In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions. Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is always true, rather than enumerate many confirmatory cases. An unproved proposition, believed to be true is known as a conjecture. Proofs employ logic but include some amount of natural language which admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory; the distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, so-called folk mathematics.
The philosophy of mathematics is concerned with the role of language and logic in proofs, mathematics as a language. The word "proof" comes from the Latin probare meaning "to test". Related modern words are the English "probe", "probation", "probability", the Spanish probar, Italian provare, the German probieren; the early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony. Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof, it is that the idea of demonstrating a conclusion first arose in connection with geometry, which meant the same as "land measurement". The development of mathematical proof is the product of ancient Greek mathematics, one of the greatest achievements thereof. Thales and Hippocrates of Chios proved some theorems in geometry. Eudoxus and Theaetetus formulated did not prove them.
Aristotle said definitions should describe the concept being defined in terms of other concepts known. Mathematical proofs were revolutionized by Euclid, who introduced the axiomatic method still in use today, starting with undefined terms and axioms, used these to prove theorems using deductive logic, his book, the Elements, was read by anyone, considered educated in the West until the middle of the 20th century. In addition to theorems of geometry, such as the Pythagorean theorem, the Elements covers number theory, including a proof that the square root of two is irrational and that there are infinitely many prime numbers. Further advances took place in medieval Islamic mathematics. While earlier Greek proofs were geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, etc. for "lines."
He used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle. Alhazen developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures. There is no longer an assumption; as practiced, a proof is expressed in natural language and is a rigorous argument intended to convince the audience of the truth of a statement. The standard of rigor has varied throughout history. A proof can be presented differently depending on the intended audience. In order to gain acceptance, a proof has to meet communal statements of rigor; the concept of a proof is formalized in the field of mathematical logic. A formal proof is written in a formal language instead of a natural language. A formal proof is defined as sequence of formulas in a formal language, in which each formula is a logical consequence of preceding formulas.
Having a definition of formal proof makes the concept of proof amenable to study. Indeed, the field of proof theory studies formal proofs and their properties, for example, the property that a statement has a formal proof. An application of proof theory is to show; the definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated proof assistants, this is done in practice. A classic question in philosophy a
Machine code is a computer program written in machine language instructions that can be executed directly by a computer's central processing unit. Each instruction causes the CPU to perform a specific task, such as a load, a store, a jump, or an ALU operation on one or more units of data in CPU registers or memory. Machine code is a numerical language, intended to run as fast as possible, may be regarded as the lowest-level representation of a compiled or assembled computer program or as a primitive and hardware-dependent programming language. While it is possible to write programs directly in machine code, it is tedious and error prone to manage individual bits and calculate numerical addresses and constants manually. For this reason, programs are rarely written directly in machine code in modern contexts, but may be done for low level debugging, program patching, assembly language disassembly; the overwhelming majority of practical programs today are written in higher-level languages or assembly language.
The source code is translated to executable machine code by utilities such as compilers and linkers, with the important exception of interpreted programs, which are not translated into machine code. However, the interpreter itself, which may be seen as an executor or processor, performing the instructions of the source code consists of directly executable machine code. Machine code is by definition the lowest level of programming detail visible to the programmer, but internally many processors use microcode or optimise and transform machine code instructions into sequences of micro-ops, this is not considered to be a machine code per se; every processor or processor family has its own instruction set. Instructions are patterns of bits that by physical design correspond to different commands to the machine. Thus, the instruction set is specific to a class of processors using the same architecture. Successor or derivative processor designs include all the instructions of a predecessor and may add additional instructions.
A successor design will discontinue or alter the meaning of some instruction code, affecting code compatibility to some extent. Systems may differ in other details, such as memory arrangement, operating systems, or peripheral devices; because a program relies on such factors, different systems will not run the same machine code when the same type of processor is used. A processor's instruction set may have all instructions of the same length, or it may have variable-length instructions. How the patterns are organized varies with the particular architecture and also with the type of instruction. Most instructions have one or more opcode fields which specifies the basic instruction type and the actual operation and other fields that may give the type of the operand, the addressing mode, the addressing offset or index, or the actual value itself. Not all machines or individual instructions have explicit operands. An accumulator machine has a combined left operand and result in an implicit accumulator for most arithmetic instructions.
Other architectures have accumulator versions of common instructions, with the accumulator regarded as one of the general registers by longer instructions. A stack machine has all of its operands on an implicit stack. Special purpose instructions often lack explicit operands; this distinction between explicit and implicit operands is important in code generators in the register allocation and live range tracking parts. A good code optimizer can track implicit as well as explicit operands which may allow more frequent constant propagation, constant folding of registers and other code enhancements. A computer program is a list of instructions. A program's execution is done in order for the CPU, executing it to solve a specific problem and thus accomplish a specific result. While simple processors are able to execute instructions one after another, superscalar processors are capable of executing a variety of different instructions at once. Program flow may be influenced by special'jump' instructions that transfer execution to an instruction other than the numerically following one.
