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Movie Title Year Distributor Notes Rev Formats Behind the Scenes of an Adult Movie 1984 Metro Clip 2 DO Blonde Next Door 1982 Metro 1 DRO Blue Vanities 278 1996 Blue Vanities DO Coed Teasers 1982 Video-X-Pix Facial D Erotic Adventures of Lolita 1982 Video-X-Pix 1 DRO Erotic Interludes 1981 Caballero Home Video Facial DRO Erotic World of Sunny Day 1984 VCR LezOnly DO Expose Me Now 1983 VCX 2 DO Forbidden Worlds 1988 Collector's Video DRO Gimme an X 1993 Vidco Entertainment Facial DRO
Golden Girls Film 32 1981 Caballero Home Video Facial O Linda Shaw Collection 2006 Alpha Blue Archives DRO Little Orphan Dusty 2 1981 Caballero Home Video O Soaked to the Skin 1994 Unknown Sorority Sweethearts 1982 Caballero Home Video Facial 1 DRO Swedish Erotica 36 1981 Caballero Home Video O Swedish Erotica Film 431 1980 Caballero Home Video Triangle of Lust 1982 VCR Triplets 1984 Cinematech Inc. Facial DO Vintage Erotica 1 2009 Blue Vanities Facial 1 DO



Computer algorithms Flowchart examples of the canonical Böhm-Jacopini structures: the SEQUENCE (rectangles descending the page), the WHILE-DO and the IF-THEN-ELSE. The three structures are made of the primitive conditional GOTO ({{{1}}}) (a diamond), the unconditional GOTO (rectangle), various assignment operators (rectangle), and HALT (rectangle). Nesting of these structures inside assignment-blocks result in complex diagrams (cf Tausworthe 1977:100, 114). In computer systems, an algorithm is basically an instance of logic written in software by software developers, to be effective for the intended "target" computer(s) to produce output from given (perhaps null) input. An optimal algorithm, even running in old hardware, would produce faster results than a non-optimal (higher time complexity) algorithm for the same purpose, running in more efficient hardware; that is why algorithms, like computer hardware, are considered technology. "Elegant" (compact) programs, "good" (fast) programs : The notion of "simplicity and elegance" appears informally in Knuth and precisely in Chaitin: Knuth: " … we want good algorithms in some loosely defined aesthetic sense. One criterion … is the length of time taken to perform the algorithm …. Other criteria are adaptability of the algorithm to computers, its simplicity and elegance, etc"[40] Chaitin: " … a program is 'elegant,' by which I mean that it's the smallest possible program for producing the output that it does"[41] Chaitin prefaces his definition with: "I'll show you can't prove that a program is 'elegant'"—such a proof would solve the Halting problem (ibid). Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist. This is true, even without expanding the available instruction set available to the programmer. Rogers observes that "It is ... important to distinguish between the notion of algorithm, i.e. procedure and the notion of function computable by algorithm, i.e. mapping yielded by procedure. The same function may have several different algorithms".[42] Unfortunately, there may be a tradeoff between goodness (speed) and elegance (compactness)—an elegant program may take more steps to complete a computation than one less elegant. An example that uses Euclid's algorithm appears below. Computers (and computors), models of computation: A computer (or human "computor"[43]) is a restricted type of machine, a "discrete deterministic mechanical device"[44] that blindly follows its instructions.[45] Melzak's and Lambek's primitive models[46] reduced this notion to four elements: (i) discrete, distinguishable locations, (ii) discrete, indistinguishable counters[47] (iii) an agent, and (iv) a list of instructions that are effective relative to the capability of the agent.[48] Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for Computability".[49] Minsky's machine proceeds sequentially through its five (or six, depending on how one counts) instructions, unless either a conditional IF–THEN GOTO or an unconditional GOTO changes program flow out of sequence. Besides HALT, Minsky's machine includes three assignment (replacement, substitution)[50] operations: ZERO (e.g. the contents of location replaced by 0: L ? 0), SUCCESSOR (e.g. L ? L+1), and DECREMENT (e.g. L ? L - 1).[51] Rarely must a programmer write "code" with such a limited instruction set. But Minsky shows (as do Melzak and Lambek) that his machine is Turing complete with only four general types of instructions: conditional GOTO, unconditional GOTO, assignment/replacement/substitution, and HALT. However, a few different assignment instructions (e.g. DECREMENT, INCREMENT, and ZERO/CLEAR/EMPTY for a Minsky machine) are also required for Turing-completeness; their exact specification is somewhat up to the designer. The unconditional GOTO is a convenience; it can be constructed by initializing a dedicated location to zero e.g. the instruction " Z ? 0 "; thereafter the instruction IF Z=0 THEN GOTO xxx is unconditional. Simulation of an algorithm: computer (computor) language: Knuth advises the reader that "the best way to learn an algorithm is to try it . . . immediately take pen and paper and work through an example".[52] But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a language that the simulator/computer/computor can effectively execute. Stone gives an example of this: when computing the roots of a quadratic equation the computor must know how to take a square root. If they don't, then the algorithm, to be effective, must provide a set of rules for extracting a square root.