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Parametrization over inductive relations of a bounded number of variables

Gregory L. McColm
1990 Annals of Pure and Applied Logic  
We present a Parametrization Theorem for (positive elementary) inductions that use a bounded number of variables.  ...  These results involve the complexity of the inductive relations-and the complexity of the structure or class of structures on which these relations live.  ...  But '8 is not inductive in a bounded number of variables. Towards contradiction, suppose that Uk = UkC1) parametrized all unary %inductive relations. For some K E w, Uk is K-inductive.  ... 
doi:10.1016/0168-0072(90)90043-2 fatcat:2tyfsumnt5b7xpu3x3j4pp6oae

Invariant Generation for Parametrized Systems Using Self-reflection [chapter]

Alejandro Sanchez, Sriram Sankaranarayanan, César Sánchez, Bor-Yuh Evan Chang
2012 Lecture Notes in Computer Science  
Parametrized systems are concurrent systems consisting of an a priori unbounded number of process instances running the same program.  ...  In our approach, the key enabler is the notion of a reflective abstraction that we prove has an important correspondence with inductive invariants.  ...  with a bounded number of resources. .  ... 
doi:10.1007/978-3-642-33125-1_12 fatcat:til3cge7qrcs7mzwnrnwitq3cu

Syntax for Free: Representing Syntax with Binding Using Parametricity [chapter]

Robert Atkey
2009 Lecture Notes in Computer Science  
To demonstrate the proof we have constructed a model of parametric polymorphism inside the Coq proof assistant. The proof of the theorem requires parametricity over Kripke relations.  ...  We show that, in a parametric model of polymorphism, the type ∀α.((α → α) → α) → (α → α → α) → α is isomorphic to closed de Bruijn terms.  ...  The preorder-indexing of the relations is used to handle the expansion of the number of meta-variables being used as object-variables as we go under binders.  ... 
doi:10.1007/978-3-642-02273-9_5 fatcat:qxwscoc3rbhprdcwgk4expkk5y

On bounding space usage of streams using interpretation analysis

Marco Gaboardi, Romain Péchoux
2015 Science of Computer Programming  
The input/output upper bounds criteria consider instead the relations between the number of elements read from the input stream and the number of elements produced on the output stream.  ...  This contribution can be seen as a first step in the development of a methodology aiming at using interpretation properties to ensure space safety properties of programs working on streams.  ...  a number of output elements related to the number and the size of input read elements.  ... 
doi:10.1016/j.scico.2015.05.004 fatcat:4tuszuio5zdkpjij7snfq66nce

A SAT-based algorithm for reparameterization in symbolic simulation

Pankaj Chauhan, Edmund M. Clarke, Daniel Kroening
2004 Proceedings of the 41st annual conference on Design automation - DAC '04  
For large circuits, the reparametrization step often results in a blowup of BDDs and is expensive due to a large number of quantifications of input variables involved.  ...  This paper presents a novel SAT-based reparameterization algorithm that is largely immune to the large number of input variables that need to be quantified.  ...  A set of Boolean vectors over the state variables represents a set of states. Consider a set S of vectors over the variables V = {v 1 , v 2 , . . . , v n }.  ... 
doi:10.1145/996566.996711 dblp:conf/dac/ChauhanCK04 fatcat:pkcmbs2j3fg4lmshzvd5yb54ve

A relational framework for higher-order shape analysis

Gowtham Kaki, Suresh Jagannathan
2014 SIGPLAN notices  
We propose the integration of a relational specification framework within a dependent type system capable of verifying complex invariants over the shapes of algebraic datatypes.  ...  Our approach is based on the observation that structural properties of such datatypes can often be naturally expressed as inductively-defined relations over the recursive structure evident in their definitions  ...  In contrast, a parametric structural relation is a structural relation over a polymorphic data type, that is parameterized by relations over type variables in the data type.  ... 
doi:10.1145/2692915.2628159 fatcat:welztkjmdvewvp7o3sunq6yxlq

A relational framework for higher-order shape analysis

Gowtham Kaki, Suresh Jagannathan
2014 Proceedings of the 19th ACM SIGPLAN international conference on Functional programming - ICFP '14  
We propose the integration of a relational specification framework within a dependent type system capable of verifying complex invariants over the shapes of algebraic datatypes.  ...  Our approach is based on the observation that structural properties of such datatypes can often be naturally expressed as inductively-defined relations over the recursive structure evident in their definitions  ...  In contrast, a parametric structural relation is a structural relation over a polymorphic data type, that is parameterized by relations over type variables in the data type.  ... 
doi:10.1145/2628136.2628159 dblp:conf/icfp/KakiJ14 fatcat:ntcadei36vhzrptsywef6hwqmi

Global and Local Space Properties of Stream Programs [chapter]

