On 'int128_t'

Every programming language has built-in integer types, both signed (representing mathematical integers) and unsigned (representing mathematical nonnegative integers). C compilers usually give these types fancy names like 'unsigned long long', and have a nasty habit of changing the sizes (and meanings) of these types on different platforms. The C type 'short' is usually 16 bits, 'int' either 16 or 32 bits, and 'long' either 32 or 64 bits, depending on platform. This eventually led to the 'stdint.h' header in C99, which provides exact-size types which can be used accross platforms. These are named 'int32_t', 'int64_t', 'uint32_t', 'uint64_t', and so on.

With stuff getting bigger, it's natural to ask the question: "Why 64?" and the answer is generally because the highest integer type most hardware can deal with is 64 bits. Can we go higher? Of course! but how? In this article, I will show you how to define 'int128_t' and 'uint128_t' in C without any compiler hacks. They can be used as parameter types and return types from functions, and they don't require any special memory management or allocation, because they're not pointer types.

First, you might say we could just make an array type:

 typedef int32_t int128_as_int32x4_t[4];
 typedef int64_t int128_as_int64x2_t[2];

but which one to we pick? What we really need is a union of each of these, so we can decide later which array type to use. However, neither arrays nor unions can be used as return values from functions, only struct's can be used as return values. So in order to have a type that can be used as a return value we need to make a struct of a union of array types, as follows:

typedef struct int128_s {
    union int128_u {
        int8_t  as_int8[16];
        int16_t as_int16[8];
        int32_t as_int32[4];
        int64_t as_int64[2];
    } value;
} int128_t;

and wrap this type in a typedef. But how do we use these new integers? First of all we need some way of constructing 'int128_t's, and in the spirit of 'stdint.h' we can make a 'INT128_C()' macro which expands to a constructed object of type 'int128_t'. We'll need a few functions for this:

int128_t int128_from_int(int from);
int128_t int128_from_str(char *from);
int int_from_int128(int128_t from);
int str_from_int128(char *to, int to_size, int128_t from);

and we can use the second one to define the macro as:

#define INT128_C(x) int128_from_str(#x)

because '(#x)' indicates to the preprocessor to turn x into a string before compile-time, which is then passed to int128_from_str which then returns an object of type 'int128_t'. For compilers that do not support compile-time constant expressions involving function calls, we can also define simpler macros as follows:

#ifdef BIG_ENDIAN
#define INT128_C64(a,b)\
    (int128_t){.value = {.as_int64 = {a, b}}}
#define INT128_C32(a,b,c,d)\
    (int128_t){.value = {.as_int32 = {a, b, c, d}}}
#else
#define INT128_C64(a,b)\
    (int128_t){.value = {.as_int64 = {b, a}}}
#define INT128_C32(a,b,c,d)\
    (int128_t){.value = {.as_int32 = {d, c, b, a}}}
#endif

Note that because we use designators (value and as_int##) this part requires a C99 compiler

Conclusion

In order to use this integer type, we also need dozens of other functions, such as add, mul, sub, div, mod, and, or, xor, lsh, rsh, pow, etc., just to match the functionality usually associated with C integer types, and from there the possibilities are endless. A future article could revisit these functions. For now, though, I just wanted to bring focus to this integer type, especially considering how many common-place datatypes fit into an 'int128_t' such as UUID's and IPv6 addresses. We may need this sooner than we think.

Tetrational-point

As you might have guessed, most of my posts are a synthesis of two or more ideas. This time, they are IEEE-754 floating point formats and composite arithmetic formats. To provide some overview, IEEE-754 has been the official standard for representing numbers in computers for over 25 years, and is widely deployed in practically every computer architecture. Composite arithmetic on the other hand is a new approach which uses a combination of integer, rational, scientific, and tetrational representation in unison to achieve greater accuracy, precision, and dynamic range.

