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

Riccardo Pucella, Prakash Panangaden
2004 arXiv   pre-print
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.  ...  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.  ... 
arXiv:cs/0405085v1 fatcat:b64r5l2ryzezvjx4d3cqaaekh4

On the expressive power of first-order boolean functions in PCF

Riccardo Pucella, Prakash Panangaden
2001 Theoretical Computer Science  
Recent results 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.  ... 
doi:10.1016/s0304-3975(00)00282-6 fatcat:u42ujfrxa5crvlluabtuzuqu7q

Totality, definability and boolean circuits [chapter]

Antonio Bucciarelli, Ivano Salvo
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.  ... 
doi:10.1007/bfb0055104 fatcat:dunvdphxfbfg3krdftdhb7ocoy

Length and Depth [article]

Shimon Garti, Saharon Shelah
2018 arXiv   pre-print
We construct, 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.  ... 
arXiv:1804.05304v1 fatcat:wclijhqa4fcxrdkkrfa7cu77pe

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

Brian T. Howard, John C. Mitchell
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.  ... 
doi:10.1145/91556.91677 dblp:conf/lfp/HowardM90 fatcat:vfqoypttzzhynj2uapkngnsr3i

A Calculus of Coroutines [chapter]

J. Laird
2004 Lecture Notes in Computer Science  
extensional 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.  ... 
doi:10.1007/978-3-540-27836-8_74 fatcat:hrcyftg22bcc3mw52qs3vbyr5y

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

Trevor Jim, Albert R. Meyer
1996 SIAM journal on computing (Print)  
A simple argument demonstrates that any denotational semantics that is adequate for 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  ... 
doi:10.1137/s0097539791224212 fatcat:dbtdasi7wrgk5ht7lj43qwu764

The Elimination of Nesting in SPCF [chapter]

J. Laird
2005 Lecture Notes in Computer Science  
Our proof is based 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.  ... 
doi:10.1007/11417170_18 fatcat:3olhgllumngzrksnyqklog5lvi

Program Logics for Sequential Higher-Order Control [chapter]

Martin Berger
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.  ... 
doi:10.1007/978-3-642-11623-0_11 fatcat:rpsyqznddjbblggxv4gsbctc3a

Full abstraction, totality and PCF

GORDON PLOTKIN
1999 Mathematical Structures in Computer Science  
Some apparently rather difficult open problems arise, essentially concerning whether 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.  ... 
doi:10.1017/s0960129598002692 fatcat:kg4f66ts25budmwdb6hqsgr57q

Can LCF be topped? Flat lattice models of typed λ-calculus

Bard Bloom
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 .  ... 
doi:10.1016/0890-5401(90)90064-o fatcat:3a3gwy6rzvhpnjygxnbklqknmq

Denotational Semantics with Nominal Scott Domains

Steffen Lösch, Andrew M. Pitts
2014 Journal of the ACM  
This is 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.  ... 
doi:10.1145/2629529 fatcat:yhpbebcghbg7lamjef27y7z7bu

Singular Cardinals [chapter]

Menachem Kojman
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  ... 
doi:10.1016/b978-0-444-51621-3.50007-4 fatcat:fr3pdoy3nbdn5pc4pz52reng24
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