Cognitive Computing: 2012 Consciousness and Computation: computing machinery & intelligence 3. EMULATION Mark Bishop

01/04/2014(c) Bishop: Consciousness and Computations2 The Universal Machine Norma Thesis: With suitable coding of data, every algorithm can be represented as a flowchart program for NORMA. But how? Some obvious criticisms of NORMA include... 1: NORMA does not define enough operations and tests: No assignment, multiply, subtract, divide etc. 2: NORMA data types are too restricted: No provision for negative or floating point numbers. 3: Access to data is too restricted: No provision for ARRAYs of numbers. 4: The restriction to flowchart programs is too restrictive: No mechanism for procedures. At the very least one needs to be able to refer to labels indirectly so that sub-routines can be constructed. 5: Other criticisms include: The lack of string handling. No logical shifts etc. The criticisms [1..4] are considered the most important.

01/04/2014(c) Bishop: Consciousness and Computations3 Informal proof of thesis To demonstrate that every algorithm can be coded as a NORMA flowchart program it is necessary to show how each of the main criticisms [1..4] can be answered. Specify a new machine NORMA+ which has the desired feature. Show how any program in NORMA+ can be –recoded as a flowchart program for NORMA.

01/04/2014(c) Bishop: Consciousness and Computations4 Answer to 1st criticism Assignment to a constant (eg. A := 0) Theorem: Every WHILE program can be mapped onto a corresponding flowchart program. WHILE (A <> 0) DO A := A -1; END; This WHILE program can be written as a macro. Similar macros can be written for (A := 1), (A := 2) etc.

01/04/2014(c) Bishop: Consciousness and Computations5 Addition A := A + B using [a]; Consider a macro to calculate (A := A + B). We need to use an extra register a, hence we write the macro as {A := A + B using a} a := 0; WHILE (B <> 0) DO {adds B into A and makes a copy of B} A := A + 1; a := a + 1; B := B - 1; END; WHILE (a <> 0) DO {resets B to original value} a := a - 1; B := B + 1; END;

01/04/2014(c) Bishop: Consciousness and Computations6 Assignment to a register A := B using [a]; A := 0; A := A + B using a;

01/04/2014(c) Bishop: Consciousness and Computations7 Subtraction A := A - B using [a, b]; a := B using b; WHILE (a <> 0) DO A := A - 1; a := a - 1; END;

01/04/2014(c) Bishop: Consciousness and Computations8 Register Test Operations 1 ?(A < 2) using [a, b]; IF ?(A < 2) THEN L2: TRUE ELSE L1: FALSE END; a := A using b; IF (a = 0) THEN GOTO L2; a := a - 1; IF (a = 0) THEN GOTO L2; L1:FALSE; L2:TRUE;

01/04/2014(c) Bishop: Consciousness and Computations9 Multiplication A := A x B using a, b, c; a := A using b; b := B using c; A := 0; WHILE (b <> 0) DO A := A + a using c; b := b - 1; END;

01/04/2014(c) Bishop: Consciousness and Computations10 Division A := A DIV B using [a..e]; {Remainder held in e} e := A using a; A := 0; WHILE (e >= B using [a..c]) DO e := e - B using d; {[a..c] still in use} INC A END;

NORMA and primes: on testing for integer division Div (A, B)? {True iff A is exactly divisible by B} a := A; WHILE (a >= B) DO a := a – B; IF (a = 0) THEN GOTO Label-Divisible ELSE GOTO Label-NOT-Divisible 01/04/2014(c) Bishop: Consciousness and Computations11

On the testing for prime aPrime (A)?{NB. Zero and one are neither prime nor composite} IF (A < 2) THEN GOTO [Label-NOT-Prime] j := A - 1; WHILE NOT (Div (A, j) DO j := j – 1; IF (j = 1) THEN GOTO [Label-Prime] ELSE GOTO [Label-NOT-Prime] 01/04/2014(c) Bishop: Consciousness and Computations12

On calculating the k th prime A := PRIME (K){Stores the K th prime in A, where the 1st prime is 2} A := 0; k := K; WHILE (k <> 0) DO k := k – 1; A := A + 1; WHILE NOT ( aPrime (A) ) DO A := A + 1; END; 01/04/2014(c) Bishop: Consciousness and Computations13

01/04/2014(c) Bishop: Consciousness and Computations14 Answer to criticism 2 Representing negative numbers An arbitrary integer m can easily be represented as the order pair (n, d) of non negative integers: n = |m| d = 0 IF (m >= 0) d = 1 otherwise

01/04/2014(c) Bishop: Consciousness and Computations15 Representing fixed length real numbers Using a similar idea to that used for negative numbers, operations on a non negative rational number r can be defined in terms of the ordered pair (a, b), where (b > 0) and (r = a/b). Since arithmetic on rationals conforms to the following rules: Addition & Subtraction (a,b) ± (c,d) = (ad ± bc, bd) Multiplication (a,b) × (c,d) = (ac, bd) Division (a,b) / (c,d) = (ad, bc) iff (c <> 0) Equality ?((a,b) = (c,d)) iff (ad = bc)... there is no problem constructing appropriate NORMA programs using fixed length reals.

