Axiomatizing Probabilistic Logic of Quantum Programs Jort Bergfeld - - PowerPoint PPT Presentation

Axiomatizing Probabilistic Logic of Quantum Programs

Jort Bergfeld and Joshua Sack Amsterdam, 2014 April 1

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Motivation and Background

Quantum Algorithms and Protocols: use logic for a better understanding. Probabilistic Logic of Quantum Programs We involve a logic that expresses probabilities of outcomes of quantum tests effects of quantum tests and unitary operations separation operations that characterize subsystems. and is decidable: PLQP & company by Baltag et al 2014 In this talk, we provide a sound proof system use it to prove that “leader election protocol” is correct

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Structures for reasoning about quantum systems

Hilbert spaces are commonly used: Quantum states are one-dimensional subspaces. Probabilities of outcomes of tests characterized by inner product of vector representatives of the states |x,y|2

|x||y| .

Composite systems are constructed from the tensor product of Hilbert spaces (subsystems) In this talk, we Involve finite dimensional Hilbert spaces Build each structure from the set of states for a basis of the Hilbert space Involve agents, each corresponding to a basis of a subsystem

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Bases and states

Let H be a Hilbert space with an orthonormal basis

• B = {

b1, . . . , bn}. For every state (one dimensional space) s, there is a unit vector s in state s, such that

1 there exists an i ∈ N such that 1

• s,

bj = 0 for all j < i, and

2

• s,

bi ∈ (0, 1],

2 |

s, bk|2 ∈ [0, 1], and

3

k∈N |

s, bk|2 = 1. The function s, · : B → C characterizes s. The function | s, ·|2 : B → [0, 1] is a probability mass function.

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Complex probability mass function

B = {bi | i ∈ N} a finite totally ordered set (called basis states). f : B → C is called a complex probability mass function on B if

1 there exists an i ∈ N such that 1

f (bj) = 0 for all j < i, and

2

f (bi) ∈ (0, 1],

2 |f (bk)|2 ∈ [0, 1], and 3

k∈N |f (bk)|2 = 1.

FB is the set of all complex probability mass functions on B.

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Complex probability mass functions are states

Proposition Given a set of basis states B = {b1, . . . , bn} there is a Hilbert space H with orthonormal basis { b1, . . . , bn} such that for any complex probability mass function s : B → C, there is a vector s, such that for each i, s(bi) = s, bi. As the complex probability mass function s uniquely identifies the state (one dimensional subspace) generated by s, we identify s with that state.

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Generating structure from FB

Define S := FB (all complex probability mass functions) inner product µ(s, t) :=

i∈n s(bi)t(bi) for all s, t ∈ S,

z is the complex conjugate of z ∈ C nonorthogonality relation R = {(s, t) ∈ S × S | µ(s, t) = 0},

• rthocomplement ∼X := {s ∈ S | (s, x) /

∈ R for all x ∈ X}, for any set X ⊆ S, testable properties T := {P ⊆ S | P = ∼∼P}, P-test relation RP := {(s, t) ∈ R | t ∈ P and |µ(s, u)|2 ≤ |µ(s, t)|2 for all u ∈ P}, unitary operators U := {U : S → S | U is a permutation and µ(s, t) = µ(Us, Ut) for all s, t ∈ S}, unitary relation RU := {(s, t) ∈ S × S | t = Us}.

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Tensor product

Definition (Tensor product) The tensor product of state bases B = {b1, . . . , bn} and C = {c1, . . . , cm} is B ⊗ C = {bicj | bi ∈ B, cj ∈ C} The elements of B ⊗ C are totally ordered by the dictionary order.

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Separability

Definition (Separable and entangled states) A complex probability mass function f ∈ FB⊗C is separable (into B and C) if there exist s ∈ FB and t ∈ FC, such that f (bc) = s(b)t(c) for all b ∈ B and c ∈ C. We write f = s ⊗ t. If f is not separable we call f entangled. Definition (Separable Unitaries) A unitary operator U if FB⊗C is separable if there exists unitaries UB and UC, such that for all s ∈ FB and t ∈ FC, U(s ⊗ t) = UB(s) ⊗ UC(t). We then write U = UB ⊗ UC.

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Multi-agent models

Let A = {0, 1, . . . , N − 1} be a finite set of agents. Let Prop be a set of atomic propositions. Definition (multi-agent probabilistic quantum model (PQM)) An A-PQF (probabilistic quantum frame) is a pair F = (B, {Bi}i∈A), where B is a basis of states and Bi is a two-state basis for each i ∈ A, such that B =

i∈A Bi.

