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25.1.23 signum(self): Z The method signum returns −1 if this rational number is less than zero, 0 if this rational number is zero, and 1 if this rational number is greater than zero. For type Q # , the signum of 0/0 is 0/0 . 25.1.24 numerator(self): Z 25.1.25 denominator(self): Z The method numerator returns the numerator of this rational number, and the method denominator returns the denominator of this rational number, when this rational number is represented in lowest terms (such that the greatest common divisor of numerator and denominator is 1). For types Q ∗ and Q # , the numerator of +∞ is 1 , the numerator of −∞ is −1 , and the numerator of 0/0 is 0 ; the denominator of +∞ , −∞ , or 0/0 is 0 . 198
25.1.26 floor(self): Z 25.1.27 opr ⌊self⌋: Z 25.1.28 ceiling(self): Z 25.1.29 opr ⌈self⌉: Z 25.1.30 round(self): Z 25.1.31 truncate(self): Z The method floor , likewise the enclosing operator ⌊. . .⌋ , returns the largest integer that is not greater than this rational number. The method ceiling , likewise the enclosing operator ⌈. . .⌉ , returns the smallest integer that is not less than this rational number. The method round returns the integer that is closest to this rational number, but if this rational number is exactly halfway between two consecutive integers, then round returns whichever of the two integers is even. The method truncate returns the ceiling of this rational number if it is negative, and otherwise returns the floor of this rational number. (This has the effect of taking the floor of the magnitude, also called “rounding toward zero.”) For types Q ∗ and Q # , all of these methods simply return the argument if it is +∞ , −∞ , or 0/0 . opr ⌊self ⌋: N opr ⌈self ⌉: N opr ⌊self ⌋: N opr ⌈self ⌉: N The hyperfloor operation ⌊x ⌋ computes 2 ( ⌊log 2 x⌋) and returns the result as a natural number. If the argument is equal to 0 , the result is 0 . If the argument is negative, an InvalidArgumentException is thrown. The hyperceiling operation ⌈x ⌉ computes 2 ( ⌈log 2 x⌉) and returns the result as a natural number. If the argument is equal to 0 , the result is 0 . If the argument is negative, an InvalidArgumentException is thrown. The hyperhyperfloor operation ⌊x ⌋ computes 2 ( ⌊log 2 x ⌋) and returns the result as a natural number. If the argument is equal to 0 or 1 , the result is the same as the argument. If the argument is negative, an InvalidArgumentException is thrown. The hyperhyperceiling operation ⌈x ⌉ computes 2 ( ⌈log 2 x ⌉) and returns the result as a natural number. If the argument is equal to 0 or 1 , the result is the same as the argument. If the argument is negative, an InvalidArgumentException is thrown. 25.1.32 realpart(self): Q The method realpart for a rational number simply returns its argument. 25.1.33 imagpart(self): Q The method imagpart for a rational number simply returns zero. 199
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The Fortress Language Specification
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4 Evaluation 36 4.1 Values . . . .
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12 Functions 85 12.1 Function Decla
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17.7 Coercions for Tuple and Arrow
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IV Fortress for Library Writers 214
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40.10The Trait Fortress.Core.Binary
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Chapter 1 Introduction The Fortress
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A note on the presented libraries i
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component HelloWorld export Executa
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foo:String → () baz: String → S
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fortress upgrade Gary with Ralph No
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halve(x : R64) : R64 = x/2 Predicta
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while x < 10 do print x x += 1 end
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It is also possible to convert a me
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factorial(n) = ∏ i←1:n i This f
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In both methods cons and append , t
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default distributions will provide
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Chapter 3 Programs A program consis
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Chapter 10), a literal (see Section
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(discussed in Section 13.13) to a s
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Chapter 5 Lexical Structure A Fortr
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U+0021 EXCLAMATION MARK ! U+0023 NU
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If a character (or any other syntac
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BIG SI unit absorbs abstract also a
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escape sequences are useful for ASC
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5.14.1 Multicharacter Enclosing Ope
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Note: the elegant way to write Avog
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• top-level variable declarations
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declares id to be a mutable variabl
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several new variables. This syntax
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Chapter 7 Names Names are used to r
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7.3 Qualified Names Fortress provid
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Fortress also allows coercion betwe
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• element types of a tuple type c
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TypeAlias ::= type Id [StaticParams
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‘}’. If such a clause contains
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Note that a trait declaration inher
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the parametric trait WrappedDiction
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Chapter 10 Objects An object is a v
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FldDef ::= FldMod ∗ Id [IsType] (
