Redundant Radix Enumeration Systems

Transcription

1 Redundant Radix Enumeration Systems CS 860: Automatic Sequences Daniel Roche University of Waterloo 21 November 2006 Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

2 Outline Outline 1 Introduction: An Example from Theoretical Definitions and Results Basic Definitions Necessary and Sufficient Conditions Automatic Sequences for Redundant Number Systems Further Results Derived from the Transducer 3 Application: Carry-Free Addition 4 Conclusions and Open Problems Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

3 Introduction Base 10 with negative digits John Colson ( ): A Short Account of Negativo-affirmative Arithmetick Still base 10, but allow any digit to be positive or negative Denote negative digits by over bar, e.g. 7 = 7 To perform arithmetic operation, convert to small figures first. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

4 Introduction Example: The Algorithm 1 Convert to small figures 2 Perform aritmnetic operation 3 Convert back to standard representation Example Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

5 Introduction Example: The Algorithm 1 Convert to small figures 2 Perform aritmnetic operation 3 Convert back to standard representation Example Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

6 Introduction Example: The Algorithm 1 Convert to small figures 2 Perform aritmnetic operation 3 Convert back to standard representation Example Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

7 Introduction Example: The Algorithm 1 Convert to small figures 2 Perform aritmnetic operation 3 Convert back to standard representation Example Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

8 Introduction Colson s arithmetic method Redundant representation (e.g = 2774) No carries in addition with small figures Biggest advantages in multiplication Fast conversion to/from standard representation Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

9 Theory Definitions Number systems and radix polynomials Definition A number system N is a (v + 1)-tuple of integers (r,d 1,d 2,...,d k ) such that v, r 1 and d 1 < d 2 < < d k. We say that r is the base and D = {d 1,d 2,...,d k } is the digit set of N. Definition Let N be a number system with base r and digit set D. The evaluation function of N is E N : D Z given by: E N (a 0 a 1 a 2 a n ) = a 0 r n + a 1 r n a n 1 r + a n. In particular, the empty string always evaluates to zero. That is, E N (ǫ) = 0. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

10 Theory Definitions Completeness and redundancy Definition A number system N = (r,d) is complete for the set S Z iff n S, w D s.t. E N (w) = n. Definition A number system N = (r,d) is redundant iff there exist two words w 1 and w 2 in D, which do not start with 0, such that E N (w 1 ) = E N (w 2 ). Goal We are interested in number systems which are redundant and complete for N or Z. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

11 Theory Definitions Examples of completeness and redundancy Example N = (2,0,1) is non-redundant and complete for N. Example N = (3,6,2) is redundant but not complete for N. Example N = (3, 1, 0, 1, 2, 3) is redundant and complete for Z. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

12 Theory Conditions A necessary condition on D Definition A set S Z is a complete residue system (CRS) mod r iff S = r and {s mod r s S} = {m mod r m Z}. Theorem If N = (r,d) is redundant and complete for N, then D contains a CRS mod r and D > r. Proof Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

13 Theory Conditions A sufficient condition for D Theorem If the following conditions hold, then N = (r,d) is redundant and complete for the set S: Proof. r 2 D D such that N = (r,d ) is complete for S D > r Just need to show that D > r guarantees redundancy. This is done by Mantua (1976), and repeated in our textbook. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

14 Theory Automaticity N-automaticity Definition Let N = (r,d) be some number system which is complete for N. The sequence (a n ) n 0 over a finite alphabet is N-automatic iff there exists a DFAO M = (Q,D,δ,q 0,,τ) such that a n = τ(δ(q 0,w)), for all n 0 and all w with E N (w) = n. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

15 Theory Example of N-automaticity Automaticity Example (The Thue-Morse Sequence is (2,1,0,1)-automatic) 1 1, ,1 1 1 Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

16 Theory Automaticity A General Transducer to (r, Σ r ) Recall that Σ r is defined to be the standard digit set for base r. Let N = (r,d) be any number system complete for N, with r 2. Define the finite-state transducer T N = (Q,D {0},δ,0,Σ r,λ) with: Q = { m, m + 1,...,0,1,...,n 1}, where m and n are the least non-negative integers such that m(r 1) d n(r 1) d D. δ : Q (D {0}) Q λ : Q (D {0}) Σ r Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

