Background

The acronym CARDIAC stands for "CARDboard Illustrative Aid to Computation." It was developed by David Hagelbarger at Bell Labs as a tool for teaching how computers work in a time when access to real computers was extremely limited. The CARDIAC kit consists of a folded cardboard "computer" and an instruction manual. In July 1969, the Bell Laboratories Record contained an article describing the system and the materials being made available to teachers for working with it.

As illustrated in the following pictures, the CARDIAC computer consisted of a left-hand CPU section and a right-hand memory section. On the CPU side there are five sliders:

One slider of input "cards"

One slider for the accumulator sign

Three sliders for the digits of an instruction

The memory side has a single slider of output "cards." Portions of the sliders show through cutouts in the card frame. The cutouts for the input and output card sliders each show the current card to be read or written. The combination of the accumulator sign and the three instruction sliders show steps through cutouts that describe the operation of the selected instruction. Effectively, the sliders and cutouts are the instruction decoder of the CPU. Finally, each memory location has a hole in it. A small carboard ladybug serves as the program counter which is moved from location to location in response to the steps described on the CPU side.

The CARDIAC manual is 50+ pages divided into 16 sections describing the basics of computers from a late 1960s perspective. The first six sections cover things like flow charts, instructions, data, addresses, and the stored program concept. Sections 7–12 discuss the CARDIAC and some basic programming techniques including loops and multiplication. Sections 13 and 14 discuss the techniques for bootstrapping and subroutines, both of which we elaborate on below. Section 15 focuses on the development of a program to play NIM. Finally, Section 16 discusses assemblers and compilers. Although there is some duplication of information, the material on this page is not intended to replace the manual. Rather, the material here expands on that in the manual, particularly from the point of view of one who is already familiar with the most basic concepts of computers and programming.

Pictures

Click on these pictures for larger versions.

Book and "Computer"

Open CARDIAC

CARDIAC Architecture

Memory

The CARDIAC has a grand total of 100 memory locations identified by the two-digit decimal numbers 00 through 99. Each memory location holds a signed three-digit decimal numer. (With the exception of a single code example, the CARDIAC book is actually silent on whether memory contains signed or unsigned values.) Locations 00 and 99 are special. Location 00 always contains the value 001, which as we see below is the instruction to read a card into location 01. This special value is used the the bootstrapping process discussed later. Location 99 always contains a value between 800 and 899. The tens and ones digits of the number are the value of the program counter after a jump instruction is executed. This provides the mechanism for a return from subroutine.

CPU

The CARDIAC CPU is a single-accumulator single-address machine. Thus each instruction operates optionally on a single memory location and the accumulator. For example, the ADD instruction reads the data in one memory location, adds it to the current value of the accumulator and stores the result back into the accumulator. The ALU supports addition, subtraction, and decimal shifting. CARDIAC's CPU architecture is illustrated in the following figure:

The CARDIAC accumulator holds a signed 4-digit number, which seems odd given that everything else is oriented around 3-digit numbers. The manual includes the statement:

Since CARDIAC's memory can store only 3-digit numbers, you may be puzzled by the inclusion of an extra square in the accumulator. It is there to handle the overflow that will result when two 3-digit numbers whose sum exceeds 999 are added.

What's not clear is under what conditions that overflow/carry digit is kept or discarded. From the discussion of the SFT instruction in Section 12 of the manual, exactly four digits are kept for the intermediate value between the left and right shift operations. However, the manual doesn't state whether all four digits are kept between instructions nor what happens when storing the accumulator to memory if the accumulator contains a number whose magnitude is greater than 999. In the case of our simulator, we retain all four digits, effectively implementing a 4-digit ALU. However, when storing the accumulator to memory, we discard the fourth digit. I.e. the number stored in memory is a mod 1000, where a is the contents of the accumulator.

I/O

The CARDIAC has exactly one input device and one output device. These are a card reader and a card punch. Unlike real punch cards, the CARDIAC input and output cards can each hold exactly one signed three-digit number. When a card is read by way of the INP instruction, it is taken into the card reader and removed from the stack of cards to be read. Similarly, on each OUT instruction, a new card is "punched" with the specified value on it, and the card moved to the output card stack.

Instruction Set

The CARDIAC's instuction set has only 10 instructions, each identified by an operation code (opcode) of 0 through 9. The instructions are as follows:

Opcode Mnemonic Operation 0 INP Read a card into memory 1 CLA Clear accumulator and add from memory (load) 2 ADD Add from memory to accumulator 3 TAC Test accumulator and jump if negative 4 SFT Shift accumulator 5 OUT Write memory location to output card 6 STO Store accumulator to memory 7 SUB Subtract memory from accumulator 8 JMP Jump and save PC 9 HRS Halt and reset

Encoding

All instructions are non-negative numbers expressed as three-digit decimal numerals. The CARDIAC manual doesn't describe what happens if an attempt is made to execute a negative instruction. In our simulator, we treat negative instructions as no-ops (i.e. they are ignored and the program continues on to the next instruction). The operation code is the most significant of those three digits, i.e., o=⌊i /100⌋, where i is the contents of the instruction register (IR) loaded from the memory location specified by the PC. For most instructions, the lower-order digits are the address of the operand, i.e. a=i mod 100. This arrangement is illustrated in the following figure.

