Automated proof search system for logic of correlated knowledge

The automated proof search system and decidability for logic of correlated knowledge is presented in this paper. The core of the proof system is the sequent calculus with the properties of soundness, completeness, admissibility of cut and structural rules, and invertibility of all rules. The proof search procedure based on the sequent calculus performs automated terminating proof search and allows us to achieve decision result for logic of correlated knowledge.


Introduction
Information about quantum systems can be handled using logical calculi. From historical point of view the research of the area went in two main directions [20]. The first was originated by J. von Neumann and G. Birkhoff [7], introducing the ideas of quantum logic. However some important impossibility results were obtained [1,19]. D. Aerts, C. Randall and D. Foulis showed that quantum logic rises problems when trying to describe compound systems consisting of more than one elementary particle that can exhibit quantum entanglement. The other direction was the Mackey-Piron way [13,14,18] -the research on an axiomatic system that can be represented as the logic of projection operators on a generalized Hilbert space. One of the latest development in this way is quantum modal logics. The research have been done by A. Baltag and S. Smets [4,5], F. Boge [8], V. Vilasini [22] and N. Nurgalieva [17]. Our study covers field of logic of correlated knowledge which is part of quantum modal logics. More approaches on logic and quantum mechanics you can find in [2,6,9].
Logic of correlated knowledge (LCK) has been introduced by A. Baltag and S. Smets in [3]. LCK is an epistemic logic enriched by observational capabilities of agents. Applications of the epistemic logic cover fields such as distributed systems, merging of knowledge bases, robotics or network security in computer science and artificial intelligence. By adding observational capabilities to agents, logic of correlated knowledge can be applied to reason about systems where knowledge correlate between spatially distributed parts of the system. This includes any social system, quantum system, distributed information system, traffic light system or any other system where knowledge is correlated.
Quantum system may consist of one or more elementary particles. Associating agent to each particle, we get multiagent system, where agents can perform observations and get results. Allowing communication between agents, correlations such as quantum entanglement can be extracted. This can not be done by traditional epistemic logic or logic of distributed knowledge.
Our main scientific result is proof search system GS-LCK-PROC for logic of correlated knowledge, which lets to reason about knowledge automatically. The core of the system is the sequent calculus GS-LCK with the properties of soundness, completeness, admissibility of cut and structural rules, and invertibility of all rules. The ideas of semantic internalization, suggested by Sara Negri in [15], are used to get such properties for the calculus. The calculus provides convenient means for backward proof search and decision procedure for logic of correlated knowledge. The procedure generates a finite model for each sequent. As a result we get termination of the proof search and decidability of logic of correlated knowledge.
We start by defining syntax, semantics, and the Hilbert style proof system for logic of correlated knowledge in section 2. In section 3 we present Gentzen style sequent calculus GS-LCK and the properties of the proof system. Soundness and completeness of the GS-LCK and the properties of admissibility of weakening, contraction and cut are proved in sections 4, 5 and 6 . Automated proof search system GS-LCK-PROC and decidability of logic of correlated knowledge are presented in the final section 7.

Syntax
Consider a set N = {a 1 , a 2 , ..., a n } of agents. Each agent can perform its local observations. Given sets O a1 , ..., O an of possible observations for each agent, a joint observation is a Each agent can know some information, and it is written as K a1 A or K {a1} A, which means that the agent a 1 knows A. A group of agents can also know some information and this is denoted by K {a1,a2,a3} A or K I A, where I = {a 1 , a 2 , a 3 }. A more detailed description about the knowledge operator K is given in [10,21].
Syntax of logic of correlated knowledge is defined as follows: Definition 1(Syntax of logic of correlated knowledge) The language of logic of correlated knowledge has the following syntax: Where p is any atomic proposition, o = (o a ) a∈I ∈ O I , r ∈ R, and I ⊆ N .

