Symmetic combinatory logic with the symmetric analogue of a combinatorially complete base (in the form of symmetric λ-calculus) is known to lack the Church-Rosser property. We prove a much stronger theorem that no symmetric combinatory logic that contains at least two proper symmetric combinators has the Church-Rosser property. Although the statement of the result looks similar to an earlier one concerning dual combinatory logic, the proof is different because symmetric (...) combinators may form redexes in both left and right associated terms. Perhaps surprisingly, we are also able to show that certain symmetric combinatory logics that include just one particular constant are not confluent. This result (beyond other differences) clearly sets apart symmetric combinatory logic from dual combinatory logic, since all dual combinatory systems with a single combinator or a single dual combinator are Church-Rosser. Lastly, we prove that a symmetric combinatory logic that contains the fixed point and the one-place identity combinator has the Church-Rosser property. (shrink)

Symmetic combinatory logic with the symmetric analogue of a combinatorially complete base (in the form of symmetric λ-calculus) is known to lack the Church-Rosser property. We prove a muchstrongertheorem that no symmetric combinatory logic that containsat least two proper symmetric combinatoryhas the Church-Rosser property. Although the statement of the result looks similar to an earlier one concerning dual combinatory logic,the proof is differentbecause symmetric combinators may form redexes in both (...) left and right associated terms. Perhaps surprisingly, we are also able to show that certain symmetric combinatory logics that include justone particular constantare not confluent. This result (beyond other differences) clearly sets apart symmetric combinatory logic from dual combinatory logic, since all dual combinatory systems with a single combinator or a single dual combinator are Church-Rosser. Lastly, we prove that a symmetric combinatory logic that contains the fixed point and the one-place identity combinator has the Church-Rosser property. (shrink)

We define dual and symmetric combinatory calculi (inequational and equational ones), and prove their consistency. Then, we introduce algebraic and set theoretical relational and operational - semantics, and prove soundness and completeness. We analyze the relationship between these logics, and argue that inequational dual logics are the best suited to model computation.

We work with symmetric extensions based on Lévy collapse and extend a few results of Apter, Cody, and Koepke. We prove a conjecture of Dimitriou from her Ph.D. thesis. We also observe that if V is a model of $$\textsf {ZFC}$$ ZFC, then $$\textsf {DC}_{<\kappa }$$ DC < κ can be preserved in the symmetric extension of V in terms of symmetric system $$\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle $$ ⟨ P, G, F ⟩, if $${\mathbb {P}}$$ P (...) is $$\kappa $$ κ -distributive and $${\mathcal {F}}$$ F is $$\kappa $$ κ -complete. Further we observe that if $$\delta <\kappa $$ δ < κ and V is a model of $$\textsf {ZF}+\textsf {DC}_{\delta }$$ ZF + DC δ, then $$\textsf {DC}_{\delta }$$ DC δ can be preserved in the symmetric extension of V in terms of symmetric system $$\langle {\mathbb {P}},{\mathcal {G}},{\mathcal {F}}\rangle $$ ⟨ P, G, F ⟩, if $${\mathbb {P}}$$ P is ($$\delta +1$$ δ + 1 )-strategically closed and $${\mathcal {F}}$$ F is $$\kappa $$ κ -complete. (shrink)

Combinatory logic is known to be related to substructural logics. Algebraic considerations of the latter, in particular, algebraic considerations of two distinct implications, led to the introduction of dual combinators in Dunn & Meyer 1997. Dual combinators are "mirror images" of the usual combinators and as such do not constitute an interesting subject of investigation by themselves. However, when combined with the usual combinators, the whole system exhibits new features. A dual combinatory system with weak equality typically (...) lacks the Church-Rosser property, and in general it is inconsistent. In many subsystems terms "unexpectedly" turn out to be weakly equivalent. The paper is a preliminary attempt to investigate some of these issues, as well as, briefly compare function application in symmetric -calculus and dual combinatory logic. (shrink)

Combinatorial operations on sets are almost never well defined on Turing degrees, a fact so obvious that counterexamples are worth exhibiting. The case we focus on is the symmetric-difference operator; there are pairs of degrees for which the symmetric-difference operation is well defined. Some examples can be extracted from the literature, e.g. from the existence of nonzero degrees with strong minimal covers. We focus on the case of incomparable r.e. degrees for which the symmetric-difference operation is well (...) defined. (shrink)

CHAPTER Addenda to Pure Combinatory Logic This chapter will treat various additions to, and modifications of, the subject matter of Chapters-7. ...

