Initial Algebra Semantics and Continuous Algebras

Journal of the ACM - Tập 24 Số 1 - Trang 68-95 - 1977
Joseph A. Goguen1, James W. Thatcher2, Eric G. Wagner2, Jesse B. Wright2
1Computer Science Department, UCLA, Los Angeles, CA
2IBM Thomas J. Watson Research Center Yorktown Heights, NY#TAB#

Tóm tắt

Many apparently divergent approaches to specifying formal semantics of programming languages are applications of initial algebra semantics. In this paper an overview of initial algebra semantics is provided. The major technical feature is an initial continuous algebra which permits unified algebraic treatment of iterative and recursive semantic features in the same framework as more basic operations.

Từ khóa


Tài liệu tham khảo

AH o. A V . AND ULLMAN , J D Properties of syntax directed translations J Computer and Syst , Scts. 3 ( 1969 ), 319 - 334 AHo. A V . AND ULLMAN, J D Properties of syntax directed translations J Computer and Syst, Scts. 3 (1969), 319-334

Ano, A B , AND ULLMAN , J D Translations of a context-free grammar Inform and Contr 19 ( 1971 ), 439 - 475 Ano, A B , AND ULLMAN, J D Translations of a context-free grammar Inform and Contr 19 (1971), 439-475

BEKI ,~ , H Definable operations in general algebra and the theory of automata and flowcharts Research Rep , IBM Laboratory , Vienna, Austria , 1969 BEKI,~, H Definable operations in general algebra and the theory of automata and flowcharts Research Rep, IBM Laboratory, Vienna, Austria, 1969

B~NABOU , J Structures algGbrlques dans les catGgorles Th~se, fac scl , Unlverslt6 de Pans , March 1966 Also, Cahters de Topologze et G~om~trte Dtff~renuelle 10 (1968), 1-126 B~NABOU, J Structures algGbrlques dans les catGgorles Th~se, fac scl , Unlverslt6 de Pans, March 1966 Also, Cahters de Topologze et G~om~trte Dtff~renuelle 10 (1968), 1-126

BII~KHOFF , G Structure of abstract algebras Proc Cambrtdge Phd Soc 31 ( 1938 ), 433 - 454 BII~KHOFF, G Structure of abstract algebras Proc Cambrtdge Phd Soc 31 (1938), 433-454

BIRKHOFF , G , AND LI e SON . J D Heterogeneous algebras J Combmatonal Theory 8 ( 1970 ), 115 - 133 BIRKHOFF, G , AND LIeSON. J D Heterogeneous algebras J Combmatonal Theory 8 (1970), 115-133

BLIKLE , A Equational languages Inform and Contr 21 ( 1972 ). 134-147 BLIKLE, A Equational languages Inform and Contr 21 (1972). 134-147

BLOOM , S L , AND ELGO r , C C The existence and construction of free iteratlve theortes. Research Rep RC-4937 , IBM Thomas J Watson Research Center , Yorktown Heights , N Y, 1974 BLOOM, S L , AND ELGOr, C C The existence and construction of free iteratlve theortes. Research Rep RC-4937, IBM Thomas J Watson Research Center, Yorktown Heights, N Y, 1974

BURSXALL , R M , ANO LANOIN , P J Programs and their proofs An algebraic approach Machine Intelhgence , Vol 4, B Meltzer and D Mlchie, Eds , Edinburgh U Press , Edinburgh, Scotland , 1969 , pp 17 - 43 BURSXALL, R M , ANO LANOIN, P J Programs and their proofs An algebraic approach Machine Intelhgence, Vol 4, B Meltzer and D Mlchie, Eds , Edinburgh U Press, Edinburgh, Scotland, 1969, pp 17- 43

BORSXALL , R M , ANO THA r CHER , J W The algebraic theory of recurslve program schemes . Lecture Notes m Computer Sctence , Vol. 25 Category Theory Apphed to Computatton and Control, Springer , Berhn. 1974 , pp 126 - 131 . BORSXALL, R M, ANO THArCHER, J W The algebraic theory of recurslve program schemes. Lecture Notes m Computer Sctence, Vol. 25 Category Theory Apphed to Computatton and Control, Springer, Berhn. 1974, pp 126-131.

