Results 1  10
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109
Church: A language for generative models
 In UAI
, 2008
"... Formal languages for probabilistic modeling enable reuse, modularity, and descriptive clarity, and can foster generic inference techniques. We introduce Church, a universal language for describing stochastic generative processes. Church is based on the Lisp model of lambda calculus, containing a pu ..."
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Cited by 58 (11 self)
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Formal languages for probabilistic modeling enable reuse, modularity, and descriptive clarity, and can foster generic inference techniques. We introduce Church, a universal language for describing stochastic generative processes. Church is based on the Lisp model of lambda calculus, containing a pure Lisp as its deterministic subset. The semantics of Church is defined in terms of evaluation histories and conditional distributions on such histories. Church also includes a novel language construct, the stochastic memoizer, which enables simple description of many complex nonparametric models. We illustrate language features through several examples, including: a generalized Bayes net in which parameters cluster over trials, infinite PCFGs, planning by inference, and various nonparametric clustering models. Finally, we show how to implement query on any Church program, exactly and approximately, using Monte Carlo techniques. 1
On The Algebraic Models Of Lambda Calculus
 Theoretical Computer Science
, 1997
"... . The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way Boolean algebras algebraize the classical propositional calculus. The equational theory of lambda abstraction algebras is intended as an alternative to combinatory ..."
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Cited by 20 (11 self)
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. The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way Boolean algebras algebraize the classical propositional calculus. The equational theory of lambda abstraction algebras is intended as an alternative to combinatory logic in this regard since it is a firstorder algebraic description of lambda calculus, which allows to keep the lambda notation and hence all the functional intuitions. In this paper we show that the lattice of the subvarieties of lambda abstraction algebras is isomorphic to the lattice of lambda theories of the lambda calculus; for every variety of lambda abstraction algebras there exists exactly one lambda theory whose term algebra generates the variety. For example, the variety generated by the term algebra of the minimal lambda theory is the variety of all lambda abstraction algebras. This result is applied to obtain a generalization of the genericity lemma of finitary lambda calculus...
Finite Family Developments
"... Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than o ..."
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Cited by 13 (6 self)
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Associate to a rewrite system R having rules l → r, its labelled version R ω having rules l ◦ m+1 → r • , for any natural number m m ∈ ω. These rules roughly express that a lefthand side l carrying labels all larger than m can be replaced by its righthand side r carrying labels all smaller than or equal to m. A rewrite system R enjoys finite family developments (FFD) if R ω is terminating. We show that the class of higher order pattern rewrite systems enjoys FFD, extending earlier results for the lambda calculus and first order term rewrite systems.
Nonmodularity Results for Lambda Calculus
 Fundamenta Informaticae
, 2001
"... The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the firstorder predicate logic. In this paper we prove that the lattice of lambda theories is not modular and that the va ..."
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Cited by 10 (7 self)
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The variety (equational class) of lambda abstraction algebras was introduced to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the firstorder predicate logic. In this paper we prove that the lattice of lambda theories is not modular and that the variety generated by the term algebra of a semisensible lambda theory is not congruence modular. Another result of the paper is that the Mal'cev condition for congruence modularity is inconsistent with the lambda theory generated by equating all the unsolvable lambdaterms.
An Analysis of the Impact of Functional Programming Techniques on Genetic Programming
, 1999
"... Genetic Programming (GP) automatically generates computer programs to solve specified problems. It develops programs through the process of a “createtestmodify ” cycle which is similar to the way a human writes programs. There are various functional programming techniques that human programmers c ..."
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Cited by 9 (0 self)
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Genetic Programming (GP) automatically generates computer programs to solve specified problems. It develops programs through the process of a “createtestmodify ” cycle which is similar to the way a human writes programs. There are various functional programming techniques that human programmers can use to accelerate the program development process. This research investigated the applicability of some of the functional techniques to GP and analyzed their impact on GP performance. Among many important functional techniques, three were chosen to be included in this research, due to their relevance to GP. They are polymorphism, implicit recursion and higherorder functions. To demonstrate their applicability, a GP system was developed with those techniques incorporated. Furthermore, a number of experiments were conducted using the system. The results were then compared to those generated by other GP systems which do not support these functional features. Finally, the program search space of the general evenparity problem was analyzed to explain how these techniques impact GP performance. The experimental results showed that the investigated functional techniques have made GP more powerful in the following ways: 1) polymorphism has enabled GP to solve problems that are very difficult for standard GP to solve, i.e. nth and map programs; 2) higherorder functions and implicit recursion have enhanced GP’s ability in solving the general evenparity problem to a greater degree than with any other known methods. Moreover, the analysis showed that these techniques directed GP to generate program solutions in a way that has never been previously reported. Finally, we provide the guidelines for the application of these techniques to other problems.
