## SLACS '97 (Oct. 29-31, 1997 at Kyushu University, Fukuoka) \$B\$N%W%m%0%i%`(B

10\$B7n(B29\$BF|(B
13\$B!'(B00 \$BF#ED(B \$B7{1Y(B \$B!J6e=#9)6HBg3X(B \$B>pJs9)3XIt!K(B

### On the Church-Rosser property of \$B&K(B_{exc}^v

We introduce a call-by-value calculus based on classical propositional logic, called \$B&K(B_{exc}^v. The classical system is simple and a simple exit mechanism can be implemented by the use of classical proofs. We will prove that \$B&K(B_{exc}^v without so-called renaming rules has the Church-Rosser property by the well-known method of parallel reductions and the lemma of Hindley-Rosen. It is observed that for \$B&K(B_ {exc}^v with renaming rules, a straightforward use of the parallel reduction method could not work to prove the Church-Rosser property.

13\$B!'(B30 Ver-Jan de Vries \$B!JEE;R5;=QAm9g8&5f=j!K(B

### Typings for infinite lambda calculus

14\$B!'(B00 \$B>.NS(B \$BAo(B \$B!JN6C+Bg3X(B \$BM}9)3XIt!K(B

### S4 \$B\$N(B realizability \$B:F9M(B

\$B9V1i\$NO@M}5-9f\$K\$D\$\$\$F\$O!"=>MhDL(B \$B\$j\$N=c4X?tE*\$J2ro\$NO@M}5-9f\$KBP\$7\$F\$b(B \$B=c4X?tE*\$G\$J\$\$2r

14\$B!'(B30 Break
14\$B!'(B40 \$BD9C+@n(B \$BN)(B \$B!JEl5~Bg3XBg3X1!(B \$B?tM}2J3X8&5f2J!K(B

### Wreath model of system F

15\$B!'(B10 \$BGr4z(B \$BM%(B \$B!J7D1~Bg3X(B \$B>&3XIt!K(B

### A coherence space semantics for linear set theory with quantifiers

We present an extension of the coherence space semantics for linear set theory to include quantifiers. The universe of sets is obtained by combining the coherence space semantics for propositional linear logic and Scott style inverse limit construction. We then define the interpretation of terms in the universe through the well-known cardinality argument from the theory of inductive definition.

15\$B!'(B40
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15\$B!'(B50 \$BN6ED(B \$B??(B \$B!J5~ETBg3X(B \$BM}3X8&5f2J!K(B

### q-realizability for Feferman's theory

Feferman \$B\$N(B constructive set theory T_0 \$B\$KBP\$9\$kJQ?t\$rFs=E2=\$7\$J(B \$B\$\$(Bq-realizability \$B\$NDj5A\$H\$=\$N7rA4@-DjM}\$OL\$2r7hLdBj\$G\$"\$C\$?\$,!"\$3\$l\$r(B \$B2r7h\$7\$?!#(BT_0 + set induction axiom \$B\$NBN7O\$KBP\$7\$F(B q-realizability \$B\$r(B \$BDj5A\$7\$=\$N7rA4@-\$r>ZL@\$9\$k!#9V1i

16\$B!'(B20 \$BHx:j(B \$B@58w(B \$B!JL>8E20Bg3XBg3X1!?M4V>pJs3X8&5f2J!K(B

### On MODkP Counting Degrees : Lattice Embedding Theorems

10\$B7n(B30\$BF|(B
10\$B!'(B00 \$BD9LnBg2p!"GO>l7r2p!J6e=#Bg3XBg3X1!%7%9%F%`>pJs2J3X8&5f2J!K(B
\$B:4DM(B \$B=(?M(B \$B!J5WN1JF9)6HBg3X!KW"@n:4@iCK!J6e=#Bg3XBg7?7W;;5!%;%s%?!

