add calgary test data set

src/main/java/org/xerial/snappy/SnappyInputStream.java

@@ -60,11 +60,12 @@ public class SnappyInputStream extends InputStream
     protected void readHeader() throws IOException {
         byte[] header = new byte[SnappyCodec.headerSize()];
         int readBytes = in.read(header, 0, header.length);
-        if (readBytes < header.length) {
+        if (header[0] != SnappyCodec.MAGIC_HEADER[0]) {
             // do the default uncompression
             readFully(header, readBytes);
             return;
         }
+
         SnappyCodec codec = SnappyCodec.readHeader(new ByteArrayInputStream(header));
         if (codec.isValidMagicHeader()) {
             // compressed by SnappyOutputStream
@@ -113,22 +114,22 @@ public class SnappyInputStream extends InputStream
 
     @Override
     public int read(byte[] b, int off, int len) throws IOException {
-        int wroteBytes = 0;
-        for (; wroteBytes < len;) {
+        int writtenBytes = 0;
+        for (; writtenBytes < len;) {
             if (uncompressedCursor >= uncompressedLimit) {
                 if (hasNextChunk())
                     continue;
                 else {
-                    return wroteBytes == 0 ? -1 : wroteBytes;
+                    return writtenBytes == 0 ? -1 : writtenBytes;
                 }
             }
             int bytesToWrite = Math.min(uncompressedLimit - uncompressedCursor, len);
-            System.arraycopy(uncompressed, uncompressedCursor, b, off + wroteBytes, bytesToWrite);
-            wroteBytes += bytesToWrite;
+            System.arraycopy(uncompressed, uncompressedCursor, b, off + writtenBytes, bytesToWrite);
+            writtenBytes += bytesToWrite;
             uncompressedCursor += bytesToWrite;
         }
 
-        return wroteBytes;
+        return writtenBytes;
     }
 
     protected boolean hasNextChunk() throws IOException {

src/test/java/org/xerial/snappy/SnappyInputStreamTest.java

@@ -98,7 +98,7 @@ public class SnappyInputStreamTest
 
     @Test
     public void biteWiseRead() throws Exception {
-        byte[] orig = readResourceFile("alice29.txt");
+        byte[] orig = readResourceFile("testdata/calgary/paper6");
         byte[] compressed = Snappy.compress(orig);
 
         SnappyInputStream in = new SnappyInputStream(new ByteArrayInputStream(compressed));

