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24.3 Top-level BooleanInterval Values 24.3.1 True: BooleanInterval 24.3.2 False: BooleanInterval 24.3.3 Uncertain: BooleanInterval 24.3.4 Impossible: BooleanInterval The immutable variables True, False, Uncertain, and Impossible have as their values the four boolean intervals. They are top-level variables declared in the Fortress standard libraries. 192
Chapter 25 Numbers 25.1 Rational Numbers The trait Q ( QQ ) encompasses all finite rational numbers, the result of dividing any integer by any nonzero integer. The trait Q ∗ ( QQ_star ) is Q with two extra elements, +∞ and −∞ . The trait Q # ( QQ_splat ) is Q ∗ with one additional element, the indefinite rational (written 0/0 ), which is used as the result of dividing zero by zero or of adding −∞ to +∞ . Often it is desirable to indicate that a variable ranges over only a subset of the rationals, such as only positive values or only nonnegative values or only nonzero values. Unfortunately, traditional notations such as Q + are not used consistently in the literature; one author may use Q + to mean the set of strictly positive rationals and another may use it to mean the set of nonnegative rationals. Fortress therefore uses a notation that is novel but unambiguous: Q ( QQ ) is the set of rationals (it is a subtype of R and Q ∗ ). Q < ( QQ_LT ) is the set of strictly negative rationals (it is a subtype of R < , Q ∗ < , Q , Q ≤ , and Q≠ ). Q ≤ ( QQ_LE ) is the set of nonpositive rationals, that is, Q < ∪ {0} (it is a subtype of R ≤ , Q ∗ ≤ , and Q ). Q ≥ ( QQ_GE ) is the set of nonnegative rationals, that is, Q > ∪ {0} (it is a subtype of R ≥ , Q ∗ ≥ , and Q ). Q > ( QQ_GT ) is the set of strictly positive rationals (it is a subtype of R > , Q ∗ > , Q , Q ≥ , and Q≠ ). Q≠ ( QQ_NE ) is the set of strictly nonzero rationals (that is, Q < ∪ Q > ) (it is a subtype of R≠ , Q ∗ ≠ , and Q ). Q ∗ ( QQ_star ) is Q with extra elements +∞ and −∞ (it is a subtype of R ∗ and Q # ). Q ∗ < ( QQ_star_LT ) is Q < with extra element −∞ (it is a subtype of R ∗ < , Q # < , Q ∗ , Q ∗ ≤ , and Q∗ ≠ ). Q ∗ ≤ ( QQ_star_LE ) is Q ≤ with extra element −∞ (it is a subtype of R ∗ ≤ , Q# ≤ , and Q∗ ). Q ∗ ≥ ( QQ_star_GE ) is Q ≥ with extra element +∞ (it is a subtype of R ∗ ≥ , Q# ≥ , and Q∗ ). Q ∗ > ( QQ_star_GT ) is Q > with extra element +∞ (it is a subtype of R ∗ > , Q # > , Q ∗ , Q ∗ ≥ , and Q∗ ≠ ). Q ∗ ≠ ( QQ_star_NE ) is Q ≠ with extra elements +∞ and −∞ (it is a subtype of R ∗ ≠ , Q# ≠ , and Q∗ ). Q # ( QQ_splat ) is Q ∗ with extra element 0/0 (it is a subtype of R # ). Q # < ( QQ_splat_LT ) is Q ∗ < with extra element 0/0 (it is a subtype of R # < , Q # , Q # ≤ , and Q# ≠ ). Q # ≤ ( QQ_splat_LE ) is Q∗ ≤ with extra element 0/0 (it is a subtype of R# ≤ and Q# ). Q # ≥ ( QQ_splat_GE ) is Q∗ ≥ with extra element 0/0 (it is a subtype of R# ≥ and Q# ). Q # > ( QQ_splat_GT ) is Q ∗ > with extra element 0/0 (it is a subtype of R # > , Q # , Q # ≥ , and Q# ≠ ). Q # ≠ ( QQ_splat_NE ) is Q∗ ≠ with extra element 0/0 (it is a subtype of R# ≠ and Q# ). The Fortress type system tracks these types closely through various arithmetic operations; for example, adding two values of type Q > produces a result of type Q > , and adding a value of type Q ∗ > and a value of type Q ≥ produces a value of type Q ∗ ≥ . 193
Page 1 and 2:
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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Page 143 and 144: trait B extends A end trait C exten
Page 145 and 146: a new coercion in a subexpression
Page 147 and 148: for tokens. For readability, plural
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
Page 159 and 160: Here the call f(a, a) ends up setti
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: 24.2.19 possibly(self): Boolean 24.
Page 195 and 196: opr ⌈self ⌉: N realpart(self):
Page 197 and 198: 25.1.11 opr (self,other: Q):Boolean
Page 199 and 200: 25.1.26 floor(self): Z 25.1.27 opr
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
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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
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Because syntax expanders are define
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Part V Fortress APIs and Documentat
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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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