Semantics
Consider a system, composed of N components or locations. Agents can be associated to locations, where they will perform observations. States (configurations) of the system are functions s : O a1 ×...×O an → R or s I : O I → R, where I ⊆ N and a set of results R is in the structure (R, Σ) together with an abstract operation Σ : P(R) → R of composing results. The operation Σ maybe partial (defined only for some subsets A ⊆ R), but it is required to satisfy the condition: If s and t are two possible states of the system and a group of agents I can make exactly the same observations in these two states, then these states are observationally equivalent to I, and it is written as s I ∼ t. Observational equivalence is defined as follows: Definition 2(Observational equivalence) Two states s and t are observationally equivalent s I ∼ t iff s I = t I . A model of logic of correlated knowledge is a multi-modal Kripke model [12], where the relations between states mean observational equivalence. It is defined as: Definition 3Model of logic of correlated knowledge The satisfaction relation |= for model M , state s and formulas o r and K I A is defined as follows: The formula K I A means that the group of agents I carries the information that A is the case, and o r means that r is the result of the joint observation o.
If formula A is true in any state of any model, then it is named as a valid formula.

Hilbert Style Salculus HS-LCK
Alexandru Baltag and Sonja Smets defined the Hilbert style calculus for logic of correlated knowledge in [3]. Fixing a finite set N = {a 1 , ..., a n } of agents, a finite result structure H10. ∧  Sets I, J may be empty in axioms H4 -H8 and in rule (K I − necessitation).
The Hilbert style calculus HS-LCK for logic of correlated knowledge is sound and complete with respect to correlation models over (R, Σ, O) [3].

Gentzen Style Sequent Calculus GS-LCK
Gerhard Gentzen introduced sequent calculus in 1934 [11]. Sequents in the system GS-LCK are statements of the form Γ ⇒ ∆, where Γ and ∆ are finite, possibly empty multisets of relational atoms s I ∼ t and labelled formulas s : A, where s, t ∈ S, I ⊆ N and A is any formula in the language of logic of correlated knowledge. The formula s : A means s |= A, and s I ∼ t is an observational equivalence or relation between the states in the model of logic of correlated knowledge.
The sequent calculus consists of axioms and rules. Applying rules to the sequents, a proof-search tree for the root sequent is constructed. If axioms are in all the leaves of the proof-search tree, then the root sequent is called as a provable sequent and ∆ follows from Γ of the root sequent.

Propositional rules:
3. Knowledge rules: t : A, s : The rule (K I ⇒) requires that I = N and t : A be not in Γ. The rule (⇒ K I ) requires that I = N and t be not in the conclusion. Set I maybe an empty set in both rules.

Observational rules: s
The rule (OE) requires that I = ∅ and formulas s The rule (OY R) requires: (a) s : o r I be not in Γ for all r ∈ R and s : o r1 I be in ∆ for some r 1 ∈ R.
The rule (CR) requires that s : e

Substitution rules:
The rules (Sub(p) ⇒) and (Sub(o r ) ⇒) require that s : p and s : o r be not in Γ, accordingly.
6. Relational rules: s The rule (Ref ) requires that s be in the conclusion and s I ∼ s be not in Γ. The rule (T rans) requires that s The rule (M on) stands for monotonicity and requires that I ⊆ J. Sets I, J may be empty. The rules (Eucl) and (M on) require that s I ∼ t and s I ∼ t be not in Γ, accordingly.
The sequent calculus GS-LCK is sound and complete with respect to correlation models over (R, Σ, O). It also has the beautiful properties of rule invertibility and admissibility of the cut and structural rules. It is crucial in making the automated proof system in the present paper. Theorem 1(Properties of GS-LCK) The sequent calculus GS-LCK has the following properties: 1) Invertibility of rules.

5) Termination.
Proofs of soundness, completeness, and the properties of GS-LCK are given in the next sections.

Proof of Soundness of GS-LCK
Definition 4(Extended syntax) Extended syntax of LCK is as follows: where p is any atomic proposition, o ∈ O I , I ⊆ N, r ∈ R and s, t ∈ S.
Definition 7 (Sequent without labels and relational atoms) If Seq is a sequent, then the sequent without labels and relational atoms of Seq is obtained removing all labels near formulas and all relational atoms from Seq.

Lemma 1 (Validity of the formula of the sequent)
If the formula of the sequent Seq is valid, then the formula of the sequent Seq without labels and relational atoms is valid, as well.
2. Propositional rules as in [16]. 3. Knowledge rules: We prove by contraposition that, if the formula of the premise (t : A, s : The formula of the conclusion (s : and t is not in the conclusion. 3) The validity of the rules (K N ⇒) and (⇒ K N ) is proved in the same way.