src/test/java/org/xerial/snappy/testdata/calgary/paper4

@@ -0,0 +1,294 @@
+.EQ
+delim $$
+.EN
+.ls 1
+.ce
+PROGRAMMING BY EXAMPLE REVISITED
+.sp
+.ce
+by John G. Cleary
+.ce
+Man-Machine Systems Laboratory
+.ce
+University of Calgary.
+.sp
+.sh "Introduction"
+.pp
+Efforts to construct an artificial intelligence have relied on
+ever more complex and carefully prepared programs.  While useful in
+themselves, these programs
+are unlikely to be useful in situations where ephemeral and
+low value knowledge must be acquired.  For example a person (or robot)
+working in a normal domestic environment knows a lot about which
+cupboards have sticky doors and where the marmalade is kept.  It seems
+unlikely that it will ever be economic to program such knowledge 
+whether this be via a language or a discourse with an expert system.
+.pp
+It is my thesis, then, that any flexible robot system working in the
+real world must contain a component of control intermediate
+between hard wired 'reflex' responses and complex intellectual 
+reasoning.  Such an intermediate system must be adaptive, be able
+to carry out complex patterned responses and be fast in operation.
+It need not, however, carry out complex forward planning or be capable
+of introspection (in the sense that expert systems are able to explain
+their actions).
+.pp
+In this talk I will examine a system that acquires knowledge by 
+constructing a model of its input behaviour and uses this to select its
+actions.  It can be viewed either as an automatic adaptive system  or
+as an instance of 'programming by example'.  Other workers have
+attempted to do this, by constructing compact models in some appropriate
+programming language:e.g. finite state automata [Bierman, 1972], 
+[Bierman and Feldman, 1972]; LISP [Bierman and Krishnaswamy, 1976]; 
+finite non-deterministic
+automata [Gaines,1976], [Gaines,1977],
+[Witten,1980]; high level languages [Bauer, 1979], [Halbert, 1981].
+These efforts, however, suffer from
+the flaw that for some inputs their computing time is 
+super-exponential in the number
+of inputs seen.  This makes them totally impractical in any system which
+is continuously receiving inputs over a long period of time.
+.pp
+The system I will examine comprises one or more simple independent
+models.  Because of their simplicity and because no attempt is made to 
+construct models which are minimal,
+the time taken to store new information and to make 
+predictions is constant and independent of the amount of information stored
+[Cleary, 1980].  This leads to a very integrated and responsive environment.
+All actions by the programmer are immediately incorporated into the program
+model. The actions are also acted upon so that their consequences are 
+immediately apparent.
+However, the amount of memory used could grow 
+linearly with time. [Witten, 1977] introduces a modelling system related
+to the one here which does not continually grow and which can be updated
+incrementally.
+.pp
+It remains to be shown that the very simple models used are capable 
+of generating any
+interestingly complex behaviour.
+In the rest of this
+talk I will use the problem of executing a subroutine to illustrate
+the potential of such systems.
+The example will also illustrate some of the techniques which have been
+developed for combining multiple models, [Cleary, 1980], [Andreae
+and Cleary, 1976], [Andreae, 1977], [Witten,1981].  It has also been
+shown in [Cleary, 1980] and in [Andreae,1977] that such systems can
+simulate any Turing machine when supplied with a suitable external memory.
+.sh "The modelling system"
+.pp
+Fig. 1 shows the general layout of the modeller.  Following the flow
+of information through the system it first receives a number of inputs
+from the external world.  These are then used to update the current
+contexts of a number of Markov models.  Note, that each Markov model
+may use different inputs to form its current context, and that they
+may be attempting to predict different inputs.  A simple robot
+which can hear and move an arm might have two models; one, say, in
+which the last three sounds it heard are used to predict the next
+word to be spoken, and another in which the last three sounds and the last
+three arm movements are used to predict the next arm movement. 
+.pp
+When the inputs are received each such context and its associated 
+prediction (usually
+an action) are added to the Markov model.  (No
+counts or statistics are maintained \(em they are not necessary.)  When the