17 Theory Automaticity Definitions of Transition and Output Functions for T N Let q Q and a (D {0}) be arbitrary. Since q,a Z and r 2, q + a = br + c, for some b,c Z with 0 c < r. Then define the transition and output functions by: δ(q,a) = b, λ(q,a) = c. Then δ(q,a) Q and λ(q,a) Σ r, so T N is a valid finite-state transducer. Proof Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

18 Theory Automaticity Examples of the transducer T N Example (N = (k,1,2,...,k)) a/a 0 a < k k/0 0 1 a/a+1 0 a < k 1 k/1 k-1/0 Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

19 Example (N = (3,2,3,5)) 0/0 2/0 0/2 5/ /0 5/2 3/2 2/1 0/1 5/0 3/2 2/0 0/2 1 5/1 2,2 1 3/1 Sample Conversion 523

20 Example (N = (3,2,3,5)) 0/0 2/0 0/2 5/ /0 5/2 3/2 2/1 0/1 5/0 3/2 2/0 0/2 1 5/1 2,2 1 3/1 Sample Conversion 523 0

21 Example (N = (3,2,3,5)) 0/0 2/0 0/2 5/ /0 5/2 3/2 2/1 0/1 5/0 3/2 2/0 0/2 1 5/1 2,2 1 3/1 Sample Conversion

22 Example (N = (3,2,3,5)) 0/0 2/0 0/2 5/ /0 5/2 3/2 2/1 0/1 5/0 3/2 2/0 0/2 1 5/1 2,2 1 3/1 Sample Conversion

23 Example (N = (3,2,3,5)) 0/0 2/0 0/2 5/ /0 5/2 3/2 2/1 0/1 5/0 3/2 2/0 0/2 1 5/1 2,2 1 3/1 Sample Conversion

24 Theory Automaticity Proof that T N maps (r, D) to (r, Σ r ) - Part 1 Lemma Let N = (r,d) be some number system and w D n. Then if T(w R ) = u = u 0 u 1 u n and the transducer ends in state q, then E N (w) = q r n+1 + u n r n + u n 1 r n u 1 r + u 0. = q r n+1 + E (r,σr)(u R ) Proof. Induction on the size of w. Detailed proof Corollary E N (w) 0 iff q 0. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

25 Theory Automaticity Proof that T N maps (r, D) to (r, Σ r ) - Part 2 Theorem Let N = (r,d) be any number system complete for N with r 2. Let N = (r,σ r ) be the standard base-r number system. Then, for all w D, E N (w) N implies E N (w) = E N (T N (w R 0 n 1 )). Proof. T N ends in state 0 on reading w R 0 n 1 whenever E N (w) 0. Detailed proof Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

26 Theory Automaticity Equivalence of N-automaticity Theorem If N = (r,d) is any number system with r 2 which is complete for N, then a sequence over a finite alphabet is r-automatic if and only if it is N-automatic. Proof. r-automaticity is the same as (r,σ r )-automaticity. Similar to proof given in class Have to be careful because of redundancy Also must show the language L = {w D E N (w) N} is regular. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

27 Theory Further Results Another sufficient condition Theorem Let N = (r,d) be any number system containing a CRS mod r, with r 2. Let the states of T N be { m,...,0,...,n 1}. Suppose that, for each q { n + 1,...,0,...,m}, there exists a word w D such that E N (w) = q. Then N is complete for N. Proof. Swap the input and output alphabets of T N. So every edge is modified, but remains. Conditions guarantee a path to state 0 with output always 0. Detailed proof Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

28 Theory Further Results A necessary and sufficient condition? Recall the sufficient condition given earlier: Theorem If the following conditions hold, then N = (r,d) is redundant and complete for the set S: r 2 D D such that N = (r,d ) is complete for S D > r Question Is this a necessary condition? Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

29 Theory Further Results Sufficient condition is not necessary Counterexample. Let N = (2,5,3,2). No 2-subset of D forms a complete number system for N with base 2. But: 1 = E N (25) 0 = E N (ǫ) 1 = E N (23) 2 = E N (2) 3 = E N (2325) 4 = E N (232) 5 = E N (2323) Then, from the previous theorem, N is complete for N. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

30 Application What is carry-free addition? Impossible to completely eliminate carries Colson s approach: renormalize before each operation This is inefficient. Instead we eliminate carry propogation. Allows for operations to be performed in parallel Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