References for composite arithmetic include: "Beyond Floating Point" (by C.W.Clenshaw and F.W.J.Olver) which focuses on extending floating point with tetration alone, "Composite Arithmetic, a Proposal for a New Standard" and "Composite Arithmetic, A Storage Form" (both by W. Neville Holmes) which both focus on combining all four approaches, "Design of a Composite Arithmetic Unit for Rational Numbers" (by Tomasz Pinkiewicz, W. Neville Holmes, and Tariq Jamil) which focuses on the implementation of these in hardware, "Design of a 32-bit Arithmetic Unit based on Composite Arithmetic and its Implementation on a Field Programmable Gate Array." (by Tomasz Hubert Pinkiewicz) which focuses on the same, and "Lecture notes on: Computer Arithmetic: Principles, Architectures, and VLSI Design" (by Reto Zimmermann) which seems to be an overview of the subject targeted at students.

The most recent edition of IEEE-754 introduced a 16-bit floating point format, called 'binary16' for short. Since this is the smallest standard floating point format, we will use it as an example of how IEEE-754 represents numbers. To begin we will see how it represents 1.5:

	s	exponent					significand
bin	0	0	1	1	1	1	1	0	0	0	0	0	0	0	0	0
hex	3				E				0				0
value	= 1.5

and this is how the standard binary16 format represents infinity:

	s	exponent					significand
bin	0	1	1	1	1	1	0	0	0	0	0	0	0	0	0	0
hex	7				C				0				0
value	= Infinity

As you can see, binary16 uses a 5-bit exponent and a 10-bit significand. One important class of representations which do not represent numbers is known collectively as Not-A-Number (NaN) representations. These are when the exponent is all 1's, regardless of the sign. Infinity is a particular example of this, since the exponent is all 1's and the significand is all 0's. If the significand is nonzero, then the remaining representations fall into two categories: signaling NaNs (sNaN) and quiet NaNs (qNaN). A signaling NaN is traditionally represented with a 0 in the MSB of the significand, and some other bit nonzero. A quiet NaN is traditionally represented with a 1 in the MSB of the significand, with anything in the other bits. Since sNaNs can cause errors and trap handlers to invoke, it is a bad idea to use them to represent numbers, since many architectures will automatically convert sNaNs to qNaNs. So, we will leave sNaNs alone, and replace qNaNs with a tetrational number representation format.

This tetrational format will use the two MSBs to represent quietness (q) and height (h) respectively. The tetrand will become the exponent at the top of the exponential tower. So a height of 0 will indicate a value of 2^2^2^2^2^(0.tetrand), and a height of 1 will indicate a value of 2^2^2^2^2^2^(0.tetrand). In general, the value associated with the tetrational format is:

(-1)^sexp₂^h+5(0.tetrand) -- HTML

${\left(-1\right)}^s{\exp }_2^{h+5}\left(\mathrm{0.tetrand}\right)$ -- MathML

We will start at 65536, which is just greater than the largest value represented by binary16 (65504). Extending this to larger formats (like binary32) should probably start here as well, because the next height is too much larger than the largest number represented by binary32. Starting from 65536, we can represent it as follows:

binary16:	s	exponent					significand
tetra16:	s	exponent					q	h	tetrand
bin	0	1	1	1	1	1	1	0	0	0	0	0	0	0	0	0
hex	7				E				0				0
value	≈ 65536

We get this value because 2^2^2^2^2^(0.0) = 2^2^2^2^1 = 2^2^2^2 = 2^2^4 = 2^16 = 65536. The next greater representation is

binary16:	s	exponent					significand
tetra16:	s	exponent					q	h	tetrand
bin	0	1	1	1	1	1	1	0	0	0	0	0	0	0	0	1
hex	7				E				0				1
value	≈ 71036

and perhaps another interesting representation is that of googolplex:

binary16:	s	exponent					significand
tetra16:	s	exponent					q	h	tetrand
bin	0	1	1	1	1	1	1	1	1	0	1	1	0	0	1	0
hex	7				F				B				2
value	≈ googolplex = 10^(10^100)

Note that we only used a single bit for the height of the exponential tower, but we could have used more. I think that 2 bits should be used for height in binary32, and 3 bits should be used for height in binary64 and binary128. All floating point formats would encode the height and tetrand in the extra bits of a qNaN so as to avoid conflicts with hardware arithmetic.