+context recurs later it will be retrieved along with all the predictions
+which have been stored with it.
+.pp
+After the contexts have been stored they 
+are updated by shifting in the new inputs. These new contexts are then
+matched against the model and all the associated predictions are retrieved.
+These independent predictions from the individual Markov models
+are then combined into a single composite 
+prediction.
+(A general theory of how to do this has been
+developed in [Cleary, 1980]).  
+.pp
+The final step is to present this 
+composite prediction to a device I have called the 'choice oracle'.
+This uses whatever information it sees fit to choose the next action.
+There are many possibilities for such a device.  One might be to choose
+from amongst the predicted actions if reward is expected and to choose
+some other random action if reward is not expected.  The whole system then 
+looks like
+a reward seeking homeostat.  At the other extreme the oracle might be
+a human programmer who chooses the next action according to his own
+principles.  The system then functions more like a programming by
+example system \(em [Witten, 1981] and [Witten, 1982] give examples of such 
+systems.
+[Andreae, 1977] gives an example of a 'teachable' system lying between
+these two extremes.
+.pp
+After an action is chosen this is
+transmitted to the external world and the resultant inputs are used
+to start the whole cycle again.  Note that the chosen action will
+be an input on the next cycle.
+.sh "Subroutines"
+.pp
+An important part of any programming language is the ability to write a 
+fragment of a program and then have it used many times without it having
+to be reprogrammed each time.  A crucial feature of such shared code is
+that after it has been executed the program should be controlled by the
+situation which held before the subroutine was called. A subroutine can be 
+visualised as a black box with an unknown and arbitrarily complex interior.
+There are many paths into the box but after passing through each splits again
+and goes its own way, independent of what happened inside the box.
+.np
+Also, if there are $p$ paths using the subroutine and $q$ different sequences
+within it then the amount of programming needed should be proportional to
+$p + q$ and not $p * q$.  The example to follow possess both these properties
+of a subroutine.
+.rh "Modelling a Subroutine."
+The actual model we will use is described in Fig. 2.  There are two Markov
+models (model-1 and model-2) each seeing and predicting different parts of
+the inputs.  The inputs are classified into four classes; ACTIONs that
+move a robot (LEFT, RIGHT, FAST, SLOW), patterns that it 'sees' (danger,
+moved, wall, stuck) and two types of special 'echo' actions, # actions
+and * actions (*home, #turn).  The # and * actions have no effect on the 
+environment,
+their only purpose is to be inputs and act as place keepers for relevant
+information.  They may be viewed as comments which remind the system of
+what it is doing.  (The term echo was used in [Andreae,1977], where the
+idea was first introduced, in analogy to spoken words of which one
+hears an echo.)
+.pp
+Model-2 is a Markov model of order 2 and uses only # actions in its
+context and seeks to predict only * actions.  Model-1 is a Markov model 
+of order 3 and uses all four classes of inputs in its context.  It
+seeks to predict ACTIONs, # actions and * actions.  However, * actions
+are treated specially.  Rather than attempt to predict the exact * action
+it only stores * to indicate that some * action has occurred.  This
+special treatment is also reflected in the procedure for combining the
+predictions of the two models.  Then the prediction of model-2 is used,
+only if model-1 predicts an *.  That is, model-1 predicts that some 
+* action will occur and model-2 is used to select which one. If model-1
+does not predict an * then its prediction is used as the combined prediction
+and that from model-2 is ignored.
+.pp
+The choice oracle that is used for this example has two modes.  In
+programmer mode a human programmer is allowed to select any action
+she wishes or to acquiesce with the current prediction, in which case
+one of the actions in the combined prediction is selected.  In
+execution mode one of the predicted actions is selected and the
+programmer is not involved at all.
+.pp
+Before embarking on the actual example some points about the predictions
+extracted from the individual Markov models should be noted.  First, if 