[53] This means that the programmer must know a "language" that is effective relative to the target computing agent (computer/computor). But what model should be used for the simulation? Van Emde Boas observes "even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains. It is at this point that the notion of simulation enters".[54] When speed is being measured, the instruction set matters. For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a "modulus" instruction available rather than just subtraction (or worse: just Minsky's "decrement"). Structured programming, canonical structures: Per the Church–Turing thesis, any algorithm can be computed by a model known to be Turing complete, and per Minsky's demonstrations, Turing completeness requires only four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. Kemeny and Kurtz observe that, while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in "spaghetti code", a programmer can write structured programs using only these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language".[55] Tausworthe augments the three Böhm-Jacopini canonical structures:[56] SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE.[57] An additional benefit of a structured program is that it lends itself to proofs of correctness using mathematical induction.[58] Canonical flowchart symbols[59]: The graphical aide called a flowchart, offers a way to describe and document an algorithm (and a computer program of one). Like the program flow of a Minsky machine, a flowchart always starts at the top of a page and proceeds down. Its primary symbols are only four: the directed arrow showing program flow, the rectangle (SEQUENCE, GOTO), the diamond (IF-THEN-ELSE), and the dot (OR-tie). The Böhm–Jacopini canonical structures are made of these primitive shapes. Sub-structures can "nest" in rectangles, but only if a single exit occurs from the superstructure. The symbols, and their use to build the canonical structures are shown in the diagram. Examples Further information: List of algorithms Algorithm example One of the simplest algorithms is to find the largest number in a list of numbers of random order. Finding the solution requires looking at every number in the list. From this follows a simple algorithm, which can be stated in a high-level description in English prose, as: High-level description: If there are no numbers in the set then there is no highest number. Assume the first number in the set is the largest number in the set. For each remaining number in the set: if this number is larger than the current largest number, consider this number to be the largest number in the set. When there are no numbers left in the set to iterate over, consider the current largest number to be the largest number of the set. (Quasi-)formal description: Written in prose but much closer to the high-level language of a computer program, the following is the more formal coding of the algorithm in pseudocode or pidgin code: Algorithm LargestNumber Input: A list of numbers L. Output: The largest number in the list L. if L.size = 0 return null largest ? L[0] for each item in L, do if item > largest, then largest ? item return largest "?" denotes assignment. For instance, "largest ? item" means that the value of largest changes to the value of item. "return" terminates the algorithm and outputs the following value. Euclid's algorithm Further information: Euclid's algorithm The example-diagram of Euclid's algorithm from T.L. Heath (1908), with more detail added. Euclid does not go beyond a third measuring and gives no numerical examples. Nicomachus gives the example of 49 and 21: "I subtract the less from the greater; 28 is left; then again I subtract from this the same 21 (for this is possible); 7 is left; I subtract this from 21, 14 is left; from which I again subtract 7 (for this is possible); 7 is left, but 7 cannot be subtracted from 7." Heath comments that "The last phrase is curious, but the meaning of it is obvious enough, as also the meaning of the phrase about ending 'at one and the same number'."(Heath 1908:300). Euclid's algorithm to compute the greatest common divisor (GCD) to two numbers appears as Proposition II in Book VII ("Elementary Number Theory") of his Elements.[60] Euclid poses the problem thus: "Given two numbers not prime to one another, to find their greatest common measure". He defines "A number [to be] a multitude composed of units": a counting number, a positive integer not including zero. To "measure" is to place a shorter measuring length s successively (q times) along longer length l until the remaining portion r is less than the shorter length s.[61] In modern words, remainder r = l - q×s, q being the quotient, or remainder r is the "modulus", the integer-fractional part left over after the division.[62] For Euclid's method to succeed, the starting lengths must satisfy two requirements: (i) the lengths must not be zero, AND (ii) the subtraction must be “proper”; i.e., a test must guarantee that the smaller of the two numbers is subtracted from the larger (or the two can be equal so their subtraction yields zero). Euclid's original proof adds a third requirement: the two lengths must not be prime to one another. Euclid stipulated this so that he could construct a reductio ad absurdum proof that the two numbers' common measure is in fact the greatest.