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Uses these interpreters as a convenient vehicle for Algorithmic Information Theory, deriving explicit upper bounds on program-size complexity.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & 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interpreters are introduced in the work?","Question",{"text":62,"@type":63},"The paper proposes binary encodings for lambda calculus and combinatory logic terms, together with concise interpreters that operate on bit-streams represented as lists of booleans.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How does the paper use these interpreters in Algorithmic Information Theory?",{"text":67,"@type":63},"It selects universal computers based on the provided interpreters and proves basic Algorithmic Information Theory results with explicit constants, including concrete upper bounds on program-size complexity.",{"name":69,"@type":60,"acceptedAnswer":70},"What is the role of bits in defining the size of a term?",{"text":71,"@type":63},"It establishes a representation-neutral notion of term size measured in bits, enabling a way to describe arbitrary data with the least number of bits possible under the reviewed description 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the 􀀌rst part we introduce binary representations of both lambda calculus and combinatory logic, together with very concise interpreters that witness their simplicity. Along the way we present a simple graphical notation for lambda calculus, a new empty list representation, improved bracket abstraction, and a new 􀀌xpoint combinator. In the second part we review Algorithmic Information Theory, for which these interpreters provide a convenient vehicle. We demonstrate this with several concrete upper bounds on program-size complexity.  \n1 Introduction  \nThe ability to represent programs as data and to map such data back to programs (known as rei􀀌cation and re􀀍ection [10]), is of both practical use in metaprogramming [16] as well as theoretical use in computability and logic [19] . It comes as no surprise that the pure lambda calculus, which represents both programs and data as functions, is well equipped to o􀀋er these features. In [7], Kleene was the 􀀌rst to propose an encoding of lambda terms, mapping them to G􀁿odel numbers, which can in turn be represented as so called Church numerals. Decoding such numbers is somewhat cumbersome, and not particularly e􀀎cient. In search of simpler constructions, various alternative encodings have been proposed using higher-order abstract syntax [9] combined with the standard lambda representation of signatures [12] . A particularly simple encoding was proposed by Mogensen [25], for which the term 􀀕m:m (􀀕x:x)(􀀕x:x) acts asa sel􀀌nterpreter. The prevalent data format, both in information theory and in practice, however, is not numbers, or syntax trees, but bits. We propose binary encodings of both lambda and combinatory logic terms, and exhibit relatively simple and e􀀎cient interpreters (using the standard representation of bit-streams as lists of booleans) .  \nThis gives us a representation-neutral notion of the size of a term, measured in bits. More importantly, it provides a way to describe arbitrary data with, in a sense, the least number of bits possible. We review the notion of how a computer reading bits and outputting some result constitutes a description method, and  \nhow universal computers correspond to optimal description methods. We then pick speci􀀌c universal computers based on our interpreters and prove several of the basic results of Algorithmic Information Theory with explicit constants.  \n2 Lambda Calculus  \nWe only summarize the basics here. For a comprehensive treatment we refer the reader to the standard reference [20] .  \nAssume a countably in􀀌nite set of variables  \na; b; : : : ; x; y; z; x0 ; x 1 ; : : :  \nThe set of lambda terms 􀀃 is built up from variables using abstraction  \n(􀀕x:M )  \nand application  \n(M N );  \nwhere x is any variable and M; N are lambda terms. (􀀕x:M ) is the function that maps x to M , while (M N ) is the application of function M to argument N. We sometimes omit parentheses, understanding abstraction to associate to the right, and application to associate to the left, e.g. 􀀕x:􀀕y:x y x denotes (􀀕x:(􀀕y:((x y)x))) . We also join consecutive abstractions as in 􀀕x y:x y x.  \nThe free variables FV (M ) of a term M are those variables not bound by an enclosing abstraction. 􀀃 0 denotes the set of closed terms, i.e. with no free variables. The simplest closed term is the identity 􀀕x:x.  \nWe consider two terms identical if they only di􀀋er in the names of bound variables, and denote this with 􀀑, e.g. 􀀕y:y x 􀀑 􀀕z:z x. The essence of 􀀕 calculus is embodied in the 􀀌-conversion rule which equates  \n(􀀕x:M )N = M [x := N ];  \nwhere M [x := N ] denotes the result of substituting N for all free occurrences of x in M (taking care to avoid variable capture by renaming bound variables in M if necessary) . For example,  \n(􀀕x y:y x)y 􀀑 (􀀕x: (􀀕z:z x))y 􀀑 (􀀕x z:z x)y = 􀀕z:z y:  \nA term with no 􀀌-redex, that is, no subterm of the form (􀀕x:M )N , is said tobe i","cbCaiiwzLQ0pvd1o","https://ap.wps.com/l/cbCaiiwzLQ0pvd1o","pdf",309077,"English","# Introduction\n# Lambda Calculus\n## Some useful lambda terms","[{\"question\":\"What new representations and interpreters are introduced in the work?\",\"answer\":\"The paper proposes binary encodings for lambda calculus and combinatory logic terms, together with concise interpreters that operate on bit-streams represented as lists of booleans.\"},{\"question\":\"How does the paper use these interpreters in Algorithmic Information Theory?\",\"answer\":\"It selects universal computers based on the provided interpreters and proves basic Algorithmic Information Theory results with explicit constants, including concrete upper bounds on program-size complexity.\"},{\"question\":\"What is the role of bits in defining the size of a term?\",\"answer\":\"It establishes a representation-neutral notion of term size measured in bits, enabling a way to describe arbitrary data with the least number of bits possible under the reviewed description framework.\"}]","Functional Bits - Lambda Calculus based Algorithmic Information Theory | PDF"]