### \$B<+A3?dO@\$K4p\$E\$/>ZL@C5:w\$HH?Nc@8@.(B

\$BD>4QZL@C5:w\$K\$*\$\$\$F!"C5:w\$NF;=g\$r5-(B \$B21\$7\$F\$*\$/\$3\$H\$G!">ZL@\$G\$-\$k>l9g\$K\$O>ZL@?^\$r!">ZL@\$G\$-\$J\$\$>l9g\$K\$O(B \$BH?Nc\$H\$7\$F\$N%/%j%W%1!&%b%G%k\$r9=@.\$9\$ke\$N(B \$B%/%j%W%1!&%b%G%k\$,@8@.\$5\$l\$k>l9g\$K\$O8EE5O@M}\$K\$*\$1\$k\$=\$NL?Bj\$N>ZL@(B \$B\$bJV\$9!#\$^\$?!"\$3\$NZL@%7%9%F%`\$r>R2p\$9\$k!#(B

10\$B!'(B30 \$B2CF#D>9T(B \$B!J6e=#Bg3XBg3X1!%7%9%F%`>pJs2J3X8&5f2J!K(B
\$B:4DM=(?M(B \$B!J5WN1JF9)6HBg3X!KW"@n:4@iCK!J6e=#Bg3XBg7?7W;;5!%;%s%?!

### \$B=iEy4v2?>ZL@%7%9%F%`\$N\$?\$a\$N@53N\$J?^\$N:n@.(B

\$B=iEy4v2?\$NLdBj\$r!"6qBNE*\$KIA\$+\$l\$??^\$N>pJs\$rMQ\$\$\$F(B \$B>ZL@\$r@8@.\$9\$k%7%9%F%`\$r3+H/\$7\$F\$\$\$k!#MxMQDj\$rK~\$?\$9\$h\$&\$J:BI8\$r5a\$a\$k!#\$3\$&\$7\$FF@\$i\$l\$k@53N\$J?^\$KBP\$7%1!<(B \$B%G%#%s%,!ZL@\$r9=@.\$9\$k!#(B

11\$B!'(B00
Break
11\$B!'(B10 \$BU*(B \$B5~(B \$B=g(B \$B!J?@8MBg3XBg3X1!<+A32J3X8&5f2J!K(B

### A Leveled IO-model for a Linear Logic Programming Language

A logic programming language Lolli which is based on intuitionistic linear logic is a superset of Prolog and I/O-model was proposed by Hodas and Miller as a execution-model for Lolli. In order to solve multiplicative goals (that is,the connection operation of the goal is multiplicative conjuntion) from the bottom up in the intuitionistic linear logic, we need to split the resource into two disjoint part. In the I/O-model ,to remove the non-determinism of such splits, each goal has its input resource and output resource and then all resource not used to prove a subgoal G1 will be used to prove the other subgoal G2 when proving the goal G1*G2. / In this paper, we propose a Leveled IO-model as an implementation-model that manage resource efficiently for Lolli. Input resource and output resource of the Leveled IO-model have an integer called consumption-level as the information that resource is either consumable or unconsumable. Our Leveled IO-model eliminate checking that each resource used to prove subgoals are same whenever proving the goal include additive conjuntion. In the Leveled IO-model, only resource used to prove a subgoal G1 will be used to prove the other subgoal G2 when proving the goal G1&G2. This idea is used at a compiler system of LLP which is a linear logic programming language and is a subset of Lolli. And also we can remove the non-determinism of \$B!H(Btop\$B!I(B by improving the idea of consumpution-level.

11\$B!'(B40 \$B@V4V(B \$BM[Fs(B \$B!JEl5~Bg3X!K(B

### Type theory for partial combinatory algebras

Partial combinatory algebra (pca, for short) is a strict generalization of combinatory algebra, and it is not necessarily embedded into a combinatory algebra. With this notion, we can prove strong normalization of higher type theories semantically. By a type-theoretic method, we will analyze the pca, and find the intrinsic relation to the set of sn terms of combinatory logic. This leads to the solution of Bethke-Klop's question.

12\$B!'(B20
Lunch
13\$B!'(B30 \$BBtB

### HOL\$B\$K\$h\$k(BRelevant logic\$B\$NDjM}>ZL@5!(B

\$B!!(BRelevant Logic\$B\$O!\$8EE5O@M}\$K8+\$i\$l\$k!V\$J\$i\$P!W\$K(B \$B4X\$9\$kO@M}E*0cOB46\$rZL@K!\$N8&5f\$O?J\$s\$G\$\$(B \$B\$J\$\$!#(B
\$B!!K\O@J8\$G\$O!"(BRelevant Logic \$B#R\$KBP\$9\$k2DG=@\$3&0UL#O@\$r(B HOL\$B\$NO@M}I=8=\$KK]Lu\$9\$k\$3\$H\$K\$h\$C\$F!"E,@ZO@M}\$NO@M}<0\$NBEEv@-\$r>ZL@(B \$B\$9\$kJ}K!\$rM?\$(\$k!#ZL@5!G=\$r;H\$C\$F>ZL@\$7!"(BRelevant logic\$BDj(B \$BM}>ZL@5!\$X\$N(BHOL\$B\$K\$h\$k%"%W%m!]%A\$NM-8z@-\$r<(\$9!#(B