In the cases of the INP and STO instructions, a is the destination address for the data coming from either an input card or the accumulator, respectively. In the cases of the CLA, ADD, OUT, and SUB instructions, a is the source address of the second operand to the ALU or the source address of the operand being written to an output card. For the TAC, JMP, and HRS instructions, a is the address to be loaded into the PC (conditionally, in the case of the TAC instruction). The remaining instruction, SFT, doesn't treat the lower-order digits as an address. Instead, each of the lower-order digits is a number of digit positions to shift first left, then right. The left shift count is given by l=⌊a /10⌋, and the right shift count is given by r=a mod 10. The instruction format for the SFT instruction is shown in the following figure:

Instruction Execution

The instructions operate as described here. In this discussion, we use the following notation:

Notation Meaning ACC Contents of the accumulator PC Contents of the program counter a Operand address as described in the previous subsection MEM[x] Contents of memory location x INPUT Contents of one card read from the input OUTPUT Contents of one card written to the output

INP The INP instruction reads a single card from the input and stores the contents of that card into the memory location identified by the operand address. (MEM[a] ← INPUT) CLA This instruction causes the contents of the memory location specified by the operand address to be loaded into the accumulator. (ACC ← MEM[a]) ADD The ADD instruction takes the contents of the accumulator, adds it to the contents of the memory location identified by the operand address and stores the sum into the accumulator. (ACC ← ACC + MEM[a]) TAC The TAC instruction is the CARDIAC's only conditional branch instruction. It tests the accumulator, and if the accumulator is negative, then the PC is loaded with the operand address. Otherwise, the PC is not modified and the program continues with the instruction following the TAC. (If ACC < 0, PC ← a) SFT This instruction causes the accumulator to be shifted to the left by some number of digits and then back to the right some number of digits. The amounts by which it is shifted are shown above in the encoding for the SFT instruction. (ACC ← (ACC × 10^l) / 10^r) OUT The OUT instruction takes the contents of the memory location specified by the operand address and writes them out to an output card. (OUTPUT ← MEM[a]) STO This is the inverse of the CLA isntruction. The accumulator is copied to the memory location given by the operand address. (MEM[a] ← ACC) SUB In the SUB instruction the contents of the memory location identified by the operand address is subtracted from the contents of the accumulator and the difference is stored in the accumulator. (ACC ← ACC − MEM[a]) JMP The JMP instruction first copies the PC into the operand part of the instruction at address 99. So if the CARDIAC is executing a JMP instruction stored in memory location 42, then the value 843 will be stored in location 99. Then the operand address is copied into the PC, causing the next instruction to be executed to be the one at the operand address. (MEM[99] ← 800 + PC; PC ← a) HRS The HRS instruction halts the CARDIAC and puts the operand address into the PC. (PC ← a; HALT)

Assembly Language

All of the code fragments and complete program examples on this page are shown in an assembly language format with each line organized into six columns:

Address: The first column shows the memory address respresented by that line. Contents: In the second column, we put the number that is stored in that memory location. In most cases, this is a instruction, but for lines with a DATA pseudo-op, it is a data value. Label: The third column contains an optional label on the memory location, allowing it to be identified by name, rather than by address. Opcode: Instruction mnemonics are places in the fourth column. In addition to the ten instructions discussed above, we will use on pseudo-op (or assembler directive), DATA. For memory locations containing a DATA item, the operand is the literal data value stored in the memory location, rather than an operand for an instruction. This pseudo-op is particularly useful when labeled for creating variables. Operand: The fifth column is the operand part of the instruction or the literal data for a DATA directive. Numerical operands are included directly in the address field of the instruction. When a label name appears as an oeprand, the memory address associated with that label is placed in the address field of the instruction. Comment: Any desired descriptive text can be placed after the operand.

Indirection, Indexing, and Pointers

Notice that the only way of specifying the address of a memory location we want to use is in the instruction itself. Most comptuer architectures provide a mechanism whereby the address we want to use can be stored in a register or another memory location. Variables which contains memory addresses are usually referred to as pointers.

Indirect Loads

Even though the CARDIAC doesn't have hardware support for using pointers directly, we can still do simple indirect addressing. Suppose we have a variable stored in a memory location called ptr and it has the value 42 in it. Now if we want to load the accumulator with the contents of memory location 42, we can do something like:

05 100 loader DATA 100 06 042 ptr DATA 042 20 105 CLA loader 21 206 ADD ptr 22 623 STO indload 23 100 indload CLA 00

Notice that even though we have specified that we will load from location 00 in the instruction at location 23, we will have changed it to load from location 42 by the time we run execute that instruction. For that matter, it doesn't matter if we've loaded anything into location 23 before starting this sequence. It will get set before we use it.

Indirect Stores

Storing the accumulator to a memory location identified by a pointer is similar. We just have to be careful not to lose the value we want to store while we're fiddling about with the store instruction and in the following bit of code:

05 600 storer DATA 600 06 042 ptr DATA 042 07 000 acc DATA 000 20 607 STO acc 21 105 CLA storer 22 206 ADD ptr 23 625 STO indstor 24 107 CLA acc 25 600 indstor STO 00

Array Indexing

Often we aren't so much interested in a pointer that identifies a single memory location as we are in an array of memory locations we can refer to by index. We will identify our array locations starting at index 0. So the first element of the array is at index 0, the second at index 1, and so on. If we have a variable called base that holds the first address of the array, then we can just add the base and the index together to get the address of a particular element. This is just a slight modification of the indirect accesses above. In particular, to load from an array element:

05 100 loader DATA 100 06 042 base DATA 042 07 000 index DATA 000 20 105 CLA loader 21 206 ADD base 22 207 ADD index 23 624 STO arrload 24 100 arrload CLA 00

and for storing to an array element:

05 600 storer DATA 600 06 042 base DATA 042 07 000 index DATA 000 08 000 acc DATA 000 20 608 STO acc 21 105 CLA storer 22 206 ADD base 23 207 ADD index 24 626 STO arrstor 25 108 CLA acc 26 600 arrstor STO 00

If we're dealing with only one array, we could eliminate one add instruction from each sequence by pre-adding the base and loader and pre-adding the base and storer.