+no context can be found stored in the memory which equals the current
+context then it is shortened by one input and a search is made for any
+recorded contexts which are equal over the reduced length.  If necessary
+this is repeated until the length is zero whereupon all possible
+allowed actions are predicted.
+.pp
+Fig. 3 shows the problem to be programmed.  If a robot sees danger it
+is to turn and flee quickly.  If it sees a wall it is to turn and return
+slowly.  The turning is to be done by a subroutine which, if it gets 
+stuck when turning left, turns right instead.
+.pp
+Fig. 4 shows the contexts and predictions stored when this is programmed.
+This is done by two passes through the problem in 'program' mode: once
+to program the fleeing and turning left; the other to program the wall
+sequence and the turning right.  Fig. 5 then shows how this programming
+is used in 'execute' mode for one of the combinations which had not been
+explicitly programmed earlier (a wall sequence with a turn left).  The
+figure shows the contexts and associated predictions for each step.
+(Note that predictions are made and new contexts are stored in both
+modes.  They have been omitted from the diagrams to preserve clarity.)
+.sh "Conclusion"
+.pp
+The type of simple modelling system presented above is of interest for a
+number of reasons.  Seen as a programing by example system, 
+it is very closely 
+integrated. Because it can update its models incrementally in real time
+functions such as input/output, programming, compilation and execution
+are subsumed into a single mechanism. Interactive languages such as LISP
+or BASIC gain much of their immediacy and usefulness by being interpretive 
+and not requiring a separate compilation step when altering the source
+program. By making execution integral with the process of program entry
+(some of) the consequencs of new programming become immediately apparent.
+.pp
+Seen as an adaptive controller, the system has the advantage of being fast
+and being able to encode any control strategy. Times to update the model
+do not grow with memory size and so it can operate continuously in real time.
+.pp
+Seen as a paradigm for understanding natural control systems, it has the
+advantage of having a very simple underlying storage mechanism. Also,
+the ability to supply an arbitrary choice oracle allows for a wide
+range of possible adaptive strategies.
+.sh "References"
+.in +4m
+.sp
+.ti -4m
+ANDREAE, J.H. 1977
+Thinking with the Teachable Machine.  Academic Press.
+.sp
+.ti -4m
+ANDREAE, J.H. and CLEARY, J.G. 1976
+A New Mechanism for a Brain.  Int. J. Man-Machine Studies
+8(1):89-119.
+.sp
+.ti -4m
+BAUER, M.A. 1979 Programming by examples. Artificial Intelligence 12:1-21.
+.sp
+.ti -4m
+BIERMAN, A.W. 1972
+On the Inference of Turing Machines from Sample Computations.
+Artificial Intelligence 3(3):181-198.
+.sp
+.ti -4m
+BIERMAN, A.W. and FELDMAN, J.A. 1972
+On the Synthesis of Finite-State Machines from Samples of
+their Behavior.  IEEE Transactions on Computers C-21, June:
+592-597.
+.sp
+.ti -4m
+BIERMAN, A.W. and KRISHNASWAMY, R. 1976 Constructing programs from example 
+computations. IEEE transactions on Software Engineering SE-2:141-153.
+.sp
+.ti -4m
+CLEARY, J.G. 1980
+An Associative and Impressible Computer. PhD thesis, University
+of Canterbury, Christchurch, New Zealand.
+.sp
+.ti -4m
+GAINES, B.R. 1976
+Behaviour/structure transformations under uncertainty.
+Int. J. Man-Machine Studies 8:337-365.
+.sp
+.ti -4m
+GAINES, B.R. 1977
+System identification, approximation and complexity.
+Int. J. General Systems, 3:145-174.
+.sp
+.ti -4m
+HALBERT, D.C. 1981
+An example of programming by example. Xerox Corporation, Palo Alto, 
+California.
+.sp
+.ti -4m
+WITTEN, I.H. 1977
+An adaptive optimal controller for discrete-time Markov
+environments.  Information and Control, 34, August: 286-295.
+.sp
+.ti -4m
+WITTEN, I.H. 1979
+Approximate, non-deterministic modelling of behaviour
+sequences.  Int. J. General Systems, 5, January: 1-12.
+.sp
+.ti -4m
+WITTEN, I.H. 1980
+Probabilistic behaviour/structure transformations using
+transitive Moore models.  Int. J. General Systems, 6(3):
+129-137.
+.sp
+.ti -4m
+WITTEN, I.H. 1981
+Programming by example for the casual user: a case study.
+Proc. Canadian Man-Computer Communication Conference, Waterloo,
+Ontario, 105-113.
+.sp
+.ti -4m
+WITTEN, I.H. 1982
+An interactive computer terminal interface which predicts user 
+entries. Proc. IEE Conference on Man-Machine Interaction,
+Manchester, England.
+.in -4m