Reader-friendly without compromising the precision of exposition, the book includes many new research results not found in the available literature.

The completeness of the axiomatization of contingency logic over symmetric frames has been thought of as a nontrivial job, the unimodal case of which cannot be generalized to the finitely multimodal case, which in turn cannot be generalized to the infinitely multimodal case. This paper deals with the completeness of symmetric contingency logic with unlimitedly many modalities, no matter whether the set of modalities is finite or infinite.

Introduction Combinatory logic deals with a class of formal systems designed for studying certain primitive ways in which functions can be combined to form ...

We characterize the polynomial time operations as those which are provably total in a first order system, which comprises (untyped) combinatory logic with extensionality, together with positive “safe induction” on the set of binary strings. The formalization of safe induction is inspired by Leivants idea of ramification. We also show how to replace ramification by means of modal logic.

In this paper, I present a modified and extended version of combinatory logic. Schönfinkel originated the study of combinatory logic (in [2]), but its development is primarily due to H. B. Curry. In the present paper, I will make use of both the symbolism (with some modification) and the results of Curry, as found in [1].What is novel about my version of combinatory logic is a kind of combinators which I call discriminators. These combinators (...) discriminate between different symbols, and yield values which are determined by the symbols which are their arguments. If the normal combinators investigated by Curry are considered as functions taking (combinations of) symbols as arguments and yielding symbols as values, these combinators can rearrange and cancel their arguments or introduce new symbols. (shrink)

The results of this paper extend some of the intimate relations that are known to obtain between combinatory logic and certain substructural logics to establish a general characterization theorem that applies to a very broad family of such logics. In particular, I demonstrate that, for every combinator X, if LX is the logic that results by adding the set of types assigned to X (in an appropriate type assignment system, TAS) as axioms to the basic positive relevant (...) logic B∘T, then LX is sound and complete with respect to the class of frames in the Routley-Meyer relational semantics for relevant and substructural logics that meet a first-order condition that corresponds in a very direct way to the structure of the combinator X itself. (shrink)

Sections 1 through 4 define, in the usual inductive style, various classes of object including one which is called the “combinatory terms of polymorphic type”. Section 5 defines a reduction relation on these terms. Section 6 shows that the weak normalizability of the combinatory terms of polymorphic type entails the weak normalizability of the lambda terms of polymorphic type. The entailment is not vacuous, because the combinatory terms of polymorphic type are indeed weakly normalizable, as is proven (...) in Sect. 7 using Tait and Girard’s computability predicates. The remainder of the paper is devoted to arguing that interesting consequences would follow from the existence of an “ordinally informative” proof, i.e. one which uses transfinite induction over recursive ordinal numbers and otherwise finitary methods, of normalizability for as large a class of the terms as possible. (shrink)

In this paper I wish to reformulate in my own way some parts of Whitehead's theory of prehensions. This reformulation will deviate in various respects from Whitehead's own detailed views and terminology, but the main inspiration is from Whitehead.

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are (...) not translated. Both translations are closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of λ-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). (shrink)

Systems TRC and TRCU of illative combinatory logic are introduced and shown to be equivalent in consistency strength and expressive power to Quine's set theory ‘New Foundations’ and the fragment NFU + Infinity of NF described by Jensen, respectively. Jensen demonstrated the consistency of NFU + Infinity relative to ZFC; the question of the consistency of NF remains open. TRC and TRCU are presented here as classical first-order theories, although they can be presented as equational theories; they are (...) not constructive. (shrink)