Stacks

Another use of indirect address is the stack data structure. If you're not familiar with a stack, think of it like a stack of plates in a cafateria. A plate is always placed on top of the stack. Likewise, the one removed is always the one on the top of the stack. We refer to the process of putting an element onto a stack as pushing and the process of taking an element off of a stack as popping. Note that we always pop that most recently pushed element. Because of this, the stack is often referred to as a last-in, first-out (LIFO) data structure. Pushing and popping are very similar to storing and loading indirectly, except that we must also adjust the value of the pointer that identifies the top of the stack. In the following code we'll use a memory location named tos (for top-of-stack) for the pointer. Also, we'll do as is often done in hardware stacks and let the stack grow downward. That is to say, as we push data onto the stack, the stack pointer moves toward lower memory addresses. With that in mind, here is a fragment of code for pushing the accumulator onto the stack:

05 600 storer DATA 600 06 100 loader DATA 100 07 089 tos DATA 089 08 000 acc DATA 000 20 608 STO acc 21 107 CLA tos 22 205 ADD storer 23 628 STO stapsh 24 107 CLA tos 25 700 SUB 00 26 607 STO tos 27 108 CLA acc 28 600 stapsh STO 00

And similarly to pop from the top of the stack:

20 107 CLA tos 21 200 ADD 00 22 607 STO tos 23 206 ADD loader 24 625 STO stapop 25 100 stapop CLA 00

These code fragments (slightly modified) are used in the example below that uses the LIFO properties of the stack to reverse the order of a list of numbers on the input cards.

Subroutines

There are many reasons why we might wish to subdivide a program into a number of smaller parts. In the context of higher level languages and methodologies, these subdivisions are often referred to by names like procedures, functions, and methods. All of these are types of subroutines, the name we usually use when working at the hardware or machine language level. In these sections, we look at the techniques for creating and using subroutines on the CARDIAC. Each subsection progressively builds from the simplest subroutine technique to more complex and advanced techiques. Don't worry if not all of it makes sense on a first reading. You can get a good sense of the general idea of subroutines without necessarily understanding the details of how recursion is implemented on a machine as limited as the CARDIAC.

Single Simple Subroutines

In the CARDIAC, the JMP instruction is effectively a jump-to-subroutine instruction, storing the return address in location 99. Because the address stored in location 99 is prefixed by the opcode 8, the instruction in that location becomes a return-from-subroutine instruction. Thus any segment of code whose last instruction is at location 99 can be called as a subroutine, simply by jumping to its first instruction. For example, a simple routine to double the value of the accumulator could be coded as:

96 000 accval DATA 000 97 696 double STO accval 98 296 ADD accval 99 800 JMP 00

and the subroutine can be called with a jump to double:

897 JMP double

Multiple Subroutines

Clearly, if our subroutine executes a JMP instruction or if it calls another subroutine, then we will lose our return address, because it will be overwritten by the JMP instruction. Along similar lines, if we have more than one subroutine in our program, only one of them can be at the end of the memory space and flow directly into location 99.

As a result, in many cases, we'll need a more involved subroutine linkage mechanism. One way to accomplish this is to save the return address somewhere and restore it when needed. If we use this method, we'll have to devise a mechansism to transfer control to location 99 with the right return address. Although location 99 can itself be used as the return from subroutine instruction, it doesn't have to be. In many cases, it will be easier to copy it to the end of our actual subroutine. Using this approach, we can write a subroutine that outputs the value of the accumulator as follows:

80 686 aprint STO 86 81 199 CLA 99 82 685 STO aexit 83 586 OUT 86 84 186 CLA 86 85 800 aexit JMP 00

Similarly, our doubling routine would look like:

90 696 double STO 96 91 199 CLA 99 92 695 STO dexit 93 196 CLA 96 94 296 ADD 96 95 800 dexit JMP 00

See below for an example of a program that uses these subroutines to produce a list of powers of two.

Recursion

There's one more limitation on subroutines still in the techniques we have developed. What happens if a subroutine calls itself? You might reasonably as, is it even useful for a function call itself? The answer is, yes, and it called recursion.

The key to making it possible for a subroutine to call itself is to realize that no matter where we're called from, we always want to return to the place from which we were most recently called that we haven't already returned to. That should sound familiar. We should use the return addresses in the same LIFO order that a stack provides. In other words, when we call a recursive subroutine, we want to push the return address onto a stack and then pop it back off when we return from that subroutine. With a little reflection, we can see that this approach applies to all subroutine calls, not just to those that are recursive. This is why pushing return addresses on a stack is the basis for hardware subroutine call support in most architectures since about the 1970s on.

On the CARDIAC, we can implement this technique with a modification of the multiple subroutine technique above. When entering a subroutine, rather than copying location 99 to the return from subroutine instruction, we push the contents of location 99 onto the stack. Then when we're about to return from the subroutine, we pop the return address off the stack into the return from subroutine instruction. So our code would look something like:

1xx CLA tos 2yy ADD storer 6zz STO stapsh 1xx CLA tos 700 SUB 00 6xx STO tos 199 CLA 99 zz 600 stapsh STO 00 . . body of the subroutine . 1xx CLA tos 200 ADD 00 6xx STO tos 2ww ADD loader 6ss STO stapop ss 100 stapop CLA 00 6rr STO rts rr 800 rts JMP 00

There's one more aspect of recursive subroutines that is also suitable for other subroutines as well. In particular, subroutines often need input data passed to them by whatever code has called them or temporary variables that are needed during the course of their operation. If a subroutine is not recursive, we can get away with just allocating some fixed memory locations for these. However, in the case of recursive subroutines, we need to make sure that we have fresh ones for each time the subroutine is called and not overwrite the ones that might still be needed by other instances we might return back to. The most natural way to handle this is to allocate them on the stack along with the return address.

