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On the Expressive Power of First-Order Boolean Functions in PCF
[article]

2004
*
arXiv
*
pre-print

Recent results

arXiv:cs/0405085v1
fatcat:b64r5l2ryzezvjx4d3cqaaekh4
*of*Bucciarelli show that*the*semilattice*of*degrees*of*parallelism*of**first*-*order**boolean**functions**in**PCF*has both infinite chains and infinite antichains. ... By considering a simple subclass*of*Sieber's sequentiality relations, we identify levels*in**the*semilattice and derive inexpressibility results concerning*functions**on*different levels. ... Thanks to*the*anonymous referees for suggestions that helped improve and tighten*the*presentation, and for various technical corrections. ...##
###
On the expressive power of first-order boolean functions in PCF

2001
*
Theoretical Computer Science
*

Recent results

doi:10.1016/s0304-3975(00)00282-6
fatcat:u42ujfrxa5crvlluabtuzuqu7q
*of*Bucciarelli show that*the*semilattice*of*degrees*of*parallelism*of*ÿrst-*order**boolean**functions**in**PCF*has both inÿnite chains and inÿnite antichains. ... By considering a simple subclass*of*Sieber's sequentiality relations, we identify levels*in**the*semilattice and derive inexpressibility results concerning*functions**on*di erent levels. ... Acknowledgements Thanks to*the*anonymous referees for suggestions that helped improve and tighten*the*presentation, and for various technical corrections. ...##
###
Totality, definability and boolean circuits
[chapter]

1998
*
Lecture Notes in Computer Science
*

*In*

*the*type frame originating from

*the*flat domain

*of*

*boolean*values, we single out elements which are hereditarily total. ...

*The*elements

*of*an equivalence class

*of*

*the*totality equivalence relation (totality class) can be seen as different algorithms for computing a given set-theoretic

*boolean*

*function*. ... ~r 9 f(x) = { k(~(a)) if 3a. x

*First*

*Order*

*Boolean*

*Functions*

*In*this section we turn to

*first*

*order*set-theoretic

*functions*, i.e.

*functions*

*of*type o --+ o --+ ... ~ o --+ o for some n 6. ...

##
###
Length and Depth
[article]

2018
*
arXiv
*
pre-print

We construct,

arXiv:1804.05304v1
fatcat:wclijhqa4fcxrdkkrfa7cu77pe
*in*ZFC, a sequence*of**Boolean*algebras for which*the*product*of*Lengths is strictly smaller than*the*Depth*of**the*product algebra. ... This gives a positive answer to Problem 30 from Monk's book "Cardinal invariants*on**Boolean*algebras". ...*In**the**first*way*one*computes I(B i ) for every i < κ, yielding a sequence*of*cardinals. Then*one*computes*the*product i<κ I(B i )/D. ...##
###
Page 6048 of Mathematical Reviews Vol. , Issue 2002H
[page]

2002
*
Mathematical Reviews
*

Norman Danner (1-UCLA; Los Angeles, CA)
2002h:68031 68N18 03B40
Pucella, Riccardo (1-CRNL-C; Ithaca, NY);
Panangaden, Prakash (3-MGL-C; Montreal, QC)

*On**the**expressive**power**of**first*-*order**Boolean**functions*... Summary: “Recent results*of*Bucciarelli show that*the*semilattice*of*degrees*of*parallelism*of**first*-*order**Boolean**functions**in**PCF*has both infinite chains and infinite antichains. ...##
###
Operational and axiomatic semantics of PCF

1990
*
Proceedings of the 1990 ACM conference on LISP and functional programming - LFP '90
*

*The*system is also computationally adequate,

*in*

*the*sense that whenever a closed term

*of*"observable" type has a

*pcf*η,sp normal form, this is also

*the*unique

*pcf*normal form. ...

*PCF*, as considered

*in*this paper, is a lazy typed lambda calculus with

*functions*, pairing, fixed-point operators and arbitrary algebraic data types. ...

*of*natural numbers and

*booleans*, as

*in*[Plo77, Sco69] , then it is easy to program any partial recursive

*function*

*on*

*the*natural numbers. ...

##
###
A Calculus of Coroutines
[chapter]

2004
*
Lecture Notes in Computer Science
*

extensional

doi:10.1007/978-3-540-27836-8_74
fatcat:hrcyftg22bcc3mw52qs3vbyr5y
*in**the*presence*of**of*errors;*in*particular adding errors to*PCF*allows*the**order**of*evaluation*of**functions*to be observed. ... Games can capture precisely*the*behaviour*of*higher-*order*programs*in*sequential languages,*in*a fashion which is very difficult or impossible for traditional forms*of*denotational semantics. ... extensional*in**the*presence*of**of*errors;*in*particular adding errors to*PCF*allows*the**order**of*evaluation*of**functions*to be observed. ...##
###
Page 4083 of Mathematical Reviews Vol. , Issue 2004e
[page]

2004
*
Mathematical Reviews
*

*First*,

*the*case

*in*which

*the*elements

*of*quasi-

*Boolean*algebras are totally

*ordered*is considered. ...

