An independent axiomatisation for free short-circuit logic

Short-circuit evaluation denotes the semantics of propositional connectives in which the second argument is evaluated only if the first argument does not suffice to determine the value of the expression. Free short-circuit logic is the equational logic in which compound statements are evaluated from left to right, while atomic evaluations are not memorised throughout the evaluation, i.e., evaluations of distinct occurrences of an atom in a compound statement may yield different truth values. We provide a simple semantics for free SCL and an independent axiomatisation. Finally, we discuss evaluation strategies, some other SCLs, and side effects.Comment: 36 pages, 4 tables. Differences with v2: Section 2.1: theorem Thm.2.1.5 and further are renumbered; corrections: p.23, line -7, p.24, lines 3 and 7. arXiv admin note: substantial text overlap with arXiv:1010.367

The solution of the Sixth Hilbert Problem: the Ultimate Galilean Revolution

I argue for a full mathematisation of the physical theory, including its axioms, which must contain no physical primitives. In provocative words: "physics from no physics". Although this may seem an oxymoron, it is the royal road to keep complete logical coherence, hence falsifiability of the theory. For such a purely mathematical theory the physical connotation must pertain only the interpretation of the mathematics, ranging from the axioms to the final theorems. On the contrary, the postulates of the two current major physical theories either don't have physical interpretation (as for von Neumann's axioms for quantum theory), or contain physical primitives as "clock", "rigid rod ", "force", "inertial mass" (as for special relativity and mechanics). A purely mathematical theory as proposed here, though with limited (but relentlessly growing) domain of applicability, will have the eternal validity of mathematical truth. It will be a theory on which natural sciences can firmly rely. Such kind of theory is what I consider to be the solution of the Sixth Hilbert's Problem. I argue that a prototype example of such a mathematical theory is provided by the novel algorithmic paradigm for physics, as in the recent information-theoretical derivation of quantum theory and free quantum field theory.Comment: Opinion paper. Special issue of Philosophical Transaction A, devoted to the VI Hilbert problem, after the Workshop "Hilbert's Sixth Problem", University of Leicester, May 02-04 201

Non-commutative propositional logic with short-circuited biconditional and NAND

Short-circuit evaluation denotes the semantics of propositional connectives in which the second argument is evaluated only if the first argument does not suffice to determine the value of the expression. In programming, short-circuit evaluation is widely used, with left-sequential conjunction and disjunction as primitive connectives. We consider left-sequential, non-commutative propositional logic, also known as MSCL (memorising short-circuit logic), and start from a previously published, equational axiomatisation. First, we extend this logic with a left-sequential version of the biconditional connective, which allows for an elegant axiomatisation of MSCL. Next, we consider a left-sequential version of the NAND operator (the Sheffer stroke) and again give a complete, equational axiomatisation of the corresponding variant of MSCL. Finally, we consider these logical systems in a three-valued setting with a constant for `undefined', and again provide completeness results.Comment: 21 pages, 6 table

A Diagrammatic Axiomatisation of Fermionic Quantum Circuits

We introduce the fermionic ZW calculus, a string-diagrammatic language for fermionic quantum computing (FQC). After defining a fermionic circuit model, we present the basic components of the calculus, together with their interpretation, and show how the main physical gates of interest in FQC can be represented in the language. We then list our axioms, and derive some additional equations. We prove that the axioms provide a complete equational axiomatisation of the monoidal category whose objects are quantum systems of finitely many local fermionic modes, with operations that preserve or reverse the parity (number of particles mod 2) of states, and the tensor product, corresponding to the composition of two systems, as monoidal product. We achieve this through a procedure that rewrites any diagram in a normal form. We conclude by showing, as an example, how the statistics of a fermionic Mach-Zehnder interferometer can be calculated in the diagrammatic language

Propositional logic with short-circuit evaluation: a non-commutative and a commutative variant

Short-circuit evaluation denotes the semantics of propositional connectives in which the second argument is evaluated only if the first argument does not suffice to determine the value of the expression. Short-circuit evaluation is widely used in programming, with sequential conjunction and disjunction as primitive connectives. We study the question which logical laws axiomatize short-circuit evaluation under the following assumptions: compound statements are evaluated from left to right, each atom (propositional variable) evaluates to either true or false, and atomic evaluations can cause a side effect. The answer to this question depends on the kind of atomic side effects that can occur and leads to different "short-circuit logics". The basic case is FSCL (free short-circuit logic), which characterizes the setting in which each atomic evaluation can cause a side effect. We recall some main results and then relate FSCL to MSCL (memorizing short-circuit logic), where in the evaluation of a compound statement, the first evaluation result of each atom is memorized. MSCL can be seen as a sequential variant of propositional logic: atomic evaluations cannot cause a side effect and the sequential connectives are not commutative. Then we relate MSCL to SSCL (static short-circuit logic), the variant of propositional logic that prescribes short-circuit evaluation with commutative sequential connectives. We present evaluation trees as an intuitive semantics for short-circuit evaluation, and simple equational axiomatizations for the short-circuit logics mentioned that use negation and the sequential connectives only.Comment: 34 pages, 6 tables. Considerable parts of the text below stem from arXiv:1206.1936, arXiv:1010.3674, and arXiv:1707.05718. Together with arXiv:1707.05718, this paper subsumes most of arXiv:1010.367