14\$B!'(B00 \$BN)LZ=(

### Gray code \$B\$K4p\$E\$\$\$? \$Bo\$N(B2\$B?JI=8=\$d(B10\$B?JI=8=\$rMQ\$\$\$?\$N\$G\$O!"(B3\$B\$r3]\$1\$k\$J\$I\$N4pK\(B \$BE*\$J4X?t\$b7W;;2DG=\$G\$J\$/\$J\$C\$F\$7\$^\$&!#\$=\$N7gE@\$rJd\$&\$?\$a\$K!"\$\$\$m\$\$\$m\$J(B \$B%3!<%I7O\$,:#\$^\$G\$K\$bDs0F\$5\$l\$F\$\$\$k\$,!"\$=\$l\$i\$O!"0lHL\$K!"9b\$\$>iD9@-\$rH<\$C(B \$B\$F\$\$\$k!#\$3\$N9V1i\$G\$O!"==J,9-\$\$7W;;2DG=@-\$r;}\$A!"\$7\$+\$b!">iD9@-\$,Dc\$\$%3!<(B \$B%I7O\$H\$7\$F!"(BGray code \$B\$H8F\$P\$l\$k(B2\$B?JI=8=\$K4p\$E\$\$\$?%3!<%I7O\$rDs0F\$9\$k!#(B 14\$B!'(B30 Break 14\$B!'(B40 \$B>._7@5D>(B \$B!JL>8E20Bg3X>pJsJ82=3XIt!K@>B<<#F;(B \$B!JBg3X1!?M4V>pJs3X8&5f2J!K(B \$BNL;R%A%e!<%j%s%05!3#\$HNL;R2sO)B2\$N7W;;NL\$NF1Ey@-\$K\$D\$\$\$F(B \$BAG0x?tJ,2rLdBj\$HN%;6E*BP?tLdBj\$O=>Mh\$N7W;;NLM}O@\$G(B \$B\$O!\$8zN(E*%"%k%4%j%:%`\$NB8:_\$7\$J\$\$LdBj\$H\$_\$J\$5\$l\$F\$-\$?!%\$\$\$/\$D\$+\$N(B \$B8x3+800E9f7O\$N0BA4@-\$b\$=\$N\$3\$H\$K0M5r\$7\$F\$\$\$k!%\$H\$3\$m\$,!\$(B1994\$BG/\$K(B Shor\$B\$K\$h\$C\$F\$3\$l\$i\$NLdBj\$r8zN(E*\$K2r\$/NL;R7W;;%"%k%4%j%:%`\$,H/8+\$5(B \$B\$l!\$NL;R7W;;5!\$N8&5f\$,\$K\$o\$+\$KCmL\\$r=8\$a\$k\$3\$H\$K\$J\$C\$?!%NL;R7W;;\$N(B \$B8&5f\$O!\$(B1980\$BG/:"!\$(BFeynman\$B\$K\$h\$C\$F!\$NL;RNO3X7O\$r8EE5NO3XE*7W;;5!\$GLO(B \$BJo\$9\$k\$3\$H\$NJ#;(@-\$+\$i!\$NL;RNO3X\$N86M}\$rMxMQ\$7\$?!\$\$h\$j8zN(\$N9b\$\$7W(B \$B;;\$N2DG=@-\$,<(:6\$5\$l\$?\$3\$H\$+\$i;O\$^\$C\$?!%(B1985\$BG/\$+\$i(B90\$BG/\$K\$+\$1\$F!\$(B Deutsch \$B\$ONL;R%A%e!<%j%s%05!3#\$HNL;R2sO)\$N#2ZL@\$r<((B \$B\$9!%(B 15\$B!'(B20 \$B3{!!9@Lw(B \$B!JF`NI=w;RBg3XM}3XIt!K(B \$B2OM8!!5*;R(B \$B!JF`NI=w;RBg3X?M4VJ82=8&5f2J!K(B \$BC]Fb!!@t(B \$B!J5~ETBg3X9)3X8&5f2J!K(B \$B!VJ,Nv \$B!!:G6a!"?t3X\$N=t35G0\$r7W;;2DG=@-\$NN)>l\$+\$iD/\$a\$?7W(B \$B;;2DG=@-?t3X\$,CmL\\$5\$l\$F\$\$\$k!#\$=\$NJ}K!\$r;H\$C\$F!"J,NvZL@\$9\$k!#\$^\$?!"R2p\$9\$k!#(B 15\$B!'(B50 Break 16\$B!'(B00 \$B55;39,5A(B \$B!J5~ETBg3X!K(B \$BItJ,7QB3\$N7W;;7O(B \$B4X?t7?%W%m%0%i%_%s%08@8l\$K\$*\$1\$k7QB3\$,8EE5O@M}\$N7W;;E*0UL#\$H\$N(B \$B4XO"\$GCmJ8\$r=8\$a\$F\$\$\$k\$,!\$\$h\$jO@M}E*\$Ke\$bM-MQ\$JItJ,7QB3\$K\$D\$\$\$F!\$7?M}O@\$NOHFb\$GDj<02=\$9\$k!%(B 16\$B!'(B30 Business Meeting 18\$B!'(B30 \$B:)?F2q(B