src/test/java/org/xerial/snappy/testdata/calgary/paper5

@@ -0,0 +1,320 @@
+.pn 0
+.EQ
+delim $$
+define RR 'bold R'
+define SS 'bold S'
+define II 'bold I'
+define mo '"\(mo"'
+define EXIST ?"\z\-\d\z\-\r\-\d\v'0.2m'\(br\v'-0.2m'"?
+define NEXIST ?"\z\-\d\z\o'\-\(sl'\r\-\d\v'0.2m'\(br\v'-0.2m'"?
+define ALL ?"\o'V-'"?
+define subset '\(sb'
+define subeq  '\(ib'
+define supset '\(sp'
+define supeq  '\(ip'
+define mo '\(mo'
+define nm ?"\o'\(mo\(sl'"?
+define li '\& sup ['
+define lo '\& sup ('
+define hi '\& sup ]'
+define ho '\& sup )'
+.EN
+.ls 1	
+.ce
+A LOGICAL IMPLEMENTATION OF ARITHMETIC 
+.sp 3
+.ce
+John G. Cleary 
+.ce
+The University of Calgary, Alberta, Canada.
+.sp 20
+\u1\dAuthor's Present Address: Man-Machine Systems Group, Department of
+Computer Science, The University of Calgary, 2500 University Drive NW
+Calgary, Canada T2N 1N4. Phone: (403)220-6087.  
+.br
+.nf
+UUCP:  ...!{ihnp4,ubc-vision}!alberta!calgary!cleary
+       ...!nrl-css!calgary!cleary
+ARPA:  cleary.calgary.ubc@csnet-relay
+CDN:   cleary@calgary
+.fi
+.sp 2
+.ls 2
+.bp 0
+.ls 2
+.ce
+Abstract
+.pp
+So far implementations of real arithmetic within logic programming
+have been non-logical.  A logical description of the behaviour of arithmetic
+on actual
+machines using finite precision numbers is not readily available.  
+Using interval analysis a simple description of real arithmetic is possible.
+This can be translated to an implementation within Prolog.
+As well as having a sound logical basis the resulting system 
+allows a very concise and powerful programming style and is potentially
+very efficient.
+.bp
+.sh "1 Introduction"
+.pp
+Logic programming aims to use sets of logical formulae as
+statements in a programming language.
+Because of many practical difficulties the full generality of logic
+cannot (yet) be used in this way.   However, by restricting the
+class of formulae used to Horn clauses practical and efficient
+languages such as PROLOG are obtained.
+One of the main problems in logic programming is to extend this area
+of practicality and efficiency to an ever wider range of formulae and 
+applications.  
+This paper considers such an implementation for arithmetic.
+.pp
+To see why arithmetic as it is commonly implemented in PROLOG systems
+is not logical consider the following example:
+.sp
+.nf
+	X = 0.67, Y = 0.45, Z is X*Y, Z = 0.30
+.fi
+.sp
+This uses the notation of the 'Edinburgh style' Prologs.
+(For the moment we assume an underlying floating point
+decimal arithmetic with two significant places.)
+The predicate 'is' assumes its righthand side is an arithmetic
+statement, computes its value, and unifies the result with its lefthand side.
+In this case the entire sequence succeeds, however, there are some serious 
+problems.
+.pp
+In a pure logic program the order of statements should be irrelevant to
+the correctness of the result (at worst termination or efficiency might be
+affected).  This is not true of the example above.  The direction of execution
+of 'is' is strictly one way so that
+.sp
+	Y = 0.45, Z = 0.30, Z is X*Y
+.sp
+will deliver an error when X is found to be uninstantiated inside 'is'.
+.pp
+The second problem is that the answer Z = 0.30 is incorrect!\ 
+The correct infinite precision answer is Z = 0.3015.  This inaccuracy
+is caused by the finite precision implemented in the floating point
+arithmetic of modern computers.
+It becomes very problematic to say what if anything it means when
+Z is bound to 0.30 by 'is'.  This problem is exacerbated by long sequences
+of arithmetic operations where the propagation of such errors can lead the
+final result to have little or no resemblence to the correct answer.
+.pp
+This is further class of errors, which is illustrated by the fact that the
+following two sequences will both succeed if the underlying arithmetic rounds:
+.sp
+	X = 0.66, Y = 0.45, Z = 0.30, Z is X*Y
+.br
+	X = 0.67, Y = 0.45, Z = 0.30, Z is X*Y
+.sp
+This means that even if some invertable form of arithmetic were devised
+capable of binding X when:
+.sp
+	Y = 0.45, Z = 0.30, Z is X*Y
+.sp
