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### Strong Completeness and Limited Canonicity for PDL

Gerard Renardel de Lavalette, Barteld Kooi, Rineke Verbrugge
2007 Journal of Logic, Language and Information
Lemma 8 For every ϕ ∈ L 0 and for all programs α ∈ L 0 : ω | ϕ iff [[ϕ]] is cofinite (5) (ω, y) ∈ R c (α) iff {x | (x, y) ∈ [[α]]} is cofinite (6) (ω, ω) ∈ R c (α) iff {x | (x, x) ∈ [[α]]} is cofinite  ...  Lemma 5 For every ϕ: ω | ϕ iff [[ϕ]] is cofinite (4) where [[ϕ]] = {x ∈ N | x | ϕ}. Proof First we show that, for every formula ϕ, [[ϕ] ] is finite or cofinite.  ...

### Complexity of Operations on Cofinite Languages [chapter]

Frédérique Bassino, Laura Giambruno, Cyril Nicaud
2010 Lecture Notes in Computer Science
We study the worst case complexity of regular operations on cofinite languages (i.e., languages whose complement is finite) and provide algorithms to compute efficiently the resulting minimal automata.  ...  We propose two algorithms, one based on usual automata constructions and another one, easier to implement, related to dynamic programming.  ...  We denote automata by tuples (A, Q, T, I, F ) where Q is a finite set of states, A is a finite set of letters called alphabet, the transition relation T is a subset of Q × A × Q, I ⊂ Q is the set of initial  ...

### Page 2667 of Mathematical Reviews Vol. , Issue 2001D [page]

2001 Mathematical Reviews
We characterise a class of programs (via their input-output relations) and a class of predicates, such that each predicate is either finite or cofinite, and programs, when viewed as predicate transformers  ...  of loops with stuttering transitions.” 2001d:68147 68Q60 03B70 03G05 Brink, Chris (SA-CAPE-MAM; Rondebosch) ; Rewitzky, Ingrid (SA-CAPE-MAM; Rondebosch) Finite-cofinite program relations.  ...

### Engineering formal metatheory

Brian Aydemir, Arthur Charguéraud, Benjamin C. Pierce, Randy Pollack, Stephanie Weirich
2008 Proceedings of the 35th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '08
induction principles of relations defined using cofinite quantification are strong enough for metatheoretic reasoning, and need not be explicitly strengthened.  ...  We propose a novel style for formalizing metatheory, combining locally nameless representation of terms and cofinite quantification of free variable names in inductive definitions of relations on terms  ...  We say that this definition of the typing relation uses "cofinite quantification" since we must show that the premise of C-TYPING-ABS ("C" for "cofinite") holds for all but finitely many names.  ...

### Engineering formal metatheory

Brian Aydemir, Arthur Charguéraud, Benjamin C. Pierce, Randy Pollack, Stephanie Weirich
2008 SIGPLAN notices
induction principles of relations defined using cofinite quantification are strong enough for metatheoretic reasoning, and need not be explicitly strengthened.  ...  We propose a novel style for formalizing metatheory, combining locally nameless representation of terms and cofinite quantification of free variable names in inductive definitions of relations on terms  ...  We say that this definition of the typing relation uses "cofinite quantification" since we must show that the premise of C-TYPING-ABS ("C" for "cofinite") holds for all but finitely many names.  ...

### Locally Nameless Permutation Types [article]

Edsko de Vries, Vasileios Koutavas
2017 arXiv   pre-print
We show that this combination is particularly useful when formalizing programming languages where bound names may become free during execution ("extrusion"), common in process calculi.  ...  We explain how to use cofinite quantification in this setting, show why reasoning about renaming is more important here than in languages without extrusion, and provide results about infinite support,  ...  Related Work and Conclusions The treatment of names is one of the first stumbling blocks towards formalizing programming language theory.  ...

### RECOGNIZING STRONG RANDOM REALS

DANIEL OSHERSON, SCOTT WEINSTEIN
2008 The Review of Symbolic Logic
For any input i, P enters a nonterminating routine that writes a nonempty, finite sequence b 1 , . . . , b m with b m = x i (m depends on i). 1 The program will not, in general, allow prediction of x  ...  Let real x be given, and suppose that (x[n]) = Yes for only finitely many n. Then, there are (cofinitely many) k with x ∈ [W f (k) ]. Suppose alternatively that (x[n]) = Yes for infinitely many n.  ...

