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In memoriam Calvin C. Elgot
TheoreticalRTKERKTYEKY;KTRYK;RRTKY;RKT;YKR;TKY;RKETY Computer Science 16 (1981) l-4 North-Holland Publishing Company
IN MEMWUAM CALVIN C. ELGQT
L* ‘Calvin C. Elgot who made many contributions to the theory of computatilon, died on July 8, 1980. He was born in New York City in 1922 and after sler%g in the KGK GKGKGKGKGKGKGKGKGKKGat City College of l\;lewYork in 1948 whiFF WEWERWERWERWERt Columb~ia University in 1951 while holding a Lectureship there. Then he wa:; a mathematician at the U.S. Naval 0rdnance Laboratory, Silver Spring, Maryland. From 1954--55 \he ‘was a Teaching Assistant at Berkeley, and from 1955-59 a Research Mathern;rtician at the University of Michigan at Ann Arbor where he received a doctorate. in 1968% For the rest of his life he was a Research Staff Member at the EIBMThcsmas J. Watson Research Center in Yorktown Heights, New Yorkc During this yer63d he also held visiting professorships at the Universities of l%ris, Bristol& Co.Gu~rbia, Yale and the Stevens Institute of Technology. ElgotS work was almost entirely tan the theory of computation, and h,e wi\ti;in it more or less at the beginning, his first paper [l] alppearing in the first volume of the 3, AeM in 1954. This was a proof that for certain pre&#ely defined simplsued types of machine, the storage requirements for arithmetic operations of a singleaddress machine were not more than those of a triple-address machine. From 1958-64 he worked mainly in the area where automatz thecry and; l~qpicmeet, He used results from logic to prove the undecidabiaity of de&ion probI+ems for l’inite automata and was one of the first to apply automata theory to logic e.g. in the note-,arorthy [ll] where he proved the decidability of various weak seco:nd order arithmetics. In the well-known paper [17] Elgot and Abraham Robinson attacked very successfully the problem of giving a mathe&matical model :not, like: Turin%, of t41e simplest conceivable computing Trrachine, but of machines with rrll the facilities, such as random-access and stored-program, of modern computers. ‘This was extended in [24] to include parallel processing. [25], which was not as Hlpellknown as it deserved to be, was one of the first papers to consider algorithms operatilkg on arbitrary data structures and to develop the analogue for them of the bzsic theorems of recursive function theory. It was also more general than si,m.ilar later treatments in considering basic c0mmand.s which were combinations of oper:atior:s arld mu exit tests, rather than the more usual division into operations autl binairy tests. The book ‘“Recursiveness” (197O)‘written with Eilenberg was a masterly lxx%entation of recursive function theory in an algebraic setting, the key being their realization that the appropriate setting was the category whose morphisms are 0304.3975/81/0000-0000/$02SO
@ North-Holland
2
In memoeiam Caltrrn C. E/got