+it is unclear which value should be given to it.
+.pp
+The problem then, is to implement arithmetic in as logical a manner
+as possible while still making use of efficient floating point arithmetic.
+The solution to this problem has three major parts.
+The first is to represent PROLOG's 
+arithmetic variables internally as intervals of real numbers.
+So the result of 'Z is 0.45*0.67' would be to bind Z to the 
+open interval (0.30,0.31).  
+This says that Z lies somewhere in the interval
+$0.30 < Z < 0.31$, which is certainly true, and probably as informative
+as possible given finite precision arithmetic.
+(Note that Z is NOT bound to the data structure (0.30,0.31), this
+is a hidden representation in much the same way that pointers are used
+to implement logical variables in PROLOG but are not explicitly visible
+to the user.  Throughout this paper brackets such as (...) or [...] will
+be used to represent open and closed intervals not Prolog data structures.)
+.pp
+The second part of the solution is to translate expressions such as
+\&'Z is (X*Y)/2' to the relational form 'multiply(X,Y,T0), multiply(2,Z,T0)'.
+Note that both the * and / operators have been translated to 'multiply'
+(with parameters in a different order).  This relational form will be seen to 
+be insensitive to which parameters are instantiated and which are not,
+thus providing invertibility.
+.pp
+The third part is to provide a small number of control 'predicates' able
+to guide the search for solutions.
+The resulting system is sufficiently powerful to be able to
+solve equations such as '0 is X*(X-2)+1' directly.
+.pp
+The next section gives a somewhat more formal description of arithmetic
+implemented this way.  Section III gives examples of its use and of the
+types of equations that are soluble within it.  Section IV compares our 
+approach here with that of other interval arithmetic systems and with
+constraint networks.  Section V notes some possibilities for a parallel 
+dataflow implementation which avoids many of the difficulties of traditional
+dataflow execution.
+.sh "II. Interval Representation"
+.pp
+Define $II(RR)$ to be the set of intervals over the real numbers, $RR$.
+So that the lower and upper bounds of each interval can be operated on as 
+single entities they will be treated as pairs of values.  
+Each value having an attribute of being open or closed 
+and an associated number.  For example the interval (0.31,0.33] will be
+treated as the the pair $lo 0.31$ and $hi 0.33$.  
+The brackets are superscripted to minimize visual confusion when writeing 
+bounds not in pairs.
+As well as the usual real numbers 
+$- inf$ and $inf$, will be used as part of bounds,
+with the properties that $ALL x mo RR~- inf < x < inf$ 
+The set of all upper bounds is defined as:
+.sp
+	$H(RR)~==~\{ x sup b : x mo RR union \{ inf \},~b mo \{ hi , ho \} \} $
+.sp
+and the set of lower bounds as:
+.sp
+	$L(RR)~==~\{ \& sup b x : x mo RR union \{ -inf \},~b mo \{ li , lo \} \} $
+.sp
+The set of all intervals is then defined by:
+.sp
+	$II(RR)~==~L(RR) times H(RR)$
+.sp
+Using this notation rather loosely intervals will be identified 
+with the apropriate subset of the reals.  For example the following 
+identifications will be made:
+.sp
+	$[0.31,15)~=~< li 0.31, ho 15 >~=~ \{ x mo RR: 0.31 <= x < 15 \}$
+.br
+	$[-inf,inf]~=~< li -inf , hi inf> ~=~ RR$
+.br
+and	$(-0.51,inf]~=~< lo -0.51 , hi inf >~=~ \{ x mo RR: 0.51 < x \}$
+.sp
+The definition above carefully excludes 'intervals' such as $[inf,inf]$
+in the interests of simplifying some of the later development.
+.pp
+The finite arithmetic available on computers is represented by a
+finite subset, $SS$, of $RR$.  It is assumed that 
+$0,1 mo SS$.  The set of intervals allowed over $SS$ is $II(SS)$ defined as 
+above for $RR$.  $SS$ might be a bounded set of integers or some more complex
+set representable by floating point numbers.
+.pp
+There is a useful mapping from $II(RR)$ to $II(SS)$ which associates
+with each real interval the best approximation to it:
+.nf
+.sp
+	$approx(<l,h>)~==~<l prime, h prime >$
+.br
