It is our pleasure to announce that the ninth workshop Computational Logic and Applications (CLA 2016) will be held on 2627 May 2016 in Novi Sad, Serbia, as a satellite event of 22nd International Conference on Types for Proofs and Programs TYPES 2016. The workshop gathers researchers interested in quantitative aspects of mathematical logic and functional programming, including, but not limited, to topics such as:
It is our pleasure to announce that
the programme is now
available!
The registration for CLA 2016 is now open! Participation is free of charge!
Important deadlines:
In case of technical difficulties, please contact:
Time  Speaker  Talk 
10:00  Open session  
12:00  Lunch break  
14:00 
Pierre Lescanne 
The problem of counting and generating simply typed lambda terms This talk will state more problems than provide solutions. After describing the role of generating typed lambda terms, I will present several methods for counting lambda terms, several results mostly experimental obtained about counting and uniformly generating simply typed lambda terms and I will conclude with open questions. 
14:45  Break  
15:00 
Sergei Soloviev 
Probabilistic distributions on formal objects with various constraints The talk is about work in progress where we are interested in probabilistic distributions on formal objects with various constraints: the case of Boolean games with random Boolean formula(e) representing pay functions, the case of isomorphisms and automorphisms of types in Type Theory etc. We are interested in facts that hold for (large) classes of distributions. 
15:45  Break  
16:00 
Łukasz Lachowski 
On the Complexity of Translations of Lambda Calculus into Combinatory We study the complexity of all possible translations of lambda calculus into combinatory logic. First, we show that the space complexity of the standard translation is $\Theta(n^3)$. Then we investigate the lower bound for worst case size for all translations. Our main result states that any translation which preserves extensional equivalence of these models produces terms of the size $\Omega(n \log{n})$. We also present an algorithm which is asymptotically optimal. 
16:45  Break  
17:00 
Agnieszka Łupińska 
Parallel standard translation between Lambda Calculus and Combinatory Logic The talk is about the parallel approach to the standard translation between Lambda Calculus and Combinatory Logic. Let L be a lambdaterm and C be a combinator produced from L by the standard translation. Each lambda abstraction occurring in L, causes the linear expansion of some paths in the tree of the C combinator. We will show that the tree expansion can be performed parallel in logarithmic time on the path length. We will also discuss whether this procedure can be performed in the constant time. 
Time  Speaker  Talk 
10:00 
Bernhard Gittenberger 
Enumeration of lambda terms according their de Bruijn index size John Tromp introduced the socalled 'binary lambda calculus' as a way to encode lambda terms in terms of binary words. Later, Grygiel and Lescanne conjectured that the number of binary lambda terms with $m$ free indices and of size $n$ (encoded as binary words of length $n$) is $O(n^{3/2} \tau^{n})$ for $\tau \approx 1.963448\ldots$. We generalize the proposed notion of size and show that for several classes of lambda terms, including binary lambda terms with $m$ free indices, the number of terms of size $n$ is $\Theta(n^{3/2} \rho^{n})$ with some class dependent constant $\rho$, which in particular disproves the above mentioned conjecture. A way to obtain lower and upper bounds for the constant near the leading term is presented and numerical results for a few previously introduced classes of lambda terms are given. 
11:30  Break  
11:45 
Katarzyna Grygiel 
A natural counting of lambda terms We study the sequence of numbers corresponding to lambda terms of given size in the model based on de Bruijn indices. It turns out that the sequence enumerates also two families of binary trees, i.e. blackwhite and zigzagfree ones. We provide a constructive proof of this fact by exhibiting appropriate bijections. Moreover, we investigate the asymptotic density of lambda terms containing an arbitrary fixed subterm, showing that strongly normalizing terms are of density zero among all lambda terms. Joint work with M. Bendkowski, P. Lescanne, M. Zaionc 
12:30  Lunch break  
14:00 
Maciej Bendkowski 
Quantitative aspects of normalorder reduction in combinatory logic In this talk, we introduce the concept of normalorder reduction grammars characterizing the set of weakly normalizing combinatory logic terms. Exploiting its algorithmic nature, we study the asymptotic density of combinatory logic terms reducting in $n \in \mathbb{N}$ normalorder reduction steps, providing an automated method of finding their relative densities. Finally, we present some supercomputer cluster experimental results for large term sizes and reduction sequences. 
14:45  Break  
15:00 
Matthieu Dien 
On the combinatorics of concurrent programs In this talk we will be interrested by two problems:

15:45  Break  
16:00 
Paul Tarau 
A Logic Programming Playground for Lambda Terms, Types and Treebased Arithmetic We overview our Prologbased "playground" that collects algorithms for generating several lambdaterm representations and families of combinators and their types, as well as computations ranging from type inference and normalization to arithmetic encodings. Our playground, built as an incrementally developed literate Prolog program is available at http://www.cse.unt.edu/~tarau/research/2015/play.pdf with extracted Prolog sources at http://www.cse.unt.edu/~tarau/research/2015/play.pro. The playground covers compact combinatorial generation algorithms for several families of lambda terms, including open, closed, simply typed and linear terms as well as type inference and normal order reduction algorithms. Among them we will focus on a Prologbased combined lambda term generator and typeinferrer for closed welltyped terms of a given size, in de Bruijn notation. We overview a compressed de Bruijn representation of lambda terms and define its bijections to standard representations. Our compressed terms facilitate derivation of sizeproportionate ranking and unranking algorithms of lambda terms and their inferred simple types. By taking advantage of Prolog's bidirectional execution model and sound unification algorithm, our generators can build ``customized'' closed terms of a given type. This "relational view" of terms and their types enables the discovery of interesting patterns about frequently used type expressions occurring in welltyped functional programs. Our study uncovers the most ``popular'' types that govern function applications among a about a million smallsized lambda terms and hints toward practical uses to combinatorial software testing. The S and K combinator expressions form a wellknown Turingcomplete subset of the lambda calculus. We specify evaluation, type inference and combinatorial generation algorithms for SKcombinator trees. In the process, we unravel properties shedding new light on interesting aspects of their structure and distribution. We study the proportion of welltyped terms among sizelimited SKexpressions as well as the typedirected generation of terms of sizes smallerthen the size of their simple types. We also introduce the em "welltyped frontier of an untypable term" and we use it to design a simplification algorithm for untypable terms taking advantage of the fact that welltyped terms are normalizable. A uniform representation, as binary trees with empty leaves, is given to expressions built with Rosser's Xcombinator, natural numbers, lambda terms and simple types. Using this shared representation, ranking/unranking algorithm of lambda terms to treebased natural numbers are described. 
17:00  Open discussion session 
Maciej Bendkowski (Jagiellonian University, Kraków, Poland)
Olivier Bodini (ParisNord University, France)
Daniele Gardy (Versailles University, France)
Antoine Genitrini (Pierre et Marie Curie University, Paris, France)
Alain Giorgetti (FrancheComté University, Besançon, France)
Bernhard Gittenberger (TU Wien, Austria)
Zbigniew Gołębiewski (Wrocław University, Poland)
Katarzyna Grygiel (Jagiellonian University, Kraków, Poland)
Pierre Lescanne (École normale supérieure de Lyon, France)
Michał Pałka (Chalmers University of Technology, Gothenburg, Sweden)
Paul Tarau (University of North Texas, USA)
Brent Yorgey (Hendrix College, Arkansas, USA)
Marek Zaionc (Jagiellonian University, Kraków, Poland)
Noam Zeilberger (INRIA, Palaiseau, France)