CHANDRA , A K Degrees of translatability and canonical forms of program schemes Pt i Proc Sixth Ann ACM Symp on Theory of Computing , Seattle , 1974 , pp 1 - 12 10.1145/800119.803879 CHANDRA, A K Degrees of translatability and canonical forms of program schemes Pt i Proc Sixth Ann ACM Symp on Theory of Computing, Seattle, 1974, pp 1-12 10.1145/800119.803879

COV l N , P M Unwersal Algebra Harper and Row . New York , 1965 COVlN, P M Unwersal Algebra Harper and Row. New York, 1965

COURCELLE , B , AND VUILLEMIN , J. Semantics and axlomatlcs of a simple recurslve language , Proc Sixth Ann ACM Symp on Theory of Computing , Seattle , 1974 , pp 13 - 26 10.1145/800119.803880 COURCELLE, B , AND VUILLEMIN, J. Semantics and axlomatlcs of a simple recurslve language, Proc Sixth Ann ACM Symp on Theory of Computing, Seattle, 1974, pp 13-26 10.1145/800119.803880

DONER , J E Tree acceptors and some of their applications J Computer and Syst Scts 4 ( 1970 ), 406 - 451 DONER, J E Tree acceptors and some of their applications J Computer and Syst Scts 4 (1970), 406- 451

EILENBERG . S , AND WRIGHT , J B Automata in general algebras Inform and Control 11 ( 1967 ), 452 - 470 EILENBERG. S , AND WRIGHT, J B Automata in general algebras Inform and Control 11 (1967), 452- 470

ELGOT , C C Monadlc computation and ~teratwe algebraic theories Research Rep. RC-4564, IBM Thomas J Watson Research Center, Yorktown Heights, N Y, 1973, also Proc Logic Colloqmum '73, Bristol, England , North-Holland Pub Co , Amsterdam , 1975 , pp 175 - 230 ELGOT, C C Monadlc computation and ~teratwe algebraic theories Research Rep. RC-4564, IBM Thomas J Watson Research Center, Yorktown Heights, N Y, 1973, also Proc Logic Colloqmum '73, Bristol, England, North-Holland Pub Co , Amsterdam, 1975, pp 175-230

ENGELFRIET , J , AND SCHMIDT , E M IO and Ol Dataloglsk Afdehng Rep , DAIM1 PB-47, Aarhus U , Aarhus, Denmark , July 1975 ENGELFRIET, J , AND SCHMIDT, E M IO and Ol Dataloglsk Afdehng Rep, DAIM1 PB-47, Aarhus U , Aarhus, Denmark, July 1975

GINSBURG , S The Mathemattcal Theory of Context-Free Languages McGraw-Hill , New York , 1962 GINSBURG, S The Mathemattcal Theory of Context-Free Languages McGraw-Hill, New York, 1962

GOGUEN , j A On homomorphlsms, correctness, terminahon, unfoldments and eqmvalence of flow diagram programs Proc 13th Ann IEEE Symp on Sw~tchlng and Automata Theory , 1972 , pp 52 - 60 A portion of this paper appears m expanded form in J Computer and Syst Scls 8 (1974), 333-365 GOGUEN, j A On homomorphlsms, correctness, terminahon, unfoldments and eqmvalence of flow diagram programs Proc 13th Ann IEEE Symp on Sw~tchlng and Automata Theory, 1972, pp 52-60 A portion of this paper appears m expanded form in J Computer and Syst Scls 8 (1974), 333-365

GO (; UEN , J A Semantics of computation Lecture Notes tn Computer Sctence , Vol 25 Category Theory Apphed to Computatton and Control , Spnnger , Berlin . 1974 , pp 151 - 163 GO(;UEN, J A Semantics of computation Lecture Notes tn Computer Sctence, Vol 25 Category Theory Apphed to Computatton and Control, Spnnger, Berlin. 1974, pp 151-163

GOGUEN , J A , AND TI-IATC~ER . I W lmttal algebra semantics Extended Abstract, Research Rep RC- 4865, IBM Thomas J Watson Research Center, Yorktown Heights, N Y, May 1974, also , Proc 15th Ann IEEE Symp on Switching and Automata Theory , 1974 , pp 63 - 77 GOGUEN, J A , AND TI-IATC~ER. I W lmttal algebra semantics Extended Abstract, Research Rep RC- 4865, IBM Thomas J Watson Research Center, Yorktown Heights, N Y, May 1974, also, Proc 15th Ann IEEE Symp on Switching and Automata Theory, 1974, pp 63-77