31 Application Basic definitions for carry-free addition Definition For any number system (r, α,...,0,...,β), the redundancy index ρ is defined by: ρ = α + β + 1 r. Let N = (r,d) be a number system on which we want to perform carry-free addition. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

32 Application Restrictions for carry-free addition Restriction 1 D = { α, α + 1,...,0,...,β 1,β}, for some non-negative integers α and β. Restriction 2 The base r is strictly greater than 2. Restriction 3 If either α or β is 1, then ρ 3. Otherwise, ρ 2. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

33 Application Carry-free addition algorithm Algorithm: Carry-Free Addition To add the numbers x and y (both in D ), we calculate three sums for each position i: position sum: p i = x i + y i interim sum: w i = p i rt i+1 final sum: s i = w i + t i Each t i+1 is the transfer digit, and we define t 0 = 0. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

34 Transfer digit selection Application Algorithm: Transfer Digit Selection First, define the following two constants: α λ = µ = r 1 Define C λ = and C µ+1 =. For λ < k µ, define C k such that: β r 1 kr (α λ) C k (k 1)r + β µ + 1. Then, for each i, we set t i+1 = k iff C k p i < C k+1. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

35 Application Correctness of carry-free addition algorithm Theorem If the final sum digits are chosen to be s i as described previously, then each s i produces no new carry. That is, α s i β. Proof. Algebraic (sometimes tedious) manipulations based on the restrictions we set and the constants we chose. Detailed proof Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

36 Application Example of carry-free addition Constants ρ = 3 λ = 2 µ = 1 C 2 = C 1 = 8 C 0 = 2 C 1 = 2 C 2 = Example (r = 5,α = 5,β = 3) t i = p i = x i + y i = t i+1 = w i = p i rt i+1 = s i = w i + t i = Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

37 Application Example of carry-free addition Constants ρ = 3 λ = 2 µ = 1 C 2 = C 1 = 8 C 0 = 2 C 1 = 2 C 2 = Example (r = 5,α = 5,β = 3) t i = 0 p i = x i + y i = 3 t i+1 = 1 w i = p i rt i+1 = 2 s i = w i + t i = Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

38 Application Example of carry-free addition Constants ρ = 3 λ = 2 µ = 1 C 2 = C 1 = 8 C 0 = 2 C 1 = 2 C 2 = Example (r = 5,α = 5,β = 3) t i = 1 p i = x i + y i = 9 t i+1 = 2 w i = p i rt i+1 = 1 s i = w i + t i = Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

39 Application Example of carry-free addition Constants ρ = 3 λ = 2 µ = 1 C 2 = C 1 = 8 C 0 = 2 C 1 = 2 C 2 = Example (r = 5,α = 5,β = 3) t i = 2 p i = x i + y i = 5 t i+1 = 1 w i = p i rt i+1 = 0 s i = w i + t i = Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

40 Application Example of carry-free addition Constants ρ = 3 λ = 2 µ = 1 C 2 = C 1 = 8 C 0 = 2 C 1 = 2 C 2 = Example (r = 5,α = 5,β = 3) t i = 1 p i = x i + y i = 0 t i+1 = 0 w i = p i rt i+1 = 0 s i = w i + t i = Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

41 Application Summary of redundant digit set arithmetic Arithmetic can be performed in parallel Biggest advantages come with multiplication Problems: subtraction, comparison Implemented in some chip designs Tradeoff of space versus time, but Higher redundancy does not give better performance Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

42 Conclusion Conclusions and summary Redundant number systems are a relatively old concept Some necessary, sufficient conditions for completeness and redundancy Redundancy preserves automaticity Can be used to perform fast, parallel arithmetic Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

43 Conclusion Open Problems Find a simple necessary and sufficient condition for a number system to be both redundant and complete for some set. Similar conditions exist for nonredundancy and completeness (e.g. Matula - basic digit sets). Is the alternate characterization of k-automaticity useful at all from an automatic sequences standpoint? Are there further applications of redundant number system arithmetic, such as in arbitrary-precision integer arithmetic? Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

