A Semantic Approach to Illative Combinatory Logic

This work introduces the theory of illative combinatory algebras, which is closely related to systems of illative combinatory logic. We thus provide a semantic interpretation for a formal framework in which both logic and computation may be expressed in a unified manner. Systems of illative combinatory logic consist of combinatory logic extended with constants and rules of inference intended to capture logical notions. Our theory does not correspond strictly to any traditional system, but draws inspiration from many. It differs from them in that it couples the notion of truth with the notion of equality between terms, which enables the use of logical formulas in conditional expressions. We give a consistency proof for first-order illative combinatory algebras. A complete embedding of classical predicate logic into our theory is also provided. The translation is very direct and natural

A general approach to define binders using matching logic

We propose a novel shallow embedding of binders using matching logic, where the binding behavior of object-level binders is obtained for free from the behavior of the built-in existential binder of matching logic. We show that binders in various logical systems such as lambda-calculus, System F, pi-calculus, pure type systems, etc., can be defined in matching logic. We show the correctness of our definitions by proving conservative extension theorems, which state that a sequent/judgment is provable in the original system if and only if it is provable in matching logic. An appealing aspect of our embedding of binders in matching logic is that it yields models to all binders, also for free. We show that models yielded by matching logic are deductively complete to the formal reasoning in the original systems. For lambda-calculus, we further show that the yielded models are representationally complete---a desired property that is not enjoyed by many existing lambda-calculus semantics.Ope

A hierarchy of languages, logics, and mathematical theories

We present mathematics from a foundational perspective as a hierarchy in which each tier consists of a language, a logic, and a mathematical theory. Each tier in the hierarchy subsumes all preceding tiers in the sense that its language, logic, and mathematical theory generalize all preceding languages, logics, and mathematical theories. Starting from the root tier, the mathematical theories in this hierarchy are: combinatory logic restricted to the identity I, combinatory logic, ZFC set theory, constructive type theory, and category theory. The languages of the first four tiers correspond to the languages of the Chomsky hierarchy: in combinatory logic Ix = x gives rise to a regular language; the language generated by S, K in combinatory logic is context-free; first-order logic is context-sensitive; and the typed lambda calculus of type theory is recursively enumerable. The logic of each tier can be characterized in terms of the cardinality of the set of its truth values: combinatory logic restricted to I has 0 truth values, while combinatory logic has 1, first-order logic 2, constructive type theory 3, and categeory theory omega_0. We conjecture that the cardinality of objects whose existence can be established in each tier is bounded; for example, combinatory logic is bounded in this sense by omega_0 and ZFC set theory by the least inaccessible cardinal. We also show that classical recursion theory presents a framework for generating the above hierarchy in terms of the initial functions zero, projection, and successor followed by composition and m-recursion, starting with the zero function I in combinatory logic This paper begins with a theory of glossogenesis, i.e. a theory of the origin of language, since this theory shows that natural language has deep connections to category theory and since it was through these connections that the last tier and ultimately the whole hierarchy were discovered. The discussion covers implications of the hierarchy for mathematics, physics, cosmology, theology, linguistics, extraterrestrial communication, and artificial intelligence

The Lambda Calculus

The λ-calculus is, at heart, a simple notation for functions and application. The main ideas are applying a function to an argument and forming functions by abstraction. The syntax of basic λ-calculus is quite sparse, making it an elegant, focused notation for representing functions. Functions and arguments are on a par with one another. The result is a non-extensional theory of functions as rules of computation, contrasting with an extensional theory of functions as sets of ordered pairs. Despite its sparse syntax, the expressiveness and flexibility of the λ-calculus make it a cornucopia of logic and mathematics. This entry develops some of the central highlights of the field and prepares the reader for further study of the subject and its applications in philosophy, linguistics, computer science, and logic