Goou EN , I A , Tn A' r CHER , J W , WAGNER , E G , ANO WRIGHT , J B A junction between computer science and category theory , I Basic definitions and examples Pt 1, Research Rep RC-4526, Pt 2, Research Rep RC-5908 , IBM Thomas J Watson Research Center , Yorktown Heights , N Y , 1973 , 1976 GoouEN, I A, TnA'rCHER, J W , WAGNER, E G, ANO WRIGHT, J B A junction between computer science and category theory, I Basic definitions and examples Pt 1, Research Rep RC-4526, Pt 2, Research Rep RC-5908, IBM Thomas J Watson Research Center, Yorktown Heights, N Y , 1973, 1976

GO~ tr EN , J A , THATCHER , J W . WAGNER , E G , AND WRIGHT , J B Abstract data-types as lmtlal algebras and correctness of data representations Proc . Conference on Computer Graphics, Pattern Recognmon and Data Structure. May 1975 , pp 89 - 93 GO~trEN, J A , THATCHER, J W. WAGNER, E G, AND WRIGHT, J B Abstract data-types as lmtlal algebras and correctness of data representations Proc. Conference on Computer Graphics, Pattern Recognmon and Data Structure. May 1975, pp 89-93

GO~UEN J A THATCHER J W WAGNER E G ANn WmGnT J B. Programs In categories (summary) in preparation GO~UEN J A THATCHER J W WAGNER E G ANn WmGnT J B. Programs In categories (summary) in preparation

GORDON , M Models of pure LISP Ph D Th , Edinburgh U , Edinburgh, Scotland , 1973 GORDON, M Models of pure LISP Ph D Th , Edinburgh U , Edinburgh, Scotland, 1973

GRAETZER , G UntversalAlgebra Van Nostrand , Princeton , N J , 1968 GRAETZER, G UntversalAlgebra Van Nostrand, Princeton, N J , 1968

HI o~ ISS , P J Algebras with a schema of operators Math Nachr 27 ( 1963 ), 115 - 132 HIo~ISS, P J Algebras with a schema of operators Math Nachr 27 (1963), 115-132

10.1145/366062.366083

KNUTH , D E Semantics of context-free languages Math Syst Theory 2 ( 1968 ), 127 - 145 KNUTH, D E Semantics of context-free languages Math Syst Theory 2 (1968), 127-145

LANDIN , P J A program machme symmetric automata theory Machlne Intelhgence 5 , B Meltzer and D MlcMe, Eds , Edinburgh U. Press, Edinburgh , Scotland , 1970 , pp 99 - 120 . LANDIN, P J A program machme symmetric automata theory Machlne Intelhgence 5, B Meltzer and D MlcMe, Eds, Edinburgh U. Press, Edinburgh, Scotland, 1970, pp 99-120.

LAWV n RE , F W Functorlal semantics of algebraic theories. Proc. Nat Acad Sct. 50 ( 1963 ), 869 - 872 . LAWVnRE, F W Functorlal semantics of algebraic theories. Proc. Nat Acad Sct. 50 (1963), 869-872.

LE wis, C H, AND ROSEN , B K Recurswely defined data types , Pt. 1 Proc ACM Symp on Principles of Programming Languages , 1973 , pp 125 - 138 , Pt 2, Research Rep. RC-4713, IBM Thomas J Watson Research Center , Yorktown Heights, N Y, 1974 10.1145/512927.512939 LEwis, C H, AND ROSEN, B K Recurswely defined data types, Pt. 1 Proc ACM Symp on Principles of Programming Languages, 1973, pp 125-138, Pt 2, Research Rep. RC-4713, IBM Thomas J Watson Research Center, Yorktown Heights, N Y, 1974 10.1145/512927.512939

10.1145/321466.321477

Mc CARTHY . J Towards a mathematical science of computation Proc IFIP Cong . North-Holland Pub Co , Amsterdam , 1962 , pp 21 - 28 McCARTHY. J Towards a mathematical science of computation Proc IFIP Cong. North-Holland Pub Co, Amsterdam, 1962, pp 21-28

MCCARTHY , J A formal description of a subset of ALGOL In "Formal Language Description Languages for Computer Programming," Proc IFIP Working Conf 1964, T B Steel, Jr, Ed , North-Holland Pub Co , Amsterdam , 1966 , pp 1 - 12 MCCARTHY, J A formal description of a subset of ALGOL In "Formal Language Description Languages for Computer Programming," Proc IFIP Working Conf 1964, T B Steel, Jr, Ed , North-Holland Pub Co , Amsterdam, 1966, pp 1-12