Here is an overview of the extensions to the binary16 format:

#x0000 = +0
#x0001 = 5.960464477539063e-8
       ... (standard binary16 floating-point) ...
#x7BFF = 65504.
#x7C00 = +Infinity
#x7C01 = +sNaN
       ... (standard binary16 signaling NaN) ...
#x7DFF = +sNaN
#x7E00 = 65536 = 2^2^2^2^2^(0)
#x7E01 = 71036.
       ... (new tetrational-point) ...
#x7E?? = 2^2^2^2^2^(0.??_16)
       ...
#x7EFE = 8.9423389054e+15721
#x7EFF = 6.2622606021e+17594
#x7F00 = 2^65536 = 2^2^2^2^2^2^(0) = 2^2^2^2^2
#x7F01 = 6.8504792165e+21383
       ... (new tetrational-point) ...
#x7F?? = 2^2^2^2^2^2^(0.??_16)
       ...
#x7FFE = 10^(2.69191e+15721) which is < 2^(2^65536)
#x7FFF = +qNaN (unique)
#x8000 = -0
#x8001 = -5.960464477539063e-8
       ... (standard binary16 floating-point) ...
#xFBFF = -65504.
#xFC00 = -Infinity
#xFC01 = -sNaN
       ... (standard binary16 signaling NaN) ...
#xFDFF = -sNaN
#xFE00 = -65536
#xFE01 = -71036.
       ... (new tetrational-point) ...
#xFE?? = -2^2^2^2^2^(0.??_16)
       ...
#xFEFE = -8.9423389054e+15721
#xFEFF = -6.2622606021e+17594
#xFF00 = -2^65536
#xFF01 = -6.8504792165e+21383
       ... (new tetrational-point) ...
#xFF?? = -2^2^2^2^2^2^(0.??_16)
       ...
#xFFFE = -10^(2.69191e+15721) which is > -2^(2^65536)
#xFFFF = -qNaN (unique)

In conclusion, we have shown exactly how to increase the dynamic range (quite dramatically) of standard IEEE-754 floating point formats by using a tetrational representation encoded in the unused bits of quiet NaNs. This provides a backwards-compatible hardware-accelerated tetrational floating point format which will encode approximate overflows as precisely as possible. So instead of overflows, you get a number! This makes it possible to be more informative when it comes to mathematical functions such as exp, log, or 1/x. With a bigger dynamic range, many of the outputs of these functions will no longer be indeterminate, but instead, finite.

GoLang Proposals

Go is a very new programming language, barely a year old. Since its initial release, it has become an instant success, almost overnight. Google designed Go primarily for high-level systems programming. Since then, Google has replaced a few of its internal servers with custom Go servers. I believe this success is driven by the simplicity of Go, and its special features which focus on concurrency.

Compared to C, it has many more features so I consider its features a strict superset of C's features. Compared to C++ however, Go is very different, possibly described by a classic Venn-diagram in which Go's features and C++'s features intersect, but both have features the other doesn't. For example, method overloading can be accomplished in both languages, but in different ways.One feature that C++ has enjoyed is that of templates, which do not exist in Go. This requires that algorithms that can work on multiple datatypes be written out each time, which negatively affects readability and maintainability. While adding full-fledged C++ templates to Go is certainly possible, I believe we need to look at other languages for guidance.

