A. TAN (1997): Full completeness for models of linear logic. PhD thesis, University of Cambridge. A Appendix: Separation Throughout this appendix # is a proof net on a cut sequent  Home/Search   Document Not in Database   Summary   Related Articles   Check

....net is an abstract representation of a proof: the translation of cut free proofs into proof nets identifies proofs modulo inessential commutations of rules. The identifications have since been verified as canonical from a semantic perspective, with numerous full completeness results for MLL, e.g. [AJ94, HO93, Loa94, Tan97, BS96, DHPP99]. Furthermore, the identifications correspond to coherences of free star autonomous categories [BCST96] The problem of finding a satisfactory extension of the theory of proof nets to unit free multiplicative additive linear logic (MALL) has remained open since the inception of linear logic ....

A. TAN (1997): Full completeness for models of linear logic. PhD thesis, University of Cambridge. 10

.... , and a Full Completeness theorem was proved for a game semantics of Multiplicative Linear Logic (with the MIX rule) This was followed by a series of papers which established full completeness results for a variety of models with respect to various versions of Multiplicative Linear Logic, e.g. [HO92, BS96, Loa94a, Loa94b, Tan97, DHPP99, Hag00]. The proofs of full completeness which have appeared to date fall into two broad classes: Proofs using decomposition arguments There were a number of signi cant precursors, as noted in [AJ92b] including representation theorems in category theory [FS91] full abstraction results in ....

A. Tan. Full completeness for models of linear logic. Ph.D. thesis, University of Cambridge, 1996.

.... 2 , and a Full Completeness theorem was proved for a game semantics of Multiplicative Linear Logic (with the MIX rule) This was followed by a series of papers which established full completeness results for a variety of models with respect to various versions of Multiplicative Linear Logic, e.g. [HO92, BS96, Loa94a, Loa94b, Tan97, DHPP99, Hag00]. The proofs of full completeness which have appeared to date fall into two broad classes: Proofs using decomposition arguments 2 There were a number of signi cant precursors, as noted in [AJ92b] including representation theorems in category theory [FS91] full abstraction results in ....

A. Tan. Full completeness for models of linear logic. Ph.D. thesis, University of Cambridge, 1996.

....In particular, the achievements of denotational completeness do not seem to throw any light on the following question Is it decidable whether a clique in the coherence space model comes from a proof Full completness results of another flavour have been obtained e.g. in A. M. Tan s PhD Thesis [8] where it is proved that every dinatural transformation on an arbitrary model of multiplicative linear logic (MLL) as e.g. in particular the coherence space model, appears as denotation of a proof in MLL. As all denotations of derivations in MLL give rise to dinatural transformations these are ....

Audrey M. Tan. Full Completeness of Models of Linear Logic. PhD thesis, Cambridge University, 1997. 14

....part so that negation can be obtained by exchanging the two. Examples for this are Player versus Opponent in games 3 , the pair of sets for Chu spaces or dialectica categories, and morphisms I A versus those A in the double glueing construction (see [9,8] employed by Tan [12]. Structurally somewhat simpler models, on the other hand, such as coherence spaces or hypercoherences, can do without this kind of built in duality the negation of any object can be de ned just via the structure it carries 4 . Our models de nitely belong to this second category. At rst sight ....

A.M. Tan. Full completeness for models of linear logic. PhD thesis, University of Cambridge, 1997. 29

....(degenerate) model for linear logic. Linear logical predicates and totality spaces. The categories LLP of linear logical predicates and Tot of totality spaces were rst considered by Loader (see [43, 44] Loader gave full completeness results for his categories and these were reconsidered by Tan [51]. The objects of LLP are sets A equipped with a pair U , X of collections of subsets of A; that is, U; X P(A) A map from (A; U; X) to (B; V; Y ) is a relation R : A B such that R u 2 V for all u 2 U and R op y 2 X for all y 2 Y: Tot is the full subcategory of LLP on objects of ....

