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Theory of computing in computer science education
by PATRICK^ C.^ FISCHER University (^) of Waterloo Waterloo, Canada
INTRODUCTION
Theory (^) of computing means^ the^ abstract^ study^ of^ the nature of^ computation^ and^ computing^ devices.^ By^ con-
vention, the terminology is usually applied^ in^ a nar-
rower sense^ to exclude^ numerical^ analysis.^ Thus,^ theory
of computing includes the theory of finite^ automata, formal languages, computability, computational com- plexity, and (^) some aspects of switching circuit theory and logical design. The deviation^ from the literal^ mean- ing of the term may have cc^l__^ red because numerical analysis (^) was already a well-established subject when the other components of^ this^ area^ were in^ their^ infancy.
On the other hand, it may^ be a^ reflection of the empha-
sis (^) on discrete mathematics by workers in theory of computing, in contrast to the preponderance^ of^ con-
tinuous methods in^ numerical^ analysis.
In Curriculum 68,^ theory of computing^ was^ repre- sented by the following courses.*
B3 Discrete^ Structures 16 Switching Theory 17 Sequential Machines (Automata Theory) Al Formal Languages and^ Syntactic^ Analysis A7 Theory^ of Computability
CURRICULUM TRENDS
Lower-level courses
The (^) courses 83,^ IG and^17 above have^ been^ remark-
ably stable, perhaps because of the relative^ distance of
their (^) content from the frontiers of research. Perhaps some (^) minor changes might be considered: Course B could include some material^ on the^ first-order^ predicate
calculus rather than just^ tho propositional^ calculus,^ and
Course 17 still lacks a textbook which is teachable and
' Comm.^ ACM^ 11, 3^ (March,^ 1968),^ pp.^ 151-197.
has (^) a good balance between intuition and^ rigor.^ Never-
theless, the 1968 descriptions^ for^ these^ courses^ are^ still
a very (^) good approximation to current course offerings in (^) a number of university and^ college departments^ of computer science.
Formal languages
Turning now to the advanced-level courses,^ one finds
more updating^ needed. Developments^ in^ formal^ lan-
guage theory have taken place in^ both^ the^ area^ of parsing methods and the study of abstract families^ of languages. An^ updated^ course^ Al^ would^ probably^ do well (^) to assimilate a (^) greater proportion of the new material in the former^ area,^ thus^ shifting^ the^ overall orientation of the course more toward^ the^ practical
side. In order to make room for this recent material,^ a
few lectures worth of^ the^ most^ basic^ material, e.g.,^ the
notion of a^ formal grammar as^ a^ generator^ of^ a^ language and some of the equivalent interpretations in^ terms^ of
abstract machines, could be introduced earlier as part
of the course I-X proposed^ below.
Theory of computability
A considerable (^) amount of activity^ has^ taken^ place within the area covered^ by^ Course^ A7,^ and^ much^ of^ the new material is already^ entering^ the^ computer^ science course offerings^ of^ several^ universities.^ The^ new^ ma- terial lies principally in three areas:
- computational^ complexity,
- analysis of algorithms,
- program schemata.
We will^ discuss^ these^ areas^ briefly^ in^ order^ to^ define their scopes, but will (^) not attempt a survey^ of the many interesting results^ which^ have^ been^ obtained.
858 Spring^ Joint^ Computer^ Conference,^1972
Computational complexity
The first^ of^ these^ areas^ dates^ back^ to^ 19G0^ when^ M. Rabin suggested^ that^ one^ could^ study^ the^ classification of computable^ functions^ according^ to^ the^ degree^ of
difficulty of computing^ thcm.^ (Until^ then,^ the^ main
efforts in computability^ theory^ were,^ in^ fact,^ directed
toward understanding^ and^ classifying^ the^ unconfut-
able functions in^ order^ to^ gain^ insight^ into^ the^ founda-
tions of mathematics.)^ Thus,^ this^ area^ is^ the^ oldest^ of
the three and^ was the^ first^ to^ split^ off^ from^ the^ tradi-
tional computability^ theory^ (often^ called^ recursive^ func-
tion theory)^ ,^ which began^ in^ the^ 1930'5.