Conditional jumps are not depending on some condition. A much more readable rendition of machine language, called assembly language, uses mnemonic codes to refer to machine code instructions, rather than using the instructions' numeric values directly. For example, on the Zilog Z80 processor, the machine code 00000101, which causes the CPU to decrement the B processor register, would be represented in assembly language as DEC B; the MIPS architecture provides a specific example for a machine code whose instructions are always 32 bits long. The general type of instruction is given by the op field. J-type and I-type instructions are specified by op. R-type instructions include an additional field funct to determine the exact operation; the fields used in the
An operating system is system software that manages computer hardware and software resources and provides common services for computer programs. Time-sharing operating systems schedule tasks for efficient use of the system and may include accounting software for cost allocation of processor time, mass storage and other resources. For hardware functions such as input and output and memory allocation, the operating system acts as an intermediary between programs and the computer hardware, although the application code is executed directly by the hardware and makes system calls to an OS function or is interrupted by it. Operating systems are found on many devices that contain a computer – from cellular phones and video game consoles to web servers and supercomputers; the dominant desktop operating system is Microsoft Windows with a market share of around 82.74%. MacOS by Apple Inc. is in second place, the varieties of Linux are collectively in third place. In the mobile sector, use in 2017 is up to 70% of Google's Android and according to third quarter 2016 data, Android on smartphones is dominant with 87.5 percent and a growth rate 10.3 percent per year, followed by Apple's iOS with 12.1 percent and a per year decrease in market share of 5.2 percent, while other operating systems amount to just 0.3 percent.
Linux distributions are dominant in supercomputing sectors. Other specialized classes of operating systems, such as embedded and real-time systems, exist for many applications. A single-tasking system can only run one program at a time, while a multi-tasking operating system allows more than one program to be running in concurrency; this is achieved by time-sharing, where the available processor time is divided between multiple processes. These processes are each interrupted in time slices by a task-scheduling subsystem of the operating system. Multi-tasking may be characterized in co-operative types. In preemptive multitasking, the operating system slices the CPU time and dedicates a slot to each of the programs. Unix-like operating systems, such as Solaris and Linux—as well as non-Unix-like, such as AmigaOS—support preemptive multitasking. Cooperative multitasking is achieved by relying on each process to provide time to the other processes in a defined manner. 16-bit versions of Microsoft Windows used cooperative multi-tasking.
32-bit versions of both Windows NT and Win9x, used preemptive multi-tasking. Single-user operating systems have no facilities to distinguish users, but may allow multiple programs to run in tandem. A multi-user operating system extends the basic concept of multi-tasking with facilities that identify processes and resources, such as disk space, belonging to multiple users, the system permits multiple users to interact with the system at the same time. Time-sharing operating systems schedule tasks for efficient use of the system and may include accounting software for cost allocation of processor time, mass storage and other resources to multiple users. A distributed operating system manages a group of distinct computers and makes them appear to be a single computer; the development of networked computers that could be linked and communicate with each other gave rise to distributed computing. Distributed computations are carried out on more than one machine; when computers in a group work in cooperation, they form a distributed system.
In an OS, distributed and cloud computing context, templating refers to creating a single virtual machine image as a guest operating system saving it as a tool for multiple running virtual machines. The technique is used both in virtualization and cloud computing management, is common in large server warehouses. Embedded operating systems are designed to be used in embedded computer systems, they are designed to operate on small machines like PDAs with less autonomy. They are able to operate with a limited number of resources, they are compact and efficient by design. Windows CE and Minix 3 are some examples of embedded operating systems. A real-time operating system is an operating system that guarantees to process events or data by a specific moment in time. A real-time operating system may be single- or multi-tasking, but when multitasking, it uses specialized scheduling algorithms so that a deterministic nature of behavior is achieved. An event-driven system switches between tasks based on their priorities or external events while time-sharing operating systems switch tasks based on clock interrupts.
A library operating system is one in which the services that a typical operating system provides, such as networking, are provided in the form of libraries and composed with the application and configuration code to construct a unikernel: a specialized, single address space, machine image that can be deployed to cloud or embedded environments. Early computers were built to perform a series of single tasks, like a calculator. Basic operating system features were developed in the 1950s, such as resident monitor functions that could automatically run different programs in succession to speed up processing. Operating systems did not exist in their more complex forms until the early 1960s. Hardware features were added, that enabled use of runtime libraries and parallel processing; when personal computers became popular in the 1980s, operating systems were made for them similar in concept to those used on larger computers. In the 1940s, the earliest electronic digital systems had no operating systems.