functions N’ + iv’, so that functions of any number of arguments and values are treated uniformly. From this point on Elgot was convinced that the best way to understand mathematics, particularly the theory of computaltion was P;oexpress it as far as possible Xnalgebraic, especiahy category theoretic, terms. TI is may have been due to hisI collaboration with Eilenberg but he had allways hat. a taste for, and skill in, algebra (e.g. [9, IO]). ‘Whatever the reason for this bias he WZN the pioneer in introducing into the theory of computation the idela of algebraic theories and,, later, of iterative theories (cf. [32, 34, 35, 36]). As well as this work on the semantics of monadic computation be worked recently on its syntax, the flowchart schemes themselves, and in [37] showed that by working with mu&i-entrance, multi-exit schemes a few simple operations suficed to build up all schemes. Several classes of ‘structural’ schemes defined by sub-classes of these operations were studied in detail from both a graph-theoretic and algebraic point of Gew [37, 42, 44, %I]. He was always conscious of the need to bridge the communication gap between those with an inritial training in pure mathematics and those who came to the theory of computation from its practice; he probably felt this particulal*ly acutely because his own work wals characterized by the utmost precision of definition and meticulous proof. At the time of his death he uas working on several different projects; it is sad that he did not live to complete them and to see the further develoament and impact of thi.3 work. J.C.. Shepherdson Universityof Bristol,England
[l] CC. Elgot, On single vs. triple-address computing machines, .I. ACM (July 1954). [2] CC. Elgot, Complex least squares, Abstract, Bull. Amer. Math. Sot., 42 (January 1956) 44. [3] J.R. Buchi, C.C. Elgot and J.B. Wright, Nonexistence atf certain algorithms in finite automata theory, Abstract, Notices Amer. Math. Sot. 1 (Feb. 1958) ‘98. [4] IM. Copi, CC. Elgot and J.B. Wright, Realization of events by logical nets, J. ACM 5 (2) (April 1958) 181-196. Ah in: E.F. Moore, Ed., Sequential Machines (Addison-Wesley, FLeading,MA, 1964). [S] J,R. B_uchiand C.C. Elgot, Decision problems offweak second-order arithmetics and finite automata, ‘Part J, Abstract, Notices Amex Math. Sot. 5 (7) (December 1958) 834. [6] CC. Elgot, Lectures on switching theory and automata theory, University of Michigan Technical Report 2755-4-P (1959). [7] CC. Elgot and J.B. Wright, lQuantifier elimination in a problem of logic design, Michigan Math. L 6 (11959)6549. [ai] CC. Elgot, Decision problems of weak second-order arithmetics and finite automata, Part II, Abstract, Notices Amer. MWz. Sot. 6 (l! (February 1959) 48. @:] CC. Elgot., On equationall!, cilefinableclasses of algebras, Notices Amer. Math. Sot. 6 (1) (February 1959) 48.
In memoriam Calvin C. EXgot
3
CC. Eigot and J.B. Wright, Series-parallel graphs and lattices, Duke M? th. 9: 2r.S(2) (June 1859) 325-338. C.C. Elgot, Decision problems of finite automata design and relaled ariahmetics, Trans. Anler. Math. Sot. 98 (1) (January 1961) 21-51. C.C. Elgot and J.D. Rutledge, RS-machines wizh almost blank tape, Abstract, Notices Amer. Math. Sot. (August 1961) 333. Full report, RC-870, IBM Research Center, Yorktown Heights, New York (December 1962). Also, J. ACM ll(3)(July 1964) 313-337. summary, Proc. 2nd r131 C.C. Elgot and J.D. Rutledge, Operation on finite automata-Extended Annual Sympo.