01/04/2014(c) Bishop: Consciousness and Computations16 Answer to 3rd criticism To answer criticism 3 we define a new machine SAM (Simple Array Machine) with more flexible access to data. SAM augments NORMA by possessing the array of registers, A[1], A [2].... A [n] in addition to the standard registers A,B.... Y, which are now referred to as index registers. The operations defined by SAM are those of NORMA plus the array operations: A [J] := A [J] + 1; A [J] := A [J] - 1; where J is any index register. A [n] := A [n] + 1; A [n] := A [n] - 1; where n is any positive integer.

01/04/2014(c) Bishop: Consciousness and Computations17 Array test operations SAM also has the array test operations: ?(A [J] = 0) ?(A [n] = 0) SAMs input and output functions are the same as NORMAs except that the input function also initialises each array register to zero.

01/04/2014(c) Bishop: Consciousness and Computations18 Theorem 2: NORMA can simulate SAM Proof: We have to show how any given program P for SAM can be translated into a NORMA program Q such that: NORMA (Q) = SAM (P) Method: Pack all of SAM array into a single NORMA register. If at some stage in the computation a SAM array contains [a1, a2,.. an] then at the equivalent stage a NORMA register will contain A. A = P 1 a1 × P 2 a2 × P 3 a3... × P n an, where P j = j th prime.

01/04/2014(c) Bishop: Consciousness and Computations19 Increment indexed array To define Q from P, we translate each SAM instruction into a sequence of NORMA instructions as follows: All index register instructions are left unchanged. Array operations of the form A [J] := A [J] + 1 are translated into: B := PRIME (J); A := A × B Where PRIME is the PRIME number function defined earlier.

01/04/2014(c) Bishop: Consciousness and Computations20 Decrement indexed array Array operations of the form A [J] := A [J] - 1 are translated into: B := PRIME (J); A := A DIV B; Where DIV is a special integer division defined by: a DIV b= a / bIf b divides exactly into a = aotherwise

01/04/2014(c) Bishop: Consciousness and Computations21 Testing an array element A test of the form ?(A [J] <> 0) is translated into: B := PRIME (J) RETURN div (A, B) Where div (A,B) is TRUE when A is exactly divisible by B and FALSE otherwise. ie. The test div (A, B) will return TRUE just in the case that A [J] 0.

01/04/2014(c) Bishop: Consciousness and Computations22 INC and DEC array using a constant index, n An operation of the form (A [n] = A [n] + 1) is translated into: B := PRIME (n); A := A × B; Similarly (A [n] = A [n] - 1) is translated into: B := PRIME (n); A := A DIV B;

01/04/2014(c) Bishop: Consciousness and Computations23 Proof of theorem 2 The test ?(A [n] = 0) uses the test div (P n, A) to RETURN the correct value, where P n = PRIME (n). Now each operation and test of Q must be replaced by the corresponding NORMA macro; And if we ensure that Q initially sets A to 1 to represent the input condition of SAMs array then … … the simulation clearly works and hence Theorem 2 is proved.

01/04/2014(c) Bishop: Consciousness and Computations24 Answer to criticism 4 - only flowcharts To answer criticism 4 we need to define a new machine SIM (Simple Indirect Machine). In a SIM program P with labels (1.. l.. k), the operations defined by SIM are those given by NORMA plus the following two Indirect Jump calls: l:GOTO (A);{GOTO a where a is the content of register A} l:IF (T) THEN GOTO (A); {IF TRUE GOTO label a}

01/04/2014(c) Bishop: Consciousness and Computations25 Theorem 3: NORMA can simulate SIM Proof: We have to show how any given program P for SIM can be translated into a NORMA program Q such that: NORMA (Q) = SIM (P) Method: Use a Jump Table. For each register (eg. A) that appears in the program we need to define an extra segment of code, tableA.

01/04/2014(c) Bishop: Consciousness and Computations26 Jump tables

01/04/2014(c) Bishop: Consciousness and Computations27 For the SIM Indirected jump This extra code can be added after instruction k of SIM program P. We can now replace the new SIM instructions by the following NORMA code: SIM l:GOTO (A); NORMA l:GOTO tableA; {Where tableA is label of jump table for the A reg}

01/04/2014(c) Bishop: Consciousness and Computations28 The SIM indirected test SIM l:IF (T) THEN GOTO (A); NORMA l:IF (T) THEN GOTO tableA; {Where tableA is label of the jump table for the A register} Now each operation and test of SIM Q must be replaced by the corresponding NORMA macro. It is now clear that the resulting NORMA program Q is equivalent to the SIM program P. The simulation clearly works and hence Theorem 3 is proved.

HOMEWORK: Implement NORMA_STACK Prove that the machine NORMA_STACK is no more powerful than the universal machine NORMA by designing two MACROs to implement: (a) X=POP which removes the top value from the STACK and places it into the X register; (b) PUSH (X) which places the contents of the X register on to the top of the STACK; and submit a short (no more than 1-page) report detailing their operation. 01/04/2014(c) Bishop: Consciousness and Computations29