Verovatnosno zaključivanje u izračunavanju i teoriji funkcionalnih tipova

This thesis investigates two different approaches for probabilistic reasoning in models of computation. The most usual approach is to extend the language of untyped lambda calculus with probabilistic choice operator which results in probabilistic computation. This approach has shown to be very useful and applicable in various fields, e.g. robotics, natural language processing, and machine learning. Another approach is to extend the language of a typed lambda calculus with probability operators and to obtain a framework for probabilistic reasoning about the typed calculus in the style of probability logic. First, we study the lazy call-by-name probabilistic lambda calculus extended with let-in operator, and program equivalence in the calculus. Since the proof of context equivalence is quite challenging, we investigate some effective methods for proving the program equivalence. Probabilistic applicative bisimilarity has proved to be a suitable tool for proving the context equivalence in probabilistic setting. We prove that the probabilistic applicative bisimilarity is fully abstract with respect to the context equivalence in the probabilistic lambda calculus with let-in operator. Next, we introduce Kripke-style semantics for the full simply typed combinatory logic, that is, the simply typed combinatory logic extended with product types, sum types, empty type and unit type. The Kripke-style semantics is defined as a Kripke applicative structure, which is extensional and has special elements corresponding to basic combinators, provided with the valuation of term variables. We prove that the full simply typed combinatory logic is sound and complete with respect to the proposed semantics. We introduce the logic of combinatory logic, that is, a propositional extension of the simply typed combinatory logic. We prove that the axiomatization of the logic of combinatory logic is sound and strongly complete with respect to the proposed semantics. In addition, we prove that the proposed semantics is the new semantics for the simply typed combinatory logic containing the typing rule that ensures that equal terms inhabit the same type. Finally, we introduce the probabilistic extension of the logic of combinatory logic. We extend the logic of combinatory logic with probability operators and obtain a framework for probabilistic reasoning about typed combinatory terms. We prove that the given axiomatization of the logic is sound and strongly complete with respect to the proposed semantics.Теза истражује два различита приступа за вероватносно закључивање у моделима израчунавања. Најчешћи приступ се састоји у проширењу ламбда рачуна вероватносним оператором избора што резултира вероватносним израчунавањем. То се показало веома корисним и примењивим у разним областима, на пример у роботици, обради природног језика и машинском учењу. Други приступ јесте да проширимо језик рачуна вероватносним операторима и добијемо модел за вероватносно закључивање о типизираном рачуну у стилу вероватносне логике. Најпре проучавамо вероватносни ламбда рачун проширен лет-ин оператором где је примењена лења позив-по-имену стратегија евалуације, и изучавамо проблем еквиваленције програма у овом окружењу. Како је проблем доказивања контекстне еквиваленције доста изазован, истраживали смо ефикасне методе за доказивање еквиваленције програма. Вероватносна апликативна бисимулација се показала као одговарајући алат за доказивање еквиваленције програма у вероватносном окружењу. Доказујемо да је вероватносна апликативна бисимулација потпуно апстрактна у односу на контекстну еквиваленцију у вероватносном ламбда рачуну са лет-ин оператором. Затим уводимо Крипкеову семантику за целу комбинаторну логику са функционалним типовима, односно комбинаторну логику са функционалним типовима проширену типовима производа, типовима суме, празним типом и јединичним типом. Крипкеову семантику дефинишемо као Крипкеову апликативну структуру, која је екстензионална и има елементе који одговарају основним комбинаторима, и којој је придружена валуација променљивих. Доказујемо да је цела комбинаторна логика са функционалним типовима сагласна и потпуна у односу на уведене семантике. Уводимо логику