Then an A-PQM (probabilistic quantum model) is a pair (F, V ), such that F = (B, {Bi}i∈A) is an A-PQF and V : Prop → P(FB) is a valuation. Given a subset I ⊆ A of agents, let BI =

i∈I Bi

SI = FBI If s is separable over BI and BA\I, let sI denote the complex probability mass function such that there exists sA\I such that s = sI ⊗ sA\I.

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Language

Let A = {0, 1, . . . , N − 1} be a finite set of agents. Let Prop be a (countable) set of atomic propositions. φ ::= ⊤I | p | t ≥ ρ | ¬φ | φ ∧ φ | φ | [α]φ α ::= ⊤I | φ? | U | U† | α ∪ α | α; α t ::= ρ Pr(φ) | t + t where p ∈ Prop, U ∈ U, I ⊆ A and ρ ∈ R. Language “L” is defined to the the set of all such φ Set “Terms” is defined to be the set of all terms t ⊤I means “I-separable” [⊤I] ranges over the “I-subsystem” (is equivalent to KA\I)

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Semantics

Let ((B, {Bi}i∈A), V ) be an A-PQM, and let S = FB. We define an extended valuation · : L → PS, and for each s ∈ S, ·s : Terms → R: ⊤I := {s ∈ S | s = sI ⊗ sA\I for some sI ∈ SI and sA\I ∈ SA\I}, p := V (p), t ≥ ρ := {s ∈ S | ts ≥ ρ} ¬φ := S \ φ, φ ∧ ψ := φ ∩ ψ, φ := {s ∈ S | R(s) ⊆ φ} (R is non-orthogonality relation) [α]φ := {s ∈ S | Rα(s) ⊆ φ} (Rα is defined on next slide). ρ Pr(φ)s := ρ

• t∈RP(s)

|µ(s, t)|2, where P = ∼∼φ, t1 + t2s := t1s + t2s

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More semantics

Here Rα can be inductively defined by R⊤I := {(s, t) | t = (UI ⊗ IdA\I)(s) for some UI ∈ UI}, Rφ? := RP, where P = ∼∼φ, RU := RU, RU† := Rc

U,

Rα∪β := Rα ∪ Rβ, and Rα;β := Rα; Rβ.

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Probabilistic abbreviations

n

k=1 ak Pr(φk)

:= a1 Pr(φ1) + · · · + an Pr(φn) ρ n

k=1 ak Pr(φk)

:= n

k=1 ρak Pr(φk)

t < ρ := ¬t ≥ ρ t ≤ ρ := −t ≥ −ρ t = ρ := t ≥ ρ ∧ t ≤ ρ t1 ≥ t2 := t1 − t2 ≥ 0

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Abbreviations

∼φ := ¬φ (orthocomplement) φ ∨ ψ := ¬(¬φ ∧ ¬ψ) (disjunction) φ ⊔ ψ := ∼(∼φ ∧ ∼ψ) (quantum join) Aφ := φ (global universal) Eφ := ♦♦φ (global existential) (φ ≤ ψ) := A(φ → ψ) (φ = ψ) := A(φ ↔ ψ) φ ⊥ ψ := φ ≤ ∼ψ (orthogonal) T(φ) := ∼∼φ = φ (testable) φI := ⊤I ∧ ⊤N\Iφ (I-component) φ =I ψ := (φ ≤ ⊤I) ∧ (ψ ≤ ⊤I) ∧ (φI = ψI) (I-equivalent) I(φ) := (φ = φI) (I-local) φ?=ρψ := Pr(φ) = ρ ∧ φ?ψ φ?>ρψ := Pr(φ) > ρ ∧ φ?ψ

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Example: Leader Election Protocol

Example Setting: There are N agents. Goal: Each should have an equal (1/N) chance of being chosen to be the leader. Strategy: Prepare a quantum state that has equal probability

• f collapsing into any of N basis elements when measured.

Solution: This state is the W -state in a 2N-dimensional Hilbert space (a subsystem for each agent).

The basis for the 2N-dimensional space is the product of the bases {0k, 1k}, for each of the N agents. The k-th agent is associated with the basis element bk = (00 ⊗ · · · ⊗ 0k−1 ⊗ 1k ⊗ 0k+1 ⊗ · · · ⊗ 0N−1). The W -state is an equally weighted superposition of the bk.

• E. D’Hondt and P. Panangaden,

The Computational Power of the W and GHZ States, Quantum Information and Computation 6 (2006), 173–83.