Observational rules:
1) Rule (OY R): If R is a set of results, and o is a joint observation, then there exists a result r ∈ R that o r is true. If there exists r that o r is true and all formulas in Γ are true and all formulas in ∆ are false, then one formula of premises ({s : o r , Γ ⇒ ∆} r∈R ) is false.
3) The soundness of rules (OE), ( Sub(p) ⇒) and ( Sub(o r ) ⇒) is proved in the same way.
5. Relational rules: The contraposition follows from condition to models of LCK: 2. If I ⊆ J then J ∼⊆ I ∼.
2) The validity of rules (Ref ), (T rans) and (Eucl) is proved in the same way.
We have proved the validity of all axioms and soundness of all the rules of GS-LCK. The statement of the theorem follows from lemma 4. 2. The cases of the remaining rules are considered similarly.

Lemma 3 (Substitution)
If a sequent (Γ ⇒ ∆) is provable in GS-LCK, then the sequent (Γ(t/s) ⇒ ∆(t/s)) is also provable with the same bound of the height of the proof in GS-LCK.
proof Lemma is proved by induction on the height < h > of the proof of the sequent (Γ ⇒ ∆).
< h > 1 > 1. The rule (⇒ K I ) was applied in the last step of the proof of the sequent.

2) Substitution (l/t).
There is no label t in the sequent Γ ⇒ ∆, s : K I A because of the requirement of the application of the rule (⇒ K I ) that t is a new label.
3) Substitution (l/s) and l = t.  proof Theorem is proved by induction on the height < h > of the proof of the sequent (Γ ⇒ ∆).
< h > 1 > 1) The rule (⇒ K I ) was applied in the last step of the proof of the sequent.
A new label t for the application of the rule (⇒ K I ) is in Π or Λ.
By Lemma , the sequent (s I ∼ t, Γ ⇒ ∆, t : A) with substitution (l/t) is provable. By the induction hypothesis, the sequent (s Here l is a new label, absent in Π, Γ, ∆ and Λ. The sequent of the theorem is proved by applying the rule (⇒ K I ): The new label t for application of the rule (⇒ K I ) is absent in Π or Λ. By the induction hypothesis, the sequent (s The sequent of the theorem is proved by applying the rule (⇒ K I ): 2) The cases of the remaining rules are considered similarly.

Theorem 4 (Invertibility of rules)
All the rules of GS-LCK are invertible with the same bound of the height of the proof.
proof Theorem is proved for each rule separately. The rule (K I ⇒) t : A, s : Invertibility is proved by induction on the height < h > of the proof of the sequent of the conclusion of the rule (K I ⇒).  2) The cases of the remaining rules are considered similarly.
proof Theorem is proved by induction on the ordered tuple pair < c, h >, where c is the complexity of formula F , and h is the sum of heights of the proof of the sequents (Γ ⇒ ∆, F ) and (F, Π ⇒ Λ).
< c ≥ 1, h = 2 > The sequents (Γ ⇒ ∆, F ) and (F, Π ⇒ Λ) are the axioms. If formula F is not principal in one at least of the sequents, then (Π, Γ ⇒ ∆, Λ) is an axiom. If formula F is principal in both sequents, then F should be in Γ and ∆ or only in Γ (the case where the axiom is of type s : o r1 , s : o r2 , Γ ⇒ ∆). Therefore the sequent (Π, Γ ⇒ ∆, Λ) is also an axiom.
a. The rule (Sub(o r ) ⇒) was applied in the last step of the proof of the sequent (Γ ⇒ ∆, F ). 2) Formula F is not principal in the sequent (F, Π ⇒ Λ). The case is considered in a similar way.

Theorem 7 (Completeness of GS-LCK)
If formula A is valid with respect to correlation models over (R, Σ, O), then sequent (⇒ s : A) is provable in GS-LCK. proof The Hilbert style proof system HS-LCK for logic of correlated knowledge is complete. Showing the provability of all valid formulas of HS-LCK in GS-LCK, the completeness of GS-LCK is proved. Theorem 6 is proved by induction on the number of steps < N Steps >, used to prove formula A in HS-LCK.
< N Steps = 1 > Formula A is an axiom of calculus HS-LCK.
5) The remaining axioms are considered in a similar way.