Java started out as a simple language. Java's success is due in part to this simplicity, however, language designers at Sun (now Oracle) decided very late in the process (2004, a full 9 years after its introduction in 1995) to add generics to the language. In addition, they chose to use C++ syntax, which has conflicts with the shift operator (>>). In my opinion, avoiding the introduction of generics has divided the community, effected incompatibilities, and complicated the standard library with multiple versions of the same function.

Depending on your definition of generics, they may have different kinds of parameters. The most intellectually challenging are Agda generics, which imply that generics are just functions which return types, and any function may take types as a parameter. C++ templates are similar to this kind of generics, but make a distinction between usual functions and functions returning types. Since this is a minority when it comes to generics implementations, we will not consider this kind of generics in the remainder of this article.

The most prevalent form of generics are found in Java and Haskell, which only allow types as parameters. From this point of view, the only generic type found in Go today is the map type, so it will be the primary example used.

---

Proposal #1: Add first-class types

This is doomed for failure, because it requires Agda-like (or C++/RTTI) semantics, but it is worth discussing. It is the proposal which requires the least syntactic changes, even if it might be the most intellectually demanding on semantics and runtime. If Go did have first-class types, then we could define the Map type as follows:

func Map(KeyT .(type), T .(type)) .(type) {
  type mapT []struct{
    hash int
    key KeyT
    value T
  }
  return mapT
}

Using this proposal, we can simulate map[KeyT]T with Map(KeyT, T). As you can see, type parameters are indicated with the ".(type)" token as is found in Go type-switch statements. One advantage of this system would be that it could be used to define Array(n, T) and Matrix(m, n, T) which would be impossible to define using more restrictive generics. Other functions that could defined using this proposal are new and make, which take a type as their first parameter. Here is a summary of the related changes to the Go specification:

Result        |= ".(type)"
ParameterDecl |= TypeName ".(type)"
Expression    |= TypeName

In addition to these syntactic changes, a few paragraphs of rules and discussion maybe required.

---

Proposal #2: Add 'generic' (or '_Generic') keyword

This is the crux of this blog article. Since first-class types require that types and objects be intermixed, it requires that every parameter be suffixed with .(type), but with the generic keyword, we can enforce that every parameter be a type parameter (Java/Haskell-style generics). This makes declarations much easier to read, and has provides a much more structured style.

generic Map[KeyT]T []struct{
    hash int
    key KeyT
    value T
}

Using this proposal, we can simulate map[KeyT]T with Map[KeyT]T. As you can see, this is very close to the standard built-in map type, with the obvious case difference. Here is an overview of the associated changes to the Go specification:

TypeLit       |= GenericType
TopLevelDecl  |= GenericDecl
GenericType    = "generic"            GSignature Type
GenericDecl    = "generic" identifier GSignature Type
GSignature     = GParameters identifier
GParameters    = "[" [ GParameterList [ "," ] ] "]"
GParameterList = GParameterDecl { "," GParameterDecl }
GParameterDecl = identifier

Having considered adding generic syntax to LiteralType, I don't think it would work well with literal syntax or semantics. What would such a literal mean? Would it simply be syntactic sugar for the base-type of the generic type with all the free variables filled in? If all that would be gained is syntax sugar, then I don't see the value of adding it to literal syntax.

This generics model also allows more than one parameter inside the brackets, but it also requires at least one parameter after the brackets, which is also very similar to how generic types work in Haskell, since the prevalence of monads has made the last parameter more important than the others. This generics model is also much closer to Go's existing map type, the only problem now is that the first letter is uppercase "M" whereas the built-in map type has a lowercase "m". This leads us to our next section, which discusses another proposal.