44 References Conclusion John Colson. A short account of negativo-affirmative arithmetick. In Philos. Trans. Royal Soc. London, volume Juha Honkala. On number systems with negative digits. Ann. Acad. Sci. Fenn. Ser. A I Math., 14(1): , David W. Matula. Radix arithmetic: digital algorithms for computer architecture. In Applied computation theory: analysis, design, modeling, pages Prentice-Hall, Englewood Cliffs, N.J., B. Parhami. Generalized signed-digit number systems: A unifying framework for redundant number representations. IEEE Transactions on Computers, 39(1):89 98, Jan Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

45 Appendix Proofs 5 Proofs Initial Necessary and Sufficient Conditions N-Automaticity Further Results 6 Carry-Free Addition Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

46 Proofs Initial Necessary and Sufficient Conditions Proof of first necessary condition Theorem statement Proof. Let N = (r,d) be some number system. First note that, for any nonempty word w D, if E N (w) m (mod r), then the last digit in w must also be congruent to m mod r. Then, since each of {1,2,...,r} is in N, it is clear that D must contain a CRS mod r. Now suppose N is complete for N and redundant. Then there exist two different, nonempty words w 1,w 2 D + such that E N (w 1 ) = E N (w 2 ). Let i be the index of the last position in which the two words differ. Then the two digits at position i must be congruent mod r. Then, since D contains a CRS and contains two digits in the same congruency class, D > r. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

47 Proofs N-Automaticity Proof that T N is a valid finite-state transducer Transducer definition Proof. Clearly c Σ r, the output alphabet. Need to show that b is a state, that is, b Q. m m(r 1) q + a (n 1) + n(r 1) mr br + c nr 1 mr (r 1) br nr 1 (m + 1)r < br < nr m b n 1 So b Q, and therefore T N is a valid finite-state transducer. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

48 Proofs N-Automaticity Proof of the lemma on evaluation of transducer output Lemma statement Proof by induction. Base case (ǫ) obvious. So assume true for n = k, k 0. Let w D k+1. Suppose T(w R ) = u 0 u 1 u k u k+1. Write w = aw. Let q be the state of T N after reading (w ) R. Then λ(q,a) = u k+1. And suppose δ(q,a) = b. Then a = br + u k+1 q. So E N (w) = a r k+1 + E N (w ) = (br + u k+1 q)r k+1 + q r k+1 + u k r k u 1 r + u 0 = b r k+2 + u k+1 r k+1 + u k r k u 1 r + u 0 And it follows directly from the definitions that E (r,σr)(u R ) = u n r n + u n 1 r n u 1 r + u 0. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

49 Proofs N-Automaticity Proof that T N maps (r, D) to (r, Σ r ) Theorem statement Proof. From the corollary, we know that T N must be in a non-negative state after reading w R. And from the transducer definition, we can see that δ(q, 0) < q whenever q 0. Then, since there are only n 1 positive states, T N must end in state 0 after reading w R 0 n 1. Now let T N (w R 0 n 1 ) = u 0 u k. Clearly E N (0 w) = E N (w), so, from the lemma, E N (w) = 0 r k+1 u k r k + + u 1 r + u 0 = E N (T N (w R 0 n 1 )) Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

50 Proofs Further Results Proof of new sufficient condition Theorem statement Proof. Define a new transducer T N, which is the same as T N but with the input and output characters swapped on each edge. Then T N maps Σ r to D. Now let n N be arbitrary. Suppose T N on input of (n) r outputs w and ends in state q. Then T N on input of w R outputs (n) R r and ends in state q. So, from the lemma earlier, E N (w) = q r w +1 + n. Let u D be a word with E N (u) = q. We know that such a word exists from the assumption of the theorem. Then Therefore N is complete for N. E N (uw) = ( q)r w +1 + q r w +1 + n = n. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40

51 Proofs Carry-Free Addition Correctness proof for carry-free addition Theorem statement Proof. From the way the transfer digits are chosen, we know that λ t i µ and C ti+1 p i < C ti And recall that for each k, kr (α λ) C k (k 1)r + β µ + 1. Then, since s i = w i + t i = p i rt i+1 + t i, we have C ti+1 rt i+1 λ s i < C ti+1 +1 rt i+1 + µ t i+1 r (α λ) rt i+1 λ s i < t i+1 r + β µ + 1 rt i+1 + µ α s i < β + 1 α s i β So s i D, and therefore no new carry is needed. Roche (University of Waterloo) Redundant Radix Enumeration Systems 21 November / 40