Adapting Logic to Physics: The Quantum-Like Eigenlogic Program

Considering links between logic and physics is important because of the fast development of quantum information technologies in our everyday life. This paper discusses a new method in logic inspired from quantum theory using operators, named Eigenlogic. It expresses logical propositions using linear algebra. Logical functions are represented by operators and logical truth tables correspond to the eigenvalue structure. It extends the possibilities of classical logic by changing the semantics from the Boolean binary alphabet { 0 , 1 } using projection operators to the binary alphabet { + 1 , − 1 } employing reversible involution operators. Also, many-valued logical operators are synthesized, for whatever alphabet, using operator methods based on Lagrange interpolation and on the Cayley–Hamilton theorem. Considering a superposition of logical input states one gets a fuzzy logic representation where the fuzzy membership function is the quantum probability given by the Born rule. Historical parallels from Boole, Post, Poincaré and Combinatory Logic are presented in relation to probability theory, non-commutative quaternion algebra and Turing machines. An extension to first order logic is proposed inspired by Grover’s algorithm. Eigenlogic is essentially a logic of operators and its truth-table logical semantics is provided by the eigenvalue structure which is shown to be related to the universality of logical quantum gates, a fundamental role being played by non-commutativity and entanglement.

How Combinatory Logic Can Limit Computing Complexity

As computing capabilities are extending, the amount of source code to manage is inevitably becoming larger and more complex. No matter how hard we try, the bewildering complexity of the source code always ends up overwhelming its own creator, to the point of giving the appearance of chaos. As a solution to the cognitive complexity of source code, we are proposing to use the framework of Combinatory Logic to construct complex computational concepts that will provide a model of description of the code that is easy and intuitive to grasp. Combinatory Logic is already known as a model of computation but what we are proposing here is to use a logic of combinators and operators to reverse engineer more and more complex computational concept up from the source code. Through the two key notions of computational concept and abstract operator, we will show that this model offers a new, meaningful and simple way of expressing what the intricate code is about.

Double Value-Ranges

From the time of Begriffsschrift onwards, Frege treated functions of two or more places on a par with those of one place. This included the treatment of relations (Beziehungen) as a special case of polyadic functions in the way that concepts (Begriffe) were a special case of monadic functions. By the time of Grundgesetze (and unlike in Begriffsschrift), Frege dealt with relations largely through their extensions, which were what he called “double value-ranges” (Doppelwerthverläufe). This is in some ways a misnomer, since double value-ranges are simply a special case of single or ordinary value-ranges, namely value-ranges of functions derived from the value-ranges of monadic functions with additional saturated places. Frege’s treatment of the extensions of relations (which he came to call simply “Relationen”) thus embodies a move analogous to the treatment of polyadic functions as functions of functions, a device invented in 1920 by Moses Schönfinkel and since (unfairly) known in combinatory logic as “currying”. This paper considers the details of Frege’s Grundgesetze treatment of relations via their extensions, exhibits its grammar, and indicates its formal elegance by comparing it with other possible treatments.

The involutions-as-principal types/application-as-unification Analogy

In 2005, S. Abramsky introduced various universal models of computation based on Affine Combinatory Logic, consisting of partial involutions over a suitable formal language of moves, in order to discuss reversible computation in a game-theoretic setting. We investigate Abramsky’s models from the point of view of the model theory of λ-calculus, focusing on the purely linear and affine fragments of Abramsky’s Combinatory Algebras.Our approach stems from realizing a structural analogy, which had not been hitherto pointed out in the literature, between the partial involution interpreting a combinator and the principal type of that term, with respect to a simple types discipline for λ-calculus. This analogy allows for explaining as unification between principal types the somewhat awkward linear application of involutions arising from Geometry of Interaction (GoI).Our approach provides immediately an answer to the open problem, raised by Abram- sky, of characterising those finitely describable partial involutions which are denotations of combinators, in the purely affine fragment. We prove also that the (purely) linear combinatory algebra of partial involutions is a (purely) linear λ-algebra, albeit not a combinatory model, while the (purely) affine combinatory algebra is not. In order to check the complex equations involved in the definition of affine λ-algebra, we implement in Erlang the compilation of λ-terms as involutions, and their execution.

Combinatory logic

Combinatory logic comprises a battery of formalisms for expressing and studying properties of operations constitutive to contemporary logic and its applications. The sole syntactic category in combinatory logic is that of the applicative term. Closed terms are called ‘combinators’; there is no binding of variables. Systems containing the basic combinators S and K exhibit the crucial property of combinatorial completeness: every routine expressible in the system can be captured by a term composed of these two combinators alone. Combinatory logic is a close relative of Church’s lambda calculus. M. Schönfinkel first introduced and defined basic combinators in 1920 in assaying foundations for mathematics that avoid bound variables and take operations, rather than sets, as fundamental. H. Curry later rediscovered the combinators (and coined the term ‘combinatory logic’) independently of Schönfinkel. Curry constructed various formal systems for combinatory logic and, throughout most of the subject’s history, was the central figure in the research. In 1969, D. Scott succeeded in constructing set-theoretic, functional models for the lambda calculus and combinatory logic. Since then semantic studies of combinatory systems, together with research on their applications to computer science and further development as foundational systems, have dominated the field.