&m sz &&citing Circuit Theory and Logical Design, AHEE (September 1961) 129-132. Also, LM Research Report, NC-168 (September 1961). u41 C.C. Elgot ‘kruth functions realizable by single threshold organs, RG-373, IBM Research Cen’ter, YorktFwn Heights, NY r:December 1960). Also, Proc. 2nd Annual Symposium on Switching Circuit Theory and Logical Des’gn, AHEE (September 1961) 225-245. C.C. Elgot and J.D. Rutledge, Machine prtilperties preserved under state minimization. RC-717, WI IBM Research Center, Yorktown Heights, NY (June 1962). Also, Yroc. 3rd Annual Syzzposium Switching Circuit Theory and Logical Design, AHEE (September 1962) 61-70. WI C.C. Elgot and J.E. Mezei, Two-sided finite-state transductions, IBM Research Report, KC-1017 (June 1963). Also, Proc 4th Annual Symposium on Switching Circuit Theory and Logical Design, AIEE (October 1963) in summary form. A modified version of this paper appeared as “On relations defined by generalized finite automata”, II3MJ. (January 1965). El71 C.C. Elgot and Abraham Robinson, Random access-stored program machines, an approach to programming languages, IBM Research Report, RC-1101 (January lFG4). Also, J. ACM Xl (4) (October 1964) 365-399. EM C.C. Elgot and M.O. Rabin, On the first-order theory of generalized successor (Abstract), presented at Meeting of AMS (January 1963). WI C.C. Elgot and M.O. Rabin, Decision problems of extensions of second-order theory of succes:;or, presented at Meeting of AMS (January 1963). CN C.C. Elgot, Direction and instruction-controlled machines, IBM Research Note, NC-500 A&, Proc. Brooklyn Polytechnical Symposium on System Theory (Pollytechnic Press of the EUytechnic Institute of Brooklyn, 1965) 121-126. c211 C.C Elgot and M.O. Rabin, Decidability a:td’ undecbdability of extensions of second ({first)order theory of (generalized successor, IBM Research Report, RC-1388 (March 1965). Also, .I. Symbolic Logic 31(2) (June 1966) 169-182. I221 C.C. Elgot, A perspective view of discrete. automata and their design, IBM Rescarl:h Report, RC-1261 (August 1964). Also, Amer. Math, Monthly 72 (2) Part II (February 1965) X15-134. c231 C.C. Elgot, Machine species and their computat:an languages, IBM Research Rleport RC-1260 (August 1964). Also in: T.B. Steel, Ed., Formal Language Description Languages fo!d Computer Programming (North-Holland, Amsterdam, 1966) 160-178. 1241 C.C. Elgot, A. Robinson and S.D. Rutledge, Multiple control computer models, IBhI Research Report, RC-1622 (March 1966). Also, Systems and Computer Science (University t,>fToronto Press, 1967) 60-76. C.C. Elgot, Algorithms abstrait et fermeture de diagrammes, Report of the lJniversIty of Paris (October 1966). Also as “Abstract’ algorithms and diagram closures”, IBM Research Ekeport, RC-1750 (January 1967) and in* F. Genuys, Ed., Programming Languages (Academic Press;, New York, 1968) l-42. WI CC. Elgot, A notion of interpretability of algorithms in algorithms, Report of IBM Laboratory, Vienna (October 1966). [271 S. Eilenberg, C.C. Elgot and J.C. Shepherdson, Sets recognized by n-tape automata, IB’VIResearch Report, RC-1944 (November 1967). Also, J. Algebra 13 t 4) (1969) 447-464. I281 S. Eilenberg and CC. Elgot, Iteration and recursion, IBM Research Report RC-2 148 ~,Julv1968). Also, Proc. Nat, Acad. Sci. U.S.A. 61(2)(October 1968) 378-379. IPI S. Eiienberg and CC. Elgot, Recursioeness, Monograph published by Academic Press (1970). C.C. Elgot, The external behavior of machines, Proc. 3rd Hawaii International Conference on iI System Sciences (1970). Also, IBM Research Report, RC-2740 (December 1969). L311 CC. Elgot, The common algebraic structure of exit-automata and mal:hines, IEM Rese:Irch Report, RC-2744 (Januarv 1970). Also. Comtwtinn 6 (Januarv 197 1) 349-370.