### On commensurable hyperbolic Coxeter groups

Rafael Guglielmetti, Matthieu Jacquemet, Ruth Kellerhals
2016 Geometriae Dedicata
For Coxeter groups acting non-cocompactly but with finite covolume on real hyperbolic space H n , new methods are presented to distinguish them up to (wide) commensurability.  ...  This commensurability relation is an equivalence relation which preserves properties such as cocompactness, cofiniteness and arithmeticity.  ...  The commensurability relation is an equivalence relation which is stable when passing to a finite index subgroup.  ...

### Representations of vertex operator algebras and braided finite tensor categories [article]

Yi-Zhi Huang
2009 arXiv   pre-print
We discuss what has been achieved in the past twenty years on the construction and study of a braided finite tensor category structure on a suitable module category for a suitable vertex operator algebra  ...  III V is C 2 -cofinite.  ...  The category of finite-dimensional modules for a finite group is a finite tensor category.  ...

### Page 1187 of Mathematical Reviews Vol. 44, Issue 6 [page]

1972 Mathematical Reviews
On large cardinals and partition relations. J. Symbolic Logic 36 (1971), 305-308.  ...  (i) For all finite subsets P of I*, for all finite subsets N of M, for all cofinite subsets Q’ of J, |A(P) A BQ’)| +|{A(P) UV BQ’)} 0 Nl 2|P|+|N| — Ducrans sr Oe — ier min{|J,|—¢, [{J\Q}OJ,|}, and (ii)  ...

### Abstract representation theorems for demonic refinement algebras

Jean-Lou De Carufel, Jules Desharnais
2010 The Journal of Logic and Algebraic Programming
Acknowledgements We thank Georg Struth and the anonymous referees of this special issue of The Journal of Logic and Algebraic Programming as well as those of the 10th International Conference on Relational  ...  Since the finite product of finite and/or cofinite subidentities is finite or cofinite (just like the finite intersection of finite and/or cofinite sets), it suffices to prove (;j | 0 ≤ j < n i : Gt i,  ...  For instance, take R to be the set of finite or cofinite subsets of the identity relation on the natural numbers.  ...

### The gamma construction and asymptotic invariants of line bundles over arbitrary fields [article]

Takumi Murayama
2019 arXiv   pre-print
To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is F-finite.  ...  Application to the minimal model program over imperfect fields. Let {X i } be a finite set of schemes essentially of finite type over A.  ...  Now let Γ ⊆ Λ be a cofinite subset, i.e., a subset Γ of Λ such that Λ Γ is a finite set.  ...

### Tensor Categories arising from the Virasoro Algebra [article]

Thomas Creutzig, Cuipo Jiang, Florencia Orosz Hunziker, David Ridout, Jinwei Yang
2021 arXiv   pre-print
We show that there is a braided tensor category structure on the category of C_1-cofinite modules for the (universal or simple) Virasoro vertex operator algebras of arbitrary central charge.  ...  For example, the quantum geometric Langlands program can be related to equivalences of tensor categories of modules of W -algebras and affine vertex algebras, see [AFO, Sec. 6] for example.  ...  if W is. (2) If W is a finite-length V -module with C 1 -cofinite composition factors, then W is also C 1 - cofinite. 2.2.  ...

### On the closure of the diagonal of a T1-space

Maria Luisa Colasante, Dominic van der Zypen
2008 The Journal of Logic and Algebraic Programming
The closure of = {(x,x) : x ∈ X} in X × X is a symmetric relation on X.  ...  We characterise those equivalence relations on an infinite set that arise as the closure of the diagonal with respect to a T 1 -topology.  ...  On an infinite set, an equivalence relation with finitely many blocks is T 1 -realisable if and only if every finite block is a singleton.  ...

### Mechanizing Metatheory Without Typing Contexts

Jonghyun Park, Jeongbong Seo, Sungwoo Park, Gyesik Lee
2013 Journal of automated reasoning
When mechanizing the metatheory of a programming language, one usually needs many lemmas proving structural properties of typing judgments, such as permutation and weakening.  ...  This technique of eliminating typing contexts, which has been around since Church [4] , is based on the view that entailment relations, such as typing judgments, are just syntactic tools for displaying  ...  Related work Previous research on mechanizing the metatheory of programming languages usually focuses on the problem of representing binders to deal with α-equivalence relations and capture-avoiding substitutions  ...
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