimization for size is necessary on byte-level such as in the implementation of boot loaders which have to fit into boot sectors . It is also sometimes used as

3381-476: The inputs, or be returned as outputs from other functions. For example, the lambda term λ x . x {\displaystyle \lambda x.x} represents the identity function , x ↦ x {\displaystyle x\mapsto x} . Further, λ x . y {\displaystyle \lambda x.y} represents the constant function x ↦ y {\displaystyle x\mapsto y} ,

3450-549: The lambda calculus are anonymous functions. They only accept one input variable, so currying is used to implement functions of several variables. The syntax of the lambda calculus defines some expressions as valid lambda calculus expressions and some as invalid, just as some strings of characters are valid C programs and some are not. A valid lambda calculus expression is called a " lambda term ". The following three rules give an inductive definition that can be applied to build all syntactically valid lambda terms: Nothing else

3519-533: The language specification, or they redefine existing constructs. As such, each extension may not be supported by all Haskell implementations. There is an ongoing effort to describe extensions and select those which will be included in future versions of the language specification. The extensions supported by the Glasgow Haskell Compiler include: An expressive static type system is one of the major defining features of Haskell. Accordingly, much of

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3588-526: The machine code of the architecture is implemented by an even more fundamental underlying layer called microcode , providing a common machine language interface across a line or family of different models of computer with widely different underlying dataflows . This is done to facilitate porting of machine language programs between different models. An example of this use is the IBM System/360 family of computers and their successors. Machine code

3657-491: The machine code to have information about the source code encoded within. The information includes a symbol table that contains debug symbols . The symbol table may be stored within the executable, or it may exist in separate files. A debugger can then read the symbol table to help the programmer interactively debug the machine code in execution . Beta reduction Lambda calculus consists of constructing lambda terms and performing reduction operations on them. In

3726-412: The other four index registers. The effective address is normally Y-C(T), where C(T) is either 0 for a tag of 0, the logical or of the selected index regisrs in multiple tag mode or the selected index register if not in multiple tag mode. However, the effective address for index register control instructions is just Y. A flag with both bits 1 selects indirect addressing; the indirect address word has both

3795-529: The particular architecture and type of instruction. Most instructions have one or more opcode fields that specify the basic instruction type (such as arithmetic, logical, jump , etc.), the operation (such as add or compare), and other fields that may give the type of the operand (s), the addressing mode (s), the addressing offset(s) or index, or the operand value itself (such constant operands contained in an instruction are called immediate ). Not all machines or individual instructions have explicit operands. On

3864-657: The point of view of a process , the code space is the part of its address space where the code in execution is stored. In multitasking systems this comprises the program's code segment and usually shared libraries . In multi-threading environment, different threads of one process share code space along with data space, which reduces the overhead of context switching considerably as compared to process switching. Various tools and methods exist to decode machine code back to its corresponding source code . Machine code can easily be decoded back to its corresponding assembly language source code because assembly language forms

3933-504: The point of view of the CPU, machine code is stored in RAM, but is typically also kept in a set of caches for performance reasons. There may be different caches for instructions and data, depending on the architecture. The CPU knows what machine code to execute, based on its internal program counter. The program counter points to a memory address and is changed based on special instructions which may cause programmatic branches. The program counter

4002-412: The registers 1 and 2 and placing the result in register 6 is encoded: Load a value into register 8, taken from the memory cell 68 cells after the location listed in register 3: Jumping to the address 1024: On processor architectures with variable-length instruction sets (such as Intel 's x86 processor family) it is, within the limits of the control-flow resynchronizing phenomenon known as

4071-416: The result of a constant expression freed up by replacing it by that constant) and other code enhancements. A much more human-friendly rendition of machine language, named assembly language , uses mnemonic codes to refer to machine code instructions, rather than using the instructions' numeric values directly, and uses symbolic names to refer to storage locations and sometimes registers . For example, on

4140-453: The resultant native code is finally linked against the GHC runtime system to produce an executable. GHC complies with the language standards, both Haskell 98 and Haskell 2010 . It also supports many optional extensions to the Haskell standard: for example, the software transactional memory (STM) library, which allows for Composable Memory Transactions . Many extensions to Haskell have been proposed. These provide features not described in

4209-439: The same architecture . Successor or derivative processor designs often include instructions of a predecessor and may add new additional instructions. Occasionally, a successor design will discontinue or alter the meaning of some instruction code (typically because it is needed for new purposes), affecting code compatibility to some extent; even compatible processors may show slightly different behavior for some instructions, but this

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4278-473: The simplest form of lambda calculus, terms are built using only the following rules: The reduction operations include: If De Bruijn indexing is used, then α-conversion is no longer required as there will be no name collisions. If repeated application of the reduction steps eventually terminates, then by the Church–Rosser theorem it will produce a β-normal form . Variable names are not needed if using

4347-443: The tag subtracts the logical or of the selected index registers and loading with multiple 1 bits in the tag loads all of the selected index registers. The 7094 and 7094 II have seven index registers, but when they are powered on they are in multiple tag mode , in which they use only the three of the index registers in a fashion compatible with earlier machines, and require a Leave Multiple Tag Mode ( LMTM ) instruction in order to access

4416-489: The variable x in the term t . The definition of a function with an abstraction merely "sets up" the function but does not invoke it. An application t {\displaystyle t}   s {\displaystyle s} represents the application of a function t to an input s , that is, it represents the act of calling function t on input s to produce t ( s ) {\displaystyle t(s)} . A lambda term may refer to

4485-637: The way, why did Church choose the notation “λ”? In [an unpublished 1964 letter to Harald Dickson] he stated clearly that it came from the notation “ x ^ {\displaystyle {\hat {x}}} ” used for class-abstraction by Whitehead and Russell , by first modifying “ x ^ {\displaystyle {\hat {x}}} ” to “ ∧ x {\displaystyle \land x} ” to distinguish function-abstraction from class-abstraction, and then changing “ ∧ {\displaystyle \land } ” to “λ” for ease of printing. This origin

4554-611: The work in extending the language has been directed towards data types and type classes . The Glasgow Haskell Compiler supports an extended type system based on the theoretical System F C . Major extensions to the type system include: Extensions relating to type classes include: Versions of GHC are available for several system or computing platform , including Windows and most varieties of Unix (such as Linux , FreeBSD , OpenBSD , and macOS ). GHC has also been ported to several different processor architectures . Machine code In computer programming , machine code

4623-406: The x86 architecture writes values into four implicit destination registers. This distinction between explicit and implicit operands is important in code generators, especially in the register allocation and live range tracking parts. A good code optimizer can track implicit and explicit operands which may allow more frequent constant propagation , constant folding of registers (a register assigned

4692-454: Was also reported in [Rosser, 1984, p.338]. On the other hand, in his later years Church told two enquirers that the choice was more accidental: a symbol was needed and λ just happened to be chosen. Dana Scott has also addressed this question in various public lectures. Scott recounts that he once posed a question about the origin of the lambda symbol to Church's former student and son-in-law John W. Addison Jr., who then wrote his father-in-law

4761-498: Was only a formalism. Thanks to Richard Montague and other linguists' applications in the semantics of natural language, the lambda calculus has begun to enjoy a respectable place in both linguistics and computer science. There is some uncertainty over the reason for Church's use of the Greek letter lambda (λ) as the notation for function-abstraction in the lambda calculus, perhaps in part due to conflicting explanations by Church himself. According to Cardone and Hindley (2006): By

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