Let $R$ be a convergent term rewriting system, and let $CR$-equality on combinatory logic terms be the equality induced by $\beta \eta R$-equality on terms of the lambda calculus under any of the standard translations between these two frameworks for higher-order reasoning. We generalize the classical notion of strong reduction to a reduction relation which generates $CR$-equality and whose irreducibles are exactly the translates of long $\beta R$-normal forms. The classical notion of strong normal form in combinatory (...) logic is also generalized, yielding yet another description of these translates. Their resulting tripartite characterization extends to the combined first-order algebraic and higher-order setting the classical combinatory logic descriptions of the translates of long $\beta$-normal forms in the lambda calculus. As a consequence, the translates of long $\beta R$-normal forms are easily seen to serve as canonical representatives for $CR$-equivalence classes of combinatory logic terms for non-empty, as well as for empty, $R$. (shrink)

This paper extends the results of an earlier paper by the author . New subsystems of the combinatory logic TRC shown in that paper to be equivalent to NF are introduced; these systems are analogous to subsystems of NF with predicativity restrictions on set comprehension introduced and shown to be consistent by Crabbé. For one of these systems, an exact equivalence in consistency strength and expressive power with the analogous subsystem of NF is established.

We study Artemov’s Reflective Combinatory Logic . We provide the explicit definition of types for and prove that every well-formed term has a unique type. We establish that the typability testing and detailed type restoration can be done in polynomial time and that the derivability relation for is decidable and PSPACE-complete. These results also formalize the intended semantics of the type t:F in . Terms store the complete information about the judgment “t is a term of type F”, (...) and this information can be extracted by the type restoration algorithm. (shrink)

Translations from Lambda calculi into combinatory logics can be used to avoid some implementational problems of the former systems. However, this scheme can only be efficient if the translation produces short output with a small number of combinators, in order to reduce the time and transient storage space spent during reduction of combinatory terms. In this paper we present a combinatory system and an abstraction algorithm, based on the original bracket abstraction operator of Schonfinkel [9]. The algorithm (...) introduces at most one combinator for each abstraction in the initial Lambda term. This avoids explosive term growth during successive abstractions and makes the system suitable for practical applications. We prove the correctness of the algorithm and establish some relations between the combinatory system and the Lambda calculus. (shrink)

The first volume of a projected two-volume treatise which will provide for the first time a systematic and comprehensive exposition of an important branch of mathematical logic. Previously published research is integrated and revised in the light of later results, and much new material is presented. Highly technical and, even in introductory sections, extraordinarily dense in style, this is a work for specialists. An important contribution.--L. K. B.

The dissertation deals with problems in "logic", more precisely, it deals with particular formal systems aiming at capturing patterns of valid reasoning. Sequent calculi were proposed to characterize logical connectives via introduction rules. These systems customarily also have structural rules which allow one to rearrange the set of premises and conclusions. In the "structurally free logic" of Dunn and Meyer the structural rules are replaced by combinatory rules which allow the same reshuffling of formulae, and additionally introduce (...) an explicit marker for the application of the rule---the combinator. The dissertation gives syntactic extensions of such systems introducing conjunction and disjunction . Cut elimination is proved for these systems. ;Syntactic systems as a rule are accompanied by a semantics which interprets them. The extended systems are supplied with semantics using algebraic methods. First, the Lindenbaum algebras are formed, and then these algebras are represented in an algebra of sets. This gives a so called Kripke-style semantics for the systems. The representation for the distributive extension utilizes well-known techniques; it uses sets of prime filters and set theoretical operations on them to deal with conjunction and disjunction, plus a ternary accessibility relation to accommodate the intensional operators as fusion and implication. The representation theorem is proven generally, i.e., for arbitrary sets of combinators using the "squeeze lemma" of Routley and Meyer. ;The non-distributive calculi require a different representation since union and intersection are distributive. Two previous representations for lattices are overviewed, and a modification of the Hartonas-Dunn representation is successfully augmented with the intensional operations. Instead of sets of filters, sets of pairs of minimally inconsistent filters and ideals are used and the representation theorem is proven. The representation results provide soundness and completeness for the systems. The dissertation also investigates properties of dual combinators , proving a series of lemmas about dual combinatory systems not being Church-Rosser and being inconsistent. Combinatorially definable classes of lambda-terms are considered as well. (shrink)