4
In memoriam Calvin C. Eligot
[32] C.C. Elgot, Algebraic theories and pro&m schemes, IIBM Research Report, KC-2925 (June 1970). Also in: E. Eng&r, Ed., Prtac. Symposium on the Semantics of Algorithmic Languages (Springer, Berlin, 197 1). [33] C.C, F,lgot, Reme riis on one argument program schemes, IBM Research Report, RC-3482 (August 1971). Also in: R. Rustin, Ed., fi~rr& Semantics ofProgrr?mming Languages, COI rantComputer Science Symposium 2 (Prentice-Hall, Englewood Cliffs, N& 1972). ~34) C.C. Elgot, l@Jnadiccomputation and iterative algebraic theories, IBM Research R port, RC-4564 (October 1973). Also in: HE. Rose and J.C. Shepherdson, ES., Logic Cobquism ‘73, Studies in Logic and the Foundations of Mathematics 80 (North-Holland, Amsterda.m, 1975) 175-230. Abstract in: J. Symbolic Logic (June 1974). [35] CC. ElIgot, Matricial theorie+s* IBM Research Report, RC-4833 (May-1974). Also, J. Algebra 42 (2) (October 1976) 391-421. 1363 S.I.,. Blloom and C.C. Elgot, Tire existence and construction of free iterative theories, IBM Research Report, RC-4937 (July 1974). Also, J. Comput. System SC!. 12 (3) (June 1976:!_305-318. [37] CC. Elgot, Structured programming with and without GO TO statements, IBM Research Report, RC-5626 (September 1975). Also, IEEE Trans. Software Engr. 2 (1) (March 19 ‘6) t;l-54. Erratum and Corrigendum (September 1976). [38] C.C. Elgot, S.L. Bloom and R. Tindell, On the algebraic structure of rooted trees, IBM Research Report, RC-6230 (October I976). Also, J. Comput. System Sci. 16 (3) (June 1478), [39] C.C. Elgot and L. Snyder, On the many facets of lists, IBM Research Repor-, RC-6449 (March 1977, revised June 1977). Also, Theowt. Comput. Sci. 5 (3) (December 1977) .175-305. 1401 C;.C. Elgot, Finite automaton from 3 flowchart scheme point of view, IBM Research Report, RC-6517 (May 1977). Also,, Proc. Mathematical Foundations of Ccmputer Science, Tatranska Lomnica, High Tatras, Czechoslovakia (September 5-9, 1977). [4f] C.C. E.lg$ Some geometrical categories associated with flowchart schemes, IBM Research Report, RC-6534 (May 1977). Also, plot. CTonferenceon Fundamentals of Computatkm Theory, PoznanKornik, Pctlantf (September 19-23, 1.977). [42] C.C. EJgot and J.C. Shepherdson, A semantically meaningful characterization of reduc :ble flowchart schemes, IBM Research Report, RC-6656 (July 1977). Also, Theoret. Comput. Sci. 8 (3) (J~Jw 1979) 325-35;‘. [43] S.L. Bloom, CC. Elgot and J.B. Wright, Solutions of the iteration equation ;nd extensions of &3e scalar iteration operation, IBM Research Report, RC-7029 (March 19X!). Also, S1AMJ. Cornput. 9 (1980) 2 S-45 and 525-540. 1441 C.C. Elgot, A representative strong equivalence clab for accessible flowchart schemes, IBM Research IReport RC-7181 (June 1978). Also, 1 ~oc. Intwrational Conference on Mathematical Studies of Akformation Processing, Kyoto, Japan (August 1978). 145) CC. Elgot, Assignment statements in the context of algebraic theories, Froc. IBM Japan Symposium, Kc:be, Japan (August 1978). Also,. IBM Research R~porf, RC-i’396 (November 1979). C%lS.L. Blo~rrn, C.C. Elgot and J.B. Wright, Vector iteration in pointed iterative theories, IBM Research I;!eport, RC-7322 (September 1978). SIAMJ. Cbmput., to appear, [47] CC. Elgot., The multiplicity semiring of a boolean ring. ‘I$$&!$ Research Report, RC-7540 (March 1979). [48] CC. Elgot mii R.E. Miller, On Coordinated sequential processes, IBM Research Report, RC-7778 (July 1979) [49] CC. Elgot, OR new roles for categori.cal algebra, IBM Research Report, RC-7931 (October 1979). To appear in volume co~=nremorating the Symposium on Algebra, held in Aspen, Colorado (May 23-27, 1979). C50] CC. Elgot tmd J.C. Shetpherdson, An equatiomal axiomatization of the algebra of reducible flow&art schemes, Speck::r Symposium, Zurich (1980); to appear in: S.L. Bloom, Ed,, Selected Papers of CaltfiinC. Elgot.
