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We present our effort to \ develop methods and tools to solve this problem in the context of the ", StyleBox["polyhedral model", FontSlant->"Italic"], " where calculations are expressed as operations on polyhedra-shaped \ collections of data. These ideas are the basis of our MMAlpha toolbox that is \ implemented using ", StyleBox["Mathematica", FontSlant->"Italic"], ". We present MMAlpha and we explain how it can be used to design parallel \ architectures and to prove properties of such architectures." }], "Abstract", TextAlignment->Left, TextJustification->1], Cell[CellGroupData[{ Cell["0. Introduction", "SectionFirst"], Cell[TextData[{ "Designing parallel programs has always been a challenge of computer \ science, and it is becoming more and more interesting, as even common \ processors tend to use several processors (see ", ButtonBox["http://en.wikipedia.org/wiki/Multicore", ButtonData:>{ URL[ "http://en.wikipedia.org/wiki/Multicore"], None}, ButtonStyle->"Hyperlink"], "). Moreover, the development of embedded devices including \ high-performance signal processing algorithms calls for parallel \ architectures, for example, in third generation mobile telephones. " }], "Text"], Cell["\<\ For more than 15 years, some of the authors (P. Quinton and T. \ Risset) have been involved in research regarding parallel integrated \ circuits. Basic mathematical ingredients of this research are program \ analysis, combinatorial optimization, polyhedral theory, and linear algebra. \ The need to develop tools and capitalize on these tools in order to exchange \ ideas among researchers lead them to find out a programming framework that \ would be flexible and powerful enough to host results of this research.\ \>", \ "Text"], Cell[TextData[{ "Using ", StyleBox["Mathematica", FontSlant->"Italic"], ", they developed together with other researchers, a framework called \ MMAlpha. This framework contains a few tens of ", StyleBox["Mathematica", FontSlant->"Italic"], " packages allowing parallelism to be extracted from programs using various \ transformations. The input of MMAlpha is a program written in a \ single-assignment language called Alpha. In Alpha, a program is written as a \ set of recurrence equations, such as for example:" }], "Text"], Cell[BoxData[ \(y\_i = \[Sum]\+\(k = 0\)\%K\( w\_k\) x\_\(i - k\)\)], "NumberedEquation"], Cell[TextData[{ "where ", Cell[BoxData[ \(x\_i\)]], " is an input signal, ", Cell[BoxData[ \(w\_k\)]], " is a vector of coefficients, and ", Cell[BoxData[ \(y\_i\)]], " is the output signal (this equation defines a finite impulse response \ filter). By applying to the textual description of such a program a set of \ transformations, one can derive a parallel architecture for its execution, by \ observing that the behavior of a synchronous processor can also be described \ using recurrence equations over time and space. This is just what is \ implemented in MMAlpha." }], "Text"], Cell[TextData[{ "In this paper, we present our effort to implement this research using ", StyleBox["Mathematica", FontSlant->"Italic"], ". Section 1 explains what MMAlpha is. Section 2 gives more details about \ how it is implemented using ", StyleBox["Mathematica.", FontSlant->"Italic"], " In Section 3, we explain how we use MMAlpha to prove properties of \ parallel circuits. Section 4 concludes the paper." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["1. What is MMAlpha ?", "Section"], Cell[TextData[StyleBox["MMAlpha is based on a single-assignment equational \ language named Alpha that we introduce in section 1.1. Section 1.2. describes \ MMAlpha, our toolbox to transform Alpha programs into parallel architectures. \ In Section 1.3., we briefly develop an example of finite impulse filter. \ Section 1.4. gives some references on the polyhedral model. ", FontColor->GrayLevel[0]]], "Text", FontColor->RGBColor[1, 0, 0]], Cell[CellGroupData[{ Cell["1.1. Alpha and the Polyhedral Model", "Subsection"], Cell["\<\ Alpha is a programming language based on recurrence equations. \ Recurrence equations are widely used in mathematics and signal processing to \ express usual calculations. In the following, we shall use the example of a \ Finite Impulse Response (FIR) Filter:\ \>", "Text"], Cell[BoxData[ \(y\_i = \[Sum]\+\(k = 0\)\%K\( w\_k\) x\_\(i - k\)\)], "NumberedEquation"], Cell["The corresponding Alpha program is as follows", "Text"], Cell[BoxData[ \("system fir : {N | 3<=N}\n (x : {i | 0<=i} of \ integer[S,16]; \n w : {n | 0<=n<=N-1} of integer[S,16])\n \ returns (y : {i | N-1<=i} of integer[S,16]);\nvar\n Y : {i,n | N-1<=i; \ -1<=n<=N-1} of integer[S,32];\nlet\n Y[i,n] = \n case\n { | \ n=-1} : 0[];\n { | 0<=n} : Y[i,n-1] + w[n] * x[i-n];\n esac;\n \ y[i] = Y[i,N-1];\ntel;"\)], "Print", CellLabel->"From In[39]:="], Cell[TextData[{ "This program (called a ", StyleBox["system", FontSlant->"Italic"], ") is parameterized by a size parameter ", Cell[BoxData[ \(TraditionalForm\`N\)]], " whose value is constrained by the inequality {", Cell[BoxData[ \(TraditionalForm\`N \[VerticalSeparator] 3 \[LessEqual] N\)]], "}. Inputs of the system are variables ", Cell[BoxData[ \(TraditionalForm\`x\)]], " and ", Cell[BoxData[ \(TraditionalForm\`w\)]], ". Variable ", Cell[BoxData[ \(TraditionalForm\`x\)]], " is an infinite sequence of integer values that represents the input of \ the filter. Variable ", Cell[BoxData[ \(TraditionalForm\`w\)]], " contains the coefficients of the filter: it is a finite sequence of \ integer values of size ", Cell[BoxData[ \(TraditionalForm\`N\)]], ". The results of this system is the variable ", Cell[BoxData[ \(TraditionalForm\`y\)]], ", again an infinite sequence of integers. All variables have the type ", Cell[BoxData[ \(TraditionalForm\`integer[S, 16]\)]], " which means 16-bit signed integers. System ", StyleBox["fir", "Output"], " contains two recurrence equations. The first one defines a local variable \ ", Cell[BoxData[ \(TraditionalForm\`Y\)]], ", a 2-dimensional array. Its first dimension is the same as that of ", Cell[BoxData[ \(TraditionalForm\`y\)]], ", and its second dimension the same as ", Cell[BoxData[ \(TraditionalForm\`x\)]], ". ", Cell[BoxData[ \(TraditionalForm\`Y[i, n]\)]], " is defined by a case expressions. When ", Cell[BoxData[ \(TraditionalForm\`n = \(-1\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`Y[i, n]\)]], " takes value ", Cell[BoxData[ \(TraditionalForm\`0\)]], " (we shall explain the ", Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[ \(TraditionalForm\`0[]\)]]]], TraditionalForm]]]]], TraditionalForm]]], "\[NonBreakingSpace]notation later on). When ", Cell[BoxData[ \(TraditionalForm\`n \[GreaterSlantEqual] 0\)]], ", ", Cell[BoxData[ \(TraditionalForm\`Y[i, n - 1]\)]], " takes the value ", Cell[BoxData[ \(TraditionalForm\`Y[i, n - 1] + w[n]\[Times]x[i - n]\)]], ". The second equation defines the output y of the system: ", Cell[BoxData[ \(TraditionalForm\`y[i]\)]], " is the ultimate value of the sequence", Cell[BoxData[ FormBox[ RowBox[{ Cell[""], \({Y[i, n]}\_\(0 \[LessEqual] n \[LessEqual] N - 1\)\)}], TraditionalForm]]], "." }], "Text"], Cell[TextData[{ "More generally, each Alpha variable is a collection of values associated \ with integral points of a polyhedron called the ", StyleBox["domain", FontSlant->"Italic"], " of this variable. Scalar variables such as the constant 0 above, are \ considered 0-dimensional array and addressed with ", StyleBox["[]", FontFamily->"Courier"], ". Alpha expressions all have a domain, whose definition depends on the way \ the expression is defined. For example, adding two expressions such as in ", Cell[BoxData[ FormBox[ RowBox[{"A", "+", RowBox[{ "B", Cell["\<\ gives an expression whose domain is the \ intersection of the domains of A and B. \ \>"]}]}], TraditionalForm]]] }], "Text"], Cell[TextData[{ "The body of the program (between the ", StyleBox["let", "Output"], " and ", StyleBox["tel", "Output"], " keywords) express the recurrence relations. The ", StyleBox["case", FontFamily->"Courier"], " operator allows one to assign different values to expressions, depending \ on domain conditions. The restriction operator, denoted :, allows one to \ restrict the domain of an expression. For example, ", StyleBox["dom : exp", "Output"], " restricts ", StyleBox["exp", "Output"], " to the domain ", StyleBox["dom", "Output"], ". " }], "Text"], Cell[TextData[{ "In Alpha, each expression has a domain, for instance the domain of \ expression ", Cell[BoxData[ \(TraditionalForm\`Y[i, n - 1]\)]], " is: ", Cell[BoxData[ \(TraditionalForm\`{i\ , n | \ 1 <= i <= N + 1}\)]], ", which corresponds to the pre-image of the domain of ", Cell[BoxData[ \(TraditionalForm\`Y\)]], " by the dependence function ", Cell[BoxData[ \(TraditionalForm\`\((i, n \[RightArrow] i, n - 1)\)\)]], ". The original Alpha notation for ", Cell[BoxData[ \(TraditionalForm\`Y[i, n - 1]\)]], " is ", Cell[BoxData[ \(TraditionalForm\`Y . \((i, n \[RightArrow] i, n - 1)\)\)]], ", but it is usually more convenient to represent Alpha programs in ", StyleBox["array notation", FontSlant->"Italic"], " as it is done above." }], "Text"], Cell[TextData[{ "Alpha programs enjoy the ", StyleBox["single assignment property", FontSlant->"Italic"], ": each array element of an Alpha variable is defined exactly once. Related \ to that, there is no predefined order of execution for the computation of the \ program: an Alpha system is therefore a definition rather than an algorithm. \ The order of execution will be determined by the scheduling process detailed \ in the next paragraph. Because of this single assignment property, an Alpha \ program contains only true dependences between computations: dependence \ analysis is greatly simplified and parallelism detection is easier. " }], "Text"], Cell[TextData[{ "Because of its restricted syntax, Alpha is well-suited for \ source-to-source program transformation. The Alpha language has been designed \ to ease the program transformation in order to, starting from a functional \ specification such as the ", StyleBox["fir", "Output"], " system, refine the specification down to a precise operational \ implementation of the same specification. The transformations of Alpha \ programs are performed thanks to transformations implemented in the MMAlpha \ programming environment." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["1.2. MMAlpha", "Subsection"], Cell[TextData[{ "MMAlpha is the programming environment used to manipulate and transform \ Alpha programs. MMAlpha consists in a set of ", StyleBox["Mathematica", FontSlant->"Italic"], " packages interfaced with C libraries for parsing Alpha programs and doing \ computations on polyhedra. " }], "Text"], Cell["\<\ From the point of view of a chip designer, MMAlpha can be seen as a \ High Level Synthesis (HLS) software: the user transforms an algorithmic \ description - an Alpha program - into a digital circuit implementing this \ description using a Hardware Description Language (HDL) such as VHDL or \ SystemC. MMAlpha is not supposed to do the refinement automatically, it only \ provides a toolbox to ease the transformations of Alpha programs.