Marco Gaboardi, Romain Péchoux
2010 Lecture Notes in Computer Science  
We study global and local upper bounds properties of both theoretical and practical interests guaranteeing that the size of each output stream element is bounded by a function in the maximal size of the  ...  In this paper, we study semantic interpretation criteria in order to ensure safety and complexity properties of first order Haskell like programs on streams.  ...  The parametrized assignment of a variable x is defined by a fresh variable X ranking over R + .  ... 
doi:10.1007/978-3-642-15331-0_4 fatcat:ynhlrebaajdu5hbs5onhzhsrha

Revisiting Reachability in Timed Automata [article]

Karin Quaas, Mahsa Shirmohammadi, James Worrell
2017 arXiv   pre-print
We revisit a fundamental result in real-time verification, namely that the binary reachability relation between configurations of a given timed automaton is definable in linear arithmetic over the integers  ...  In this paper we give a new and simpler proof of this result, building on the well-known reachability analysis of timed automata involving difference bound matrices.  ...  Given a parameter valuation ξ, we define a satisfaction relation |ù ξ between configurations of A and PTRL formulas by induction over the structure of formulas.  ... 
arXiv:1702.03450v2 fatcat:jfqgafejabebzkz53uqueq5e4a

The Girard–Reynolds isomorphism

Philip Wadler
2003 Information and Computation  
It also extends to a correspondence between second-order intuitionistic logic with quantifiers over proposition variables and second-order typed lambda calculus with quantifiers over type variables, which  ...  However, we show that in the presence of Reynolds's parametricity property that this is indeed the case, for propositions corresponding to inductive definitions of naturals or other algebraic types.  ...  Acknowledgments My thanks to Andrew Pitts, Jon Riecke, and referees of POPL 1999, LICS 2000, TACS 2001, and Information and Computation; with particular thanks to Andrew Pitts and some of the referees  ... 
doi:10.1016/s0890-5401(03)00141-x fatcat:aubcjde6avgolmbve7kbkr465y

Page 6391 of Mathematical Reviews Vol. , Issue 91M [page]

1991 Mathematical Reviews  
Bulitko (Odessa) 91m:03048 03D70 03D75 McColm, Gregory L. (1-SFL) Parametrization over inductive relations of a bounded number of variables. Ann. Pure Appl. Logic 48 (1990), no. 2, 103-134.  ...  A key complexity measure is the property of being k-inductive, i.e. inductive via a definition using at most k variables, free and bound.  ... 

The Girard–Reynolds isomorphism (second edition)

Philip Wadler
2007 Theoretical Computer Science  
Reynolds proved an Abstraction Theorem: every term in F2 satisfies a suitable notion of logical relation; and formulated a notion of parametricity satisfied by well-behaved models.  ...  Reynolds additionally proved an Abstraction Theorem: every term in F2 satisfies a suitable notion of logical relation; and formulated a notion of parametricity satisfied by well-behaved models.  ...  Acknowledgements My thanks to Rasmus Møgelberg, Martin Odersky, Andrew Pitts, Gordon Plotkin, Jon Riecke, Colin Runciman, Clemens Szyperski, René Vestergaard, to referees of POPL 1999, LICS 2000, TACS  ... 
doi:10.1016/j.tcs.2006.12.042 fatcat:qp3znp5lpraitaygmw3ay763ye

Variant parametric types

Atsushi Igarashi, Mirko Viroli
2006 ACM Transactions on Programming Languages and Systems  
For instance, a method that only reads the elements of a container of numbers can be easily modified so as to accept containers of integers, floating point numbers, or any subtype of the number type.  ...  By exploiting variant parametric types, a programmer can write generic code abstractions working on a wide range of parametric types in a safe way.  ...  ACKNOWLEDGMENTS We thank anonymous reviewers for useful comments, Gilad Bracha and Benjamin Pierce for helping us clarify about related work, and David Griswold and people involved in the development of  ... 
doi:10.1145/1152649.1152650 fatcat:lttuadsjlvgkbfh3gz3zmwigea

Parameter Synthesis Algorithms for Parametric Interval Markov Chains [chapter]

Laure Petrucci, Jaco van de Pol
2018 Lecture Notes in Computer Science  
In particular, we introduce a co-inductive definition of consistency, which improves and simplifies previous inductive definitions considerably.  ...  We provide a sharper upper bound, based on the longest simple path in the IMC.  ...  They acknowledge the support of the Van Gogh project PAMPAS, that partly covered their mutual research visits.  ... 
doi:10.1007/978-3-319-92612-4_7 fatcat:2k754m6janhgrdcjbowld7kcci

SPEED: Symbolic Complexity Bound Analysis [chapter]

Sumit Gulwani
2009 Lecture Notes in Computer Science  
The SPEED project addresses the problem of computing symbolic computational complexity bounds of procedures in terms of their inputs.  ...  We discuss some of the challenges that arise and present various orthogonal/complementary techniques recently developed in the SPEED project for addressing these challenges.  ...  We thank Varun Aggarwala and Aditya Nori for providing useful feedback on a draft of this paper.  ... 
doi:10.1007/978-3-642-02658-4_7 fatcat:h5hf2uaohzaf7oonfqlkcfjh7y
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