Syntactic approaches to negative results in process algebras and modal logics

Concurrency as a phenomenon is observed in most of the current computer science trends. However the inherent complexity of analyzing the behavior of such a system is incremented due to the many different models of concurrency, the variety of applications and architectures, as well as the wide spectrum of specification languages and demanded correctness criteria. For the scope of this thesis we focus on state based models of concurrent computation, and on modal logics as specification languages. First we study syntactically the process algebras that describe several different concurrent behaviors, by analyzing their equational theories. Here, we use well-established techniques from the equational logic of processes to older and newer setups, and then transition to the use of more general and novel methods for the syntactical analysis of models of concurrent programs and specification languages. Our main contributions are several positive and negative axiomatizability results over various process algebraic languages and equivalences, along with some complexity results over the satisfiability of multi-agent modal logic with recursion, as a specification language.Samhliða sem fyrirbæri sést í flestum núverandi tölvunarfræði stefnur. Hins vegar er eðlislægt flókið að greina hegðun slíks kerfis- tem er aukið vegna margra mismunandi gerða samhliða, fjölbreytileikans af forritum og arkitektúr, svo og breitt svið forskrifta mælikvarða og kröfðust réttmætisviðmiða. Fyrir umfang þessarar ritgerðar leggjum við áherslu á ástandsbundin líkön af samhliða útreikningum og á formlegum rökfræði sem forskrift tungumálum. Fyrst skoðum við setningafræðilega ferlialgebrurnar sem lýsa nokkrum mismunandi samhliða hegðun, með því að greina jöfnukenningar þeirra. Hér notum við rótgróin tækni mynda jöfnunarrökfræði ferla til eldri og nýrri uppsetningar, og síðan umskipti yfir í notkun almennari og nýrra aðferða fyrir setningafræðileg greining á líkönum samhliða forrita og forskriftartungumála. Helstu framlög okkar eru nokkrar jákvæðar og neikvæðar niðurstöður um axiomatizability yfir ýmis ferli algebrumál og jafngildi, ásamt nokkrum samSveigjanleiki leiðir af því að fullnægjanleiki fjölþátta formrökfræði með endurkomu, sem a forskrift tungumál.RANNIS: `Open Problems in the Equational Logic of Processes’ (OPEL) (grant No 196050-051) Reykjavik University research fund: `Runtime and Equational Verification of Concurrent Programs' (ReVoCoP) (grant No 222021

Representations and Completions for Ordered Algebraic Structures

The primary concerns of this thesis are completions and representations for various classes of poset expansion, and a recurring theme will be that of axiomatizability. By a representation we mean something similar to the Stone representation whereby a Boolean algebra can be homomorphically embedded into a field of sets. So, in general we are interested in order embedding posets into fields of sets in such a way that existing meets and joins are interpreted naturally as set theoretic intersections and unions respectively. Our contributions in this area are an investigation into the ostensibly second order property of whether a poset can be order embedded into a field of sets in such a way that arbitrary meets and/or joins are interpreted as set theoretic intersections and/or unions respectively. Among other things we show that unlike Boolean algebras, which have such a ‘complete’ representation if and only if they are atomic, the classes of bounded, distributive lattices and posets with complete representations have no first order axiomatizations (though they are pseudoelementary). We also show that the class of posets with representations preserving arbitrary joins is pseudoelementary but not elementary (a dual result also holds). We discuss various completions relating to the canonical extension, whose classical construction is related to the Stone representation. We claim some new results on the structure of classes of poset meet-completions which preserve particular sets of meets, in particular that they form a weakly upper semimodular lattice. We make explicit the construction of \Delta_{1}-completions using a two stage process involving meet- and join-completions. Linking our twin topics we discuss canonicity for the representation classes we deal with, and by building representations using a meet-completion construction as a base we show that the class of representable ordered domain algebras is finitely axiomatizable. Our method has the advantage of representing finite algebras over finite bases