In computational complexity^ studies,^ the^ most^ com-
mon overall^ approach^ is^ to^ consider^ programs^ which
are "efficient"^ in^ the^ sense^ that they^ either^ do^ not^ ex- ceed a^ bound^ on^ resources available^ at^ execution^ time or do^ not^ exceed^ a^ bound^ on^ those^ available^ at^ compile
time. Thus, one either considers^ all^ programs^ which^ are
sufficiently "fast" or^ sufficiently^ "small"^ and^ deter- mines which functions can be computed^ by such^ pro- grams. The classes^ of^ functions^ obtained^ relative^ to^ a given resource bound arp^ calbd^ "complexity^ classes," and (^) one principal aim of this^ area is^ to^ study^ their mathematical properties. Examples^ of^ resources^ related to execution of a (^) program which have been studied^ are:
number of steps on a Turing^ machine,^ memory^ cycles
on a random-access machine,^ total^ memory^ space
needed on (^) a typical computer, number^ of^ bitwise^ opera-
tions on^ any^ computer.^ Examples^ of^ resources^ related
to compilation of a program are:^ program size^ (in^ bits), depth of nesting of DO^ loops^ in^ a^ FORTRAN-like language, number of blocks in a program.
In addition,^ the^ features^ of all^ computations^ with^ re-
source bounds have been^ axiomatized^ by^ M.^ Blum,
and a number of machine-independent^ results^ have been proved. Thus,^ mathematical^ rigor^ has^ reinforced the programmer's intuition in verifying^ that^ for^ any computable function there are arbitrarily bad^ (e.g.,^ slow or huge) programs which^ do,^ in^ fact,^ compute^ the
function. On the other^ hand,^ the^ "best"^ programs^ for
a function cannot always be ciTectively^ found.^ Further- more, some functions simply have no^ best programs. The notion^ of^ "time-space^ trade^ off"^ is^ beginning^ to be the object of^ study^ via^ the^ consideration^ of^ com- plexity of finite functions^ (i.e.,^ functions^ which^ need^ to be evaluated for only a^ finite number of^ inputs).^ Until
recently, finite functions had been dismissed as^ trivial,
since all finite functions can be^ computed^ quickly^ via programs with large tables^ built^ into^ them.^ By^ taking program size^ into^ account,^ a^ rich^ enough^ structure^ is
available to permit the investigations of this subarea.
Another related concept is that of randomness,^ since^ a sequence (^) can be considered to be essentially random
if a program to^ print it involves^ the^ equivalent^ of
storing the entire sequence in the program.^ Conversely,
a non-random^ sequence^ is^ one^ which^ can^ be^ predicted and therefore computed^ by^ a^ relatively^ short^ program. The techniques used^ to^ prove^ theorems^ in^ computa-
tional complexity^ theory^ tend^ to^ be^ closely^ related^ to
techniques used in automata theory^ and^ recursive
function theory. Thus,^ proofs may^ involve diagonal
constructions, counting arguments or analysis^ of ratc-
of-growth of resource needs.^ The^ results^ in^ this^ area have had^ relatively^ little^ direct^ practical^ application^ in the sense that the algorithms^ produced^ generally^ com-
pute functions of only theoretical^ interest.^ However,^ a
great deal of understanding about the^ nature^ of^ com- putation and the relationship between machine^ organi- zation and^ computational^ efficiency^ is^ being^ obtained.
Analysis (^) of algorithms
This area has evolved as a^ branch of^ computational complexity but has recently acquired its own distinctive character as well as a sizable^ group^ of^ converts.^ (It^ is still new enough to have no standard^ name; other names include^ "efficient^ algorithms,"^ "optimal^ algo-
rithms," and "concrete computational^ complexity.")
Again, interest is in the degree of difficulty of^ computing
functions. However, one now tends^ to^ focus^ on^ a^ given
function (or (^) a relatively small^ class^ of^ functions)^ and to-attempt to (^) determine the complexity of^ the^ best^ algo-
rithms for computing^ it,^ if^ such^ exist^ and^ can^ be^ found.