IN MEMWUAM CALVIN C. ELGQT
L* ‘Calvin C. Elgot who made many contributions to the theory of computatilon, died on July 8, 1980. He was born in New York City in 1922 and after sler%g in the KGK GKGKGKGKGKGKGKGKGKKGat City College of l\;lewYork in 1948 whiFF WEWERWERWERWERt Columb~ia University in 1951 while holding a Lectureship there. Then he wa:; a mathematician at the U.S. Naval 0rdnance Laboratory, Silver Spring, Maryland. From 1954--55 \he ‘was a Teaching Assistant at Berkeley, and from 1955-59 a Research Mathern;rtician at the University of Michigan at Ann Arbor where he received a doctorate. in 1968% For the rest of his life he was a Research Staff Member at the EIBMThcsmas J. Watson Research Center in Yorktown Heights, New Yorkc During this yer63d he also held visiting professorships at the Universities of l%ris, Bristol& Co.Gu~rbia, Yale and the Stevens Institute of Technology. ElgotS work was almost entirely tan the theory of computation, and h,e wi\ti;in it more or less at the beginning, his first paper [l] alppearing in the first volume of the 3, AeM in 1954. This was a proof that for certain pre&#ely defined simplsued types of machine, the storage requirements for arithmetic operations of a singleaddress machine were not more than those of a triple-address machine. From 1958-64 he worked mainly in the area where automatz thecry and; l~qpicmeet, He used results from logic to prove the undecidabiaity of de&ion probI+ems for l’inite automata and was one of the first to apply automata theory to logic e.g. in the note-,arorthy [ll] where he proved the decidability of various weak seco:nd order arithmetics. In the well-known paper [17] Elgot and Abraham Robinson attacked very successfully the problem of giving a mathe&matical model :not, like: Turin%, of t41e simplest conceivable computing Trrachine, but of machines with rrll the facilities, such as random-access and stored-program, of modern computers. ‘This was extended in [24] to include parallel processing. [25], which was not as Hlpellknown as it deserved to be, was one of the first papers to consider algorithms operatilkg on arbitrary data structures and to develop the analogue for them of the bzsic theorems of recursive function theory. It was also more general than si,m.ilar later treatments in considering basic c0mmand.s which were combinations of oper:atior:s arld mu exit tests, rather than the more usual division into operations autl binairy tests. The book ‘“Recursiveness” (197O)‘written with Eilenberg was a masterly lxx%entation of recursive function theory in an algebraic setting, the key being their realization that the appropriate setting was the category whose morphisms are 0304.3975/81/0000-0000/$02SO
@ North-Holland
2
In memoeiam Caltrrn C. E/got
functions N’ + iv’, so that functions of any number of arguments and values are treated uniformly. From this point on Elgot was convinced that the best way to understand mathematics, particularly the theory of computaltion was P;oexpress it as far as possible Xnalgebraic, especiahy category theoretic, terms. TI is may have been due to hisI collaboration with Eilenberg but he had allways hat. a taste for, and skill in, algebra (e.g. [9, IO]). ‘Whatever the reason for this bias he WZN the pioneer in introducing into the theory of computation the idela of algebraic