Related papers: PBNF-transform as a formulation of Propositional C…
Here we show, in the second paper in a series of articles, methods to calculate propositional statements with algebraic polyno mials as symbols for the connectives, which here are named operators. In the first article, we explained this…
We introduce power term polynomial algebra, a representation language for Boolean formulae designed to bridge conjunctive normal form (CNF) and algebraic normal form (ANF). The language is motivated by the tiling mismatch between these…
We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege---yielding a semantic way to define a…
The univalence axiom expresses the principle of extensionality for dependent type theory. However, if we simply add the univalence axiom to type theory, then we lose the property of canonicity - that every closed term computes to a…
This article introduces probabilistic disjunctive normal forms (PDNFs) as a framework for representing and reasoning about uncertainty in logical systems. Unlike classical DNFs, PDNFs assign real-valued weights to variables, encoding…
In Weighted Model Counting (WMC), we assign weights to literals and compute the sum of the weights of the models of a given propositional formula where the weight of an assignment is the product of the weights of its literals. The current…
A Pseudo-Boolean (PB) constraint is a linear arithmetic constraint over Boolean variables. PB constraints are convenient and widely used in expressing NP-complete problems. We introduce a new, two step, method for transforming PB…
Quantified Boolean formulas (QBFs) generalize propositional formulas by admitting quantifications over propositional variables. QBFs can be viewed as (restricted) formulas of first-order predicate logic and easy translations of QBFs into…
Does every Boolean tautology have a short propositional-calculus proof? Here, a propositional calculus (i.e. Frege) proof is a proof starting from a set of axioms and deriving new Boolean formulas using a set of fixed sound derivation…
In this short note we relate some known properties of propositional calculus to purely algebraic considerations of a Boolean algebra. Classes of formulas of propositional calculus are considered as elements of a Boolean algebra. As such…
Configurable systems typically consist of reusable assets that have dependencies between each other. To specify such dependencies, feature models are commonly used. As feature models in practice are often complex, automated reasoning is…
We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal)…
We design a proof system for propositional classical logic that integrates two languages for Boolean functions: standard conjunction-disjunction-negation and binary decision trees. We give two reasons to do so. The first is…
In this paper we present methods for the synthesis of polynomial invariants for probabilistic transition systems. Our approach is based on martingale theory. We construct invariants in the form of polynomials over program variables, which…
A polynomial transform is the multiplication of an input vector $x\in\C^n$ by a matrix $\PT_{b,\alpha}\in\C^{n\times n},$ whose $(k,\ell)$-th element is defined as $p_\ell(\alpha_k)$ for polynomials $p_\ell(x)\in\C[x]$ from a list…
The minimization problem for propositional formulas is an important optimization problem in the second level of the polynomial hierarchy. In general, the problem is Sigma-2-complete under Turing reductions, but restricted versions are…
It is well-known that every quantified boolean formula (QBF) can be transformed into a prenex QBF whose only boolean operators are negation, conjunction, and disjunction. It is also well-known that the transformation is polynomial if the…
This work, shows how propositional resolution can be generalized to obtain a resolution proof system for constrained pseudo-propositional logic (CPPL), which is an extension resulted from inserting the natural numbers with few constraints…
The question whether a set of formulae G implies a formula f is fundamental. The present paper studies the complexity of the above implication problem for propositional formulae that are built from a systematically restricted set of Boolean…
This paper extends the dual calculus with inductive types and coinductive types. The paper first introduces a non-deterministic dual calculus with inductive and coinductive types. Besides the same duality of the original dual calculus, it…