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### Pebble Games with Algebraic Rules [chapter]

Anuj Dawar, Bjarki Holm
2012 Lecture Notes in Computer Science
The general framework we introduce includes as special cases the pebble games for finite-variable logics with and without counting.  ...  We define a general framework of partition games for formulating twoplayer pebble games over finite structures.  ...  It turns out that the games for ≡ L k and ≡ C k can be formulated as partition games, by replacing the algebraic rules of the matrix-equivalence game with weaker conditions that the partitions need to  ...

### Pebble games with algebraic rules [article]

Anuj Dawar, Bjarki Holm
2012 arXiv   pre-print
The general framework we introduce includes as special cases the pebble games for finite-variable logics with and without counting.  ...  We define a general framework of partition games for formulating two-player pebble games over finite structures.  ...  It turns out that the games for ≡ L k and ≡ C k can be formulated as partition games, by replacing the algebraic rules of the matrix-equivalence game with weaker conditions that the partitions need to  ...

### Relation algebras with n-dimensional relational bases

Robin Hirsch, Ian Hodkinson
2000 Annals of Pure and Applied Logic
We study relation algebras with n-dimensional relational bases in the sense of Maddux. Fix n with 3 ≤ n ≤ ω.  ...  Write B n for the class of non-associative algebras with an ndimensional relational basis, and RA n for the variety generated by B n .  ...  The game EF p r (A, B) is defined as follows. (Here, we use bold font for EF. Soon we will define a similar game, denoted EF .) It is a game played by ∀, ∃, with r rounds and p pairs of 'pebbles'.  ...

### Page 9538 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews
Summary: “Existential k-pebble games, k > 2, are combinato- rial games played between two players, called the Spoiler and the Duplicator, on two structures.  ...  Summary: “We define process algebras with a generalised oper- ation 5° for choice.  ...

### Page 3259 of Mathematical Reviews Vol. , Issue 81H [page]

1981 Mathematical Reviews
Authors’ summary: “We study a one-person game played by plac- ing pebbles, according to certain rules, on the vertices of a directed graph. J. E. Hopcroft, Paul and L.  ...  The pebbling problem is: Given a pebble game G and a specified vertex, v, in G, can v be pebbled using no more than s pebbles?  ...

### An Infinite Pebble Game and Applications

A.J. Kfoury, A.P. Stolboushkin
1997 Information and Computation
We scan this derivation in reverse order (starting with the last production), and play a pebble game as follows: If at the current step the derivation applies the rule S ;!  ...  Some of the generalizations of the game on nite dag's are the \black-and-white pebble game", the \pebble game with auxiliary pushdown stores", the \edging game", and the \parallel pebble game" (see 13]  ...

### Page 6454 of Mathematical Reviews Vol. , Issue 2004h [page]

2004 Mathematical Reviews
with exactly 3 pebbles.  ...  Contrary to the bounded case studied by Blum and Hewitt, the hierarchy collapses: there are families recognized with 0,1,2, and 3 pebbles, but each family recognized with more than three pebbles is recognized  ...

### Page 1194 of Mathematical Reviews Vol. , Issue 81C [page]

1981 Mathematical Reviews
The authors consider games where a set of pebbles is placed ona subset of nodes and the rules have the form “a pebble can be moved from occupied node x to unoccupied node z provided node y is also occupied  ...  As a one-player game, the player tries to manipulate the pebbles to reach a specified terminal node.  ...

### Constraint Satisfaction, Databases, and Logic

Phokion G. Kolaitis
2003 International Joint Conference on Artificial Intelligence
The existential k-pebble game (or, in short, the -pebble game) is played between two players, the Spoiler and the Duplicator, on two relational structures A and B according to the follow ing rules: each  ...  A different family of games, known as k-pebble games, has been used to study fixed-point logics and infinitary logics with finitely many variables (see [KV901.  ...  Moreover, they reveal that all tractable cases of the form CSP(A, All) can be solved by the same polynomial-time algorithm, namely, the algorithm for determining the winner in the -pebble game.  ...

### Page 5267 of Mathematical Reviews Vol. , Issue 86k [page]

1986 Mathematical Reviews
Paral- lelism in the first case corresponds to multiple processors and is modeled by the parallel pebble game.  ...  We show the effect of parallelism on the minimum space required to pebble a graph in the parallel pebble game and derive lower bounds on tradeoffs that must be satisfied for problems such as superconcen  ...

### Proof Complexity Meets Algebra

Albert Atserias, Joanna Ochremiak
2018 ACM Transactions on Computational Logic
This class is also known to coincide with the class of languages with Sums-of-Squares refutations of sublinear degree, a fact for which we provide an alternative proof.  ...  We hence ask for the existence of a natural proof system with good behaviour with respect to reductions and simultaneously small size refutations beyond bounded width.  ...  Then Spoiler wins the existential l-pebble game on A and B, and hence also the existential (k + 2)-pebble game on A and B, since k + 2 ≥ l.  ...

### Constraint Satisfaction, Complexity, and Logic [chapter]

Phokion G. Kolaitis
2004 Lecture Notes in Computer Science
k)-pebble game.  ...  k)-pebble game on A and B?  ...  Concluding Remarks Constraint Satisfaction is a meeting point of • Computational Complexity • Database Theory • Logic • Universal Algebra.  ...

### First-order axiomatisations of representable relation algebras need formulas of unbounded quantifier depth [article]

Rob Egrot, Robin Hirsch
2021 arXiv   pre-print
Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any first-order relation  ...  algebra theory of bounded quantifier depth.  ...  First a straightforward adaptation of the well known Ehrenfeucht-Fraïssé pebble game to relation algebras, and also a new colouring game which we call the Seurat game for sets, as it is a special case  ...

### Time and space complexity of reversible pebbling

Richard Královič
2004 RAIRO - Theoretical Informatics and Applications
Hence, the rules of the pebble game with black and white pebbles are following: Definition 2.11.  ...  The pebble game with black and white pebbles is a pebbling game that uses two kind of pebbles -black and white. One move of the game consists of applying one of the following rules: 1.  ...  Then G can be pebbled by a reversible computation with p · S(K) pebbles in time at most T(K). Proof.  ...

### Time and Space Complexity of Reversible Pebbling [chapter]

Richard Král'ovič
2001 Lecture Notes in Computer Science
Hence, the rules of the pebble game with black and white pebbles are following: Definition 2.11.  ...  The pebble game with black and white pebbles is a pebbling game that uses two kind of pebbles -black and white. One move of the game consists of applying one of the following rules: 1.  ...  Then G can be pebbled by a reversible computation with p · S(K) pebbles in time at most T(K). Proof.  ...
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