If optimal algorithms are not known,^ lower^ bounds
on complexity measures^ are^ investigated^ and^ near- optimal algorithms sought. Examples^ of^ functions^ or
procedures considered include: sorting,^ determination
of the median of a sample,^ polynomial^ evaluation, matrix multiplication, checking^ graph^ isomorphism. In analysis^ of^ algorithms,^ the^ measures^ of^ complexity
are all oriented toward execution^ efficiency^ rather^ than
program structure.^ Examples^ of^ measures^ commonly used are:^ the number of^ multiplications^ and/or divisions
executed, number^ of^ all^ arithmetic^ operations,^ number
of data references, number of^ data^ comparisons.^ Th<" measures thus^ involve^ quantities^ which^ arc^ basic^ in most programming^ systems.^ Furthermore,^ by^ consider- ing only programs which^ compute^ a^ given^ function, rather than studying^ all^ programs possessing^ a^ certain degree of efficiency (regardless of what^ they^ compute efficiently), (^) one limits himself to programs computmß functions of practical^ interest,^ and^ his^ search^ for^ an algorithm which is both efficient^ and^ useful follows^ mon* closely that of the programmer^ than^ the^ theoretician. For these^ reasons,^ analysis^ of^ algorithms^ has^ a^ con- siderably more practical flavor than (other)^ computa- tional complexity^ theory.
Course I-X: Introduction^ to^ computability^ theory^ and formal languages
Approach
This course is^ designed^ to^ introduce^ the^ advanced
undergraduate or first-year^ graduate^ to^ formal^ systems of computation^ (including^ formal^ languages)^ and^ to give him^ a^ basic^ understanding^ of^ the^ nature^ of^ the unsolvability phenomena which^ pervade^ attempts^ to deduce correctness of^ programs,^ equivalence^ of^ algo- rithms, etc. Prerequisite: Course B3 or a course^ in^ logic^ or^ algebra.
Content
1. The intuitive^ notion^ of^ effective^ computability^ as
involving the existence of^ an^ algorithm.^ The^ Turing machine as one precise^ formal^ model^ for^ the^ notion.
Equivalence of the computing^ power^ of^ the^ standard
class of Turing^ machines^ with^ classes^ obtained^ by^ vary-
ing the control structure^ of^ the^ mpchme^ (e^ g.,^ Wang's
single-address serial^ memory^ machines)^ and^ the^ data structure of^ the^ machine^ (e.g.,^ more^ than^ one^ tape,
multiheaded tapes,^ ..-dimensional^ tapes,^ nonerasing
tapes, random-access^ storage,^ etc.).
2. Universal^ Turing^ machines,^ i.e.,^ the^ construction
of (^) an interpretive^ program^ to^ simulate^ other^ Turing
machine programs.
3. The unsolvability^ of^ the^ halting^ problem;^ the
effective production^ of^ counterexamples^ for^ alleged " halting^ problem^ solvers.^ Some^ other^ unsolvable^ prob-
lems (blank^ tape^ halting^ problem,^ printing^ problem,
etc.). Some^ solvable^ problems^ (computations^ on
bounded space,^ etc.)^.
4. Numerical^ approaches^ to^ computability.^ Primi-
tive recursive and^ recursive^ functions.^ Encodings^ of lists of^ integers^ into^ single^ integers^ (pairing^ functions)
and encodings^ of^ computations^ (arithmctization).
- Other^ models^ for^ digital^ computation:^ program
machines, loop machines,^ machines^ with^ pushdown
stores, stacks,^ counters,^ etc.^ Simulation^ of^ Turing^ ma- chines by^ machines^ with^ only^ two^ counters.
6. Post^ combinatorial^ systems^ and^ formal^ grammars
as a special^ case.^ Production^ notation^ for^ specifying
grammars. Recursively^ enumerable,^ context-sensitive,
context-free and^ finite-state^ grammars.^ Examples^ of
languages of^ varying^ complexity,^ e.g.,^ a
nb n (^) , (^) a"b n^ C.
Proof that an^ b"c" is^ not^ context-free.
7. Grammars^ vs.^ abstract^ machines.^ Representations
of the above^ in^ terms^ of^ Turing^ machines,^ linear
bounded automata, one-pushdown^ machines,^ finite-
state automata,^ respectively.