---

Proposal #3: Add 'attrib' (or '_Attrib') keyword

This proposal combines two issues into a single solution. The first issue is nonessential attributes (such as alignment, packing, etc.) for which GCC uses the __attribute__((id)) syntax. The second issue is the public/private convention in Go which may be an issue in the future if it is used to implement existing APIs such as POSIX, which require lowercase external symbols. The solution I propose to both of these issues are a new syntax: _Attrib(id), which allows simple annotation of declarations in such a way as to override normal Go semantics. Consider two such attributes: public and private, which would allow us to define the built-in map type as follows:

generic attrib(public) map[KeyT]T []struct{
    hash int
    key KeyT
    value T
}

Using this proposal, there would be no difference between this defined type and the built-in map type. This would allow very small Go compilers which need only handle arrays and slices, and leave the map type, and possibly other built-in functions such as cap and len to a library implementation.

The syntax was designed to force the attrib specifier immediately before the identifier, to make it clear which identifier it is referring to. The changes to the MethodDecl production are questionable, and require more consideration and discussion. I have also considered moving AttribSpec to before the keywords which would simplify the grammar significantly. This is the summary of all the required extensions to the Go specification.

ConstSpec    |= AttribSpec IdentifierList 
                [ [ Type ] "=" ExpressionList ]
TypeSpec     |= AttribSpec identifier Type
VarSpec      |= AttribSpec IdentifierList 
                ( Type [ "=" ExpressionList ] 
                | "=" ExpressionList )
MethodDecl   |= "func" Receiver AttribSpec 
                MethodName Signature [ Body ]
FunctionDecl |= "func" AttribSpec identifier 
                Signature [ Body ]
GenericDecl  |= "generic" AttribSpec identifier 
                GSignature Type
AttribSpec    = "attrib" "(" identifier 
                [ "(" ExpressionList ")" ] ")"

---

Proposal #4: Add 'pragma' (or '_Pragma') keyword

Another method used for nonessential attributes are top-level pragmas. Using pragmas, you can specify information to be used on a per-file or per-package basis. This could be a useful and simple extension to the language that could bring existing GCC syntax to Go compilers. Here is an example of using pragmas in a Go source file:

pragma("GCC poison strdup")
pragma("DSGO prefix pthread_")

Other possibilities for pragmas are to use Go with OpenMP, even though it might sound funny at first, because most of the features of OpenMP are already in Go! However, we should never underestimate what people will do with compilers, given the chance. Here is a summary of the associated extensions to the Go specification:

TopLevelDecl |= PragmaDecl
PragmaDecl    = "pragma" "(" string_lit ")"

---

Conclusion

We have reviewed four possible extensions to the Go programming language, which may not seem in the spirit of Go right now, but over time may become necessary. If Go is to be used for serious projects, developers may come to expect these features, rather than hope for them. As we have learned from Java, if we wait for too long to introduce new features to Go, it may polarize the community, which would be an undesirable outcome for everyone.

Fuzzy Scrollbars

After several weeks away from this blog, I would like to start up again with the exciting field of fuzzy logic. Most introductions to fuzzy logic talk about how everything is different, and overview the three major types: Gödel, product, and Lukasiewicz, but I'm going to take a different approach. I'm going to stick with product fuzzy logic for the remainder of this post.

Product Fuzzy Logic

It is important to discuss domains when regarding operations, and fuzzy logic is somewhat consistent in its use of the domain [0, 1], which includes 0, 1 and all the real numbers in between. Let's call this the Clamp datatype. There are obvious mappings to and from Clamp and Bool: 0 maps to False, 1 maps to True, and the in betweens can be handled on a case-by-case basis (possibly to throw and error or something).

The operations on the Clamp datatype would be both arithmetic and logic operations, which would include addition, subtraction, multiplication, division, conjunction (and), and disjuction (or). How would these be defined in product fuzzy logic? According to the product T-norm (the fancy name for how "and" is defined), the logic operations would be defined as

• not(x) = 1 - x
• and(x, y) = xy
• or(x, y) = x + y - xy
• implies(x, y) = 1 - x(1 - y)

One interesting thing about product fuzzy logic is that there are multiple ways of defining equiv, depending on what properties you are looking for. One possibility is that equiv(x, y) = and(implies(x, y), implies(y, x)), which would mean that it would have to be defined as equiv(x, y) = (1 - x - y + 3xy - yx² - xy² + x²y²). This is quite a mess, and the same effect (in terms of mapping to Bool) can be accomplished by the definition equiv(x, y) = (1 - x - y + 2xy).