Putting all these things together, we can summarize the steps for calling a subroutine in the most general cases:

Before calling the subroutine, we push any inputs (also called arguments or parameters) onto the stack. Transfer control to the first instruction of the subroutine, saving the PC (which holds the return address) in the process. If the hardware has not already saved the PC onto the stack, the first thing we do in the subroutine is copy it to the stack. Move the stack pointer to resever space on the stack for any temporary (local) variables the subroutine will need. Before returning, the subroutine readjusts the stack pointer to remove the temporary variables it allocated. If the hardware does not already expect the return address to be on the stack, we need to pop it off the stack and copy it back to where it does need to be. Return control from the subroutine back to the code that called it. Finally, the calling code adjusts the stack pointer to remove the arguments it pushed onto the stack before calling the subroutine.

Bootstrapping

Like many of the early system designs, the mechanism for loading an initial program into the CARDIAC and getting it running involves a small amount of hardware support and a lot of cleverness. The whole enterprise is often somewhat remenescent of the image of a person attempting to lift themselves off the ground by pulling on their own bootstraps. This is why we usually refrer to the process as bootstrapping or often just booting.

The CARDIAC's hardware support for bootstrapping is the fixed value of memory location 00. The fixed contents of this memory location are 001 which is the instruction to load a single card into location 01. Naturally, after executing this instruction, the PC moves to location 01 and executes the instruction on the card just read. But what do we put on that card to load? The answer is 002, which is the instruction to load a card into location 02. This causes us to load a second card into location 02 and execut it. At first glance, it would seem we haven't really improved things any, because we're right back where we're still just loading a single card and executing it. But here's where the first bit of cleverness comes in. The card we load into location 02 has the value 800 on it which causes us to jump back to location 00 which will load another card into location 01. We now have a loop that loads cards into location 01 and executes them. If the instructions we put into location 01 are reads of cards into other memory locations, we now have a little program that reads a whole set of cards into memory. Conveniently, a card containing just a memory address also contains the instruction to read a card into that memory address. So if the next card we read after the 800 is, say, 010, then location 01 will be changed to an instruction to read a card into location 10, after which we'll execut the 800 instruction to jump back to location 00 and do it all over again. This means that after the 002 and 800 cards, we can have pairs of cards where the first of the pair is the address where we want to put some data, and the second of the pair is the data to put there.

If this is all we did, we'd read all the remaining cards into memory and then the computer would halt when there were no more cards to read. But there's another trick we can play. If we make the last address-data card pair change location 02 from 800 to a jump to the first instruction of our program, the loader loop will stop and control will transfer to the program we just loaded. So after all of our address-data card pairs, we'll append the cards 002 and 8xx where xx is the address of the first instruction of our program. The net effect is that we can now load a program and start running it without any manual intervention.

The last piece of this puzzle is how do we include the data we want the program to operate on? It turns out, that's a simple as just appending the data after the 002 and 8xx cards. When control transfers to the program we loaded, any remaining cards will still be in the reader waiting to be read. When the program executes its first INP instruction, it will happily read the next card, not knowing that there were a bunch of other cards read ahead of it.

So putting all the pieces together, we bootstrap the CARDIAC by putting together a card deck that looks like:

002 800 . . address-data card pairs . 002 8xx where xx is address of the first instruction . . data cards .

Then we put that deck into the card reader, and start the computer at address 00. The CARDIAC will first load the two-card bootstrap loader, then load the program into memory, then transfer control to the newly loaded program. If the program itself also includes INP instructions, they read the remaining data cards.

Simulator

We have developed a CARDIAC simulator suitable for running the code discussed on this page. All of the examples in the next section have been tested using this simulator.

To avoid any unnecessary requirements on screen layout, the simulator is laid out a little differently than the physical CARDIAC. At the top of the screen is the CARDIAC logo from a photograph of the actual unit. This picture is also a link back to this page. The next section of the screen is the CARDIAC memory space as appears on the right hand side of the physical device. When the simulator starts up, the value 001 in location 00 and the value 8-- in location 99 are preloaded. As a simplification, we don't use a picture of a ladybug for the program counter, but instead highlight the memory location to which the PC points with a light green background. Each memory location is editable (including the ones that are intended to be fixed), and the tab key moves focus down each column in memory address order.

The bottom section of the simulator is the I/O and CPU. Input is divided into two text areas. The first is the card deck and is editable. The second area is the card reader, and as cards are consumed by the reader they are removed from the listing in the reader. Cards in the deck are loaded into the reader with the Load button. Output cards appear in the Output text area as they are generated with the OUT instruction.

The CPU section of the simulator has four parts showing the status of the CPU and buttons for control. On the top of the CPU section, the Program Counter is shown in an editable text box. Below that is the instruction decoder with non-editable text boxes showing the contents of the Instruction Register and a breakdown of the instruction decoding in the form of an opcode mnemonic and numeric operand. The Accumulator is shown below the instuction decoder. Below the register display are six buttons that control the operation of the simulator:

Reset The Reset button clears the instruction register, resets the PC and accumulators to 0 and clears the output card deck. Clear Mem This button resets all memory locations to blank and re-initializes location 00 to 001 and location 99 to 8--. Step Clicking on the Step button causes the simulator to execute the single instruction highlighted in the memory space as pointed to by the program counter. Upon completion of the instruction, the screen is updated to show the state of the computer after the instruction. Slow The Slow button causes the simulator to begin executing code starting at the current PC. Instructions are executed at the rate of 10 per second with the screen being updated after each instruction. When the program is run in this way, the movement of the highlighted memory shows the flow of control in the program very clearly. Run In the current version of the simulator, the Run button causes the program to be executed beginning from the current PC at the full speed of the JavaScript interpreter. Because of the way JavaScript is typically implemented, the screen contents will not show the effects of code execution until the simulator executes the HRS instruction and the program halts. Halt Pressing the Halt button while the program is running in slow mode causes the simulator to stop after the current instruction. The state of the machine remains intact and can be continued with any of the Step , Slow , or Run buttons.

