Algebra of Concurrent Games

1 Introduction

The combination of games and logic (especially computational logic) always exists two ways [3]

: one is the usual usage of game theory to interpret logic, such as the well-known game semantics; the other is to use logic to understand game theory, such as game logic

[1] [2] [4]. The basic algebra of games [5] [6] is also a way to use logic to interpret games algebraically.

Since concurrency is a fundamental activity pattern in nature, it exists not only in computational systems, but also in any process in nature. Concurrent games [7] [8] [9] make the game theory to capture concurrency. But the logic of concurrent games is still missing.

In this paper, we extend the basic algebra of games [5] [6] to discuss the logic of concurrent games algebraically. This paper is organized as follows. In Section 2, we repeat the main concepts and conclusions of the basic algebra of games for the convenience of the reader. We give the algebra of concurrent games in Section 3. And finally, in Section 4, we conclude this paper.

2 Basic Algebra of Games

For the convenience of the readers, in this section, we repeat the definitions and main results of Basic Algebra of Games (abbreviated BAG) in [5] and [6], for they are used in the following sections.

Definition 2.1 (Game language).

The game language $G L$ consists of:

a set of atomic games $G_{a t} = {g_{a}}_{a \in A}$ , and a special idle atomic game $ι = g_{0} \in G_{a t}$ ;
game operations choice of first player $\lor$ , choice of second player $\land$ , dualization $^{d}$ and composition of games $\circ$ .

Atomic games and their duals are called literals.

Definition 2.2 (Game terms).

The game terms are defined inductively as follows:

every atomic game $g_{a}$ is a game term;
if $G, H$ are game terms, then $G^{d}, H^{d}$ , $G \lor H$ , $G \land H$ and $G \circ H$ are all game terms.

Definition 2.3 (Outcome conditions).

$G L = ⟨ S, {ρ_{a}^{i}}_{a \in A; i = 1, 2} ⟩$ are called game boards, where $S$ is the set of states and $ρ_{a}^{i} \subseteq S \times P (S)$ are outcome relations, which satisfy the following two forcing conditions:

monotonicity (MON): for any $s \in S$ , and $X \subseteq Y \subseteq S$ , if $s ρ_{a}^{i} X$ , then $s ρ_{a}^{i} Y$ ;
consistency (CON): for any $s \in S, X \subseteq S$ , if $s ρ_{a}^{1} X$ , then not $s ρ_{a}^{2} (S - X)$ .

And the following optional conditions:

termination (FIN): for any $s \in S$ , then $s ρ_{a}^{i} S$ , and the class of terminating game boards are denoted FIN;
determinacy (DET): $s ρ_{a}^{2} (S - X)$ iff $s ρ_{a}^{1} X$ , and the class of determined game boards are denoted DET.

The outcome relation $ρ_{G}^{i}$ for any game term $G$ can be defined inductively according to the structure of $G$ .

Definition 2.4 (Included).

For game terms $G_{1}$ and $G_{2}$ on game board $B$ , if $ρ_{G_{1}}^{i} \subseteq ρ_{G_{2}}^{i}$ , then $G_{1}$ is i-included in $G_{2}$ on $B$ , denoted $G_{1} \subseteq_{i} g_{2}$ ; if $G_{1} \subseteq_{1} G_{2}$ and $G_{1} \subseteq_{2} G_{2}$ , then $G_{1}$ is included in $G_{2}$ on $B$ , denoted $B ⊨ G_{1} ⪯ G_{2}$ ; if $B ⊨ G_{1} ⪯ G_{2}$ for any $B$ , then $G_{1} ⪯ G_{2}$ is called a valid term inclusion, denoted $⊨ G_{1} ⪯ G_{2}$ .

If $G_{1}$ and $G_{2}$ are assigned the same outcome relation in $B$ , then they are equivalent on $B$ , denoted $B ⊨ G_{1} = G_{2}$ ; if $B ⊨ G_{1} = G_{2}$ for any game board $B$ , then $G_{1} = G_{2}$ is a valid term identity, denoted $⊨ G_{1} = G_{2}$ .

The axioms of BAG are shown in Table 1.