\ \>", "Text"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " manipulates internal representations of Alpha programs. These internal \ representations, called Abstract Syntax Trees (AST), are ", StyleBox["Mathematica", FontSlant->"Italic"], " symbolic expressions. The MMAlpha packages provide analysis and \ transformations of AST. The user loads an Alpha program, performs various \ transformations, eventually save the transformed program and translates the \ final result into a HDL. At any time, the user can simulate the Alpha \ program by generating C code and executing it." }], "Text", CellDingbat->None], Cell["\<\ In general, the transformation process consists in the three main \ following steps, once the program is loaded:\ \>", "Text"], Cell[TextData[{ "Schedule the program: finds out when the calculation of some variable \ instance, say ", Cell[BoxData[ FormBox[ RowBox[{\(Y[i, n]\), Cell[" is computed. T"]}], TraditionalForm]]], "his is done automatically using one of the available schedulers of \ MMAlpha." }], "Text", CellDingbat->"\[GraySquare]"], Cell[TextData[{ "Place the computations: once a schedule has been found, rewrite the \ program in such a way that variables now have time and space indexes. For \ example, ", Cell[BoxData[ \(TraditionalForm\`Y[i, n]\)]], " is replaced by ", Cell[BoxData[ \(TraditionalForm\`Y[t, p]\)]], " where ", Cell[BoxData[ \(TraditionalForm\`t\)]], " represents the time at which, and ", Cell[BoxData[ \(TraditionalForm\`p\)]], " the processor number where this expression is computed. A placed system \ can therefore be interpreted as an architecture." }], "Text", CellDingbat->"\[GraySquare]"], Cell[TextData[{ "Transform to hardware description language: from the skeleton produced by \ schedule and place, derive a ", StyleBox["real", FontSlant->"Italic"], " architecture in a Hardware Description Language. " }], "Text", CellDingbat->"\[GraySquare]"], Cell["\<\ To illustrate the use of MMAlpha, we provide the derivation that a \ user, usually a hardware designer, will execute on a simple example: the FIR \ filter\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["1.3. Example of the FIR filter", "Subsection"], Cell[TextData[{ "Consider the program given in section 1.1. A hardware description \ executing such a program can be obtained using the following ", StyleBox["Mathematica", FontSlant->"Italic"], " program:" }], "Text"], Cell[BoxData[ \(\(\( (*\ 1\ *) \)\(\ \)\(\(load["\"];\)\[IndentingNewLine] (*\ 2\ *) \ \(scd[addConstraints \[Rule] {TxD1 == 1, TyD1 == 1}, optimizationType \[Rule] Null];\)\[IndentingNewLine] (*\ 3\ *) \ \(appSched[];\)\[IndentingNewLine] (*\ 4\ *) \ toAlpha0v2[]; alpha0ToAlphard[];\[IndentingNewLine] (*\ 5\ *) \ \(fixParameter["\", 8];\)\ \ \[IndentingNewLine] (*\ 6\ *) \(a2v[];\)\)\)\)], "Input", CellLabel->"In[99]:="], Cell[TextData[{ "Commands are explained as follows:\n\t1. Parse and load the Alpha program.\ \n\t2. Schedule it. Some constraints and options are given to the scheduler \ in order to reach a ", StyleBox["good", FontSlant->"Italic"], " solution.", "\n\t3. Place calculations using the schedule.\n\t4. Translate program into \ a hardware description, still using Alpha syntax.\n\t5. Set the value of the \ N parameter.\n\t6. Translate to VHDL." }], "Text"], Cell["\<\ The following presents a small part of the resulting VHDL \ file:\ \>", "Text"], Cell[BoxData[ \("-- VHDL Model Created for \"system cellfirModule1\" \n-- 31/3/2006 \ 7:44:9.477505\n-- Alpha2Vhdl Version 0.9 \n\nlibrary IEEE;\nuse \ IEEE.std_logic_1164.all;\nuse IEEE.std_logic_signed.all;\nuse \ IEEE.numeric_std.all;\n\nlibrary work;\nuse work.definition.all;\n\n\nENTITY \ cellfirModule1 IS\nPORT(\n clk: IN STD_LOGIC;\n CE : IN STD_LOGIC;\n Rst : \ IN STD_LOGIC;\n wXMirr1 : IN SIGNED (15 DOWNTO 0);\n xXMirr1 : IN SIGNED \ (15 DOWNTO 0);\n YReg1Xloc : IN SIGNED (31 DOWNTO 0);\n Y : OUT SIGNED \ (31 DOWNTO 0)\n);\nEND cellfirModule1;\n\nARCHITECTURE behavioural OF \ cellfirModule1 IS\n SIGNAL YReg1 : SIGNED (31 DOWNTO 0) := \ \"00000000000000000000000000000000\";\n\n\n -- Insert missing components \ here!