- Unsolvability^ of^ the^ Post^ correspondence^ problem. Applications to unsolvability^ problems^ in^ formal languages.
References — Course^ I-X
The richest sources of^ material^ for^ this^ course^ are Minsky and Hopcroft-Ullman.^ ABA6^ The^ former^ has
good, readable coverage^ of^ 1-5,^ above,^ and^ part^ of^ G.
The latter has^ a^ more^ formal^ approach^ and^ covers^ G-S
very well.
Al S AANDERAA P^ C^ FISCHER The solvability^ of the^ halting^ problem^ for^ 2-stale^ Post machines J ACM^14 1907 pp^ 677- A2 (^) M DAVIS Computability and^ unsolvability McGraw-Hill New^ York^1958 210 pp A3 P C FISCHER On (^) formalisms for Turing^ machines J ACM 12 1905 pp^ 570- A4 (^) S GINSBURG Mathematical theory^ of^ context-free^ languages McGraw-Hill New^ York^ 190G^243 pp A5 II (^) HERMES Enumcrability, decidability, computability Academic Press^ New^ York^1905 245 pp A6 J E HOPCROFT J^ D^ ULLMAN Formal languages^ and^ their^ relation^ to^ automata Addison-Wesley (^) Pub Co lnc 1969 242 pp A7 S (^) C KLEENE Mathematical logic Wiley (^) New York 1967 398 pp A8 (^) M L MINSKY Computation: Finite and^ infinite machines Prentice-Hall Englewood^ Cliffs^ NJ^ 1967 317^ pp A9 J MYIIILL Linear bounded^ automata WADD Tech Note^ 60-165^ Wright-Patterson^ Air^ Force Base Ohio^1960 AlO E L POST Recursive unsolvability^ of a^ problem^ of^ Thue J Symbol^ Logic.^11 1947 pp^ 1- All (^) II ROGERS JR Theory (^) of recursive (^) functions and^ effective computability McGraw-Hill New^ York^1967 482 pp Al2 J (^) C SHEPIIERDSON II^ E.^ STURGIS
Computability of recursive functions
J ACM 10 1903^ pp^ 217- Al3 B A (^) TRAKUTENBROT Algorithms and automatic computing^ machines Translation by^ J Kristian^ J^ D^ McCawley^ and^ S^ A^ SchmiU D C Heath Boston 1903 101 pp Al4 A M TURING On computable^ numbers,^ with^ an^ application^ to^ the Entscheidungsprobli m Proc London^ Math^ Soc^ Ser^ 2 42^ 1930-1937^ pp^ 230-lw Corr ibid 43 1937 pp^ 544- Al5 II (^) WANG A variant to Turing's^ theory^ of computing^ machines J ACM 4 1957 pp^ 61-
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I I
Advanced computability^ theory i
No single^ solution^ is^ proposed^ for^ taking^ Course^ A7, now bereft^ of^ both^ basic^ computability^ theory^ and^ re-
cursive function^ theory,^ and^ refurbishing^ it^ with^ more
relevant material^ while^ retaining^ the^ computational
complexity theory^ previously^ contained^ there.^ One
reasonable approach^ would^ be^ to^ convert^ the^ course
into a^ survey-in-depth^ of^ all^ three^ areas^ described
above. The^ most^ generous^ answer^ (for^ devotees^ of^ com- putability theory)^ would^ be^ to^ convert^ A7^ into^ a^ course on computational^ complexity^ theory^ and^ to^ establish new courses^ in^ both^ algorithm^ analysis^ and^ program
schemata. This,^ in^ fact,^ is^ the^ current^ situation^ in^ the
graduate program at^ Waterloo.^ However,^ graduate
course offerings are obviously^ highly^ dependent^ upon the interests^ and^ quantity of^ faculty^ members.^ (Until
the current^ academic^ year^ Waterloo^ offered^ courses
equivalent to I-X^ and^ a^ single^ graduate^ term^ course^ in computability theory.^ The^ latter^ course^ took^ the survey-in-depth approach.) Activity in analysis^ of^ algorithms^ has^ grown^ so
rapidly that the accretion of^ new^ results^ and^ interest-
ing algorithms has far^ exceeded^ progress^ in^ determining underlying principles. Thus,^ it^ will^ be^ difficult^ to^ keep
a course in^ this^ area^ from^ becoming^ a^ collection^ of
programming tricks.^ Nevertheless,^ courses^ in^ analysis
of algorithms have already^ sprung^ up^ at^ a^ number^ of universities and are attracting^ respectable^ enrollment figures. Furthermore,^ the^ potential^ value^ of^ such^ a course for persons who^ think^ they^ like^ programming and dislike all theory outweighs^ the^ difficulties^ due^ to the (^) roughness of the area and^ lack^ of^ adequate^ peda- gogical material.^ Thus,^ it^ is^ quite^ likely^ that^ many schools will choose to^ offer^ a^ course^ in^ analysis^ of^ algo- rithms regardless of whether^ or not^ the^ computational