theories and,, later, of iterative theories (cf. [32, 34, 35, 36]). As well as this work on the semantics of monadic computation be worked recently on its syntax, the flowchart schemes themselves, and in [37] showed that by working with mu&i-entrance, multi-exit schemes a few simple operations suficed to build up all schemes. Several classes of ‘structural’ schemes defined by sub-classes of these operations were studied in detail from both a graph-theoretic and algebraic point of Gew [37, 42, 44, %I]. He was always conscious of the need to bridge the communication gap between those with an inritial training in pure mathematics and those who came to the theory of computation from its practice; he probably felt this particulal*ly acutely because his own work wals characterized by the utmost precision of definition and meticulous proof. At the time of his death he uas working on several different projects; it is sad that he did not live to complete them and to see the further develoament and impact of thi.3 work. J.C.. Shepherdson Universityof Bristol,England
[l] CC. Elgot, On single vs. triple-address computing machines, .I. ACM (July 1954). [2] CC. Elgot, Complex least squares, Abstract, Bull. Amer. Math. Sot., 42 (January 1956) 44. [3] J.R. Buchi, C.C. Elgot and J.B. Wright, Nonexistence atf certain algorithms in finite automata theory, Abstract, Notices Amer. Math. Sot. 1 (Feb. 1958) ‘98. [4] IM. Copi, CC. Elgot and J.B. Wright, Realization of events by logical nets, J. ACM 5 (2) (April 1958) 181-196. Ah in: E.F. Moore, Ed., Sequential Machines (Addison-Wesley, FLeading,MA, 1964). [S] J,R. B_uchiand C.C. Elgot, Decision problems offweak second-order arithmetics and finite automata, ‘Part J, Abstract, Notices Amex Math. Sot. 5 (7) (December 1958) 834. [6] CC. Elgot, Lectures on switching theory and automata theory, University of Michigan Technical Report 2755-4-P (1959). [7] CC. Elgot and J.B. Wright, lQuantifier elimination in a problem of logic design, Michigan Math. L 6 (11959)6549. [ai] CC. Elgot, Decision problems of weak second-order arithmetics and finite automata, Part II, Abstract, Notices Amer. MWz. Sot. 6 (l! (February 1959) 48. @:] CC. Elgot., On equationall!, cilefinableclasses of algebras, Notices Amer. Math. Sot. 6 (1) (February 1959) 48.
In memoriam Calvin C. EXgot
3
CC. Eigot and J.B. Wright, Series-parallel graphs and lattices, Duke M? th. 9: 2r.S(2) (June 1859) 325-338. C.C. Elgot, Decision problems of finite automata design and relaled ariahmetics, Trans. Anler. Math. Sot. 98 (1) (January 1961) 21-51. C.C. Elgot and J.D. Rutledge, RS-machines wizh almost blank tape, Abstract, Notices Amer. Math. Sot. (August 1961) 333. Full report, RC-870, IBM Research Center, Yorktown Heights, New York (December 1962). Also, J. ACM ll(3)(July 1964) 313-337. summary, Proc. 2nd r131 C.C. Elgot and J.D. Rutledge, Operation on finite automata-Extended Annual Sympo.