+where	$l prime mo L(SS), l prime <= l, and NEXIST x mo L(SS)~l prime <x<l$
+.br
+	$h prime mo H(SS), h prime >= h, and NEXIST x mo H(SS)~h prime >x>h$.
+.pp
+The ordering on the bounds is defined as follows:
+.sp
+	$l < h, ~ l,h mo II(RR)~ <->~l= \& sup u x and h = \& sup v y$
+			and $x<y$ or $x=y$ and $u<v$
+where 	$ ho, li, hi, lo$ occur in this order and $x<y$ is the usual ordering 
+on the reals extended to include $-inf$ and $inf$.  
+The ordering on the brackets is carefully chosen so that intervals such as
+(3.1,3.1) map to the empty set.
+Given this definition it is easily verified that 'approx' gives
+the smallest interval in $II(SS)$ enclosing the original interval in $II(RR)$.
+The definition also allows the intersection of two intervals to be readily 
+computed:
+.sp
+	$<l sub 1,h sub 1> inter <l sub 2, h sub 2>~=~$
+		$< max(l sub 1 , l sub 2), min(h sub 1 , h sub 2 )>$
+.sp
+Also and interval $<l,h>$ will be empty if $l > h$.  For example, according
+to the definition above $lo 3.1 > ho 3.1$ so (3.1,3.1) is correctly computed
+as being empty.
+.pp
+Intervals are introduced into logic by extending the notion of 
+unification.  A logical variable I can be bound to an interval $I$,
+written I:$I$.  Unification of I to any other value J gives the following
+results:
+.LB
+.NP
+if J is unbound then it is bound to the interval, J:$I$;
+.NP
+if J is bound to the interval J:$J$ then
+I and J are bound to the same interval $I inter J$.
+The unification fails if $I inter J$ is empty.
+.NP
+a constant C is equivalent to $approx([C,C])$;
+.NP
+if J is bound to anything other than an interval the unification fails.
+.LE
+.pp
+Below are some simple Prolog programs and the bindings that result when
+they are run (assuming as usual two decimal places of accuracy).
+.sp
+.nf
+	X = 3.141592
+	X:(3.1,3.2)
+
+	X > -5.22, Y <= 31, X=Y
+	X:(-5.3,32]  Y:(-5.3,31]
+.fi
+.sp
+.rh "Addition"
+.pp
+Addition is implemented by the relation 'add(I,J,K)'
+which says that K is the sum of I and J.
+\&'add' can be viewed as a relation on $RR times RR times RR$ defined
+by:
+.sp
+	$add ~==~ \{<x,y,z>:x,y,z mo  RR,~x+y=z\}$
+.sp
+Given that I,J, and K are initially bound to the intervals $I,J,K$ 
+respectively, the fully correct set of solutions with the additional
+constrain 'add(I,J,K)' is given by all triples in the set 
+$add inter I times J times K$.  
+This set is however infinite, to get an effectively computable procedure
+I will approximate the additional constraint by binding I, J and K
+to smaller intervals.  
+So as not to exclude any possible triples the new bindings, 
+$I prime, J prime roman ~and~ K prime$ must obey:
+.sp
+	$add inter I times J times K ~subeq~ I prime times J prime times K prime$
+.sp
+Figure 1 illustrates this process of
+.ul
+narrowing.
+The initial bindings are I:[0,2], J:[1,3]
+and K:[4,6].  After applying 'add(I,J,K)' the smallest possible bindings
+are I:[1,2], J:[2,3] and K:[4,5].  Note that all three intervals have been
+narrowed.
+.pp
+It can easily be seen that:
+.sp
+	$I prime supeq \{x:<x,y,z> ~mo~ add inter I times J times K \}$
+.br
+	$J prime supeq \{y:<x,y,z> ~mo~ add inter I times J times K \}$
+.br
+	$K prime supeq \{z:<x,y,z> ~mo~ add inter I times J times K \}$
+.sp
+If there are 'holes' in the projected set then $I prime$ will be a strict
+superset of the projection, however, $I prime$ will still 
+be uniquely determined by the projection.  This will be true of any
+subset of $RR sup n$ not just $add$.
+.pp
+In general for
+.sp
+	$R subeq RR sup n,~ I sub 1 , I sub 2 , ... , I sub n mo II(RR)$
+and $I prime  sub 1 , I prime  sub 2 , ... , I prime  sub n mo II(RR)$
+.sp
+I will write 
+.br
+	$R inter I sub 1 times I sub 2 times ... times I sub n nar 
+I prime sub 1 times I prime sub 2 times ... times I prime sub $
+.br 
+when the intervals $I prime sub 1 , I prime sub 2 , ... , I prime sub $
+are the uniquelly determined smallest intervals including all solutions.
+
+.sh "IV. Comparison with Interval Arithmetic"
+.pp
+.sh "V.  Implementation"
+.pp
+.sh "VI. Summary"
+.sh "Acknowledgements"
+.sh "References"
+.ls 1
+.[
+$LIST$
+.]