complexity and program schemata^ theory^ arc^ covered
as well. If this turns out to be the^ case,^ the^ best^ solution
for (^) Course A7 would be to embrace both of^ these^ latter areas. A list of topics and^ references^ in^ each^ of^ the^ three areas is given below. In view^ of^ the^ discussion^ just
concluded, these are not to be interpreted^ as^ specific
recommendations for courses^ but^ only^ as^ a^ sample^ of material in each of the areas.
ADVANCED TOPICS^ AND^ REFERENCES
As stated previously, the topics^ and^ references^ pre- sented (^) below arc (^) not all inclusive and^ are not^ suggestions for (^) throe separate courses.
Computational complexity
Topics
1. Complexity^ classes^ defined^ in^ terms^ of^ Turing
machine time bounds,^ space^ bounds,^ and^ tape
reversals (^) 'B21^ ' B::
- Models^ for^ real-time^ computation,^ Turing^ ma- chines with many^ tapes^ vs.^ one^ or^ two^ tapes,^ Turing machines with many heads per tape^ vs.^ one^ head^ per taPC 8!3.8_4,8_9,
3. Random-access^ register^ machines,^ iterative^ arrays,
n-counter machines^ and^ real-time^ countability,^ other computing devices.^82 ' 89 ' 815 '^816 -^836
- Axiomatic approaches^ to^ machine^ structure, bounded action and bounded^ activity^ machines.^810 '^835
5. Complexity classification by^ functional^ definition,
primitive recursive^ functions,^ the^ Grzegorczyk^ classes, honest functions,^ loop program^ machines.^88 -^826 - 830 -^831
- Axioms^ for^ machine-dependent^ complexity theory, existence^ of^ arbitrarily^ bad^ programs^ and^ arbi- trarily hard-to-compute^ functions,^ speed-up^ theorem.^83
7. Machine-independent^ complexity^ classes,^ uniform
hierarchj (compression) theorem,^ honesty^ theorem,
gap theorem,^ union^ theorem.^820
8. Complexity^ of^ finite^ functions,^ expressive^ power
of programs, random^ sequences.^84 '^87 -^825
References— Computational^ complexity
In general, only^ material^ appearing^ in^ standard
journals and written in^ English^ is^ given.^ Thus,^ a^ num- ber of recent^ results^ and^ papers^ in^ foreign^ languages,
principally Russian, containing^ relevant^ material^ have
been omitted. Fairly^ comprehensive^ coverage^ of^ ma-
chine-independent complexity^ theory^ as^ well^ as^ some additional references are found^ in^ the^ paper^ by^ Hart- manis and Hopcroft. 820 Two previous^ references^ are also relevant to^ this^ area.^ A9^ - Al
An extensive "Bibliography^ on^ computational^ com-
plexity" was compiled^ by^ M. I.^ Irland^ and^ the^ author in October,^ 1970,^ but^ has^ not^ been^ published.^ A^ limited number of copies^ are^ available^ as^ Research^ Report CSRR 2025,^ Department of^ Computer^ Science,^ Uni- versity of Waterloo. Bl A V^ AHO^ J^ E^ HOPCROFT^ J^ D^ ULLMAN Time and^ tape^ complexity^ of^ pushdown^ automaton languages Information and^ Control^13 pp^ 180- B2 A J ATRUBIN A onc-dimcn.'iional real-time^ ikrativc^ multiplier J ACM^14 1905 pp^ 394- B3 M BLUM A machine-independent^ theory^ of^ the^ complexity^ of^ recursive functions J ACM 14 1967 pp^ 322-
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Theory of (^) Computing (^) in Computer (^) Science Education 863
2. Polynomial^ evaluation. Optimality of Horner's
rule in^ general^ case,^ preconditioning. (^) Evaluation of (^) a polynomial at (^) more than (^) one point. 07
3. Lower bounds for addition and multiplication of
integers. Methods for multiplying in fewer than n 2 bit-
wise operations.^ Fast Fourier transforms and their
application to multiplication. osoBo
012
i. Sorting and searching methods. Treesort, radix
sorts, polyphase (^) sorts. (^) Lower bounds (^) on the (^) number (^) of comparisons (^) necessary. (^) Finding (^) the ith largest (^) in a (^) set
of elements. Evaluation cf binary searching, AVL
trees. (^) '
5. Graph algorithms. Planarity testing, connected
components (^) of a graph, (^) transitive (^) closure.
