J. A. Goguen
Skip Abstract Section
Abstract
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.
- 1 AHo. A V . AND ULLMAN, J D Properties of syntax directed translations J Computer and Syst, Scts. 3 (1969), 319-334Google Scholar
- 2 Ano, A B , AND ULLMAN, J D Translations of a context-free grammar Inform and Contr 19 (1971), 439-475Google Scholar
- 3 BAKER, B S Tree transductlons and families of tree languages Tech Rep TR9-73, Center for Research m Computing Technology, Harvard U , Cambridge, Mass , 1973Google Scholar
- 4 BEKI,~, H Definable operations in general algebra and the theory of automata and flowcharts Research Rep, IBM Laboratory, Vienna, Austria, 1969Google Scholar
- 5 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), 1-126Google Scholar
- 6 BII~KHOFF, G Structure of abstract algebras Proc Cambrtdge Phd Soc 31 (1938), 433-454Google Scholar
- 7 BIRKHOFF, G , AND LIeSON. J D Heterogeneous algebras J Combmatonal Theory 8 (1970), 115-133Google Scholar
- 8 BLIKLE, A Equational languages Inform and Contr 21 (1972). 134-147Google Scholar
- 9 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, 1974Google Scholar
- 10 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- 43Google Scholar
- 11 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. Google Scholar
- 12 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 Google Scholar
- 13 COVlN, P M Unwersal Algebra Harper and Row. New York, 1965Google Scholar
- 14 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 Google Scholar
- 15 DONER, J E Tree acceptors and some of their applications J Computer and Syst Scts 4 (1970), 406- 451Google Scholar
- 16 EILENBERG. S , AND WRIGHT, J B Automata in general algebras Inform and Control 11 (1967), 452- 470Google Scholar
- 17 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 175-230Google Scholar
- 18 ENGELFRIET, J , AND SCHMIDT, E M IO and Ol Dataloglsk Afdehng Rep, DAIM1 PB-47, Aarhus U , Aarhus, Denmark, July 1975Google Scholar
- 19 GINSBURG, S The Mathemattcal Theory of Context-Free Languages McGraw-Hill, New York, 1962 Google Scholar
- 20 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), 333-365Google Scholar
- 21 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 Google Scholar
- 22 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 63-77Google Scholar
- 23 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, 1976Google Scholar
- 24 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 89-93Google Scholar
- 25 GO~UEN, J A , THATCHER, J W, WAGNER, E G , ANn WmGnT, J B. Programs In categories (summary), in preparationGoogle Scholar
- 26 GORDON, M Models of pure LISP Ph D Th , Edinburgh U , Edinburgh, Scotland, 1973Google Scholar
- 27 GRAETZER, G UntversalAlgebra Van Nostrand, Princeton, N J , 1968Google Scholar
- 28 HIo~ISS, P J Algebras with a schema of operators Math Nachr 27 (1963), 115-132Google Scholar
- 29 IRONS, E T A syntax dtrected compder for ALGOL 60 Comm ACM 4 (1961), 51-55 Google Scholar
- 30 KNUTH, D E Semantics of context-free languages Math Syst Theory 2 (1968), 127-145Google Scholar
- 31 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.Google Scholar
- 32 LAWVnRE, F W Functorlal semantics of algebraic theories. Proc. Nat Acad Sct. 50 (1963), 869-872.Google Scholar
- 33 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 Google Scholar
- 34 LEWIS, P.M II,AND STEARNS, R E Syntax-directed transduction J ACM 15, 3 (July 1968), 465-488 Google Scholar
- 35 LUCAS, P, LAUER, P, AND STIGLEITNER. H Method and notation for the formal definition of programmmg languages Tech Rep TR 25 087, IBM Laboratory, Vienna, Austria, 1968Google Scholar
- 36 McCARTHY. J Towards a mathematical science of computation Proc IFIP Cong. North-Holland Pub Co, Amsterdam, 1962, pp 21-28Google Scholar
- 37 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 1-12Google Scholar
- 38 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 33-41Google Scholar
- 39 MACLANE, S Category Theory for the Worklng Mathematician Springer, New York, 1971Google Scholar
- 40 MAGIDOR, i , AND MORGAN, G Finite automata over finite trees Tech Rep 30, Hebrew U , Jerusalem, Israel, 1969Google Scholar
- 41 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. 224-230Google Scholar
- 42 MAIBAUM, T S E Generalized grammars and homomorphic images of regular sets Research Rep CS-73- 30, U of Waterloo, Waterloo, Ontario, Canada, 1973Google Scholar
- 43 MANNA, Z Properties of programs and the first-order predicate calculus J ACM 16, 2 (April 1969), 244-255 Google Scholar
- 44 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 1974Google Scholar
