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Chapter 18 Dimensions and Units Syntax: DimType ::= DimRef | TypeRef DimRef | TypeRef · DimRef | TypeRef / DimRef | TypeRef per DimRef | TypeRef UnitRef | TypeRef · UnitRef | TypeRef / UnitRef | TypeRef per UnitRef | TypeRef in DimRef DimRef ::= Unity | DottedId | DimRef DimRef | DimRef · DimRef | DimRef / DimRef | DimRef per DimRef | DimRef ˆ NatRef | 1 / DimRef | (DimRef ) | DUPreOp DimRef | DimRef DUPostOp DUPreOp ::= square | cubic | inverse DUPostOp ::= squared | cubed UnitExpr ::= UnitRef | Expr UnitRef | Expr · UnitRef | Expr / UnitRef | Expr per UnitRef | Expr in UnitRef UnitRef ::= dimensionless | DottedId | UnitRef UnitRef | UnitRef · UnitRef | UnitRef / UnitRef | UnitRef per UnitRef | UnitRef ˆNatRef | 1/ UnitRef | (UnitRef ) | DUPreOp UnitRef | UnitRef DUPostOp There are special type-like constructs called dimensions that are separate from other types (described in Chapter 8). There are also special constructs that modify types and values called units that are instances of dimensions. These are used to describe physical quantities. The Fortress standard libraries define dimensions and units for the standard SI system of measurement based on meters, kilograms, seconds, amperes, and so on (as described in Section 29.1). The Fortress standard libraries also provide supplemental units of measurement, such as feet and miles (as described in Section 29.2). For example, the Fortress standard libraries provide a dimension named Length whose default unit is named meter and abbreviated m_. By rendering convention, this abbreviation is rendered in roman type without the underscore: m . In contrast, the variable m is rendered as in standard mathematical notation: m . See Section 5.17 for a discussion of formatting conventions 146
for tokens. For readability, plural forms of the unit names are defined as equivalent to the corresponding singular forms; thus one can write meters per second , for example. Standard SI prefixes may be used on both the name and the symbol, so that nanometer and nm are also units of the dimension named Length, related to meter and m by a conversion factor of 10 −9 . Every dimension may have a default unit that is used for representing values of quantities of that dimension if no unit is specified explicitly. The Fortress standard libraries define these default dimensions and units: Length meter MagneticFlux weber Mass kilogram MagneticFluxDensity tesla Time second Inductance henry Frequency hertz Velocity meters per second Force newton Acceleration meters per second squared Pressure pascal Angle radian Energy joule SolidAngle steradian Power watt LuminousIntensity candela Temperature kelvin LuminousFlux lumen Area square meter Illuminance lux Volume cubic meter RadionuclideActivity becquerel ElectricCurrent ampere AbsorbedDose gray ElectricCharge coulomb DoseEquivalent sievert ElectricPotential volt AmountOfSubstance mole Capacitance farad CatalyticActivity katal Resistance ohm MassDensity kilograms per cubic meter Conductance siemens In addition, the Fortress standard libraries define the dimension Information with units bit and byte (and the plurals bits and bytes), a byte being equal to 8 bits. To avoid confusion, SI prefixes are not provided for these units; instead, programmers must use appropriate powers of 2 or 10, for example 10 6 bits or 2 20 bits . Here are some examples of the use of dimensions and units: x: R64 Length = 1.3 m t: R64 Time = 5 s v: R64 Velocity = x/t w: R64 Velocity in nm/s = 17 nm/s x: R64 Velocity in furlongs per fortnight = v in furlongs per fortnight Dimensions and units can be multiplied, divided, and raised to rational powers to produce new dimensions and units. Both the numerator and the denominator of a rational power of a dimension or a unit must be a valid nat parameter instantiation (as described in Section 11.2). Multiplication can be expressed using juxtaposition or · ; division can be expressed using / or per. The syntactic operators square and cubic may be applied to a dimension or unit in order to raise it to the second power, third power, respectively; the special postfix syntactic operators squared and cubed may be used for the same purpose. The syntactic operator inverse may be applied to a dimension or unit to divide it into 1. All of these syntactic operators are merely syntactic sugar, expanded before type checking. grams per cubic centimeter meter per second squared inverse ohms One can also write 1/X as a synonym for X −1 if X is either a dimension or a unit. Most numerical values in Fortress are dimensionless quantity values. Multiplying or dividing a dimensionless value by a unit produces a dimensioned value; thus 5 s is the dimensioned value equal to five seconds, which has numerical 147
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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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Page 97 and 98: 13.10 Assignments Syntax: Expr ::=
Page 99 and 100: treeSum(t : TreeLeaf) = 0 treeSum(t
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Page 109 and 110: 13.26 Try Expressions Syntax: Flow
Page 111 and 112: 13.27.2 Static Expressions and Valu
Page 113 and 114: A = [1 2 3; 4 5 6; 7 8 9] then A 1,
Page 115 and 116: evaluates to the set {0, 4, 16} Map
Page 117 and 118: ignore(f x) is equivalent to: f x;
Page 119 and 120: trait CheckedException extends { Ex
Page 121 and 122: Chapter 15 Overloading and Multiple
Page 123 and 124: e called with. In other words, we e
Page 125 and 126: Chapter 16 Operators Operators are
Page 127 and 128: • “ordinary” operators: + −
Page 129 and 130: left context whitespace operator fi
Page 131 and 132: The rules below are designed to for
Page 133 and 134: 16.8.3 Enclosing Operators When use
Page 135 and 136: 16.8.9 Intersection, Union, and Set
Page 137 and 138: Chapter 17 Conversions and Coercion
Page 139 and 140: 17.3 Coercion Invocations One may i
Page 141 and 142: arguments nor with variable number
Page 143 and 144: trait B extends A end trait C exten
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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
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
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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.
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25.1.11 opr (self,other: Q):Boolean
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25.1.26 floor(self): Z 25.1.27 opr
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Chapter 26 Negated Relational Opera
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26.1.26 opr ⊀T extends BinaryPred
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27.3 The Trait Fortress.Standard.Un
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Chapter 29 Dimensions and Units 29.
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unit secondOfAngle secondsOfAngle:
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Chapter 30 Tests 30.1 The Object Fo
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31.2 Convenience Types An optional
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Chapter 32 Parallelism and Locality
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y evaluating the two elements of th
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There is a DefaultDistribution whic
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v = spawn at a.region(i) do a i end
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expr ∑ type wrapper ∑ body R,
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32.8.3 Using Overloading to Adapt G
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Chapter 33 Overloaded Functional De
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coercions: T U ⇐⇒ T ♦ U ∧
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The More Specific Rule for Function
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An infix/multifix operator declarat
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may be so defined except ordinary p
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and here some of these computed dim
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unit name 1 name 2 name 3 : . . . u
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Chapter 36 Support for Domain-Speci
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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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