MCCARTHY , J , AND PAINTER , J Correctness of a compiler for arithmetic expressions In "Mathematical Aspects of Computer Science," Proc of Symposia In Applied Mathematics , Vol 19, J T Schwartz, Ed , Amer Math Soc , Providence , R I , 1967 , pp 33 - 41 MCCARTHY, J , AND PAINTER, J Correctness of a compiler for arithmetic expressions In "Mathematical Aspects of Computer Science," Proc of Symposia In Applied Mathematics, Vol 19, J T Schwartz, Ed , Amer Math Soc, Providence, R I , 1967, pp 33-41

MACLANE , S Category Theory for the Worklng Mathematician Springer , New York , 1971 MACLANE, S Category Theory for the Worklng Mathematician Springer, New York, 1971

MA i BAUM , T S E The characterization of the derivation trees of context-free sets of terms as regular sets Proc 13th Ann IEEE Symp on Switching and Automata Theory , 1972 , pp. 224 - 230 MAiBAUM, T S E The characterization of the derivation trees of context-free sets of terms as regular sets Proc 13th Ann IEEE Symp on Switching and Automata Theory, 1972, pp. 224-230

MAIBAUM , T S E Generalized grammars and homomorphic images of regular sets Research Rep CS-73- 30, U of Waterloo , Waterloo , Ontario, Canada , 1973 MAIBAUM, T S E Generalized grammars and homomorphic images of regular sets Research Rep CS-73- 30, U of Waterloo, Waterloo, Ontario, Canada, 1973

10.1145/321510.321516

MARKOWSKY , G Chain-complete posets and directed sets with applications Research Rep RC-5024 , IBM Thomas J Watson Research Center , Yorktown Heights , N Y, Aug 1974 MARKOWSKY, G Chain-complete posets and directed sets with applications Research Rep RC-5024, IBM Thomas J Watson Research Center, Yorktown Heights, N Y, Aug 1974

MEZEI , J , AND WRIGHT , J B Algebraic automata and context-free sets Inform and Contr 11 ( 1967 ), 3 - 29 MEZEI, J , AND WRIGHT, J B Algebraic automata and context-free sets Inform and Contr 11 (1967), 3- 29

MORRIS , F L Correctness of translations of programming languages Ph D Th, Computer Science Memo CS72-303 , Stanford U , Stanford, Cahf , 1972 MORRIS, F L Correctness of translations of programming languages Ph D Th, Computer Science Memo CS72-303, Stanford U , Stanford, Cahf, 1972

MORRIS , F.L Advice on structuring compders and proving them correct Proc Symp on Pnnciples of Programming Lanugages , Boston , 1973 , pp 144 - 152 10.1145/512927.512941 MORRIS, F.L Advice on structuring compders and proving them correct Proc Symp on Pnnciples of Programming Lanugages, Boston, 1973, pp 144-152 10.1145/512927.512941

NWAT , M Languages algGbralc sur le magna hbre et sGmantique des schdmas de programme In Automata, Languages and Programming, M Nlvat (Ed) , North-Holland Pub Co , Amsterdam , 1972 , pp 293 - 308 NWAT, M Languages algGbralc sur le magna hbre et sGmantique des schdmas de programme In Automata, Languages and Programming, M Nlvat (Ed), North-Holland Pub Co , Amsterdam, 1972, pp 293-308

PARK , D F~xpomt induction and proofs of program properties Machine lntelhgence , Vol 5, B Meltzer and D Mlchie, Eds , Edinburgh U Press , Edmburgh, Scotland , 1970 . pp 59 - 78 ' PARK, D F~xpomt induction and proofs of program properties Machine lntelhgence, Vol 5, B Meltzer and D Mlchie, Eds , Edinburgh U Press, Edmburgh, Scotland, 1970. pp 59-78 '

PETRONE , L Syntactic mappings of context-free languages Proc IFIP Cong 1965 , Vol 2, North- Holland Pub Co, Amsterdam , pp 590 - 591 PETRONE, L Syntactic mappings of context-free languages Proc IFIP Cong 1965, Vol 2, North- Holland Pub Co, Amsterdam, pp 590-591

Porr ENGER , R A RBOL , a system for defining functions on trees A I Memo 3 , UCLA , 1976 PorrENGER, R ARBOL, a system for defining functions on trees A I Memo 3, UCLA, 1976

RABIN , M P , AND SCOT , D Finite automata and their decision problems IBM J Res Develop 3 ( 1959 ), 114 - 125 RABIN, M P, AND SCOT, D Finite automata and their decision problems IBM J Res Develop 3 (1959), 114-125