No.	Axiom
$G 1$	$x \lor x = x x \land x = x$
$G 2$	$x \lor y = y \lor x x \land y = y \land x$
$G 3$	$x \lor (y \lor z) = (x \lor y) \lor z x \land (y \land z) = (x \land y) \land z$
$G 4$	$x \lor (x \land y) = x x \land (x \lor y) = x$
$G 5$	$x \lor (y \land z) = (x \lor y) \land (x \lor z) x \land (y \lor z) = (x \land y) \lor (x \land z)$
$G 6$	$(x^{d})^{d} = x$
$G 7$	$(x \lor y)^{d} = x^{d} \land y^{d} (x \land y)^{d} = x^{d} \lor y^{d}$
$G 8$	$(x \circ y) \circ z = x \circ (y \circ z)$
$G 9$	$(x \lor y) \circ z = (x \circ z) \lor (y \circ z) (x \land y) \circ z = (x \circ z) \land (y \circ z)$
$G 10$	$x^{d} \circ y^{d} = (x \circ y)^{d}$
$G 11$	$y ⪯ z \to x \circ y ⪯ x \circ z$
$G 12$	$x \circ ι = ι \circ x = x$
$G 13$	$x \lor ι = x x \land ι = x$

Table 1: Axioms of BAG

Definition 2.5 (Canonical terms).

The canonical terms can be defined as follows:

$ι$ and $g_{a}$ are canonical terms;
$⋁_{i \in I} ⋀_{k \in K_{i}} g_{i k} \circ G_{i k}$ is a canonical term, where $g_{i k}$ is a literal and $G_{i k}$ is a game term.

Theorem 2.6 (Elimination theorem 1).

Every game term $G$ is equivalent to a canonical game term in ${BAG}^{ι}$ .

Definition 2.7 (Isomorphic).

Two canonical terms $G$ and $H$ , if one can be obtained from the other by means of successive permutations of conjuncts (and disjuncts) within the same $⋀$ ’s (and $⋁$ ’s) in subterms.

Definition 2.8 (Embedding).

The embedding of canonical terms is defined inductively as follows:

$ι ↣ ι$ ;
if $g, h$ are literals and $G, H$ are canonical terms, $g \circ G$ embeds into $h \circ H$ iff $g = h$ and $G ↣ H$ ;
$⋀_{k \in K} g_{k} \circ G_{k}$ embeds into $⋀_{m \in M} h_{m} \circ H_{m}$ , if for every $m \in M$ there is some $k \in K$ such that $g_{k} \circ G_{k} ↣ h_{m} \circ H_{m}$ ;
for $G = ⋁_{i \in I} ⋀_{k \in K_{i}} g_{i k} \circ g_{i k}$ and $H = ⋁_{j \in J} ⋀_{m \in M_{j}} h_{j m} \circ H_{j m}$ , $G ↣ H$ iff every disjunct of $G$ embeds into some disjunct of $H$ .

Definition 2.9 (Minimal canonical terms).

The minimal canonical terms are defined inductively as follows:

$ι$ is a minimal canonical term;
for a canonical term $G = ⋁_{i \in I} ⋀_{k \in K_{i}} g_{i k} \circ G_{i k}$ with all $G_{i k}$ are minimal canonical terms, then $G$ is minimal canonical term if:
1. $ι^{d}$ does not occur in $G$ ;
2. None of $g_{i k}$ is $ι$ unless $G_{i k}$ is $ι$ ;
3. No conjunct occurring in $⋀_{k \in K} g_{i k} \circ G_{i k}$ embeds into another conjunct from the same conjunction;
4. No disjunct in $G$ embeds into another disjunct of $G$ .

Theorem 2.10 (Elimination theorem 2).

Every game term $G$ is equivalent to a minimal canonical game term in ${BAG}^{ι}$ .

Theorem 2.11 (Completeness theorem).

The minimal canonical terms $G$ and $H$ are equivalent iff they are isomorphic.

Proof.

See the proof in [5] and [6], the proof in [5] is based on modal logic and that in [6] is in a pure algebraic way. ∎

3 Algebra of Concurrent Games

Concurrent games [7] [8] [9] mean that players can play games simultaneously. For two atomic games $g_{a}$ and $g_{b}$ in parallel denoted $g_{a} ∥ g_{b}$ , the outcomes may be $g_{a} \circ g_{b}$ , or $g_{b} \circ g_{a}$ , or they are played simultaneously. Since $g_{a}$ and $g_{b}$ may be played simultaneously, $∥$ must be treated as fundamental game operation. In this section, we will add $∥$ game operation into the basic algebra of games, and the new formed algebra is called Algebra of Concurrent Games, abbreviated ACG.

Definition 3.1 (Game language with parallelism).

The new kind of game operation $∥$ is added into the game language in Definition 2.1.

Definition 3.2 (Game terms with parallelism).

If $G$ and $H$ are game terms, then $G ∥ H$ is also a game term in Definition 2.2.

The definitions of Outcome conditions and Included are the same as those in Section 2.

For concurrent games, the following axioms in Table 2 should be added into Table 1.