6. Miscellaneous algorithms. Matrix transposing in a
paging (^) environment, garbage (^) collection, (^) g.c.d. (^) of poly-
nomials, power of an expression, etc. 01
References— Analysis of algorithms
A considerable amount of current material is avail-
able only (^) in the form (^) of (^) course notes or (^) technical re-
ports. Such items have been omitted here although
courses (^) being (^) taught in analysis (^) of (^) algorithms (^) are draw-
ing up to half of their material from such sources. Thus,
the (^) references (^) below (^) serve (^) more to give (^) the flavor of this (^) area (^) than to (^) constitute any (^) approximation (^) to a
complete set of source material. Presumably much of
the (^) current (^) material will (^) appear in (^) standard (^) journals
within a year.
Many informal working papers in this area can be
found in recent Proceedings of the ACM Symposia on
Theory of Computing (available from ACM) and
Proceedings (^) of (^) the lEEE (^) Symposia (^) on (^) Switching and Automata Theory (^) (available (^) from (^) the lEEE (^) or the lEEE (^) Computer (^) Group). Also, a (^) "Bibliography on the analysis (^) of algorithms" (^) has been (^) compiled (^) by Professor
E- M. Reingold, Department of Computer Science,
University (^) of (^) Illinois at (^) Urbana. (^) As of (^) the (^) time of writing (^) it has (^) not been published.
Cl A (^) BRAUER On (^) addition chains Bull (^) Amer (^) Math Soc (^45 1939) pp (^) 736- C2 (^) C (^) A R (^) HOARE
Computer (^) J 5 1962 pp (^) 10- C3 (^) J E (^) HOPCROFT L (^) KERR On (^) minimizing the (^) number (^) of multiplications (^) necessary matrix (^) multiplication for SIAM J Appl^ Math 20 1971 pp (^) 30- c (^4) R M (^) KARP W L (^) MIRANKEU Parallel (^) search (^) for a (^) maximum
J Combinatorial Theory 4 1968 pp 19-
C5 (^) D E KNUTH The (^) art (^) of computer (^) programming, (^) volume. (^2) Addison-Wesley (^) Reading (^) Massachusetts (^1969) sections 4.3.3, 4.6.3, (^) and (^) 4.6. C6 (^) D (^) E KNUTH The (^) art (^) of computer programming, (^) volume (^3) Addison-Wesley (^) Reading (^) Massachusetts to appear (^1972) section 5. C7 V Ya PAN Methods (^) of computing (^) values (^) of polynomials Uspehi (^) Mat Nauk (^21 1966) PP 103-131 (^) Russian English translation in (^) Russian Math (^) Surveys 21 1966 pp (^) 105- CS A (^) SCHONHAGE Multiplication (^) of large (^) numbers Computing (^1 1906) pp (^) 182-196 (German (^) with English summary) (^) See Comput (^) Rev (^8 1967) review (^) number 11544 C9 P (^) M SPIRA The time required (^) for group multiplication J ACM (^16 1969) pp (^) 235- ClO (^) V STRASSEN Gaussian elimination is (^) not optimial Numer Math 13 1969 pp (^) 354- Cll S (^) WINOGRAD On ihe time required (^) to perform (^) addition J ACM 12 1965 pp (^) 277- Cl2 S (^) WINOGRAD On the (^) time required (^) lo perform multiplication J ACM 14 1967 pp (^) 793-SO Cl3 S^ WINOGRAD On the^ number (^) of multiplications (^) necessary (^) to compute certainfunctions Comm (^) Pure Appl (^) Math 23 1970 pp (^) 165-
