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70
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 54 (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.
Callbyvalue is dual to callbyname, reloaded
 In Rewriting Technics and Application, RTA’05, volume 3467 of LNCS
, 2005
"... Abstract. We consider the relation of the dual calculus of Wadler (2003) to the λµcalculus of Parigot (1992). We give translations from the λµcalculus into the dual calculus and back again. The translations form an equational correspondence as defined by Sabry and Felleisen (1993). In particular, ..."
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Cited by 11 (0 self)
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Abstract. We consider the relation of the dual calculus of Wadler (2003) to the λµcalculus of Parigot (1992). We give translations from the λµcalculus into the dual calculus and back again. The translations form an equational correspondence as defined by Sabry and Felleisen (1993). In particular, translating from λµ to dual and then ‘reloading ’ from dual back into λµ yields a term equal to the original term. Composing the translations with duality on the dual calculus yields an involutive notion of duality on the λµcalculus. A previous notion of duality on the λµcalculus has been suggested by Selinger (2001), but it is not involutive. Note This paper uses color to clarify the relation of types and terms, and of source and target calculi. If the URL below is not in blue please download the color version from
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.
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 ...
Gödel on computability
"... Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of t ..."
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Cited by 5 (0 self)
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Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of the second theorem was explicitly based on Turing’s 1936 reduction of finite procedures to machine computations. Turing gave in his 1954 an understated analysis of finite procedures in terms of Post production systems. This analysis, prima facie quite different from that given in 1936, served as the basis for an exposition of various unsolvable problems. Turing had addressed issues of mentality and intelligence in contemporaneous essays, the best known of which is of course Computing machinery and intelligence. Gödel’s and Turing’s considerations from this period intersect through their attempt, on the one hand, to analyze finite, mechanical procedures and, on the other hand, to approach mental phenomena in a scientific way. Neuroscience or brain science was an important component of the latter for both: Gödel’s remarks in the Gibbs Lecture as well as in his later conversations with Wang and Turing’s Intelligent Machinery can serve as clear evidence for that. 2 Both men were convinced that some mental processes are not mechanical, in the sense that Turing machines cannot mimic them. For Gödel, such processes were to be found in mathematical experience and he was led to the conclusion that mind is separate from matter. Turing simply noted that for a machine or a brain it is not enough to be converted into a universal (Turing) machine in order to become intelligent: “discipline”, the characteristic
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
Canonical typing and Πconversion in the Barendregt Cube
, 1996
"... In this article, we extend the Barendregt Cube with \Piconversion (which is the analogue of betaconversion, on product type level) and study its properties. We use this extension to separate the problem of whether a term is typable from the problem of what is the type of a term. ..."
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Cited by 4 (3 self)
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In this article, we extend the Barendregt Cube with \Piconversion (which is the analogue of betaconversion, on product type level) and study its properties. We use this extension to separate the problem of whether a term is typable from the problem of what is the type of a term.