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19
Step by Step  Building Representations in Algebraic Logic
 Journal of Symbolic Logic
, 1995
"... We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defini ..."
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Cited by 28 (15 self)
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We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterised according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Countable relation algebras with homogeneous representations are characterised by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is !categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another twoplayer game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this ap...
Canonical Varieties with No Canonical Axiomatisation
 Trans. Amer. Math. Soc
, 2003
"... We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or firstorder sentences. We then show that the variety RRA of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every ..."
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Cited by 12 (7 self)
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We give a simple example of a variety V of modal algebras that is canonical but cannot be axiomatised by canonical equations or firstorder sentences. We then show that the variety RRA of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many noncanonical sentences. Using probabilistic methods...
Relation Algebras with nDimensional Relational Bases
 Annals of Pure and Applied Logic
, 1999
"... We study relation algebras with ndimensional relational bases in the sense of Maddux. Fix n with 3 n !. Write Bn for the class of nonassociative algebras with an n dimensional relational basis, and RAn for the variety generated by Bn . We de ne a notion of representation for algebras in RAn , ..."
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Cited by 9 (2 self)
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We study relation algebras with ndimensional relational bases in the sense of Maddux. Fix n with 3 n !. Write Bn for the class of nonassociative algebras with an n dimensional relational basis, and RAn for the variety generated by Bn . We de ne a notion of representation for algebras in RAn , and use it to give an explicit (hence recursive) equational axiomatisation of RAn , and to reprove Maddux's result that RAn is canonical. We show that the algebras in Bn are precisely those that have a complete representation. Then we prove that whenever 4 n < l !, RA l is not nitely axiomatisable over RAn . This con rms a conjecture of Maddux. We also prove that Bn is elementary for n = 3; 4 only.
Finite Variable Logics
, 1993
"... In this survey article we discuss some aspects of finite variable logics. We translate some wellknown fixedpoint logics into the infinitary logic L ! 1! , discussing complexity issues. We give a game characterisation of L ! 1! , and use it to derive results on Scott sentences. In this conne ..."
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Cited by 7 (0 self)
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In this survey article we discuss some aspects of finite variable logics. We translate some wellknown fixedpoint logics into the infinitary logic L ! 1! , discussing complexity issues. We give a game characterisation of L ! 1! , and use it to derive results on Scott sentences. In this connection we consider definable linear orderings of types realised in finite structures. We then show that the Craig interpolation and Beth definability properties fail for L ! 1! . Finally we examine some connections of finite variable logic to temporal logic. Credits and references are given throughout. 1 Some extensions of firstorder logic Quisani: Hello. Who are you? I am Yuri's imaginary student, and I usually talk to him at this time. Author: I'm afraid he may be a bit late. I am a computer scientist from London, England. I have some imaginary students myself, so maybe I can help. I was reading your earlier conversation on 01 laws [Gu3]. Quisani: I remember it. We examined the...
Combinatorial Game Theory Foundations Applied to Digraph Kernels
 Electronic Journal Combinatorics
, 1996
"... Known complexity facts: the decision problem of the existence of a kernel in a digraph G =( V,E) is NPcomplete; if all of the cycles of G have even length, then G has a kernel; and the question of the number of kernels is #Pcomplete even for this restricted class of digraphs. In the opposite direc ..."
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Cited by 7 (1 self)
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Known complexity facts: the decision problem of the existence of a kernel in a digraph G =( V,E) is NPcomplete; if all of the cycles of G have even length, then G has a kernel; and the question of the number of kernels is #Pcomplete even for this restricted class of digraphs. In the opposite direction, we construct game theory tools, of independent interest, concerning strategies in the presence of draw positions, to show how to partition V ,inO ( E ) time, into 3 subsets S 1 ,S 2 ,S 3 , such that S 1 lies in all the kernels; S 2 lies in the complements of all the kernels; and on S 3 the kernels may be nonunique. Thus, in particular, digraphs with a "large" number of kernels are those in which S 3 is "large"; possibly S 1 = S 2 = #.
Methods for the transformation of ωautomata: Complexity and connection to secondorder logic
, 1998
"... ..."