A Complete Transformation System for Polymorphic HigherOrder Unification
, 1991
"... Polymorphic higherorder unification is a method for unifying terms in the polymorphically typed calculus, that is, given a set of pairs of terms S 0 = fs 1 ? = t 2 ; : : : ; s n ? = t n g, called a unification problem, finding a substitution oe such that oe(s i ) and oe(t i ) are equivalent u ..."
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Cited by 6 (0 self)
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Polymorphic higherorder unification is a method for unifying terms in the polymorphically typed calculus, that is, given a set of pairs of terms S 0 = fs 1 ? = t 2 ; : : : ; s n ? = t n g, called a unification problem, finding a substitution oe such that oe(s i ) and oe(t i ) are equivalent under the conversion rules of the calculus for all i, 1 i n. I present the method as a transformation system, i.e. as a set of schematic rules U =) U 0 such that any unification problem ffi (U ) can be transformed into ffi (U 0 ) where ffi is an instantiation of the metalevel variables in U and U 0 . By successive use of transformation rules one possibly obtains a solved unification problem with obvious unifier. I show that the transformation system is correct and complete, i.e. if ffi (U ) =) ffi (U 0 ) is an instance of a transformation rule, then the set of all unifiers of ffi (U 0 ) is a subset of the set of all unifiers of ffi (U ) and if U is the set of all unification ...
What is a Categorical Model of the Differential and the Resource λCalculi?
"... The differential λcalculus is a paradigmatic functional programming language endowed with a syntactical differentiation operator that allows to apply a program to an argument in a linear way. One of the main features of this language is that it is resource conscious and gives the programmer suitab ..."
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Cited by 5 (1 self)
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The differential λcalculus is a paradigmatic functional programming language endowed with a syntactical differentiation operator that allows to apply a program to an argument in a linear way. One of the main features of this language is that it is resource conscious and gives the programmer suitable primitives to handle explicitly the resources used by a program during its execution. The differential operator also allows to write the full Taylor expansion of a program. Through this expansion every program can be decomposed into an infinite sum (representing nondeterministic choice) of ‘simpler’ programs that are strictly linear. The aim of this paper is to develop an abstract ‘model theory ’ for the untyped differential λcalculus. In particular, we investigate what should be a general categorical definition of denotational model for this calculus. Starting from the work of Blute, Cockett and Seely on differential categories we provide the notion of Cartesian closed differential category and we prove that linear reflexive objects living in such categories constitute sound models of the untyped differential λcalculus. We also give sufficient conditions for Cartesian closed differential categories to model the Taylor expansion. This entails that every model living in such categories equates all programs having the same full Taylor expansion. We then
CurryHoward Term Calculi for GentzenStyle Classical Logic
, 2008
"... This thesis is concerned with the extension of the CurryHoward Correspondence to classical logic. Although much progress has been made in this area since the seminal paper by Griffin, we believe that the question of finding canonical calculi corresponding to classical logics has not yet been resolv ..."
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Cited by 5 (1 self)
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This thesis is concerned with the extension of the CurryHoward Correspondence to classical logic. Although much progress has been made in this area since the seminal paper by Griffin, we believe that the question of finding canonical calculi corresponding to classical logics has not yet been resolved. We examine computational interpretations of classical logics which we keep as close as possible to Gentzen’s original systems, equipped with general notions of reduction. We present a calculus X i which is based on classical sequent calculus and the stronglynormalising cutelimination procedure defined by Christian Urban. We examine how the notion of shallow polymorphism can be adapted to the moregeneral setting of this calculus. We show that the intuitive adaptation of these ideas fails to be sound, and give a novel solution. In the setting of classical natural deduction, we examine the lambdamu calculus of Parigot. We show that the underlying logic is incomplete in various ways, compared with a standard Gentzenstyle presentation of classical natural deduction. We relax the identified