Other considerations for operations on the Clamp datatype are that when performing arithmetic operations, the output of the normal operation must be minimized or maximized to fall within the range [0, 1]. This may have the unfortunate side effect of many run-time errors, or unexpected mathematical outputs. If all conversions to and from the Clamp datatype are excplicit, then this is not a problem.

Scrollbars

Just as the String is the basis for the text field widget, the Clamp datatype can be thought of as the basis for the scrollbar widget. Scrollbars are usually tied to a viewport, which can contain an image, a document, or a webpage. Usually the viewport has all the information that the scrollbar needs to do its job. It has height and width information of the entire image, and the height and width information of the part that is showing. Using this information, the only additional information that needs to be stored in the scrollbar is the percentage of how far down on the scrollbar you are, which is equivalent to what the Clamp datatype provides.

Colors

Why is it that such a fundamental datatype like Clamp is not found in more programming languages? Usually it is defined as equivalent to Float or Double, and intended to be used with values between 0.0 and 1.0, but no strict validation or bounds checking are done to ensure that this is so. One example where this kind of usage is found is in OpenGL, which we will talk about next.In OpenGL, one of the datatypes that is pervasive throughout the API is the GLclamp datatype. This is also a value between 0 and 1, and also where I got the name from. With this datatype, OpenGL defines some texture and image datatypes as structure with members of this type, for example a color in OpenGL is a structure with 3 GLclamps.

With so many applications, the Clamp datatype is a prime example of a design goal that I've been trying to find a name for. When abstractions are too low-level, then you may have few abstractions (like bit and byte), but using these abstractions is a nightmare (just look at assembly code). When abstractions are too high-level, then you may be able to use them easily (like drag-and-drop database form designers), but the number of abstractions makes a steep learning curve, and a high pressure on the discovery of abstractions other people have made (for example: some game libraries have 3 types of fog, and dozens of types of meshes, depending on whether you want to add skeletons or water ripples, etc.). The design goal I'm trying to acheive is a balance of these two extremes that balances low-level abstraction and high-level abstraction in such a way that there are few abstractions, and using them is easy. I think Clamp datatype meets both of these requirements.

From fuzzy logic to colors to scollbars, Clamp seems to form the basis of an unrecognized basis for computation that may be useful in the future.

Embedded OpenGL

OpenGL is a graphics library that has provided a foundation for many other high-level graphics libraries built on top of it. Because it is so versitle and low-level, it has been proven to stand the test of time, and has evolved to handle the changing demands that library authors have been placed on it. As demands have changed, new functions are added, and one trend between all versions of OpenGL is that more specific functions are being replaced by more general functions. For example, the functions glColorPointer, glNormalPointer, and glTexCoordPointer have been augmented with glVertexAttribPointer, which can be used to replace all of the previous functions. This has led to a completely different view of OpenGL that has led to the development of smaller profile of OpenGL functions called the Embedded Specification, or OpenGL-ES for short. One side effect of this is that the more abstract functions are increasingly similar to an object-oriented library, which means there are fewer and fewer functions that do not fit into an object model of OpenGL. If we partition the functions found in OpenGL-ES 2.0 into an object-oriented hierarchy, then we arrive at the following classes:

• Blend
• Buffer -- pixel buffers, vertex buffers, and index buffers
• Capability -- abstract class with glEnable and glDisable
• Clear -- clearing the background to a solid color
• Color
• CullFace -- consists of glCullFace and glFrontFace
• Depth
• FrameBuffer
• Get -- reading global state with functions like glGetString etc.
• Mipmap -- utilities for the Texture class
• Program
• RenderBuffer
• Shader -- vertex and fragment shaders
• Stencil
• Texture
• Uniform -- uniform variables
• VertexAttrib -- vertex attributes

after removing these functions from the OpenGL-ES 2.0 API, we are left with the miscellaneous functions:

• glDrawArrays(mode, first, count)
• glDrawElements(mode, count, type, indices)
• glFinish()
• glFlush()
• glLineWidth(width)
• glPolygonOffset(factor, units)
• glSampleCoverage(value, invert)
• glPixelStorei(name, param)
• glReadPixels(x, y, width, height, format, type, pixels)
• glScissor(x, y, width, height)
• glViewport(x, y, width, height)

This says a lot about OpenGL-ES, especially the fact that glDrawPixels has been removed. This means the only way to copy pixels into an OpenGL-ES context is to apply a texture with the pixel data to a rectangle. However, OpenGL-ES forbids GL_QUADS, so you have to use GL_TRIANGLE_FAN to accomplish the same result. glReadPixels, glScissor, and glViewport can be related in that they all accept a rectangle as an argument.

Although OpenGL-ES allows one to do everything that can be done in OpenGL 2.0, there is more of a suggestion to use more general interfaces, like shaders and buffer objects, since all of the fixed functionality has been removed. This means that it is possible to reimplement all of OpenGL in OpenGL-ES. However, since ES is still rather new, there have not been any such implementations of fixed functionality in terms of shaders or the like. There are some tutorials that show how to duplicate such simple functionality as colors in terms of shaders, but a complete reimplementation is still nowhere to be found.

Semantic MathML

I don't know what they were thinking, but there is a much better way to encode MathML in RDF. One can be tempted to assign an RDF property to every element in MathML, but that wouldn't be the OpenMath way. Since MathML3 is getting more and more dependent on OpenMath, it seems appropriate to encode as much as possible using this combined system. MathML3 is split up into several sections: Presentation, Pragmatic Content, and Strict Content. The last one requires only 10 XML Elements to be understood by MathML3 processors, namely: m:apply, m:bind, m:bvar, m:csymbol, m:ci, m:cn, m:cs, m:share, m:semantics, m:cerror, and m:cbytes. This provides for a great economy of thought, and a chance to easily define a total mapping from MathML to RDF. MathML3 already defines a mapping from MathML2 to Strict Content MathML3, so this is the only set of Elements we need to consider. These are the prefixes we will use:

@prefix ari: <http://www.openmath.org/cd/arith1#> .
@prefix fns: <http://www.openmath.org/cd/fns1#> .
@prefix sts: <http://www.openmath.org/cd/sts#> .
@prefix sm: <http://example.com/SemanticMath/> .
@prefix m: <http://www.w3.org/1998/Math/MathML> .

These are the rdfs:Class's we will define:

• sm:Content (all MathML Content)
• sm:Number (for <cn/>, subclass of sts:NumericalValue)
• sm:String (for <cs/>, subclass of xs:string)

and these are the rdf:Property's we will define:

• sm:apply :: Property * sm:Content
• sm:applyTo :: Property * (List sm:Content)
• sm:bind :: Property * (List sm:Content)
• sm:bindOp :: Property * sm:Content
• sm:bindIn :: Property * sm:Content
• sm:error :: Property sm:Error sm:Content
• sm:errorWas :: Property sm:Error (List sm:Content)

Strict Content MathML3 can be translated with the following algorithm:

• <m:cn> NUM </m:cn>
"NUM"^^sm:Number
• <m:cs> TEXT </m:cs>
" TEXT "^^sm:String
• <m:ci> Name </m:ci>
Use blank node identifier (like _:Name)
• <m:csymbol> Symbol </m:csymbol>
Use URIs (http://www.openmath.org/cd/arith1#plus for <m:csymbol cd="arith1">plus</m:csymbol>, or the value of the definitionURL for MathML2)
• <m:share/>
Use URIs
• <m:cbytes> DATA </m:cbytes>
" DATA "^^xs:base64Binary
• <m:cerror> Symbol Content* </m:cerror>

[] rdf:type sm:Error ;
   sm:error Symbol ;
   sm:errorWas LIST(Content) .