*The*

*expressive*

*power*

*of*

*the*branching-time temporal logic CTL* has been long studied.

*One*

*of*

*the*early papers (not referred to

*in*

*the*paper under review) was by M. C. Browne, E. M. Clarke, Jr. and O. ...

##
###
Full Abstraction and the Context Lemma

1996
*
SIAM journal on computing (Print)
*

A simple argument demonstrates that any denotational semantics that is adequate for

doi:10.1137/s0097539791224212
fatcat:dbtdasi7wrgk5ht7lj43qwu764
*PCF*, and*in*which certain simple*Boolean**functionals*exist, cannot be fully abstract for any extension*of**PCF*satisfying ...*In*particular, we de ne a general notion*of*rewriting system*of**the*kind used for evaluating simply typed -terms*in*Scott's*PCF*. ... For example, with*the*semantics based*on*cpo's,*PCF*must be extended with a parallel-or" combinator*in**order*to*express*enough computations to be fully abstract, i.e., semantical distinctions and computational ...##
###
The Elimination of Nesting in SPCF
[chapter]

2005
*
Lecture Notes in Computer Science
*

Our proof is based

doi:10.1007/11417170_18
fatcat:3olhgllumngzrksnyqklog5lvi
*on**the*observation*of*Longley -already used to prove universality and full abstraction results for models*of*SPCFthat every type*of*SPCF is a retract*of*a*first*-*order*type. ...*In**the*case*of*finitary SPCF,*the*retraction is based*on*a simple induction, which yields bounds for*the*size*of**the*resulting term. ... Introduction*One**of**the*important sources*of**expressive**power**in**functional*languages is*the*nesting*of**function*calls, both explicitly, and*in*recursive definitions. ...##
###
Program Logics for Sequential Higher-Order Control
[chapter]

2010
*
Lecture Notes in Computer Science
*

*In*contrast,

*in*our approach programs or program parts that do not feature higher-

*order*control can be reasoned about

*in*simpler logics for conventional

*functional*languages. ... This enables

*the*assertion language to capture precisely

*the*intensional and extensional effects

*of*jumping by internalising rely/guarantee reasoning, leading to simple proof rules for higher-

*order*

*functions*...

*expressive*

*power*. ...

##
###
Full abstraction, totality and PCF

1999
*
Mathematical Structures in Computer Science
*

Some apparently rather difficult open problems arise, essentially concerning whether

doi:10.1017/s0960129598002692
fatcat:kg4f66ts25budmwdb6hqsgr57q
*the*sequential and parallel versions*of**PCF*have*the*same*expressive**power*,*in**the*sense*of*total equivalence. ... Inspired by a question*of*Riecke, we consider*the*interaction*of*totality and full abstraction, asking whether full abstraction holds for Scott's model*of*cpos and continuous*functions*if*one*restricts ... This research was supported by*the*UK EPSRC. ...##
###
Can LCF be topped? Flat lattice models of typed λ-calculus

1990
*
Information and Computation
*

*In*

*the*

*first*case l+M-+l+T-+l+(n-l)-+n and otherwise l+M-++ l+(n-1)-n.

*The*rest

*of*

*the*

*first*-

*order*constants are polite by a similar case analysis. Y is nontrivial. ... Most reduction rules proposed for LCF

*in*

*the*literature are simple; e.g., all

*first*-

*order*

*functions*(taking only arguments

*of*ground type) are definable by simple rules. ...

*The*

*first*reduction sequence is justified by operand evaluation rules;

*the*next part by rule p; and

*the*last by Eq. 9. If M= (Ax. RS . ...

##
###
Denotational Semantics with Nominal Scott Domains

2014
*
Journal of the ACM
*

This is

doi:10.1145/2629529
fatcat:yhpbebcghbg7lamjef27y7z7bu
*the**first*full abstraction result we know*of*for languages combining higher-*order**functions*with some form*of*locally scoped names which uses a domain theory based*on*ordinary extensional*functions*... for binders can conveniently*express**the*wide range*of**first*-*order*syntax-manipulating algorithms commonly employed*in*informal practice; see Savary-Belanger et al. [2013], for example. ... It is a*boolean*-valued operation that evaluates two*boolean**expressions*"*in*parallel" and is true whenever*one**of**the**expressions*evaluates to true, no matter if*the*other*one*diverges. ...##
###
Singular Cardinals
[chapter]

2012
*
Handbook of the History of Logic
*

*In*

*the*very

*first*section

*of*

*the*chapter Shelah proves a

*pcf*generalization

*of*

*the*Galvin-Hajnal theorem for his pp

*function*. ...

*In*[1983] Shelah had already proved that for

*the*

*first*fixed point

*of*

*order*ω 1

*of*

*the*aleph

*function*

*of*cofinality ℵ 1 27 this was

*the*best upper bound. ... ., 15 almost free group, 23 Axiom

*of*Choice, 6 ...

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