No.	Axiom
$C G 1$	$(x ∥ y) ∥ z = x ∥ (y ∥ z)$
$C G 2$	$g_{a} ∥ (g_{b} \circ y) = (g_{a} ∥ g_{b}) \circ y$
$C G 3$	$(g_{a} \circ x) ∥ g_{b} = (g_{a} ∥ g_{b}) \circ x$
$C G 4$	$(g_{a} \circ x) ∥ (g_{b} \circ y) = (g_{a} ∥ g_{b}) \circ (x ∥ y)$
$C G 5$	$(x \lor y) ∥ z = (x ∥ z) \lor (y ∥ z)$
$C G 6$	$x ∥ (y \lor z) = (x ∥ y) \lor (x ∥ z)$
$C G 7$	$(x \land y) ∥ z = (x ∥ z) \land (y ∥ z)$
$C G 8$	$x ∥ (y \land z) = (x ∥ y) \land (x ∥ z)$
$C G 9$	$(x ∥ y)^{d} = x^{d} ∥ y^{d}$
$C G 10$	$ι ∥ x = x$
$C G 11$	$x ∥ ι = x$

Table 2: Axioms of ACG

Definition 3.3 (Canonical terms with parallelism).

The canonical terms can be defined as follows:

$ι$ and $g_{a}$ are canonical terms;
$⋁_{i \in I} ⋀_{k \in K_{i}} | |_{l \in L} g_{i k l} \circ | |_{l \in L} G_{i k l}$ is a canonical term, where $g_{i k l}$ is a literal and $G_{i k l}$ is a game term.

Theorem 3.4 (Elimination theorem 1 with parallelism).

Every game term $G$ is equivalent to a canonical game term in ${ACG}^{ι}$ .

Proof.

Similarly to the proof of Theorem 2.6 in [5], it is sufficient to induct on the structure of game terms, and the new case is $∥$ . ∎

Definition 3.5 (Isomorphic with parallelism).

Two canonical terms $G$ and $H$ , if one can be obtained from the other by means of successive permutations of conjuncts (and disjuncts) within the same $⋀$ ’s (and $⋁$ ’s) in subterms.

Definition 3.6 (Embedding with parallelism).

The embedding of canonical terms is defined inductively as follows:

$ι ↣ ι$ ;
if $g_{1}, g_{2}, h_{1}, h_{2}$ are literals and $G, H$ are canonical terms, $(g_{1} ∥ g_{2}) \circ G$ embeds into $(h_{1} ∥ h_{2}) \circ H$ iff $g_{1} = h_{1}$ , $g_{2} = h_{2}$ , and $G ↣ H$ ;
if $g, h$ are literals and $G, H$ are canonical terms, $g \circ G$ embeds into $h \circ H$ iff $g = h$ and $G ↣ H$ ;
$⋀_{k \in K} g_{k} \circ G_{k}$ embeds into $⋀_{m \in M} h_{m} \circ H_{m}$ , if for every $m \in M$ there is some $k \in K$ such that $g_{k} \circ G_{k} ↣ h_{m} \circ H_{m}$ ;
for $G = ⋁_{i \in I} ⋀_{k \in K_{i}} g_{i k} \circ g_{i k}$ and $H = ⋁_{j \in J} ⋀_{m \in M_{j}} h_{j m} \circ H_{j m}$ , $G ↣ H$ iff every disjunct of $G$ embeds into some disjunct of $H$ .

Definition 3.7 (Minimal canonical terms with parallelism).

The minimal canonical terms are defined inductively as follows:

$ι$ is a minimal canonical term;
for a canonical term $G = ⋁_{i \in I} ⋀_{k \in K_{i}} | |_{l \in L_{k}} g_{i k l} \circ G_{i k l}$ with all $G_{i k l}$ are minimal canonical terms, then $G$ is minimal canonical term if:
1. $ι^{d}$ does not occur in $G$ ;
2. None of $g_{i k l}$ is $ι$ unless $G_{i k l}$ is $ι$ ;
3. No conjunct occurring in $⋀_{k \in K} | |_{l \in L_{k}} g_{i k l} \circ G_{i k l}$ embeds into another conjunct from the same conjunction;
4. No disjunct in $G$ embeds into another disjunct of $G$ .

Theorem 3.8 (Elimination theorem 2).

Every game term $G$ is equivalent to a minimal canonical game term in ${ACG}^{ι}$ .

Proof.

Similarly to the proof of Theorem 2.10 in [5], it is sufficient to induct on the structure of game terms according to the definition of minimal canonical game terms, the new case is $∥$ . ∎

Since the Elimination theorem holds, the completeness of ACG only need to discuss the relationship among the minimal canonical terms.

Theorem 3.9 (Completeness theorem).

The minimal canonical terms $G$ and $H$ are equivalent iff they are isomorphic.

Proof.

Similarly to the proof of Theorem 2.11 in [5]:

firstly, translate ACG to the same modal logic in [5];
secondly, prove the translation preserve validity;
Finally, get the completeness result.