• <m:apply> Symbol Content* </m:apply>

[] sm:apply Symbol ;
   sm:applyTo LIST(Content) .

• <m:bind> Symbol Vars* Content* </m:bind>

[] sm:bindOp Symbol ;
   sm:bind LIST(Vars) ;
        sm:bindIn LIST(Content) .

This allows all of Strict Content MathML to be encoded in a set of RDF triples. In order to have the full semantics of MathML and OpenMath there would have to be an OM-interpretation, OM-entailment, and so on, just as there is with every other application of RDF. Although I have not worked out the details of this, an OM-interpretation of an RDF graph that contains these properties would have to treat sts:mapsTo special, and maybe map it to an appropriate rdfs:Class. As an example of how this might be used, this is how the Haskell code (\x -> x + 2) would be translated into RDF using the properties listed above as follows:

_:f sm:bindOp fns:lambda ;
    sm:bind LIST(_:x) ;
    sm:bindIn LIST(
    [ sm:apply ari:plus ; 
      sm:applyTo LIST(_:x "2"^^sm:Number) ]).

Now that RDF has lambdas, the possibilities are endless!

Subtly Different Linked Lists

If you don't know what a linked list is, then you probably shouldn't be reading this. However, the idea is very simple: a linked list is a list constructed from smaller lists of length two. So a list of length 3 would be constructed as (A, B, C) = (A, (B, (C, Nothing))), where Nothing would indicate that we have reached the end of the list. Most programming languages have some kind of linked list datatype. C++ has the list<T> type in STL, Lisp has the 'list type, Haskell has the [T] type, and RDF has the rdf:List type. We will not consider sequences here, so C types will be left for a future article.

Without any type restrictions, such a pair need not be required to make a list. For example, we could make the structure ((A, B), C), but we could not interpret it as a list. This is exactly how the Common Lisp cl:cons constructor works. The Common Lisp HyperSpec calls anything constructed with cl:cons a list. The special case where the second element of cl:cons is either another cl:cons or a cl:nil is called a proper list. This restriction makes a subtype which can always be interpreted as a list. This subtype corresponds to the lists found in scripting languages such as Perl, Python, and Ruby. RDF lists can also be described as proper lists, since an RDFS-interpretation requires that the range of rdf:rest is rdf:List, so any attempt to make an improper list with rdf:rest will result in an inconsistent RDF graph.

With the restriction that the second part of a pair is a list, we obtain so-called heterogeneous lists, because the members of the list (encoded as the first part of each pair) can be of any type. If we also enforce the restriction that each member of the list is of the same type, then we obtain what is called homogeneous lists for obvious reasons. This subtype is what is found in more strict languages, such as list<T> in C++ and [T] in Haskell. This is an overview of the different kinds of lists we have talked about:

• Improper list (a, (b, c))
• Proper list (a, (b, (c, ())))
• Heterogeneous list [1, "message"]
• Homogeneous list [1, 2, 3]

While we have talked about lists before, restricting RDF's heterogeneous lists to obtain homogeneous lists. However, in this article we are going to consider generalizing RDF's proper lists to obtain improper lists as well. In order to do this, we will make a distinction between cl:Cons the Class and cl:cons the constructor. First we need rdf:List rdf:subClassOf cl:Cons so that they can work together, and then cl:Cons will represent both improper lists and proper lists, and rdf:List will represent proper lists only. To represent an improper list, would would have to be an instance of cl:Cons but not rdf:List. For a heterogeneous list, one would have to be missing the ex:listType property, and for a homogeneous list, one would require the presence of the ex:listType property. This covers all the different list types discussed above.