< N Steps > 1 >
One of the rules (M odus ponens) or (K I − necessitation) of calculus HS-LCK was applied in the last step of the proof of the formula. 7) The rule (K I − necessitation) was applied.
By the induction hypothesis, the sequent (⇒ s : A) is provable in GS-LCK. By Lemma "Substition", the sequent (⇒ t : A) is provable. By Theorem "Admissibility of weakening", the sequent (s I ∼ t ⇒ t : A) is provable. The sequent of the theorem is proved by applying the rule (⇒ K I ):

Automated Proof Search System GS-LCK-PROC
Having sound and complete sequent calculus GS-LCK for logic of correlated knowledge we can model automated

Finish. END
Procedure GS-LCK-PROC gets the sequent, T ableLK, T ableRK, starting Output and returns "True", if the sequent is provable. Otherwise -"False", if it is not provable. Procedure is constructed in such a way, that it produces proofs, where number of applications of the knowledge rules of sequent calculus GS-LCK is finite. Also number of applications of other rules are bounded by requirements to rules and finite initial sets of agents, observations and results, which allows procedure to perform terminating proof search.
Lemma 4 (Permutation of the rule (K I ⇒)) Rule (K I ⇒) permutes down with respect to all rules of GS-LCK, except rules (⇒ K I ) and (OE). Rule (K I ⇒) permutes down with rules (⇒ K I ) and (OE) in case the principal atom of (K I ⇒) is not active in it.
proof The Lemma 6 is proved in the same way as the Lemma 6.3. in [15].
Lemma 5 (Number of applications of the rule (K I ⇒)) If a sequent S is provable in GS-LCK, then there exists the proof of S such that rule (K I ⇒) is applied no more than once on the same pair of principal formulas on any branch.
proof The Lemma 6 is proved by induction on the number N of pairs of applications of rule (K I ⇒) on the same branch with the same principal pair.
< N = 0 > The proof of the lemma is obtained. < N > 0 > We diminish the inductive paramater in the same way as in the proof of Corollary 6.5. in [15], using Lemma 4. QED Lemma 6 (Number of applications of the rule (⇒ K I )) If a sequent S is provable in GS-LCK, then there exists the proof of S such that for each formula s : K I A in its positive part there are at most n(K I ) applications of (⇒ K I ) iterated on a chain of accessible worlds s I ∼ s 1 , s 1 I ∼ s 2 , ..., with principal formula s i : K I A. The latter proof is called regular.
proof The Lemma 6 is proved by induction on the number N of series of applications of rule (⇒ K I ), which make the initial proof non-regular.
< N = 0 > The proof of the lemma is obtained. < N > 0 > We diminish the inductive paramater in the same way as in the proof of Proposition 6.9. in [15]. QED According to finite number of applications of all rules, the procedure GS-LCK-PROC performs the terminating proof search for any sequent. QED Theorem 9 (Soundness and completeness of GS-LCK-PROC) The procedure GS-LCK-PROC is sound and complete over (R, Σ, O).
proof From construction of the procedure GS-LCK-PROC follows that if procedure returns "True" for a sequent S, then S is provable in GS-LCK. If procedure returns "False", then sequent S is not provable in GS-LCK, according to Lemma 6 and Lemma 6. QED Theorem 10 (Decidibility of LCK) Logic LCK is decidable. proof From Theorem 6 and Theorem 6 follows that GS-LCK-PROC is a decision procedure for logic LCK. QED

Conclusions
Sequent calculus GS-LCK has properties of soundness, completeness, admissibility of cut and structural rules, and invertibility of all rules. Procedure GS-LCK-PROC performs automated terminating proof search for logic of correlated knowledge and also has properties of soundness and completeness.
Using GS-LCK-PROC, the validity of the formula of any sequent can be determined and inferences can be checked if they follow from some knowledge base. Modelling the knowledge of distributed systems in the logic of correlated knowledge, questions about the systems can be answered automatically. Also soundness, completeness and termination of GS-LCK-PROC show that GS-LCK-PROC is a desicion procedure for logic of correlated knowledge and LCK is decidable logic, which means asking questions about the system we will always get the answer.
Logic of correlated knowledge expands the range of the applications of family of epistemic logics and captures deeper knowledge of the group of agents in the distributed systems. GS-LCK-PROC allows to reason about correlated knowledge automatically, without human interaction in the reasoning process.