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Chapter 11 Static Parameters Trait,
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11.5 Operator and Identifier Parame
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Chapter 12 Functions Functions are
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swap(x :Object, y : Object) = (y, x
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A function declaration may be separ
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Chapter 13 Expressions Fortress is
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Negating the literal 0 produces a s
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the function is evaluated with the
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13.10 Assignments Syntax: Expr ::=
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treeSum(t : TreeLeaf) = 0 treeSum(t
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Several common mathematical operato
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The body of each iteration is run i
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to the value of the condition expre
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An atomic expression consists of th
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13.26 Try Expressions Syntax: Flow
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13.27.2 Static Expressions and Valu
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A = [1 2 3; 4 5 6; 7 8 9] then A 1,
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evaluates to the set {0, 4, 16} Map
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ignore(f x) is equivalent to: f x;
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trait CheckedException extends { Ex
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Chapter 15 Overloading and Multiple
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e called with. In other words, we e
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Chapter 16 Operators Operators are
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• “ordinary” operators: + −
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left context whitespace operator fi
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The rules below are designed to for
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16.8.3 Enclosing Operators When use
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16.8.9 Intersection, Union, and Set
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Chapter 17 Conversions and Coercion
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17.3 Coercion Invocations One may i
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arguments nor with variable number
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trait B extends A end trait C exten
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a new coercion in a subexpression
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Page 149 and 150: Chapter 19 Tests and Properties For
Page 151 and 152: 19.5 Test Suites In order to allow
Page 153 and 154: Chapter 20 Type Inference Type infe
Page 155 and 156: 3. If a i is a functional applicati
Page 157 and 158: 21.2 Programming Discipline If Fort
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Page 161 and 162: 2. Ancestors, dynamically ordered b
Page 163 and 164: The ways in which fortresses are ac
Page 165 and 166: Component equivalence is determined
Page 167 and 168: 22.5 Type Inference for Components
Page 169 and 170: Fortress.IO Fortress.Crypto IronLin
Page 171 and 172: execute(componentName:String, args:
Page 173 and 174: 3. If the replacement does not expo
Page 175 and 176: This method runs all test functions
Page 177 and 178: Fortress.IO Fortress.Crypto Fortres
Page 179 and 180: Part III Fortress APIs and Document
Page 181 and 182: 23.1.3 hash(maxval: N64): N64 23.1.
Page 183 and 184: opr =(self,other: Boolean):Boolean
Page 185 and 186: 24.1.20 getter hashCode(): N64 24.1
Page 187 and 188: False: BooleanInterval Uncertain: B
Page 189 and 190: 24.2.11 opr ∧(self,other: Boolean
Page 191 and 192: 24.2.19 possibly(self): Boolean 24.
Page 193 and 194: Chapter 25 Numbers 25.1 Rational Nu
Page 195 and 196: opr ⌈self ⌉: N realpart(self):
Page 197: 25.1.11 opr (self,other: Q):Boolean
Page 201 and 202: Chapter 26 Negated Relational Opera
Page 203 and 204: 26.1.26 opr ⊀T extends BinaryPred
Page 205 and 206: 27.3 The Trait Fortress.Standard.Un
Page 207 and 208: Chapter 29 Dimensions and Units 29.
Page 209 and 210: unit secondOfAngle secondsOfAngle:
Page 211 and 212: Chapter 30 Tests 30.1 The Object Fo
Page 213 and 214: 31.2 Convenience Types An optional
Page 215 and 216: Chapter 32 Parallelism and Locality
Page 217 and 218: y evaluating the two elements of th
Page 219 and 220: There is a DefaultDistribution whic
Page 221 and 222: v = spawn at a.region(i) do a i end
Page 223 and 224: expr ∑ type wrapper ∑ body R,
Page 225 and 226: 32.8.3 Using Overloading to Adapt G
Page 227 and 228: Chapter 33 Overloaded Functional De
Page 229 and 230: coercions: T U ⇐⇒ T ♦ U ∧
Page 231 and 232: The More Specific Rule for Function
Page 233 and 234: An infix/multifix operator declarat
Page 235 and 236: may be so defined except ordinary p
Page 237 and 238: and here some of these computed dim
Page 239 and 240: unit name 1 name 2 name 3 : . . . u
Page 241 and 242: Chapter 36 Support for Domain-Speci
Page 243 and 244: Because syntax expanders are define
Page 245 and 246: Part V Fortress APIs and Documentat
Page 247 and 248: property ∀(a:T) ¬(a ∼ a) end A
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trait PartialOrderOperatorsT extend
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The HasMinimalElement trait specifi
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The trait Commutative requires the
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A value e is a left identity for a
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trait ApproximatelyHasInverses T ex
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end extends { ApproximateCommutativ
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A default definition is provided fo
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trait RationalQuantityunit U absorb
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(self,other:RationalQuantityU,ninf
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38.2.4 getter hashCode(): Z64 38.2.