REYNOLDS , J C Definitional interpreters for higher-order programming languages Proc 25th National ACM Conference , Boston , Mass , Aug 1972 , pp 717 - 740 10.1145/800194.805852 REYNOLDS, J C Definitional interpreters for higher-order programming languages Proc 25th National ACM Conference, Boston, Mass, Aug 1972, pp 717-740 10.1145/800194.805852

REYNOLDS , J C On the relation between direct and continuation semantics Lecture Notes tn Computer Science , Vol 14 Automata , Languages and Programming, Springer , Berlin , 1974 , pp 141 - 156 REYNOLDS, J C On the relation between direct and continuation semantics Lecture Notes tn Computer Science, Vol 14 Automata, Languages and Programming, Springer, Berlin, 1974, pp 141-156

REYNOLDS , J C . Semantics of the lattice of flow diagrams. Manuscript , Syracuse U , Syracuse, N Y , submitted for publication , July 1975 REYNOLDS, J C. Semantics of the lattice of flow diagrams. Manuscript, Syracuse U , Syracuse, N Y , submitted for publication, July 1975

ROSEN , B K Program equivalence and context-free grammars Proc 13th Ann IEEE Symp on Switching and Automata Theory , 1972 , pp 7 - 18 , revised as Research Rep RC-4822, IBM Thomas J Watson Research Center , Yorktown Heights. N Y, 1974 ROSEN, B K Program equivalence and context-free grammars Proc 13th Ann IEEE Symp on Switching and Automata Theory, 1972, pp 7-18, revised as Research Rep RC-4822, IBM Thomas J Watson Research Center, Yorktown Heights. N Y, 1974

ROUNDS , W C Mappings and grammars on trees Math Syst Theory 4 ( 1970 ), 256 - 287 ROUNDS, W C Mappings and grammars on trees Math Syst Theory 4 (1970), 256-287

SCHUTZENBERGER , M P Context-free languages and push down automata Inform and Contr 6 ( 1963 ), 246 - 264 SCHUTZENBERGER, M P Context-free languages and push down automata Inform and Contr 6 (1963), 246-264

SCHWARTZ , J T Semantic defmmon methods In Formal Semantics of Programmmg Languages , R Rustm, Ed , Prentice-Hall , Englewood Cliffs , N J , 1972 , pp 1 - 23 SCHWARTZ, J T Semantic defmmon methods In Formal Semantics of Programmmg Languages, R Rustm, Ed , Prentice-Hall, Englewood Cliffs, N J , 1972, pp 1-23

SCOTT , D Outline of a mathematical theory of computation Proc 4th Ann Princeton Conf on Information Sciences and Systems , 1970 , pp 169 - 176 SCOTT, D Outline of a mathematical theory of computation Proc 4th Ann Princeton Conf on Information Sciences and Systems, 1970, pp 169-176

Scoyr , D The lattice of flow diagrams Tech Monograph PRG 3 , Oxford U Computing Lab , Oxford U , Oxford, England, also, Lecture Notes tn Mathematics, Vol 182 Semantics of Algortthmtc Languages, E Engeler. Ed , Springer . Berlin, 1971 , pp 311 - 366 Scoyr, D The lattice of flow diagrams Tech Monograph PRG 3, Oxford U Computing Lab , Oxford U , Oxford, England, also, Lecture Notes tn Mathematics, Vol 182 Semantics of Algortthmtc Languages, E Engeler. Ed , Springer. Berlin, 1971, pp 311-366

ScoTt , D Continuous lattices Tech. Monograph PRG 7 , Oxford U Computing Lab , Oxford U, Oxford, England , 1971 , also, Lecture Notes m Mathemattcs, Vol 274, Springer , Berhn, 1971, pp 97 - 136 ScoTt, D Continuous lattices Tech. Monograph PRG 7, Oxford U Computing Lab, Oxford U, Oxford, England, 1971, also, Lecture Notes m Mathemattcs, Vol 274, Springer, Berhn, 1971, pp 97-136

Scorr , D Data types as lattices Unpubhshed notes , Amsterdam , 1972 Scorr, D Data types as lattices Unpubhshed notes, Amsterdam, 1972

Sco TT , D Data types as latUces Unpubhshed notes , Oxford , 1974 ScoTT, D Data types as latUces Unpubhshed notes, Oxford, 1974