10\$B7n(B31\$BF|(B
10\$B!'(B00 \$BEOn4(B \$B9((B \$B!JKL3\$F;Bg3XBg3X1!M}3X8&5f2J!K(B

### The subobject classifier of the category of labeled transition systems and simulation maps

\$BA+0\7O\$H8F\$P\$l\$k%i%Y%kIU\$-%0%i%U\$O!\$%W%m%;%9Be?t\$"\$k\$\$\$OMMAjO@M}\$N0UL#NN(B \$B0h!&7W;;\$N%b%G%k\$J\$I!\$\$5\$^\$6\$^\$JJ,Ln\$GMxMQ\$5\$l\$F\$\$\$k!%(B
\$BK\8&5f\$G\$O!\$A+0\7O\$NLOJoNJDC17w\$N9=B\$\$@\$1\$G\$J\$/!\$J,N`BP>](B(subobject classifier)\$B\$b;}\$D\$3\$H\$r8+\$\$\$@\$7\$?!%J,N`BP>]\$NB8:_>ZL@\$O!\$LZ\$N7A\$r\$7\$?A+(B \$B0\7O\$+\$i\$J\$kcGL)\$J>.ItJ,7w\$,B8:_\$9\$k\$3\$H!\$\$3\$N>.ItJ,7w>e\$NA0AX7w\$KA+0\7O(B \$B\$N7w\$rKd\$a9~\$a\$k\$3\$H!\$\$=\$7\$F\$3\$NKd\$a9~\$_\$,(Breflection\$B\$r;}\$D\$3\$H!\$\$K4p\$E\$/!%(B \$BB8:_>ZL@\$N80\$H\$J\$kJdBj\$O!\$ItJ,7w>e\$NA0AX7w\$K4X\$9\$k0lHLE*;v\$N(B \$B7w\$X\$N1~MQ\$b\$"\$k\$H;W\$o\$l\$k!%(B
\$B0J>e\$N9M;!\$K\$h\$j!\$A+0\7O\$NLOJo]\$NB8:_\$9\$kBP>N(B \$BJDC17w\$H\$J\$k!%\$3\$N\$h\$&\$J%?%\$%W\$N7w\$NM}O@E*0U5A\$O:#8e\$N2]Bj\$G\$"\$k!%(B

10\$B!'(B40 Stefanescu, Gheorghe \$B!J6e=#Bg3X!&(B University of Bucharest\$B!K(B