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Chapter 39 Components and APIs We d
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Chapter 40 Memory Sequences and Bin
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updatenat k(j: IntegerStatick, v: T
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40.1.12 update(j:IndexInt, v: T): L
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40.3.2 opr nat m[r:RangeOfStaticSiz
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40.5 The Trait Fortress.Core.Binary
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40.5.5 bit(m:IndexInt): Bit The res
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40.6 The Trait Fortress.Core.Binary
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40.6.4 opr nat m[r:RangeOfStaticSiz
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40.6.16 opr ‖ nat m,bool bigEndia
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countLeadingZeroBits():IndexInt cou
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40.7.41 signedShift(j:IndexInt):T 4
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40.8.6 signedDivide(other: T,overfl
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littleEndian():BinaryEndianLinearEn
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40.9.16 opr ‖ nat m,nat radix,nat
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v: BinaryEndianWordb,bigEndianBytes
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40.10.9 opr nat m[r:RangeOfStaticSi
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40.10.24 splitnat b ′ () : Binary
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itAnd(selfStart: IndexInt,source:T,
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40.13.2 wrappingAdd(selfStart: Inde
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40.13.30 unsignedMax(selfStart: Ind
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40.13.55 gatherBits(selfStart:Index
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Appendix A Fortress Calculi A.1 Bas
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Values, evaluation contexts, redexe
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Expression typing: p; ∆; Γ ⊢ e
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α, β type variables f method name
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Function/method name lookup: Fname(
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One owner for all the visible metho
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Traits defining a method: defining
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Values, evaluation contexts, and re
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Valid function declarations: p ⊢
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Method definition lookup: visible p
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Values, intermediate expressions, e
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Field typing: p; ∆; Γ ⊢ vd ok
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Appendix B Overloaded Functional De
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Proof. We prove this for δ, but th
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Upgrade A predicate upg? takes two
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a is rendered as a foobar is render
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• If a portion is sub and another
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supercalifragilisticexpialidocious
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any alternative names, with undersc
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Appendix F Operator Precedence, Cha
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U+007C VERTICAL LINE | | U+2016 DOU
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The following two operators have th
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U+003D EQUALS SIGN = = EQ U+2243 AS
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U+22DD EQUAL TO OR GREATER-THAN U+2
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U+2289 NEITHER A SUPERSET OF NOR EQ
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F.4.8 Miscellaneous Relational Oper
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U+21A5 UPWARDS ARROW FROM BAR U+21A
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U+2224 DOES NOT DIVIDE ∤ U+2225 P
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U+27F8 LONG LEFTWARDS DOUBLE ARROW
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U+2960 UPWARDS HARPOON WITH BARB LE
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U+29ED BLACK CIRCLE WITH DOWN ARROW
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Appendix G Concrete Syntax In this
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Extends ::= extends TraitTypes Excl
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StaticParam ::= Id [Extends] [absor
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Value ::= Literal | fn ValParam [Is
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Appendix H Generated Concrete Synta
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-> fn_decl -> object_decl -> var_de
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-> id extends_opt absorbs_opt -> "n
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-> w "excludes" w type_ref_list com
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-> obj_decl_body_elem br obj_decl_b
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-> op wr op_follows_op_w -> op wr e
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-> no_space_op_expr_result no_space
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id_opt -> -> w id with_opt -> -> w
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-> unpasting_elem rect_separator un
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-> "}" comprehension -> set_compreh
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-> do_expr -> for_expr -> spawn_exp
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-> type_ref -> "(" w type_ref w ","
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enc -> enc_literal op_or_enc -> op
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Bibliography [1] O. Agesen, L. Bak,
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