комбинаторне логике, то јест исказно проширење комбинаторне логике са функционалним типовима. Доказујемо да је аксиоматизација логике комбинаторне логике сагласна и потпуна у односу на предложену семантику. Даље, показујемо да је уведена семантика нова семантика за комбинаторну логику са функционалним типовима проширену правилом типизирања које осигурава да једнаки терми имају исти тип. На крају, уводимо вероватносно проширење логике комбинаторне логике. Логику комбинаторне логике смо проширили са вероватносним операторима и добили модел за вероватносно закључивање о типизираним комбинаторним термима. Показујемо да је аксиоматизација логике сагласна и јако потпуна у односу на предложену семантику.Teza istražuje dva različita pristupa za verovatnosno zaključivanje u modelima izračunavanja. Najčešći pristup se sastoji u proširenju lambda računa verovatnosnim operatorom izbora što rezultira verovatnosnim izračunavanjem. To se pokazalo veoma korisnim i primenjivim u raznim oblastima, na primer u robotici, obradi prirodnog jezika i mašinskom učenju. Drugi pristup jeste da proširimo jezik računa verovatnosnim operatorima i dobijemo model za verovatnosno zaključivanje o tipiziranom računu u stilu verovatnosne logike. Najpre proučavamo verovatnosni lambda račun proširen let-in operatorom gde je primenjena lenja poziv-po-imenu strategija evaluacije, i izučavamo problem ekvivalencije programa u ovom okruženju. Kako je problem dokazivanja kontekstne ekvivalencije dosta izazovan, istraživali smo efikasne metode za dokazivanje ekvivalencije programa. Verovatnosna aplikativna bisimulacija se pokazala kao odgovarajući alat za dokazivanje ekvivalencije programa u verovatnosnom okruženju. Dokazujemo da je verovatnosna aplikativna bisimulacija potpuno apstraktna u odnosu na kontekstnu ekvivalenciju u verovatnosnom lambda računu sa let-in operatorom. Zatim uvodimo Kripkeovu semantiku za celu kombinatornu logiku sa funkcionalnim tipovima, odnosno kombinatornu logiku sa funkcionalnim tipovima proširenu tipovima proizvoda, tipovima sume, praznim tipom i jediničnim tipom. Kripkeovu semantiku definišemo kao Kripkeovu aplikativnu strukturu, koja je ekstenzionalna i ima elemente koji odgovaraju osnovnim kombinatorima, i kojoj je pridružena valuacija promenljivih. Dokazujemo da je cela kombinatorna logika sa funkcionalnim tipovima saglasna i potpuna u odnosu na uvedene semantike. Uvodimo logiku kombinatorne logike, to jest iskazno proširenje kombinatorne logike sa funkcionalnim tipovima. Dokazujemo da je aksiomatizacija logike kombinatorne logike saglasna i potpuna u odnosu na predloženu semantiku. Dalje, pokazujemo da je uvedena semantika nova semantika za kombinatornu logiku sa funkcionalnim tipovima proširenu pravilom tipiziranja koje osigurava da jednaki termi imaju isti tip. Na kraju, uvodimo verovatnosno proširenje logike kombinatorne logike. Logiku kombinatorne logike smo proširili sa verovatnosnim operatorima i dobili model za verovatnosno zaključivanje o tipiziranim kombinatornim termima. Pokazujemo da je aksiomatizacija logike saglasna i jako potpuna u odnosu na predloženu semantiku

Simultaneous Abstraction and Semantic Theories

Institute for Communicating and Collaborative SystemsI present a simple Simultaneous Abstraction Calculus, where the familiar lambda-abstraction over single variables is replaced by abstraction over whole sets of them. Terms are applied to partial assignments of objects to variables. Variants of the system are investigated and compared, with respect to their semantic and proof theoretic properties. The system overcomes the strict ordering requirements of the standard lambda-calculus,and is shown to provide the kind of "non-selective" binding needed for Dynamic Montague Grammar and Discourse Representation Theory. It is closely related to a more complex system, due to Peter Aczel and Rachel Lunon, and can be used for Situation Theory in a similar way. I present versions of these theories within an axiomatic, property-theoretic framework, based on Aczels Frege Structures. The aim of this work is to provide the means for integrating various semantic theories within a formal framework,so that they can share what is common between them, and adopt from each other what is compatible with them