Electronic systems of this time were programmed on rows of mechanical switches or by jumper wires on plug boards. These were special-purpose systems that, for example, generated ballistics tables for the military or controlled the pri
Matt Kaufmann is a Senior Research Scientist in the Department of Computer Sciences at the University of Texas at Austin, United States. He was a recipient of the 2005 ACM Software System Award along with Robert S. Boyer and J Strother Moore, for his work on the Boyer-Moore Theorem Prover. Matt Kaufmann homepage Matt Kaufmann at DBLP Bibliography Server
Free and open-source software
Free and open-source software is software that can be classified as both free software and open-source software. That is, anyone is licensed to use, copy and change the software in any way, the source code is shared so that people are encouraged to voluntarily improve the design of the software; this is in contrast to proprietary software, where the software is under restrictive copyright licensing and the source code is hidden from the users. FOSS maintains the software user's civil liberty rights. Other benefits of using FOSS can include decreased software costs, increased security and stability, protecting privacy and giving users more control over their own hardware. Free and open-source operating systems such as Linux and descendants of BSD are utilized today, powering millions of servers, desktops and other devices. Free-software licenses and open-source licenses are used by many software packages; the free-software movement and the open-source software movement are online social movements behind widespread production and adoption of FOSS.
"Free and open-source software" is an umbrella term for software, considered both Free software and open-source software. FOSS allows the user to inspect the source code and provides a high level of control of the software's functions compared to proprietary software; the term "free software" does not refer to the monetary cost of the software at all, but rather whether the license maintains the software user's civil liberties. There are a number of related terms and abbreviations for free and open-source software, or free/libre and open-source software. Although there is a complete overlap between free-software licenses and open-source-software licenses, there is a strong philosophical disagreement between the advocates of these two positions; the terminology of FOSS or "Free and Open-source software" was created to be a neutral on these philosophical disagreements between the FSF and OSI and have a single unified term that could refer to both concepts. As the Free Software Foundation explains the philosophical difference between free software and open-source software: "The two terms describe the same category of software, but they stand for views based on fundamentally different values.
Open-source is a development methodology. For the free-software movement, free software is an ethical imperative, essential respect for the users' freedom. By contrast, the philosophy of open-source considers issues in terms of how to make software “better”—in a practical sense only." In parallel to this the Open Source Initiative considers many free-software licenses to be open source. These include the latest versions of the FSF's three main licenses: the GPL, the Lesser General Public License, the GNU Affero General Public License. Richard Stallman's Free Software Definition, adopted by the Free Software Foundation, defines free software as a matter of liberty not price, it upholds the Four Essential Freedoms; the earliest-known publication of the definition of his free-software idea was in the February 1986 edition of the FSF's now-discontinued GNU's Bulletin publication. The canonical source for the document is in the philosophy section of the GNU Project website; as of August 2017, it is published there in 40 languages.
To meet the definition of "free software", the FSF requires the software's licensing rights what the FSF respect the civil liberties / human rights of what the FSF calls the software user's "Four Essential Freedoms". The freedom to run the program as you wish, for any purpose; the freedom to study how the program works, change it so it does your computing as you wish. Access to the source code is a precondition for this; the freedom to redistribute copies. The freedom to distribute copies of your modified versions to others. By doing this you can give the whole community a chance to benefit from your changes. Access to the source code is a precondition for this; the open-source-software definition is used by the Open Source Initiative to determine whether a software license qualifies for the organization's insignia for Open-source software. The definition was based on the Debian Free Software Guidelines and adapted by Bruce Perens. Perens did not base his writing on the Four Essential Freedoms of free software from the Free Software Foundation, which were only available on the web.
Perens subsequently stated that he felt Eric Raymond's promotion of Open-source unfairly overshadowed the Free Software Foundation's efforts and reaffirmed his support for Free software. In the following 2000s, he spoke about open source again. In the 1950s through the 1980s, it was common for computer users to have the source code for all programs they used, the permission and ability to modify it for their own use. Software, including source code, was shared by individuals who used computers as public domain software. Most companies had a business model based on hardware sales, provided or bundled software with hardware, free of charge. By the late 1960s, the prevailing business model around software was changing. A growing and evolving software industry was competing with the hardware manufacturer's bundled software products. Leased machines required software support while providing n
Nqthm is a theorem prover sometimes referred to as the Boyer–Moore theorem prover. It was a precursor to ACL2; the system was developed by Robert S. Boyer and J Strother Moore, professors of computer science at the University of Texas, Austin, they began work on the system in 1971 in Scotland. Their goal was to make a automatic, logic-based theorem prover, they used a variant of Pure LISP as the working logic. Definitions are formed as recursive functions, the system makes extensive use of rewriting and an induction heuristic, used when rewriting and something that they called symbolic evaluation fails; the system was built on top of Lisp and had some basic knowledge in what was called "Ground-zero", the state of the machine after bootstrapping it onto a Common Lisp implementation. This is an example of the proof of a simple arithmetic theorem; the function TIMES is part of the BOOT-STRAP and is defined to be The formulation of the theorem is given in a Lisp-like syntax: Should the theorem prove to be true, it will be added to the knowledge basis of the system and can be used as a rewrite rule for future proofs.