Scott, D AND SXRACHEV , C Towards a mathematacal semantics for computer languages Tech Monograph PRG 6 , Oxford U. Computing Lab. , Oxford U., Oxford, England , 1971 , also, Computers and Automata, J Fox, Ed , Wdey, New York , 1971, pp 19 - 46 Scott, D AND SXRACHEV, C Towards a mathematacal semantics for computer languages Tech Monograph PRG 6, Oxford U. Computing Lab., Oxford U., Oxford, England, 1971, also, Computers and Automata, J Fox, Ed , Wdey, New York, 1971, pp 19-46

STRACHEV , C , ANO WADSWORTH , C P Contmuatlons-A mathematical semantics for handhng full jumps Tech Monograph PRG-11 , Programming Research Group , Oxford U Computing Lab, Oxford, England, 1974 STRACHEV, C, ANO WADSWORTH, C P Contmuatlons-A mathematical semantics for handhng full jumps Tech Monograph PRG-11, Programming Research Group, Oxford U Computing Lab, Oxford, England, 1974

THATCHER , J.W. Characterizing derwat~on trees of context free grammars through a generahzatlon of fimte automata theory J Computer and Syst Scts 1 ( 1967 ), 317 - 322 THATCHER, J.W. Characterizing derwat~on trees of context free grammars through a generahzatlon of fimte automata theory J Computer and Syst Scts 1 (1967), 317-322

THATC r IER , J W Generahzed~- sequential machines J Computer and Syst. Scts 4 ( 1970 ), 339 - 367 THATCrIER, J W Generahzed~- sequential machines J Computer and Syst. Scts 4 (1970), 339-367

THATCHER , J W Tree automata' An informal survey In Currents tn Computing , A V, Aho, Ed , Prentice-Hall , Englewood Chffs , N J , 1973 , 143-172 THATCHER, J W Tree automata' An informal survey In Currents tn Computing, A V, Aho, Ed, Prentice-Hall, Englewood Chffs, N J , 1973, 143-172

THATCHER , J W ., AND WRIGHT , J B . Generahzed flmte automata theory with an application to a decision problem of second-order logic Math Systems Theory 2 ( 1968 ), 57 - 81 THATCHER, J W., AND WRIGHT, J B. Generahzed flmte automata theory with an application to a decision problem of second-order logic Math Systems Theory 2 (1968), 57-81

TURNE~ , R Doctoral Dlss , U of London , London, England , 1973 TURNE~, R Doctoral Dlss, U of London, London, England, 1973

VUILLEMIN , J Syntaxe , S6mant~que et Axlomat~que d'un Langage de Programmation Simple These d'Etat , Umverslt6 Pans 6, France , 1974 VUILLEMIN, J Syntaxe, S6mant~que et Axlomat~que d'un Langage de Programmation Simple These d'Etat, Umverslt6 Pans 6, France, 1974

WAGNER , E.G An algebraic theory of recurstve defmtttons and recurstve languages . Proc. Third Annual ACM Symposium on Theory of Computing , 1971 , pp 12 - 23 10.1145/800157.805034 WAGNER, E.G An algebraic theory of recurstve defmtttons and recurstve languages. Proc. Third Annual ACM Symposium on Theory of Computing, 1971, pp 12-23 10.1145/800157.805034

WA 6 NER , E G Languages for defining sets in arbitrary algebras . Proc llth Ann IEEE Syrup on Switching and Automata Theory , 1971 , pp 191 - 201 WA6NER, E G Languages for defining sets in arbitrary algebras. Proc llth Ann IEEE Syrup on Switching and Automata Theory, 1971, pp 191-201

WAND , M A concrete approach to abstract recurswe deflmt~ons inAutomata, Languages and Programmmg, M Ntvat, Ed , North-Holland Pub Co. , Amsterdam , 1972 , pp 331 - 341 . WAND, M A concrete approach to abstract recurswe deflmt~ons inAutomata, Languages and Programmmg, M Ntvat, Ed , North-Holland Pub Co., Amsterdam, 1972, pp 331-341.

WAND , M An algebraic formulation of the Chomsky hierarchy Lecture Notes zn Computer Science , Vol 25 Category Theory Apphed to Computation and Control , Springer , Berhn , 1974 , pp 209 - 213 WAND, M An algebraic formulation of the Chomsky hierarchy Lecture Notes zn Computer Science, Vol 25 Category Theory Apphed to Computation and Control, Springer, Berhn, 1974, pp 209-213

VAN WIJNGAARDEN , A. , Ed Report on the algonthmlc language ALGOL 68 Numer Math 14 ( 1969 ), 79 - 218 VAN WIJNGAARDEN, A., Ed Report on the algonthmlc language ALGOL 68 Numer Math 14 (1969), 79-218