- 45 MEZEI, J , AND WRIGHT, J B Algebraic automata and context-free sets Inform and Contr 11 (1967), 3- 29Google Scholar
- 46 MORRIS, F L Correctness of translations of programming languages Ph D Th, Computer Science Memo CS72-303, Stanford U , Stanford, Cahf, 1972 Google Scholar
- 47 MORRIS, F.L Advice on structuring compders and proving them correct Proc Symp on Pnnciples of Programming Lanugages, Boston, 1973, pp 144-152 Google Scholar
- 48 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 293-308Google Scholar
- 49 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 'Google Scholar
- 50 PETRONE, L Syntactic mappings of context-free languages Proc IFIP Cong 1965, Vol 2, North- Holland Pub Co, Amsterdam, pp 590-591Google Scholar
- 51 PorrENGER, R ARBOL, a system for defining functions on trees A I Memo 3, UCLA, 1976Google Scholar
- 52 RABIN, M P, AND SCOT, D Finite automata and their decision problems IBM J Res Develop 3 (1959), 114-125Google Scholar
- 53 REYNOLDS, J C Definitional interpreters for higher-order programming languages Proc 25th National ACM Conference, Boston, Mass, Aug 1972, pp 717-740 Google Scholar
- 54 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 Google Scholar
- 55 REYNOLDS, J C. Semantics of the lattice of flow diagrams. Manuscript, Syracuse U , Syracuse, N Y , submitted for publication, July 1975Google Scholar
- 56 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, 1974Google Scholar
- 57 ROUNDS, W C Mappings and grammars on trees Math Syst Theory 4 (1970), 256-287Google Scholar
- 58 SCHUTZENBERGER, M P Context-free languages and push down automata Inform and Contr 6 (1963), 246-264Google Scholar
- 59 SCHWARTZ, J T Semantic defmmon methods In Formal Semantics of Programmmg Languages, R Rustm, Ed , Prentice-Hall, Englewood Cliffs, N J , 1972, pp 1-23Google Scholar
- 60 SCOTT, D Outline of a mathematical theory of computation Proc 4th Ann Princeton Conf on Information Sciences and Systems, 1970, pp 169-176Google Scholar
- 61 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 311-366Google Scholar
- 62 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 97-136Google Scholar
- 63 Scorr, D Data types as lattices Unpubhshed notes, Amsterdam, 1972Google Scholar
- 64 ScoTT, D Data types as latUces Unpubhshed notes, Oxford, 1974Google Scholar
- 65 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 19-46Google Scholar
- 66 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, 1974Google Scholar
- 67 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), 317-322Google Scholar
- 68 THATCrIER, J W Generahzed~- sequential machines J Computer and Syst. Scts 4 (1970), 339-367Google Scholar
- 69 THATCHER, J W Tree automata' An informal survey In Currents tn Computing, A V, Aho, Ed, Prentice-Hall, Englewood Chffs, N J , 1973, 143-172Google Scholar
- 70 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), 57-81Google Scholar
- 71 TURNE~, R Doctoral Dlss, U of London, London, England, 1973Google Scholar
- 72 VUILLEMIN, J Syntaxe, S6mant~que et Axlomat~que d'un Langage de Programmation Simple These d'Etat, Umverslt6 Pans 6, France, 1974Google Scholar
- 73 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 Google Scholar
- 74 WA6NER, E G Languages for defining sets in arbitrary algebras. Proc llth Ann IEEE Syrup on Switching and Automata Theory, 1971, pp 191-201Google Scholar
- 75 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.Google Scholar
- 76 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 Google Scholar
- 77 VAN WIJNGAARDEN, A., Ed Report on the algonthmlc language ALGOL 68 Numer Math 14 (1969), 79-218Google Scholar
Index Terms
- Initial Algebra Semantics and Continuous Algebras
Recommendations
Dualising initial algebras
Whilst the relationship between initial algebras and monads is well understood, the relationship between final coalgebras and comonads is less well explored. This paper shows that the problem is more subtle than might appear at first glance: final ...
Initial algebra semantics
SWAT '74: Proceedings of the 15th Annual Symposium on Switching and Automata Theory (swat 1974)Many apparently divergent approaches to specifying formal semantics for programming languages are applications of initial algebra semantics. Here we provide an overview of the concept of initial algebra semantics.
Commutative pseudo-equality algebras
Pseudo-equality algebras were initially introduced by Jenei and Kóródi as a possible algebraic semantic for fuzzy-type theory, and they have been revised by Dvureăźenskij and Zahiri under the name of JK-algebras. In this paper, we define and study the ...
Comments