arXiv subjects

Alban Ponse

Publications and source records attributed to Alban Ponse.

At least 19 recordsLinked to original sources

arXiv2025

Fracterm Calculus for Partial Meadows

Partial algebras and datatypes are discussed with the use of signatures that allow partial functions, and a three-valued short-circuit (sequential) first order logic with a Tarski semantics. The propositional part of this logic is also known as McCarthy calculus and has been studied extensively. Axioms for the fracterm calculus of partial meadows are given. The case is made that in this way a rather natural formalisation of fields with division operator is obtained. It is noticed that the logic thus obtained cannot express that division by zero must be undefined. An interpretation of the three-valued sequential logic into $\bot$-enlargements of partial algebras is given, for which it is concluded that the consequence relation of the former logic is semi-computable, and that the $\bot$-enlargement of a partial meadow is a common meadow.

cs.LO↗

arXiv2024

Fully Evaluated Left-Sequential Logics

We consider a family of two-valued "fully evaluated left-sequential logics" (FELs), of which Free FEL (defined by Staudt in 2012) is most distinguishing (weakest) and immune to atomic side effects. Next is Memorising FEL, in which evaluations of subexpressions are memorised. The following stronger logic is Conditional FEL (inspired by Guzm\'an and Squier's Conditional logic, 1990). The strongest FEL is static FEL, a sequential version of propositional logic. We use evaluation trees as a simple, intuitive semantics and provide complete axiomatisations for closed terms (left-sequential propositional expressions). For each FEL except Static FEL, we also define its three-valued version, with a constant U for "undefinedness" and again provide complete, independent axiomatisations, each one containing two additional axioms for U on top of the axiomatisations of the two-valued case. In this setting, the strongest FEL is equivalent to Bochvar's strict logic.

cs.LO↗

arXiv2023

Conditional logic as a short-circuit logic

Three-valued conditional logic (CL) is defined by Guzm\'an and Squier (1990), and based on McCarthy's noncommutative connectives, axiomatises a short-circuit logic (SCL) that defines more identities than three-valued MSCL (Memorising SCL, which also has a two-valued variant). This follows from the fact that the definable connective that prescribes full left-sequential conjunction is commutative in CL. We show that in CL, the full left-sequential connectives and negation define Bochvar's three-valued strict logic. We observe that CL also has a two-valued variant of which the full left-sequential connectives and negation define a commutative logic that is weaker than propositional logic because the absorption laws do not hold. Next, we show that the original, equational axiomatisation of CL is not independent and give several alternative, independent axiomatisations.

cs.LO↗

arXiv2022

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

cs.LO↗

arXiv2021

Division by zero in common meadows

Common meadows are fields expanded with a total inverse function. Division by zero produces an additional value denoted with "a" that propagates through all operations of the meadow signature (this additional value can be interpreted as an error element). We provide a basis theorem for so-called common cancellation meadows of characteristic zero, that is, common meadows of characteristic zero that admit a certain cancellation law.

math.RA↗

arXiv2018

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.

cs.LO↗

arXiv2018

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.

cs.LO↗

arXiv2016

Probability functions in the context of signed involutive meadows

The Kolmogorov axioms for probability functions are placed in the context of signed meadows. A completeness theorem is stated and proven for the resulting equational theory of probability calculus. Elementary definitions of probability theory are restated in this framework.

math.LO↗

arXiv2016

Datatype defining rewrite systems for naturals and integers

A datatype defining rewrite system (DDRS) is an algebraic (equational) specification intended to specify a datatype. When interpreting the equations from left-to-right, a DDRS defines a term rewriting system that must be ground-complete. First we define two DDRSs for the ring of integers, each comprising twelve rewrite rules, and prove their ground-completeness. Then we introduce natural number and integer arithmetic specified according to unary view, that is, arithmetic based on a postfix unary append constructor (a form of tallying). Next we specify arithmetic based on two other views: binary and decimal notation. The binary and decimal view have as their characteristic that each normal form resembles common number notation, that is, either a digit, or a string of digits without leading zero, or the negated versions of the latter. Integer arithmetic in binary and decimal notation is based on (postfix) digit append functions. For each view we define a DDRS, and in each case the resulting datatype is a canonical term algebra that extends a corresponding canonical term algebra for natural numbers. Then, for each view, we consider an alternative DDRS based on tree constructors that yields comparable normal forms, which for that view admits expressions that are algorithmically more involved. For all DDRSs considered, ground-completeness is proven.

cs.LO↗

arXiv2016

Three Datatype Defining Rewrite Systems for Datatypes of Integers each extending a Datatype of Naturals

Integer arithmetic is specified according to three views: unary, binary, and decimal notation. The binary and decimal view have as their characteristic that each normal form resembles common number notation, that is, either a digit, or a string of digits without leading zero, or the negated versions of the latter. The unary view comprises a specification of integer arithmetic based on 0, successor function $S$, and predecessor function, with negative normal forms $-S^i(0)$. Integer arithmetic in binary and decimal notation is based on (postfix) digit append functions. For each view we define a ground-confluent and terminating datatype defining rewrite system (DDRS), and in each case the resulting datatype is a canonical term algebra that extends a corresponding canonical term algebra for natural numbers. Then, for each view, we consider an alternative DDRS based on tree constructors that yield comparable normal forms, which for that binary and decimal view admits expressions that are algorithmically more involved. These DDRSes are incorporated because they are closer to existing literature. For these DDRSes we also provide ground-completeness results. Finally, we define a DDRS for the ring of Integers (comprising fifteen rewrite rules) and prove its ground-completeness.