---------\nBEGIN\n\n PROCESS(clk) BEGIN IF (clk = '1' AND clk'EVENT) \ THEN\n IF CE='1' THEN YReg1 <= YReg1Xloc; END IF;\n END \ IF;\n END PROCESS;\n\n Y <= (YReg1 + (wXMirr1 * xXMirr1));\n\nEND \ behavioural;\n\n"\)], "Print", CellLabel->"From In[3]:="], Cell[TextData[{ "For the reader who is not familiar with the VHDL language, the few final \ lines (between the ", StyleBox["BEGIN", "Output"], " and the ", StyleBox["END behavioural", "Output"], " keywords) describe a register (as a VHDL ", StyleBox["PROCESS", "Output"], ") and a multiply and add operator that constitute the basic element of \ this architecture." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["1.4. The Polyhedral Model", "Subsection"], Cell[TextData[{ "MMAlpha is based on the so-called ", StyleBox["polyhedral model", FontSlant->"Italic"], " of computation where computations are expressed as operations on data \ collections associated with integral points of polyhedra. Polyhedra are an \ interesting structure from the algebraic point of view, as they are stable \ under many interesting operations: intersection, (convex) union, pre-image by \ an affine function. (In fact, the data structure that used in Alpha is the \ finite union of polyhedra, which is also stable by union.)" }], "Text"], Cell["\<\ Finding out the optimum of a linear form on a polyhedron is a \ well-known problem tackled by linear programming methods. In MMAlpha, \ polyhedra are represented by their dual representation [Schrijver86]. A \ polyhedron is either represented by a finite set of inequalities or by its \ system of generators including its vertices and rays (directions of \ half-lines contained in the polyhedron).\ \>", "Text"], Cell[TextData[{ "For example, the polyhedron ", Cell[BoxData[ FormBox[Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`{i, j | 0 \[LessEqual] i \[And] 0 \[LessEqual] j \[LessEqual] 10}\)]], " " }]], TraditionalForm]]], "has two vertices ", Cell[BoxData[ \(TraditionalForm\`v\_1 = \((0, 10)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`v\_2 = \((0, 10)\)\)]], " and one ray ", Cell[BoxData[ \(TraditionalForm\`r = \((1, 0)\)\)]], " as shown here. 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Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output", CellLabel->"Out[33]="] }, Open ]], Cell["\<\ This representation is well-suited to the analysis of loops in \ imperative languages, as the iteration space of a loop, that is to say, the \ range of the loop indexes, is usually a polyhedron. Therefore, it is quite \ natural to represent loop statements as a single operation on sets of values \ indexed by the iteration indexes.\ \>", "Text"], Cell["\<\ The use of recurrence equations to model parallel computations was \ first described in [Karp67]. Leslie Lamport [Lamport74] was the first to \ model loop parallelization for parallelization. Later on [Moldovan83] and \ [Quinton84] developed techniques to represent parallel architectures using \ recurrence equations and dependence analysis. Feautrier [Feautrier86] \ contributed notably to the polyhedral model in various way, among which the \ invention of parameterized integer programming. The definition of the Alpha \ language was due to Mauras [Mauras89].\ \>", "Text"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["2. The Implementation of MMAlpha", "Section"], Cell[TextData[{ "In this section, we briefly describe how MMAlpha is implemented using ", StyleBox["Mathematica", FontSlant->"Italic"], "." }], "Text"], Cell[CellGroupData[{ Cell["2.1. Packages and external libraries", "Subsection"], Cell[TextData[{ "MMAlpha is implemented as a set of about 50 ", StyleBox["Mathematica", FontSlant->"Italic"], " packages that describe all the transformations. The software is freely \ available (", ButtonBox["http://www.irisa.fr/cosi/ALPHA/welcome.html", ButtonData:>{ URL[ "http://www.irisa.fr/cosi/ALPHA/welcome.html"], None}, ButtonStyle->"Hyperlink"], ")." }], "Text", TextAlignment->Left, TextJustification->1], Cell[TextData[{ "These ", StyleBox["Mathematica", FontSlant->"Italic"], " packages make use of a few external C programs that perform