### Network Algebra: Past, Present, and Future

Kleene's calculus of regular expressions and the associated regular algebra describe in an elegant way cyclic processes with global states. This calculus has a deep mathematical theory and many applications to sequential programs, automata, formal languages, circuits, etc. but less in parallel/distributed computation.
The ``Network Algebra'' and the associated ``Calculus of Flownomials'' may be seen as an extension of the above calculus to cope with processes having multiple entries and exits, i.e., pins for connections. Each pin may be seen as a local state and this makes the extension well-suited for distributed computation. Applications to dataflow networks, process graphs, or action calculi have already been done. Technically speaking, the main ingredient is a new axiomatic looping operation, called ``feedback''; it connects an output to an input and then both pins are hidden.
This new setting allows for a simple algebra (BNA) for flowgraphs modulo graph isomorphism. We claim that whenever a cyclic process is implicitely of explicitely present, a semantic algebra fulfilling the BNA axioms may be constructed.
On top of this algebra one may add simple axioms to cope with correct and complete axiomatisations for certain classical equivalences, e.g., the equivalences associated to the input behaviour (or equivalently, those given by unfolding flowgraphs into trees), bisimulation or input-output behaviour. The so called `enzymatic rule' which produces these equivalences is a rule to reason in a cyclic environment.
The first part of the talk includes a short introduction to network algebra. Then we present some new results, mainly related to the mixed network algebra project. The BNA axioms seem to be very basic. After their introduction (Stefanescu 1986) the BNA axioms were rediscovered in various setting, e.g by Bartha (systolic systems), Stark (dataflow networks), Milner (action calculi), Joyal-Street-Verity (traced monoidal categories), etc. The last part of the talk includes some comments on these papers.

11\$B!'(B10
Break
11\$B!'(B20 \$BBgDM(B \$B42(B \$B!J6e=#Bg3XBg3X1!?tM}3X8&5f2J!K(B

### \$B%V%m!<%I%-%c%9%H\$r9MN8\$7\$?%W%m%;%9Be?t\$K\$*\$1\$k%G%C%I%m%C%/(B

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11\$B!'(B50 \$BEDJU(B \$B@?(B \$B!J2J3X5;=Q?66=;v6HCD(B \$B\$5\$-\$,\$1(B21\$B!K(B

### \$B;~Aj@~7?O@M}\$H;~4V%Z%H%j%M%C%H(B

It is well known that Petri nets constitute the algebraic structure of quantales, which can be models of linear logic. As a timed extension to quantales, timed R-monoids are defined, which are constructed from timed Petri nets.
Next, temporal linear logic is introduced, which has timed Petri nets as its models, i.e., whose formulas can be interpreted as sets of timed markings of a timed Petri net.
Finally, examples show how to express properties of timed Petri nets by temporal linear logic.

12\$B!'(B20
Lunch
13\$B!'(B30 \$BCf>>(B \$BOBVa(B \$B!JJ<8K8)N)I1O)C;4|Bg3X(B \$B7P1D>pJs3X2J!K(B

### \$B%Y%/%H%k??M}CMIU\$-O@M}\$K\$D\$\$\$F(B(On the Vector Annotated Logic)

defeasible\$B?dO@!"%^%k%A%(%\$%8%'%s%H%7%9%F%`\$N%b%G%k\$X\$N1~MQ(B \$B\$rL\E*\$H\$7\$F!"%Y%/%H%k??M}CMIU\$-O@M}\$rDs>'\$9\$k!#!!\$3\$l\$O!"(B V.S.Subrahmanian, Newton,C.A.Da Costa\$B\$i\$K\$h\$C\$FDs>'\$5\$l\$?(B \$B??M}CMIU\$-O@M}(B(annotated logic)\$B\$N??M}CM(B(annotation)\$B\$r%Y%/%H%k\$K(B \$B3HD%\$7\$?\$b\$N\$G\$"\$k!#(B

14\$B!'(B10 \$B2

### \$B%U%!%8%#=89g\$N4pK\1i;;\$K\$D\$\$\$F(B

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14\$B!'(B50 \$B8E_7(B \$B?N(B \$B!J6e=#Bg3XBg3X1!%7%9%F%`>pJs2J3X8&5f2J!K(B

### A Representation Theorem for Relation Algebras: Concepts of Scalar Relations and Point Relations

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```SLACS '97 \$B!w(B Kyushu University \$B44;v(B
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E-mail:masa@i.kyushu-u.ac.jp
http://www.i.kyushu-u.ac.jp/~masa/
Tel:092-642-2693
```