Strongly representable atom structures of relation algebras
 PROC. AMER. MATH. SOC
, 2001
"... A relation algebra atom structure α is said to be strongly representable if all atomic relation algebras with that atom structure are representable. This is equivalent to saying that the complex algebra Cm α is a representable relation algebra. We show that the class of all strongly representable r ..."
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Cited by 3 (3 self)
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A relation algebra atom structure α is said to be strongly representable if all atomic relation algebras with that atom structure are representable. This is equivalent to saying that the complex algebra Cm α is a representable relation algebra. We show that the class of all strongly representable relation algebra atom structures is not closed under ultraproducts and is therefore not elementary. This answers a question of Maddux (1982). Our proof is based on the following construction. From an arbitrary undirected, loopfree graph Γ, we construct a relation algebra atom structure α(Γ) and prove, for infinite Γ, that α(Γ) is strongly representable if and only if the chromatic number of Γ is infinite. A construction of Erdös shows that there are graphs Γr (r <ω) with infinite chromatic number, with a nonprincipal ultraproduct � D Γr whose chromatic number is just two. It follows that α(Γr) is strongly representable (each r<ω) but � D (α(Γr)) is not.
A Fractal which violates the Axiom of Determinacy
, 1994
"... By use of the axiom of choice I construct a symmetrical and selfsimilar subset A ` [0; 1] ` R. Then by an elementary strategy stealing argument it is shown that A is not determined. The (possible) existence of fractals like A clarifies the status of the controversial Axiom of Determinacy. ..."
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Cited by 2 (0 self)
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By use of the axiom of choice I construct a symmetrical and selfsimilar subset A ` [0; 1] ` R. Then by an elementary strategy stealing argument it is shown that A is not determined. The (possible) existence of fractals like A clarifies the status of the controversial Axiom of Determinacy.
A MultiAgent GraphGame Approach to Theoretical Foundations of Linguistic Geometry
 Proc. of the Second World Conference on the Fundamentals of Artificial Intelligence (WOCFAI 95
, 1995
"... The Linguistic Geometry (LG) approach to discrete systems was introduced by B. Stilman in early 80s. It employed competing/cooperating agents for modeling and controlling of discrete systems. The approach was applied to a variety of problems with huge state spaces including control of aircraft, batt ..."
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Cited by 1 (0 self)
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The Linguistic Geometry (LG) approach to discrete systems was introduced by B. Stilman in early 80s. It employed competing/cooperating agents for modeling and controlling of discrete systems. The approach was applied to a variety of problems with huge state spaces including control of aircraft, battlefield robots, and chess. One of the key innovations of LG is the use of almost winning strategies, rather than truly winning strategies for the participating agents. There are many cases where the winning strategies have so high time complexity that they are not computable in practice, whereas the almost winning strategies can be applied and they beat the opposing agent almost guaranteed. Independently of LG the idea of competing/ cooperating agents was employed in the late 80s by A. Nerode, A. Yakhnis, and V. Yakhnis (NYY) within their approach to modeling concurrent systems and, more recently, within the “Strategy Approach to Hybrid Systems ” developed for continuous systems by A. Nerode, W. Kohn, A. Yakhnis, and others.
Blackwell Games
, 1996
"... Blackwell games are infinite games of imperfect information. The two players simultaneously make their moves and are then informed of each other's moves. Payoff is determined by a Borel measurable function f on the set of possible resulting sequences of moves. A standard result in Game Theory is th ..."
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Cited by 1 (0 self)
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Blackwell games are infinite games of imperfect information. The two players simultaneously make their moves and are then informed of each other's moves. Payoff is determined by a Borel measurable function f on the set of possible resulting sequences of moves. A standard result in Game Theory is that finite games of this type are determined. Blackwell proved that infinite games are determined, but only for the cases where the payoff function is the indicator function of an open or G ffi set [2, 3]. For games of perfect information, determinacy has been proven for games of arbitrary Borel complexity [6, 7, 8]. In this paper I prove the determinacy of Blackwell games over a G ffioe set, in a manner similar to Davis' proof of determinacy of games of G ffioe complexity of perfect information [5]. There is also extensive literature about the consequences of assuming AD, the axiom that all such games of perfect information are determined [9, 11]. In the final section of this paper I formula...