A.M. Tan. Full completeness for models of linear logic. PhD thesis, University of Cambridge, 1997. 47

....and a contra variant part so that negation can be obtained by exchanging the two. Examples for this are Player versus Opponent in games 3 , the pair of sets for Chu spaces or dialectica, and morphisms I A versus those A in the double glueing construction (see [8,7] employed by Tan [11]. Structurally somewhat simpler models, on the other hand, such as coherence spaces or hypercoherences, can do without this kind of built in duality the negation of any object can be de ned just via the structure it carries 4 . Our models de nitely belong to this second category. At rst sight ....

A.M. Tan. Full completeness for models of linear logic. PhD thesis, University of Cambridge, 1997. 27

....begins with semisimple (par of tensors) MLL formulas A, and associates a MIX proof net with every Chu logical transformation j of A. We pull j back along the Lafont Streicher embedding (LS) of coherence spaces [LS91] to yield a dinatural b j, then appeal to full completeness for MLL with MIX [Tan97] to obtain (Section 3.1) Next we show that determines j not only in the LS image but also beyond, by using logical relations to tie down its behaviour at an arbitrary Chu space via the LS image of its coherence space simulation (Section 3.3) To refute MIX we show that, in Chu, ....

....to prove full completeness of MLL, without units but with the MIX rule A Omega B A . B, for coherence spaces [Tan97]. For Chu spaces however dinaturality is not strong enough in that it admits certain spurious transformations corresponding to no MLL proof [Pra97] A further drawback of dinatural transformations is that they do not always compose. In this paper we eliminate all spurious transformations with the ....

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A. Tan. Full completeness for models of linear logic. PhD thesis, King's College, University of Cambridge, October 1997.

....Linear Logic) 2. symmetric monoidal adjunctions (for interpreting the modality ) and 3. autonomous categories (models of Multiplicative Linear Logic) The glueing construction for autonomous categories is a mild generalization of the double glueing construction due to Hyland and Tan [42]. Each of them can be used for creating interesting models of linear logic. For instance, though not central for our development, we demonstrate how phase semantics [21] and its variants can be derived systematically from the glueing techniques (Example 3.6, 3.9, 3.11, 3.18 and 3.23) Then we are ....

....P(jNj) X ; in such cases T gives a sound interpretation of the modality . In [21] M is chosen to be the submonoid fu 2 I 0 j u u = ug of N, with t : M N the inclusion. ut 3. 3 Glueing Autonomous Categories We give a mild generalization of the double glueing construction of Hyland and Tan [42] (see Example 3.24 below) The essential idea is that we double the objects of the glued category so that the duality of the underlying autonomous category scales up to the glued category. While an object in the glued category considered so far is essentially a predicate on an object of the ....

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Tan, A.M. (1997), "Full Completeness for Models of Linear Logic", Ph.D. thesis, University of Cambridge.

....categorical models is likely to apply to many other linear type theories as well. In fact it is routine to modify our technique for non commutative linear logic and monoidal (bi)closed categories (see [17] Furthermore, by combining our approach with Hyland and Tan s double glueing construction [23] (see Example 5) we can deal with a classical linear type theory (MLL) These results, proofs and further category theoretic analysis are reported in the full paper [13] Also it might be fruitful to adapt our method to programming languages, see for example the complexity parameterized logical ....

....linear type theories. It is natural to expect that our construction works equally well in the settings with duality, i.e. classical linear theories. Here is a relevant construction explored by Tan: Example 5 (Double Glueing) An attractive use of categorical glueing is developed in Tan s thesis [23]. Let C be a autonomous category (typically a compact closed category) Because of the duality, C op is also autonomous and we have subscones (Example 1) e C and g C op with projections p 1 : e C C and p 2 : g C op C op . Hyland noticed that the category GC obtained by the ....

Tan, A.M. (1997), \Full Completeness for Models of Linear Logic", Ph.D. thesis, University of Cambridge.

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A. TAN (1997): Full completeness for models of linear logic. PhD thesis, University of Cambridge. A Appendix: Separation Throughout this appendix # is a proof net on a cut sequent

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