Equational theory of ordinals with addition and left multiplication by $ω$

Cantor's Views on the Foundations of Mathematics

The class of Horn clause sets in propositional logic is extended to a larger class for which the satisfiability problem can still be solved by unit resolution in linear time. It is shown that to every arborescence there corresponds a family of extended Horn sets, where ordinary Horn sets correspond to stars with a root at the center. These results derive from a theorem of Chandresekaran that characterizes when an integer solution of a system of inequalities can be found by rounding a real solution in a certain way. A linear-time procedure is provided for identifying “hidden” extended Horn sets (extended Horn but for complementation of variables) that correspond to a specified arborescence. Finally, a way to interpret extended Horn sets in applications is suggested.

Guarded Kleene Algebra with Tests (GKAT) is a variation on Kleene Algebra with Tests (KAT) that arises by restricting the union (+) and iteration (*) operations from KAT to predicate-guarded versions. We develop the (co)algebraic theory of GKAT and show how it can be efficiently used to reason about imperative programs. In contrast to KAT, whose equational theory is PSPACE-complete, we show that the equational theory of GKAT is (almost) linear time. We also provide a full Kleene theorem and prove completeness for an analogue of Salomaa’s axiomatization of Kleene Algebra.

We consider two standard notions in formal security protocol analysis: message deducibility and static equivalence under equational theories. We present new polynomial-time algorithms for deciding both notions under subterm convergent equational theories and under a theory representing symmetric encryption with the prex property. For these equational theories, polynomial-time algorithms for the decision problems associated to both notions are well-known (although this has not been proven for static equivalence under the prex theory). However, our algorithms have a signicantly better asymptotic complexity than existing approaches. As an application, we use our algorithm for static equivalence to discover off-line guessing attacks on the Kerberos protocol when implemented using a symmetric encryption scheme for which the prex property holds.

This paper contributes to the study of the equational theory of the semantics in van Glabbeek’s linear time - branching time spectrum over the language BCCSP, a basic process algebra for the description of finite synchronization trees. It offers an algorithm for producing a complete (respectively, ground-complete) equational axiomatization of any behavioral congruence lying between ready simulation equivalence and partial traces equivalence from a complete (respectively, ground-complete) inequational axiomatization of its underlying precongruence—that is, of the precongruence whose kernel is the equivalence. The algorithm preserves finiteness of the axiomatization when the set of actions is finite.

This paper contributes to the study of the equational theory of the semantics in van Glabbeek's linear time - branching time spectrum over the language BCCSP, a basic process algebra for the description of finite synchronization trees. It offers an algorithm for producing a complete (respectively, ground-complete) equational axiomatization of a behavioral congruence lying between ready simulation equivalence and partial traces equivalence from a complete (respectively, ground-complete) inequational axiomatization of its underlying precongruence--that is, of the precongruence whose kernel is the equivalence. The algorithm preserves finiteness of the axiomatization when the set of actions is finite. It follows that each equivalence in the spectrum whose discriminating power lies in between that of ready simulation and partial traces equivalence is finitely axiomatizable over the language BCCSP if so is its defining preorder.

We give an algorithm for deciding E-unification problems for linear standard equational theories (linear equations with all shared variables at a depth less than two) and varity 1 goals (linear equations with no shared variables). We show that the algorithm halts in quadratic time for the non-uniform E-unification problem, and linear time if the equational theory is varity 1. The algorithm is still polynomial for the uniform problem. The size of the complete set of unifiers is exponential, but membership in that set can be determined in polynomial time. For any goal (not just varity 1) we give a NEXPTIME algorithm.

Finite model and counter model generation is a potential alternative in automated theorem proving. In this paper, we introduce a system called FMSET which generates finite structures representing models of equational theories. FMSET performs a satisfiability test over a set of special first order clauses called “simple clauses”. The algorithm's originality stems from the combination of a standard enumeration technique and the use of first order resolution. Our objective is to obtain more propagations and still achieve good space and time complexity. Several experiments over a variety of problems have been pursued. FMSET uses symmetry to prune from the search tree unwanted isomorphic branches.