Program (^) schemata
Topics
1. Flowchart schemata. Interpretations of schemata.
Herbrand Interpretations. Undecidability of halting
and equivalence problems. Various notions of
equivalence.o4olo
2. Special^ classes of flowchart schemata: lanov
schemata, two-location schemata, nested-loop ('while')
schemata. Translation of flowchart schemata to 'while'
schemata.
3. Recursive schemata. Translation of flowchart
schemata to recursive schemata. Recursive schemata
with no equivalent flowchart schemata. Dccidable
properties (^) of monadic (^) recursive schemata. (^) Equivalence
methods for recursive schemata; truncation induction." 5
4. Models of parallel computation. Determinancy;
deadlocks. (^) Equivalence-preserving (^) transformations; in- creasing (^) and decreasing (^) the (^) 'parallelism'. o
5. Correctness of programs, choice of predicates asso-
ciated with locations in a program. m-D "-D»
6. Formal definitions of program structure. 01 ' Dl
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Spring Joint^ Computer^ Conference,^1972
References—Program^ schemata
Some papers^ concerned^ with^ program^ definition^
and
programming semantics^ are^ included^ as^ well^
as (^) those dealing directly^ with^ program^ schemata.^ As^
before,
much current^ material^ has^ not^ made^ its^ way^ into^ the
accessible literature.
Dl E ENGELER Algorithmic approximations J Computer^ Syst^ Sci^5 1971 pp^ 67- D2 (^) R W FLOYD Assigning meaning lo programs In Proc Symposia^ in^ Applied^ Mathematics^19 American Mathematical Society^1907 pp^ 19- D3 R M KARP^ R^ E^ MILLER Parallel program^ schemata J Computer^ Syst^ Sci^3 1969 pp^ 147- D4 D C LUCKHAM^ D^ M^ R^ PARK M S PATERSON On (^) formalised computer^ programs J Computer^ Syst^ Sci^4 1970 pp^ 220- D5 j McCarthy A basis for a^ mathematical^ theory^ of^ compulation In Computer^ Programming^ and^ Formal^ Systems North-Holland Amsterdam^1963 pp^ 33-
D6 Z MANNA '
The correctness^ of^ programs J Computer^ Syst^ Sci^3 1909 pp^ 119-
D7 Z^ MANNA^ R^ WALDINGER
Toward automatic^ program^ synthesis Comm ACM^14 1971 pp^ 151- D8 R MILNER Program schemes and^ recursive^ function^ theory In Machine Intelligence^5 Edinburgh^ University^ Press 1970 pp^ 39- D9 D^ PARK Fixpoint induction^ and^ proofs^ of^ program^ properties Ibid pp^ 59- DlO M S PATERSON Program schemata In Machine^ Intelligence^3 Edinburgh^ University^ Press 190S pp^ 18- Dll D SCOTT Outline of a^ mathematical^ theory^ of^ computation Tech Monograph^ PRG-2^ Oxford^ University^ Computing Laboratory Programming^ Research^ Group^19 <0^24 pp Dl2 D (^) SCOTT The lattice^ of flow diagrams Tech Monograph^ PRG-3^ Oxford^ University^ Computing Laboratory Programming^ Research^ Group^ 19/0^ o/^ pp
ACKNOWLEDGMENTS
The author^ gratefully^ acknowledges^ the^ material^
and
suggestions provided^ by^ E.^ A.^ Ashcroft,^ M.^ J.^
Fischer,
A°R. Meyer,^ J.^ I.^ Munro^ and^ E.^ M.^ Reingold.