Examples

The remainder of this page are a number of examples of programs written for the CARDIAC. They have all been tested using the simulator described above. Because the memory space of the CARDIAC is so limited, none of the programs are particularly complex. You won't find a compiler, operating system, or web browser here. However, we do have a few of more complexity than you might expect. There's a pretty simple program for generating a list of the powers of 2. There's one that recursively solves the Towers of Hanoi problem. For each of them, we include the assembly language source code with assembled machine language code and a card deck suitable for bootstrapping on the CARDIAC.

Note that most of these examples aren't the most compact way of solving the problem. Rather, they illustrate techniques as described through this page. The primary exception is the Towers of Hanoi solution which requried some effort to squeeze it into the limited memory space of the CARDIAC.

When we take these programs and turn them into decks of cards to be bootstrapped on the CARDIAC, we get the card decks listed below the program listings. If you cut and paste the list into the input deck of the simulator, hit load, and hit slow, you can see the program get loaded into memory and run.

Count from 1 to 10

This is sort of our CARDIAC version of "Hello World." Our objective is simply to print out a set of output cards with the values 1 to 10. We keep two variables to control the process. One, called n keeps track of how many cards we still have left to print. At any point in time it represents that we need to print n+1 more cards. We also have a variable called cntr wich is the number to print out. Each time through the loop, we check to see if n is negative and if so, we're done. If not, we decrement it, print cntr and then increment cntr.

Program Listing

04 009 n DATA 009 05 000 cntr DATA 000 10 100 CLA 00 Initialize the counter 11 605 STO cntr 12 104 loop CLA n If n < 0, exit 13 322 TAC exit 14 505 OUT cntr Output a card 15 105 CLA cntr Increment the card 16 200 ADD 00 17 605 STO cntr 18 104 CLA n Decrement n 19 700 SUB 00 20 604 STO n 21 812 JMP loop 22 900 exit HRS 00

Card Deck

002 800 010 100 011 605 012 104 013 322 014 505 015 105 016 200 017 605 018 104 019 700 020 604 021 812 022 900 004 009 002 810

List Reversal

Our next example uses the stack techniques described above to take in a list of cards and output the same list in reverse order. The first card in the input deck (after the bootstrapping and the program code) is the count of how many cards we're operating on. The remainder of the input deck are the cards to reverse. In the example card deck, we are reversing the first seven Fibonacci numbers.

Program Listing

04 600 storer DATA 600 05 100 loader DATA 100 06 089 tos DATA 089 Stack pointer 07 000 acc DATA 000 Temp for saving accumulator 08 000 n1 DATA 000 Write counter 09 000 n2 DATA 000 Read counter 10 008 IN n1 Get the number of cards to reverse 11 108 CLA n1 Initialize a counter 12 609 STO n2 13 109 rdlp CLA n2 Check to see if there are any more cards to read 14 700 SUB 00 15 327 TAC wrlp 16 609 STO n2 17 007 IN acc Read a card 18 106 CLA tos Push it onto the stack 19 204 ADD storer 20 625 STO stapsh 21 106 CLA tos 22 700 SUB 00 23 606 STO tos 24 107 CLA acc 25 600 stapsh STO 00 26 813 JMP rdlp 27 108 wrlp CLA n1 Check to see if there are any more cards to write 28 700 SUB 00 29 339 TAC done 30 608 STO n1 31 106 CLA tos Pop a card off the stack 32 200 ADD 00 33 606 STO tos 34 205 ADD loader 35 636 STO stapop 36 100 stapop CLA 00 37 890 JMP aprint Output a card 38 827 JMP wrlp 39 900 done HRS 00 90 696 aprint STO 96 Write a card containing the contents of the accumulator 91 199 CLA 99 92 695 STO aexit 93 596 OUT 96 94 196 CLA 96 95 800 aexit JMP 00

Card Deck

002 800 004 600 005 100 006 089 007 000 008 000 009 000 010 008 011 108 012 609 013 109 014 700 015 327 016 609 017 007 018 106 019 204 020 625 021 106 022 700 023 606 024 107 025 600 026 813 027 108 028 700 029 339 030 608 031 106 032 200 033 606 034 205 035 636 036 100 037 890 038 827 039 900 090 696 091 199 092 695 093 596 094 196 095 800 002 810 007 001 001 002 003 005 008 013

Powers of 2

This is a slightly more interesting version of the list from 1 to 10. In this case, we are printing the powers of 2 from 0 to 9. The main difference is that instead of incrementing the number to output, we call a subroutine that doubles it. The program illustrates the use of multiple subroutines as discussed above.

Program Listing

04 000 n DATA 000 05 009 cntr DATA 009 10 100 CLA 00 Initialize the power variable with 2^0 11 880 JMP aprint 12 604 loop STO n 13 105 CLA cntr Decrement the counter 14 700 SUB 00 15 321 TAC exit Are we done yet? 16 605 STO cntr 17 104 CLA n 18 890 JMP double Double the power variable 19 880 JMP aprint Print it 20 812 JMP loop 21 900 exit HRS 00 80 686 aprint STO 86 Print a card with the contents of the accumulator 81 199 CLA 99 82 685 STO aexit 83 586 OUT 86 84 186 CLA 86 85 800 aexit JMP 00 90 696 double STO 96 Double the contents of the accumulator 91 199 CLA 99 92 695 STO dexit 93 196 CLA 96 94 296 ADD 96 95 800 dexit JMP 00

Card Deck

002 800 005 009 010 100 011 880 012 604 013 105 014 700 015 321 016 605 017 104 018 890 019 880 020 812 021 900 080 686 081 199 082 685 083 586 084 186 090 696 091 199 092 695 093 196 094 296 002 810

Towers of Hanoi

By far the most complex example we include is a solution to the Towers of Hanoi problem. The puzzle consists of three posts on which disks can be placed. We begin with a tower of disks on one post with each disk smaller than the one below it. The other two posts are empty. The objective is to move all of the disks from one post to another subject to the following rules:

Only one disk at a time may be moved. No disk may be placed on top of a smaller disk.