A formalisation of nominal α-equivalence with A, C, and AC function symbols

Ordinal Conditional Functions (OCFs) are one of the predominant frameworks to define belief change operators. In his original paper Spohn defines OCFs as functions from the set of worlds to the set of ordinals. But in subsequent paper by Spohn and others, OCFs are just used as functions from the set of worlds to natural numbers (plus eventually + ***). The use of transfinite ordinals in this framework has never been studied. This paper opens this way. We study generalisations of transmutations operators to transfinite ordinals. Using transfinite ordinals allows to represent different of beliefs, that naturally appear in real applications. This can be viewed as a generalisation of the usual two levels of beliefs framework: knowledge versus beliefs ; or rules base versus facts base, issued from expert systems works.

Expansion is an operation on typings pairs of type environments and result types in type systems for the λ-calculus. Expansion was originally introduced for calculating possible typings of a λ-term in systems with intersection types. This paper aims to clarify expansion and make it more accessible to a wider community by isolating the pure notion of expansion on its own, independent of type systems and types, and thereby make it easier for non-specialists to obtain type inference with flexible precision by making use of theory and techniques that were originally developed for intersection types. We redefine expansion as a simple algebra on terms with variables, substitutions, composition, and miscellaneous constructors such that the algebra satisfies 8 simple axioms and axiom schemas: the 3 standard axioms of a monoid, 4 standard axioms or axiom schemas of substitutions including one that corresponds to the usual “substitution lemma” that composes substitutions, and 1 very simple axiom schema for expansion itself. Many of the results on the algebra have also been formally checked with the Coq proof assistant. We then take System E, a λ-calculus type system with intersection types and expansion variables, and redefine it using the expansion algebra, thereby demonstrating how a single uniform notion of expansion can operate on both typings and proofs. Because we present a simplified version of System E omitting many features, this may be independently useful for those seeking an easier-to-comprehend version.

We compare two kinds of unification problems: Asymmetric Unification and Disunification, which are variants of Equational Unification. Asymmetric Unification is a type of Equational Unification where the instances of the right-hand sides of the equations are in normal form with respect to the given term rewriting system. In Disunification we solve equations and disequations with respect to an equational theory for the case with free constants. We contrast the time complexities of both and show that the two problems are incomparable: there are theories where one can be solved in polynomial time while the other is NP-hard. This goes both ways. The time complexity also varies based on the termination ordering used in the term rewriting system.

We present a new manifestation of Gödel’s second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert’s program. Specifically, we consider a proper extension of Peano arithmetic ( $\mathbf {PA}$ ) by a mathematically meaningful axiom scheme that consists of $\Sigma ^0_2$ -sentences. These sentences assert that each computably enumerable ( $\Sigma ^0_1$ -definable without parameters) property of finite binary trees has a finite basis. Since this fact entails the existence of polynomial time algorithms, it is relevant for computer science. On a technical level, our axiom scheme is a variant of an independence result due to Harvey Friedman. At the same time, the meta-mathematical properties of our axiom scheme distinguish it from most known independence results: Due to its logical complexity, our axiom scheme does not add computational strength. The only known method to establish its independence relies on Gödel’s second incompleteness theorem. In contrast, Gödel’s theorem is not needed for typical examples of $\Pi ^0_2$ -independence (such as the Paris–Harrington principle), since computational strength provides an extensional invariant on the level of $\Pi ^0_2$ -sentences.

Of all the branches of mathematics that we could choose to formalize, arithmetic is no doubt the most natural choice. This is what we undertake in the present chapter. In Section 6.1, we describe the language of arithmetic and present the set of its axioms, commonly known as Peano’s axioms, which we denote by P. The purpose of some of these axioms ( A1 through A7) is to force addition and multiplication to behave correctly; the others (the axiom scheme IS) are to sanction the well known proofs by induction. Superficially, these are very simple axioms and we could even ask ourselves whether they are not too simple.

Characterizing the interpretation of set theory in Martin-Löf type theory