The proof itself is given in a quasi-natural language manner. The authors randomly choose typical mathematical phrases for embedding the steps in the mathematical proof, which does make the proofs quite readable. There are macros for LaTeX that can transform the Lisp structure into more or less readable mathematical language; the proof of the commutativity of times continues: Give the conjecture the name *1. We will appeal to induction. Two inductions are suggested by terms in the conjecture. We limit our consideration to the two suggested by the largest number of nonprimitive recursive functions in the conjecture. Since both of these are likely, we will choose arbitrarily. We will induct according to the following scheme:. Linear arithmetic, the lemma COUNT-NUMBERP, the definition of ZEROP inform us that the measure decreases according to the well-founded relation LESSP in each induction step of the scheme; the above induction scheme produces the following two new conjectures: Case 2.. and after winding itself through a number of induction proofs concludes that Case 1..
This simplifies, expanding the definitions of ZEROP, TIMES, PLUS, EQUAL, to: T. That finishes the proof of *1.1, which finishes the proof of *1. Q. E. D. COMMUTATIVITY-OF-TIMES Many proofs have been done or confirmed with the system list concatenation insertion sort a binary adder an expression compiler for a stack machine uniqueness of prime factorizations invertibility of the RSA encryption algorithm unsolvability of the halting problem for Pure Lisp FM8501 microprocessor Gödel's incompleteness theorem CLI Stack Gauss' law of quadratic reciprocity Byzantine Generals and Clock Synchronization A compiler for a subset of the Nqthm language bi-phase mark asynchronous communications protocol Motorola MC68020 and Berkeley C String Library Paris–Harrington Ramsey theorem The equivalence of NFSA and DFSA A more powerful version, called PC-Nqthm was developed by Matt Kaufmann; this gave the proof tools that the system uses automatically to the user, so that more guidance can be given to the proof. This is a great help, as the system has an unproductive tendency to wander down infinite chains of inductive proofs.
A Computational Logic Handbook, R. S. Boyer and J S. Moore, Academic Press, 1997; the Boyer-Moore Theorem Prover and Its Interactive Enhancement, with M. Kaufmann and R. S. Boyer and Mathematics with Applications, 29, 1995, pp. 27–62. In 2005 Robert S. Boyer, Matt Kaufmann, J Strother Moore received the ACM Software System Award for their work on the Nqthm theorem prover; the Automated Reasoning System, Nqthm The Boyer-Moore Theorem Prover Even though the system is no longer being supported, it is still available at Runnable version on GitHub
J Strother Moore
J Strother Moore is a computer scientist, he is a co-developer of the Boyer–Moore string search algorithm, Boyer–Moore majority vote algorithm, the Boyer–Moore automated theorem prover, Nqthm. He made pioneering contributions to structure sharing including the piece table data structure and early logic programming. An example of the workings of the Boyer–Moore string search algorithm is given in Moore's website. Moore received his SB in mathematics at Massachusetts Institute of Technology in 1970 and his Ph. D in computational logic at University of Edinburgh in Scotland in 1973. In addition, Moore is a co-author of the ACL2 automated theorem prover, he and others used ACL2 to prove the correctness of the floating point division operations of the AMD K5 microprocessor in the wake of the Pentium FDIV bug. For his contributions to automated deduction, Moore received the 1999 Herbrand Award with Robert S. Boyer, in 2006 he was inducted as a Fellow of the Association for Computing Machinery. Moore was elected to the National Academy of Engineering in 2007, is a Fellow of the AAAI.
He was elected a Corresponding Fellow of the Royal Society of Edinburgh in 2015. He is the Admiral B. R. Inman Centennial Chair in Computing Theory at The University of Texas at Austin, was Chair of the Department of Computer Science from 2001-2009. Before joining the Department of Computer Sciences as the chair, he formed a company, Computational Logic Inc. along with others including his close friend at the University of Texas at Austin and one of the regarded professors in the field of Automated Reasoning, Robert S. Boyer. Moore enjoys rock climbing. Boyer–Moore majority vote algorithm J Strother Moore's home page ``My Best Ideas Boyer-Moore fast string search algorithm