According to legend, there is a set of 64 disks which a group of monks are responsible for moving from one post to another. When the puzzle with 64 disks is finally solved, the world will end.

Although the puzzle sounds like it would be difficult to solve, it's very easy if we think recursively. Moving n disks from Post a to Post b using Post c as a spare can be done as follows:

Move n−1 disks from Post a to Post c. Move one disk from Post a to Post b. Move n−1 disks from Post c to Post b.

The CARDIAC doesn't have enough memory to solve a 64-disk puzzle, but we can solve smaller instances of the problem. In particular, the program we show here can solve up to six disks. The actual number of disks to solve is given by the first data card, and the initial assignment of source destination and spare posts is given on the second data card. The post assignments as well as the output encoding are shown in the following table.

Output Disk Move 000 1 → 3 001 2 → 3 002 3 → 2 003 3 → 1 004 2 → 1 005 1 → 2

For example, the post assignments indicated by a card with the value 3 are that Post 3 is a, Post 2 is c and Post 1 is b. Similarly, an output card with 3 indicates that we are to move a disk from Post 3 to Post 1.

Before trying to understand the details of this program, note that there are several tricks used to reduce the memory usage. The amount of memory available for the stack allows for a puzzle of up to six disks to be solved with this program. Be aware, however, that slow running this program on six disks takes the better part of a half hour to run.

Program Listing

03 031 tos DATA 031 04 100 loader DATA 100 05 600 storer DATA 600 06 107 r2ld DATA r2 07 001 r2 DATA 001 08 000 DATA 000 09 005 five DATA 005 10 004 DATA 004 11 003 three DATA 003 12 002 DATA 002 34 033 INP 32 Get the number of disks from the cards 35 032 INP 31 Get the column ordering from the cards 36 838 JMP tower Call the tower solver 37 900 HRS 38 199 tower CLA 99 Push the return address on the stack 39 890 JMP push 40 111 CLA three Fetch n from the stack 41 870 JMP stkref 42 700 SUB 00 Check for n=0 43 366 TAC towdone 44 890 JMP push Push n-1 for a recursive call 45 111 CLA three Get the first recursive order 46 870 JMP stkref 47 669 STO t1 48 109 CLA five 49 769 SUB t1 50 890 JMP push 51 838 JMP tower Make first recursive call 52 880 JMP pop 53 111 CLA three Get move to output 54 870 JMP stkref 55 669 STO t1 56 569 OUT t1 57 111 CLA three Get second recursive order 58 870 JMP stkref 59 206 ADD r2ld 60 661 STO t2 61 100 t2 CLA 00 62 890 JMP push 63 838 JMP tower Make second recursive call 64 880 JMP pop 65 880 JMP pop 66 880 towdone JMP pop 67 668 STO towret 68 800 towret JMP 00 70 679 stkref STO refsav Replace the accumulator with the contents 71 199 CLA 99 of the stack indexed by the accumulator 72 678 STO refret 73 179 CLA refsav 74 203 ADD tos 75 204 ADD loader 76 677 STO ref 77 100 ref CLA 00 78 800 refret JMP 00 80 199 pop CLA 99 Pop the stack into the accumulator 81 688 STO popret 82 103 CLA tos 83 200 ADD 00 84 603 STO tos 85 204 ADD loader 86 687 STO popa 87 100 popa CLA 00 88 800 popret JMP 00 90 689 push STO pshsav Push the accumulator on to the stack 91 103 CLA tos 92 205 ADD storer 93 698 STO psha 94 103 CLA tos 95 700 SUB 00 96 603 STO tos 97 189 CLA pshsav 98 600 psha STO 00

Card Deck

002 800 003 031 004 100 005 600 006 107 007 001 008 000 009 005 010 004 011 003 012 002 034 033 035 032 036 838 037 900 038 199 039 890 040 111 041 870 042 700 043 366 044 890 045 111 046 870 047 669 048 109 049 769 050 890 051 838 052 880 053 111 054 870 055 669 056 569 057 111 058 870 059 206 060 661 061 100 062 890 063 838 064 880 065 880 066 880 067 668 068 800 070 679 071 199 072 678 073 179 074 203 075 204 076 677 077 100 078 800 080 199 081 688 082 103 083 200 084 603 085 204 086 687 087 100 088 800 090 689 091 103 092 205 093 698 094 103 095 700 096 603 097 189 098 600 002 834 003 000

Pythagorian Triples

The next example comes courtesy of Mark and Will Tapley. It finds sets of three integers which satisfy the Pythagorian property of x2+y2=z2.

Discussion

There is much motivation and explanation for this program at:

https://www.khanacademy.org/math/recreational-math/vi-hart/vi-cool-stuff/v/what-was-up-with-pythagoras

Subroutine to calculate square of a number:

In finding pythagorean triplets, the operation of squaring a number occurs very often, so the program uses a subroutine to perform this function.

Addresses 076–099 are loaded with the subroutine to utilize the return function hard-wired at address 099.

Addresses 072–075 are used for data storage for the subroutine.

Address 072 is loaded with the value 32, one larger than the largest allowable input. The calling program can test an input by subtracting this value from the prospective input and branching if the result is negative. (Negative value means legal input.)

Address 073 accepts the input to the subroutine. On return, the absolute value of the input will be in this location.

Address 074 is used as a counter during routine execution.

Address 075 will contain the calculated square, an integer between 0 and 961 inclusive.