The only difference is the translation of $∥$ , because two game terms $G_{1}$ and $G_{2}$ may be in race condition, denoted $G_{1} % G_{2}$ , for a game term $G$ and the corresponding modal logic formula $m (G)$ , the translation of $∥$ is:

if $G_{1} % G_{2}$ , then $m (G_{1} ∥ G_{2}) = m (G_{1}) (m (G_{2}))$ or $m (G_{1} ∥ G_{2}) = m (G_{2}) (m (G_{1}))$ ;
else $m (G_{1} ∥ G_{2}) = m (G_{1}) \lor m (G_{2})$ .

∎

4 Conclusions

We introduce parallelism in the basic algebra of games [5] [6] to model concurrent game algebraically. The resulted algebra ACG can be used reasoning parallel systems with game theory support.

References

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R. Parikh. (1985). The logic of games and its applications. Selected Papers of the International Conference on Foundations of Computation Theory on Topics in the Theory of Computation. Elsevier North-Holland, Inc.
[3] J. Van Benthem. (2014). Logic in Games. The MIT Press.
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[8] E. Asarin, O. Maler, A. Pnueli, and J. Sifakis. Controller synthesis for timed automata. (1998). In Proc. IFAC Symposium on System Structure and Control. Elsevier Science, 469-474.
[9] P. Bouyer, R. Brenguier, and N. Markey. (2010). Nash equilibria for reachability objectives in multi-player timed games. In Proc. 21th International Conference on Concurrency Theory (CONCUR 10), LNCS 6269, p. 192-206.

Algebra of Concurrent Games

Operational Semantics of Games

Truly Concurrent Bisimilarities are Game Equivalent

Ergodic Mean-Payoff Games for the Analysis of Attacks in Crypto-Currencies

EigenGame Unloaded: When playing games is better than optimizing

Fast Concurrent Data Sketches

Concurrent Kleene Algebra with Observations: from Hypotheses to Completeness

The Quantitative Collapse of Concurrent Games with Symmetry

1 Introduction

2 Basic Algebra of Games

Definition 2.1 (Game language).

Definition 2.2 (Game terms).

Definition 2.3 (Outcome conditions).

Definition 2.4 (Included).

Definition 2.5 (Canonical terms).

Theorem 2.6 (Elimination theorem 1).

Definition 2.7 (Isomorphic).

Definition 2.8 (Embedding).

Definition 2.9 (Minimal canonical terms).

Theorem 2.10 (Elimination theorem 2).

Theorem 2.11 (Completeness theorem).

Proof.

3 Algebra of Concurrent Games

Definition 3.1 (Game language with parallelism).

Definition 3.2 (Game terms with parallelism).

Definition 3.3 (Canonical terms with parallelism).

Theorem 3.4 (Elimination theorem 1 with parallelism).

Proof.

Definition 3.5 (Isomorphic with parallelism).

Definition 3.6 (Embedding with parallelism).

Definition 3.7 (Minimal canonical terms with parallelism).

Theorem 3.8 (Elimination theorem 2).

Proof.

Theorem 3.9 (Completeness theorem).

Proof.

4 Conclusions

References

Algebra of Concurrent Games

Related Research

Operational Semantics of Games

Truly Concurrent Bisimilarities are Game Equivalent

Ergodic Mean-Payoff Games for the Analysis of Attacks in Crypto-Currencies

EigenGame Unloaded: When playing games is better than optimizing

Fast Concurrent Data Sketches

Concurrent Kleene Algebra with Observations: from Hypotheses to Completeness

The Quantitative Collapse of Concurrent Games with Symmetry

1 Introduction

2 Basic Algebra of Games

Definition 2.1 (Game language).

Definition 2.2 (Game terms).

Definition 2.3 (Outcome conditions).

Definition 2.4 (Included).

Definition 2.5 (Canonical terms).

Theorem 2.6 (Elimination theorem 1).

Definition 2.7 (Isomorphic).

Definition 2.8 (Embedding).

Definition 2.9 (Minimal canonical terms).

Theorem 2.10 (Elimination theorem 2).

Theorem 2.11 (Completeness theorem).

Proof.

3 Algebra of Concurrent Games

Definition 3.1 (Game language with parallelism).

Definition 3.2 (Game terms with parallelism).

Definition 3.3 (Canonical terms with parallelism).

Theorem 3.4 (Elimination theorem 1 with parallelism).

Proof.

Definition 3.5 (Isomorphic with parallelism).

Definition 3.6 (Embedding with parallelism).

Definition 3.7 (Minimal canonical terms with parallelism).

Theorem 3.8 (Elimination theorem 2).

Proof.

Theorem 3.9 (Completeness theorem).

Proof.

4 Conclusions

References