cs.LO↗

arXiv2016

Fracpairs and fractions over a reduced commutative ring

In the well-known construction of the field of fractions of an integral domain, division by zero is excluded. We introduce "fracpairs" as pairs subject to laws consistent with the use of the pair as a fraction, but do not exclude denominators to be zero. We investigate fracpairs over a reduced commutative ring (a commutative ring that has no nonzero nilpotent elements) and provide these with natural definitions for addition, multiplication, and additive and multiplicative inverse. We find that modulo a simple congruence these fracpairs constitute a "common meadow", which is a commutative monoid both for addition and multiplication, extended with a weak additive inverse, a multiplicative inverse except for zero, and an additional element "a" that is the image of the multiplicative inverse on zero and that propagates through all operations. Considering "a" as an error-value supports the intuition. The equivalence classes of fracpairs thus obtained are called common cancellation fractions (cc-fractions), and cc-fractions over the integers constitute a homomorphic pre-image of the common meadow Qa, the field Q of rational numbers expanded with an a-totalized inverse. Moreover, the initial common meadow is isomorphic to the initial algebra of cc-fractions over the integer numbers. Next, we define canonical term algebras for cc-fractions over the integers and some meadows that model the rational numbers expanded with a totalized inverse, and provide some negative results concerning their associated term rewriting properties. Then we consider reduced commutative rings in which the sum of two squares plus one cannot be a zero divisor: by extending the equivalence relation on fracpairs we obtain an initial algebra that is isomorphic to Qa. Finally, we express negative conjectures concerning alternative specifications for these (concrete) datatypes.

math.RA↗

arXiv2015

Poly-infix operators and operator families

Poly-infix operators and operator families are introduced as an alternative for working modulo associativity and the corresponding bracket deletion convention. Poly-infix operators represent the basic intuition of repetitively connecting an ordered sequence of entities with the same connecting primitive.

math.HO↗

arXiv2015

Evaluation trees for proposition algebra

Proposition algebra is based on Hoare's conditional connective, which is a ternary connective comparable to if-then-else and used in the setting of propositional logic. Conditional statements are provided with a simple semantics that is based on evaluation trees and that characterizes so-called free valuation congruence: two conditional statements are free valuation congruent if, and only if, they have equal evaluation trees. Free valuation congruence is axiomatized by the four basic equational axioms of proposition algebra that define the conditional connective. Valuation congruences that identify more conditional statements than free valuation congruence are repetition-proof, contractive, memorizing, and static valuation congruence. Each of these valuation congruences is characterized using a transformation on evaluation trees: two conditional statements are C-valuation congruent if, and only if, their C-transformed evaluation trees are equal. These transformations are simple and natural, and only for static valuation congruence a slightly more complex transformation is used. Also, each of these valuation congruences is axiomatized in proposition algebra. A spin-off of our approach can be called "normalization functions for proposition algebra": for each valuation congruence C considered, two conditional statements are C-valuation congruent if, and only if, the C-normalization function returns equal images.

cs.LO↗

arXiv2015

Equations for formally real meadows

We consider the signatures $Σ_m=(0,1,-,+, \cdot, \ ^{-1})$ of meadows and $(Σ_m, {\mathbf s})$ of signed meadows. We give two complete axiomatizations of the equational theories of the real numbers with respect to these signatures. In the first case, we extend the axiomatization of zero-totalized fields by a single axiom scheme expressing formal realness; the second axiomatization presupposes an ordering. We apply these completeness results in order to obtain complete axiomatizations of the complex numbers.

math.RA↗

arXiv2013

Cancellation Meadows: a Generic Basis Theorem and Some Applications

Let Q_0 denote the rational numbers expanded to a "meadow", that is, after taking its zero-totalized form (0^{-1}=0) as the preferred interpretation. In this paper we consider "cancellation meadows", i.e., meadows without proper zero divisors, such as $Q_0$ and prove a generic completeness result. We apply this result to cancellation meadows expanded with differentiation operators, the sign function, and with floor, ceiling and a signed variant of the square root, respectively. We give an equational axiomatization of these operators and thus obtain a finite basis for various expanded cancellation meadows.

math.RA↗

arXiv2013

Periodic Single-Pass Instruction Sequences

A program is a finite piece of data that produces a (possibly infinite) sequence of primitive instructions. From scratch we develop a linear notation for sequential, imperative programs, using a familiar class of primitive instructions and so-called repeat instructions, a particular type of control instructions. The resulting mathematical structure is a semigroup. We relate this set of programs to program algebra (PGA) and show that a particular subsemigroup is a carrier for PGA by providing axioms for single-pass congruence, structural congruence, and thread extraction. This subsemigroup characterizes periodic single-pass instruction sequences and provides a direct basis for PGA's toolset.

cs.PL↗

arXiv2010

On Hoare-McCarthy algebras

We discuss an algebraic approach to propositional logic with side effects. To this end, we use Hoare's conditional [1985], which is a ternary connective comparable to if-then-else. Starting from McCarthy's notion of sequential evaluation [1963] we discuss a number of valuation congruences and we introduce Hoare-McCarthy algebras as the structures that characterize these congruences.

cs.LO↗

arXiv2009

An Instruction Sequence Semigroup with Involutive Anti-Automorphisms

We introduce an algebra of instruction sequences by presenting a semigroup C in which programs can be represented without directional bias: in terms of the next instruction to be executed, C has both forward and backward instructions and a C-expression can be interpreted starting from any instruction. We provide equations for thread extraction, i.e., C's program semantics. Then we consider thread extraction compatible (anti-)homomorphisms and (anti-)automorphisms. Finally we discuss some expressiveness results.

cs.PL↗