Subroutine INPUT:

Store the number to be squared in address 073

Jump to address 077 (label SQmem in assembly listing)

-OR-

Load the number to be squared into the accumulator

Jump to address 076 (label SQacc in assembly listing)

Subroutine OUTPUT:

On return, the square of the input number is in address 075.

The subroutine has a single loop (addresses 090–098). In each loop, it subtracts one from a counter which is initially set to one greater than the input number N, then adds a copy of N into the output address. When the counter reaches 1, the output address contains the sum of N copies of N=N2 and the loop exits, returning program control to the location from which it was called (per the return capability special function of location 99).

Limitations:

The square of the input number must have 3 or fewer digits to comply with cell storage limitations. Therefore the input number is checked to be 31 or less (since 322=1024). Violating this condition will cause the subroutine to terminate execution (HRS) with the program counter pointing at location 086. The input number is converted from negative to positive if it was negative, so if the calling program needs a copy of the input, it should store it in some location other than Address 073 (SQIN). After the subroutine executes, that location will contain the absolute value of the input.

Main Program:

The main program searches over all allowable lengths of the shortest side S of the right triangles corresponding to pythagorean triplets. For each shortest side, it then searches over all possible lengths of the intermediate side L. For each combination of short and intermediate sides, it checks whether there is a hypotenuse H that satisfies the condition S2+L2=H2. The short side (S) search starts at 0, to avoid missing any triplets with very small values. (This results in identifying the degenerate triplet (0,1,1) which does satisfy 02+12=12 but does not really correspond to a right triangle.) The long side (L) search for each value of S starts at S+1, because L cannot equal S for an integer triplet (see URL above) and if L<S, the corresponding triplet should already have been found with a smaller S. (So, this program will identify (3,4,5) but will not identify (4,3,5).) The hypotenuse (H) search starts at 1.4 times S, since the minimum possible length of the hypotenuse is greater than the square root of 2 (1.404...) times S. (Note: 1.4 times S is calculated by shifting S right and then adding four copies of the result, which is truncated to an integer, to S. For S<10, the result is just S, so the search takes needlessly long until S≥10.)

With the starting values for S, L, and H, the program calculates S2 + L2−H2. If the result is <0, H is too long. In this case, the program increments L and tries again. If the result is =0, a triplet has been found and is printed out. The program then increments L and tries again. If the result is >0, H is too short. In this case, H is incremented and the program tries again. When H is long enough that no more triplets can be found for this value of S, the value of S is incremented, new L and H starting values are calculated, and the loop repeats.

Addresses 010–067 are loaded with the main program.

Addresses 004–009 are used for data storage.

Address 004 contains S, the smallest member of the triplet (length of the short leg of the triangle) and is initially set to 0.

Address 005 contains S 2 , calculated each time S is changed.

, calculated each time S is changed. Address 006 contains L, the intermediate member of the triplet (length of the long "leg" of the triangle) and is re-initialized for each smallest member loop to one greater than the smallest member (which is always the minimum possible value for L; see above)

Address 007 contains L 2 , calculated each time L is changed.

, calculated each time L is changed. Address 008 contains H, the largest member of the triplet (length of the hypotenuse of the triangle) and is initialized for each smallest member to a value <1.4×(the smallest value) (which is always shorter than the minimum possible value for H)

Address 009 contains H2, calculated each time H is changed. The same address also contains S/10 (S shifted right by one place), used to initialize H each time S is changed. This value is used to set the initial value of H to 1.4 S, which is just less than √ 2 S.

The "outside" loop of the program (addresses 010–067) tests for all possible sets of triplets with the smallest value S stored in 004. After each loop, it increments the value of S and tries again. This loop will terminate when the value of 1.4×S exceeds 31, since the subroutine will no longer be able to calculate correct squares for any possible hypotenuse value (H). The subroutine will halt execution when this input is sent to it. (The outer loop also contains a check to verify that the value of S itself doesn't exceed 31, but this check is never reached.)

The next-inner loop (addresses 032–061) starts with a value of L=S+1. Any smaller, and L would take the role of S (and hence, the resulting triplet would have already been found with a smaller S) or would be qual to S (and the length of the corresponding hypotenuse would be irrational). This loop terminates on one of two conditions: first, when the value of H exceeds 31 (in which case the subroutine to calculate squares can no longer work); or second, when 2L>S2. This latter condition applies because once L exceeds S2/2, L2 and H2 cannot differ by as little as S2 even if H=L+1. At that point, H2−L2 = (L+1)2−L2=2L+1>S2.

The innermost section (addresses 032–044) calculates the difference S2+L2−H2. If the difference is positive, H is incremented and the loop repeats. If the difference is zero, a triplet has been found and the values of S, L, and H are printed out. If the difference is negative or zero, L is then incremented and the loop repeats. In any case where H is incremented, its new value is checked against the limit for inputs to the subroutine, and if it exceeds that limit, the inner two loops terminate and the outer loop progresses to the next value of S.