calculations \ that would be inefficient if written in ", StyleBox["Mathematica", FontSlant->"Italic"], " or that were easier to implement in an imperative language. " }], "Text"], Cell[TextData[{ "The ", StyleBox["Polylib", FontSlant->"Italic"], " library is such a basic external tool: it allows operations on polyhedra \ to be done efficiently. Polylib is interfaced to MMAlpha using ", StyleBox["Domlib", FontSlant->"Italic"], ", a ", StyleBox["Mathematica", FontSlant->"Italic"], " Package that uses Mathlink to call the Polylib programs. " }], "Text"], Cell["Parsers and un-parsers of Alpha are also written in C. ", "Text"], Cell[TextData[{ "Finally, MMAlpha uses another C tool named Pip (", ButtonBox["http://www.piplib.org/", ButtonData:>{ URL[ "http://www.piplib.org/"], None}, ButtonStyle->"Hyperlink"], ") that performs parametric integer programming[Feautrier86]. Pip allows \ one to find out the lexicographic minimum (or maximum) of a parameterized \ polyhedron." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell[TextData[{ "2.2. Programming Style and ", StyleBox["Mathematica", FontSlant->"Italic"], " features used" }], "Subsection"], Cell["\<\ MMAlpha packages are mainly written using the functionnal style. As \ many MMAlpha functions are transformations of the abstract syntactic tree of \ the program, patterns are extensively used to locate parts of the tree that \ have to be transformed. A few packages are written directly as a set of \ replacement rules. \ \>", "Text"], Cell[TextData[{ "One main concern while writing programs using ", StyleBox["Mathematica", FontSlant->"Italic"], " was to find out a programming style that would be safe enough to be used \ for such a big application. We more or less adhered to principles presented \ in Roman Maeder's book [Maeder97]. " }], "Text"], Cell[TextData[{ "The initial decision to use ", StyleBox["Mathematica", FontSlant->"Italic"], " (in 1989!) was based on the foreseen interest of using symbolic libraries \ and also rewriting systems. Actually, MMAlpha uses less possibilities than we \ expected. Still, MMAlpha uses linear solvers, some optimization tools, and \ some Graphics facilities. " }], "Text"], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " has been found to be a very stable, always upwards compatible, and \ portable software. These are very interesting qualities for hosting a \ long-term research effort. Its main drawback is that is does not allow ", StyleBox["Mathematica-", FontSlant->"Italic"], "independent applications to be produced: this makes it difficult to draw \ the interest of researchers in communities where this software is not \ available." }], "Text"], Cell[TextData[{ "Below is an example of using the graphics capabilities of ", StyleBox["Mathematica", FontSlant->"Italic"], " in order to vizualize domains. The following Alpha program defines a \ 3-dimensional variable A whose domain is a cube of size 10. However, the \ definition of A is broken into four different regions. " }], "Text"], Cell[BoxData[ \("system strange (a : {i,j | 1<=i<=10; 1<=j<=10} of integer)\n \ returns (c : integer);\nvar\n A : {i,j,k | 1<=i<=10; 0<=j<=10; 1<=k<=10} of \ integer;\nlet\n A[i,j,k] = \n case\n { | j=0} : a[i,k];\n \ { | j+1<=i; 0<=j} : A[i,j-1,k];\n { | i<=j-1; 0<=j} : A[i,j-1,k];\n \ { | i=j} : a[i,i];\n esac;\ntel;"\)], "Print", CellLabel->"From In[18]:="], Cell["\<\ The following MMAlpha expression allows one to create an animation \ to visualize the domain of variable A (open the cell, select the group of \ graphic cells and type command-y to animate it.)\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(vshow["\", minR \[Rule] 0.2, stepR \[Rule] 0.2, maxR \[Rule] 3]\)], "Input", Editable->False, Evaluatable->False], Cell[CellGroupData[{ Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics3D 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