&m sz &&citing Circuit Theory and Logical Design, AHEE (September 1961) 129-132. Also, LM Research Report, NC-168 (September 1961). u41 C.C. Elgot ‘kruth functions realizable by single threshold organs, RG-373, IBM Research Cen’ter, YorktFwn Heights, NY r:December 1960). Also, Proc. 2nd Annual Symposium on Switching Circuit Theory and Logical Des’gn, AHEE (September 1961) 225-245. C.C. Elgot and J.D. Rutledge, Machine prtilperties preserved under state minimization. RC-717, WI IBM Research Center, Yorktown Heights, NY (June 1962). Also, Yroc. 3rd Annual Syzzposium Switching Circuit Theory and Logical Design, AHEE (September 1962) 61-70. WI C.C. Elgot and J.E. Mezei, Two-sided finite-state transductions, IBM Research Report, KC-1017 (June 1963). Also, Proc 4th Annual Symposium on Switching Circuit Theory and Logical Design, AIEE (October 1963) in summary form. A modified version of this paper appeared as “On relations defined by generalized finite automata”, II3MJ. (January 1965). El71 C.C. Elgot and Abraham Robinson, Random access-stored program machines, an approach to programming languages, IBM Research Report, RC-1101 (January lFG4). Also, J. ACM Xl (4) (October 1964) 365-399. EM C.C. Elgot and M.O. Rabin, On the first-order theory of generalized successor (Abstract), presented at Meeting of AMS (January 1963). WI C.C. Elgot and M.O. Rabin, Decision problems of extensions of second-order theory of succes:;or, presented at Meeting of AMS (January 1963). CN C.C. Elgot, Direction and instruction-controlled machines, IBM Research Note, NC-500 A&, Proc. Brooklyn Polytechnical Symposium on System Theory (Pollytechnic Press of the EUytechnic Institute of Brooklyn, 1965) 121-126. c211 C.C Elgot and M.O. Rabin, Decidability a:td’ undecbdability of extensions of second ({first)order theory of (generalized successor, IBM Research Report, RC-1388 (March 1965). Also, .I. Symbolic Logic 31(2) (June 1966) 169-182. I221 C.C. Elgot, A perspective view of discrete. automata and their design, IBM Rescarl:h Report, RC-1261 (August 1964). Also, Amer. Math, Monthly 72 (2) Part II (February 1965) X15-134. c231 C.C. Elgot, Machine species and their computat:an languages, IBM Research Rleport RC-1260 (August 1964). Also in: T.B. Steel, Ed., Formal Language Description Languages fo!d Computer Programming (North-Holland, Amsterdam, 1966) 160-178. 1241 C.C. Elgot, A. Robinson and S.D. Rutledge, Multiple control computer models, IBhI Research Report, RC-1622 (March 1966). Also, Systems and Computer Science (University t,>fToronto Press, 1967) 60-76. C.C. Elgot, Algorithms abstrait et fermeture de diagrammes, Report of the lJniversIty of Paris (October 1966). Also as “Abstract’ algorithms and diagram closures”, IBM Research Ekeport, RC-1750 (January 1967) and in* F. Genuys, Ed., Programming Languages (Academic Press;, New York, 1968) l-42. WI CC. Elgot, A notion of interpretability of algorithms in algorithms, Report of IBM Laboratory, Vienna (October 1966). [271 S. Eilenberg, C.C. Elgot and J.C. Shepherdson, Sets recognized by n-tape automata, IB’VIResearch Report, RC-1944 (November 1967). Also, J. Algebra 13 t 4) (1969) 447-464. I281 S. Eilenberg and CC. Elgot, Iteration and recursion, IBM Research Report RC-2 148 ~,Julv1968). Also, Proc. Nat, Acad. Sci. U.S.A. 61(2)(October 1968) 378-379. IPI S. Eiienberg and CC. Elgot, Recursioeness, Monograph published by Academic Press (1970). C.C. Elgot, The external behavior of machines, Proc. 3rd Hawaii International Conference on iI System Sciences (1970). Also, IBM Research Report, RC-2740 (December 1969). L311 CC. Elgot, The common algebraic structure of exit-automata and mal:hines, IEM Rese:Irch Report, RC-2744 (Januarv 1970). Also. Comtwtinn 6 (Januarv 197 1) 349-370.