Independent Verification:

The code below is instructions to Mathematica (tested on versions 8 and 3) which should compute the same output as the above program, but using a more general (and slower) algorithm. It will also generate a plot of triplets by (short side) against (intermediate side).

candid = Table[ Table[ Table[ {i, j, k}, {k, j, i^2/2 + 2} ], {j, i+1, i^2/2 + 1} ], {i, 0, 31} ]; trips = Select[Flatten[candid, 2], #1[[1]]^2 + #1[[2]]^2 == #1[[3]]^2 & ]; smalltrips = Select[trips, #1[[3]] < 32 & ] ListPlot[(Take[#1, 2] & ) /@ trips]

Program Listing

Symbol map:

Address Variable 04 S short side = 0 initially 05 S2 square of short side 06 L long side 07 L2 square of long side 08 H hypotenuse 09 H2 square of hypotenuse. (Also used to store S/10 in picking initial value of H each loop.) -- ---- 72 SQLIM maximum input to Square = 30 initially 73 SQIN input to square subroutine 74 SQCNT counter for square subroutine 75 SQOUT output for square subroutine Address Name (as referenced by JMP instruction) 00 BootLp 10 S_Loop 32 L_Loop 45 Next_H 49 PrintTr 52 Inc_L 62 Next_S -- ----- 76 SQacc 77 SQmem 83 SQpos 87 SQgood 90 SQloop Load Instr. Address Opcode Instruction Comment BootLp: 002 800 JMP BootLp Bootstrap loop. Code self-modifies to load memory locations. 004 000 (variable) S Initial value for Short side = 0 072 032 (constant) SQLIM Limit on input to square = 32 S_Loop: 010 104 CLA S 011 673 STO SQIN Input to square subroutine 012 200 ADD 1 (Using ROM value) 013 606 STO L Save long side L 014 877 JMP SQmem Square subroutine (saved entry) 015 175 CLA SQOUT Retrieve result of subroutine 016 605 STO S2 Store square of S 017 106 CLA L Load L 018 876 JMP SQacc Square subroutine, entry using ACC 019 175 CLA SQOUT Retrieve result of subroutine 020 607 STO L2 Store square of L 021 104 CLA S Load S 022 401 SFT 01 Divide by 10 023 609 STO H2 Save S/10 temporarily in H2 location 024 209 ADD H2 Sum into accumulator 025 209 ADD H2 Sum into accumulator 026 209 ADD H2 Sum into accumulator 027 204 ADD S Sum is now between S and 1.4 S ~ S sqrt(2) 028 608 STO H Store initial hypotenuse H 029 876 JMP SQacc Square subroutine (accumulator entry) 030 175 CLA SQOUT 031 609 STO H2 Store square of H L_Loop: 032 105 CLA S2 Load short side squared 033 207 ADD L2 Add long side squared 034 709 SUB H2 Subtract hyp. squared 035 352 TAC Inc_L if H2 too big, increment L 036 700 SUB 1 Subtract 1 (ROM) 037 349 TAC PrintTr H was just right - print 038 108 CLA H H too small, so load H 039 200 ADD 1 Add 1 (ROM) 040 608 STO H Store back 041 673 STO SQIN Save in input to Square routine 042 772 SUB SQLIM Subtract limit for input 043 345 TAC Next_H Go on if negative (input < 32) 044 862 JMP Next_S Branch to next value of S if not. Next_H: 045 877 JMP SQmem (saved entry) 046 175 CLA SQOUT Get result 047 609 STO H2 048 832 JMP L_Loop Try again PrintTr: 049 504 OUT S Print S 050 506 OUT L Print L 051 508 OUT H Print H Inc_L: 052 106 CLA L Load L 053 200 ADD 1 Increment 054 606 STO L Store 055 876 JMP SQacc Square subr. 056 175 CLA SQOUT get result 057 607 STO L2 Store new L squared 058 106 CLA L Load new L 059 206 ADD L Double it 060 705 SUB S2 Subtract S^2 061 332 TAC L_Loop If S^2 still bigger, keep looking Next_S: 062 104 CLA S Load short side S 063 200 ADD 1 Increment 064 604 STO S Store short side S 065 772 SUB SQLIM Subtract upper limit for Square 066 310 TAC S_Loop If result is negative, new S is low enough to loop again 067 900 HRS Else, S is longer than Square can handle, so Done - exit. --- --- -------- SQacc: 076 673 STO SQIN Jump here if input value is in ACC SQmem: 077 173 CLA SQIN Jump here if input is already in SQIN 078 773 SUB SQIN Input was in both accumulator and SQIN, so this gets 0 079 675 STO SQOUT initialize output to 0 for use later 080 773 SUB SQIN This gets negative of SQIN 081 383 TAC SQpos If the negative is negative, SQIN is positive - good. 082 673 STO SQIN If the negative is positive, store that in SQIN. SQpos: 083 173 CLA SQIN Load Absolute value of input 084 772 SUB SQLIM Compare against limit value 085 387 TAC SQgood Quit if number to square > limit 086 986 HRS Halt if error on input. SQgood: 087 173 CLA SQIN Retrieve number 088 200 ADD 0 Add one 089 674 STO SQCNT Count is input + 1 SQloop: 090 174 CLA SQCNT load counter 091 700 SUB 0 subtract 1 092 674 STO SQCNT save new counter value 093 175 CLA SQOUT load output 094 273 ADD SQIN add a copy of input 095 675 STO SQOUT store cumulative sum 096 100 CLA 0 load 1 (from ROM) 097 774 SUB SQCNT subtract counter 098 390 TAC SQloop loop again if counter was > 1 Jump out of boot loop to 10 (skips initial increment to S) 002 810 JMP S_Loop

Card Deck

002 800 004 000 072 032 010 104 011 673 012 200 013 606 014 877 015 175 016 605 017 106 018 876 019 175 020 607 021 104 022 401 023 609 024 209 025 209 026 209 027 204 028 608 029 876 030 175 031 609 032 105 033 207 034 709 035 352 036 700 037 349 038 108 039 200 040 608 041 673 042 772 043 345 044 862 045 877 046 175 047 609 048 832 049 504 050 506 051 508 052 106 053 200 054 606 055 876 056 175 057 607 058 106 059 206 060 705 061 332 062 104 063 200 064 604 065 772 066 310 067 900 076 673 077 173 078 773 079 675 080 773 081 383 082 673 083 173 084 772 085 387 086 986 087 173 088 200 089 674 090 174 091 700 092 674 093 175 094 273 095 675 096 100 097 774 098 390 002 810

Other CARDIAC Resources

Related Work