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 wellknown 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 consists of:

a set of atomic games , and a special idle atomic game ;

game operations choice of first player , choice of second player , dualization and composition of games .
Atomic games and their duals are called literals.
Definition 2.2 (Game terms).
The game terms are defined inductively as follows:

every atomic game is a game term;

if are game terms, then , , and are all game terms.
Definition 2.3 (Outcome conditions).
are called game boards, where is the set of states and are outcome relations, which satisfy the following two forcing conditions:

monotonicity (MON): for any , and , if , then ;

consistency (CON): for any , if , then not .
And the following optional conditions:

termination (FIN): for any , then , and the class of terminating game boards are denoted FIN;

determinacy (DET): iff , and the class of determined game boards are denoted DET.
The outcome relation for any game term can be defined inductively according to the structure of .
Definition 2.4 (Included).
For game terms and on game board , if , then is iincluded in on , denoted ; if and , then is included in on , denoted ; if for any , then is called a valid term inclusion, denoted .
If and are assigned the same outcome relation in , then they are equivalent on , denoted ; if for any game board , then is a valid term identity, denoted .
The axioms of BAG are shown in Table 1.
No.  Axiom 

Definition 2.5 (Canonical terms).
The canonical terms can be defined as follows:

and are canonical terms;

is a canonical term, where is a literal and is a game term.
Theorem 2.6 (Elimination theorem 1).
Every game term is equivalent to a canonical game term in .
Definition 2.7 (Isomorphic).
Two canonical terms and , 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:

;

if are literals and are canonical terms, embeds into iff and ;

embeds into , if for every there is some such that ;

for and , iff every disjunct of embeds into some disjunct of .
Definition 2.9 (Minimal canonical terms).
The minimal canonical terms are defined inductively as follows:

is a minimal canonical term;

for a canonical term with all are minimal canonical terms, then is minimal canonical term if:

does not occur in ;

None of is unless is ;

No conjunct occurring in embeds into another conjunct from the same conjunction;

No disjunct in embeds into another disjunct of .

Theorem 2.10 (Elimination theorem 2).
Every game term is equivalent to a minimal canonical game term in .
Theorem 2.11 (Completeness theorem).
The minimal canonical terms and are equivalent iff they are isomorphic.
3 Algebra of Concurrent Games
Concurrent games [7] [8] [9] mean that players can play games simultaneously. For two atomic games and in parallel denoted , the outcomes may be , or , or they are played simultaneously. Since and 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 and are game terms, then is also a game term in Definition 2.2.
The definitions of Outcome conditions and Included are the same as those in Section 2.
No.  Axiom 

Definition 3.3 (Canonical terms with parallelism).
The canonical terms can be defined as follows:

and are canonical terms;

is a canonical term, where is a literal and is a game term.
Theorem 3.4 (Elimination theorem 1 with parallelism).
Every game term is equivalent to a canonical game term in .
Proof.
Definition 3.5 (Isomorphic with parallelism).
Two canonical terms and , 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:

;

if are literals and are canonical terms, embeds into iff , , and ;

if are literals and are canonical terms, embeds into iff and ;

embeds into , if for every there is some such that ;

for and , iff every disjunct of embeds into some disjunct of .
Definition 3.7 (Minimal canonical terms with parallelism).
The minimal canonical terms are defined inductively as follows:

is a minimal canonical term;

for a canonical term with all are minimal canonical terms, then is minimal canonical term if:

does not occur in ;

None of is unless is ;

No conjunct occurring in embeds into another conjunct from the same conjunction;

No disjunct in embeds into another disjunct of .

Theorem 3.8 (Elimination theorem 2).
Every game term is equivalent to a minimal canonical game term in .
Proof.
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 and are equivalent iff they are isomorphic.
Proof.

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 and may be in race condition, denoted , for a game term and the corresponding modal logic formula , the translation of is:

if , then or ;

else .
∎
4 Conclusions
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