[63] While Nicomachus' algorithm is the same as Euclid's, when the numbers are prime to one another, it yields the number "1" for their common measure. So, to be precise, the following is really Nicomachus' algorithm. A graphical expression of Euclid's algorithm to find the greatest common divisor for 1599 and 650. 1599 = 650×2 + 299 650 = 299×2 + 52 299 = 52×5 + 39 52 = 39×1 + 13 39 = 13×3 + 0 Computer language for Euclid's algorithm Only a few instruction types are required to execute Euclid's algorithm—some logical tests (conditional GOTO), unconditional GOTO, assignment (replacement), and subtraction. A location is symbolized by upper case letter(s), e.g. S, A, etc. The varying quantity (number) in a location is written in lower case letter(s) and (usually) associated with the location's name. For example, location L at the start might contain the number l = 3009. An inelegant program for Euclid's algorithm "Inelegant" is a translation of Knuth's version of the algorithm with a subtraction-based remainder-loop replacing his use of division (or a "modulus" instruction). Derived from Knuth 1973:2–4. Depending on the two numbers "Inelegant" may compute the g.c.d. in fewer steps than "Elegant". The following algorithm is framed as Knuth's four-step version of Euclid's and Nicomachus', but, rather than using division to find the remainder, it uses successive subtractions of the shorter length s from the remaining length r until r is less than s. The high-level description, shown in boldface, is adapted from Knuth 1973:2–4: INPUT: 1 [Into two locations L and S put the numbers l and s that represent the two lengths]: INPUT L, S 2 [Initialize R: make the remaining length r equal to the starting/initial/input length l]: R ? L E0: [Ensure r = s.] 3 [Ensure the smaller of the two numbers is in S and the larger in R]: IF R > S THEN the contents of L is the larger number so skip over the exchange-steps 4, 5 and 6: GOTO step 6 ELSE swap the contents of R and S. 4 L ? R (this first step is redundant, but is useful for later discussion). 5 R ? S 6 S ? L E1: [Find remainder]: Until the remaining length r in R is less than the shorter length s in S, repeatedly subtract the measuring number s in S from the remaining length r in R. 7 IF S > R THEN done measuring so GOTO 10 ELSE measure again, 8 R ? R - S 9 [Remainder-loop]: GOTO 7. E2: [Is the remainder zero?]: EITHER (i) the last measure was exact, the remainder in R is zero, and the program can halt, OR (ii) the algorithm must continue: the last measure left a remainder in R less than measuring number in S. 10 IF R = 0 THEN done so GOTO step 15 ELSE CONTINUE TO step 11, E3: [Interchange s and r]: The nut of Euclid's algorithm. Use remainder r to measure what was previously smaller number s; L serves as a temporary location. 11 L ? R 12 R ? S 13 S ? L 14 [Repeat the measuring process]: GOTO 7 OUTPUT: 15 [Done. S contains the greatest common divisor]: PRINT S DONE: 16 HALT, END, STOP. An elegant program for Euclid's algorithm The following version of Euclid's algorithm requires only six core instructions to do what thirteen are required to do by "Inelegant"; worse, "Inelegant" requires more types of instructions.[clarify] The flowchart of "Elegant" can be found at the top of this article. In the (unstructured) Basic language, the steps are numbered, and the instruction LET [] = [] is the assignment instruction symbolized by ?. 5 REM Euclid's algorithm for greatest common divisor 6 PRINT "Type two integers greater than 0" 10 INPUT A,B 20 IF B=0 THEN GOTO 80 30 IF A > B THEN GOTO 60 40 LET B=B-A 50 GOTO 20 60 LET A=A-B 70 GOTO 20 80 PRINT A 90 END How "Elegant" works: In place of an outer "Euclid loop", "Elegant" shifts back and forth between two "co-loops", an A > B loop that computes A ? A - B, and a B = A loop that computes B ? B - A. This works because, when at last the minuend M is less than or equal to the subtrahend S (Difference = Minuend - Subtrahend), the minuend can become s (the new measuring length) and the subtrahend can become the new r (the length to be measured); in other words the "sense" of the subtraction reverses. The following version can be used with object-oriented languages: // Euclid's algorithm for greatest common divisor int euclidAlgorithm (int A, int B){ A=Math.abs(A); B=Math.abs(B); while (B!=0){ if (A>B) A=A-B; else B=B-A; } return A; } Testing the Euclid algorithms Does an algorithm do what its author wants it to do? A few test cases usually give some confidence in the core functionality. But tests are not enough. For test cases, one source[64] uses 3009 and 884. Knuth suggested 40902, 24140. Another interesting case is the two relatively prime numbers 14157 and 5950. But "exceptional cases"[65] must be identified and tested. Will "Inelegant" perform properly when R > S, S > R, R = S? Ditto for "Elegant": B > A, A > B, A = B? (Yes to all). What happens when one number is zero, both numbers are zero? ("Inelegant" computes forever in all cases; "Elegant" computes forever when A = 0.) What happens if negative numbers are entered? Fractional numbers? If the input numbers, i.e. the domain of the function computed by the algorithm/program, is to include only positive integers including zero, then the failures at zero indicate that the algorithm (and the program that instantiates it) is a partial function rather


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