4
In memoriam Calvin C. Eligot
[32] C.C. Elgot, Algebraic theories and pro&m schemes, IIBM Research Report, KC-2925 (June 1970). Also in: E. Eng&r, Ed., Prtac. Symposium on the Semantics of Algorithmic Languages (Springer, Berlin, 197 1). [33] C.C, F,lgot, Reme riis on one argument program schemes, IBM Research Report, RC-3482 (August 1971). Also in: R. Rustin, Ed., fi~rr& Semantics ofProgrr?mming Languages, COI rantComputer Science Symposium 2 (Prentice-Hall, Englewood Cliffs, N& 1972). ~34) C.C. Elgot, l@Jnadiccomputation and iterative algebraic theories, IBM Research R port, RC-4564 (October 1973). Also in: HE. Rose and J.C. Shepherdson, ES., Logic Cobquism ‘73, Studies in Logic and the Foundations of Mathematics 80 (North-Holland, Amsterda.m, 1975) 175-230. Abstract in: J. Symbolic Logic (June 1974). [35] CC. ElIgot, Matricial theorie+s* IBM Research Report, RC-4833 (May-1974). Also, J. Algebra 42 (2) (October 1976) 391-421. 1363 S.I.,. Blloom and C.C. Elgot, Tire existence and construction of free iterative theories, IBM Research Report, RC-4937 (July 1974). Also, J. Comput. System SC!. 12 (3) (June 1976:!_305-318. [37] CC. Elgot, Structured programming with and without GO TO statements, IBM Research Report, RC-5626 (September 1975). Also, IEEE Trans. Software Engr. 2 (1) (March 19 ‘6) t;l-54. Erratum and Corrigendum (September 1976). [38] C.C. Elgot, S.L. Bloom and R. Tindell, On the algebraic structure of rooted trees, IBM Research Report, RC-6230 (October I976). Also, J. Comput. System Sci. 16 (3) (June 1478), [39] C.C. Elgot and L. Snyder, On the many facets of lists, IBM Research Repor-, RC-6449 (March 1977, revised June 1977). Also, Theowt. Comput. Sci. 5 (3) (December 1977) .175-305. 1401 C;.C. Elgot, Finite automaton from 3 flowchart scheme point of view, IBM Research Report, RC-6517 (May 1977). Also,, Proc. Mathematical Foundations of Ccmputer Science, Tatranska Lomnica, High Tatras, Czechoslovakia (September 5-9, 1977). [4f] C.C. E.lg$ Some geometrical categories associated with flowchart schemes, IBM Research Report, RC-6534 (May 1977). Also, plot. CTonferenceon Fundamentals of Computatkm Theory, PoznanKornik, Pctlantf (September 19-23, 1.977). [42] C.C. EJgot and J.C. Shepherdson, A semantically meaningful characterization of reduc :ble flowchart schemes, IBM Research Report, RC-6656 (July 1977). Also, Theoret. Comput. Sci. 8 (3) (J~Jw 1979) 325-35;‘. [43] S.L. Bloom, CC. Elgot and J.B. Wright, Solutions of the iteration equation ;nd extensions of &3e scalar iteration operation, IBM Research Report, RC-7029 (March 19X!). Also, S1AMJ. Cornput. 9 (1980) 2 S-45 and 525-540. 1441 C.C. Elgot, A representative strong equivalence clab for accessible flowchart schemes, IBM Research IReport RC-7181 (June 1978). Also, 1 ~oc. Intwrational Conference on Mathematical Studies of Akformation Processing, Kyoto, Japan (August 1978). 145) CC. Elgot, Assignment statements in the context of algebraic theories, Froc. IBM Japan Symposium, Kc:be, Japan (August 1978). Also,. IBM Research R~porf, RC-i’396 (November 1979). C%lS.L. Blo~rrn, C.C. Elgot and J.B. Wright, Vector iteration in pointed iterative theories, IBM Research I;!eport, RC-7322 (September 1978). SIAMJ. Cbmput., to appear, [47] CC. Elgot., The multiplicity semiring of a boolean ring. ‘I$$&!$ Research Report, RC-7540 (March 1979). [48] CC. Elgot mii R.E. Miller, On Coordinated sequential processes, IBM Research Report, RC-7778 (July 1979) [49] CC. Elgot, OR new roles for categori.cal algebra, IBM Research Report, RC-7931 (October 1979). To appear in volume co~=nremorating the Symposium on Algebra, held in Aspen, Colorado (May 23-27, 1979). C50] CC. Elgot tmd J.C. Shetpherdson, An equatiomal axiomatization of the algebra of reducible flow&art schemes, Speck::r Symposium, Zurich (1980); to appear in: S.L. Bloom, Ed,, Selected Papers of CaltfiinC. Elgot.