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Expressing the Leader Election Protocol

Let A = {0, 1, . . . , N − 1} be a finite set of agents. Basis(B) := (

• i∈2N

bi = ⊤) ∧

• i=j

(bi ⊥ bj). Separable(B) :=

• i∈2N

(bi ≤

• a∈A

⊤a). Let W = {Wi | i ∈ {0, . . . , N}} ⊆ B. Think of WN as (00 ⊗ · · · ⊗ 0N−1). QLE(W) :=

• a∈A

 ((Wa)a =a (WN)a) ∧

• b∈A\a

((Wa)b =b (WN)b)   . The correctness of the quantum leader election is expressed by Basis(B) ∧ Separable(B) ∧ QLE(W) → E

• a∈A

(Pr(Wa) = 1 N ).

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Duality result

Theorem Each quantum dynamic frame is dual to a Piron lattice. A quantum dynamic frame is a special Kripke frame that satisfies Atomicity, Intersection, Orthocomplement, Adequacy, Repeatability, Partial functionality, Self-adjointness, Proper superposition, Cover law.

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Duality result

Theorem Each quantum dynamic frame is dual to a Piron lattice. A quantum dynamic frame is a special Kripke frame that satisfies Atomicity, Intersection, Orthocomplement, Adequacy, Repeatability, Partial functionality, Self-adjointness, Proper superposition, Cover law.

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Some axioms

We base some axioms on the properties of quantum dynamic frame. Adequacy For all P ∈ T and for all s ∈ P we have s

P?

− → s. p → (q → p?q) Orthocomplement s ∈ ∼P iff s t for all t ∈ P. p?⊤ ↔ ♦p (= ¬∼p)

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Some axioms

We base some axioms on the properties of quantum dynamic frame. Adequacy For all P ∈ T and for all s ∈ P we have s

P?

− → s. p → (q → p?q) Orthocomplement s ∈ ∼P iff s t for all t ∈ P. p?⊤ ↔ ♦p (= ¬∼p)

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Some rules

1 φ, φ → ψ =

⇒ ψ Modus ponens

2 φ =

⇒ [α]φ, φ Necessitation

3 φ → [p?]ψ =

⇒ φ → ψ if p / ∈ φ, ψ

4 φ =

⇒ φσ Uniform substitution for some σ : Prop → L (p → q) → (p → q)

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Reflexivity

Lemma ⊢ p → ♦p (p → ¬∼p) Proof.

1 p → (⊤ → p?⊤) 2 p?⊤ ↔ ♦p 3 p → p?⊤ 4 p → ♦p 21/32

Reflexivity

Lemma ⊢ p → ♦p (p → ¬∼p) Proof.

1 p → (⊤ → p?⊤) 2 p?⊤ ↔ ♦p 3 p → p?⊤ 4 p → ♦p 21/32

Reflexivity

Lemma ⊢ p → ♦p (p → ¬∼p) Proof.

1 p → (⊤ → p?⊤) 2 p?⊤ ↔ ♦p 3 p → p?⊤ 4 p → ♦p 21/32

Reflexivity

Lemma ⊢ p → ♦p (p → ¬∼p) Proof.

1 p → (⊤ → p?⊤) 2 p?⊤ ↔ ♦p 3 p → p?⊤ 4 p → ♦p 21/32

Quantum join with orthocomplement

Lemma ⊢ ∼(p ∧ ∼p) Proof.

1 p → ¬∼p 2 (p ∧ ∼p) → (¬∼p ∧ ∼p) 3 (p ∧ ∼p) → ⊥ 4 ¬(p ∧ ∼p) 5 ¬(p ∧ ∼p) 6 ∼(p ∧ ∼p) 22/32

Quantum join with orthocomplement

Lemma ⊢ ∼(p ∧ ∼p) Proof.

1 p → ¬∼p 2 (p ∧ ∼p) → (¬∼p ∧ ∼p) 3 (p ∧ ∼p) → ⊥ 4 ¬(p ∧ ∼p) 5 ¬(p ∧ ∼p) 6 ∼(p ∧ ∼p) 22/32

Quantum join with orthocomplement

Lemma ⊢ ∼(p ∧ ∼p) Proof.

1 p → ¬∼p 2 (p ∧ ∼p) → (¬∼p ∧ ∼p) 3 (p ∧ ∼p) → ⊥ 4 ¬(p ∧ ∼p) 5 ¬(p ∧ ∼p) 6 ∼(p ∧ ∼p) 22/32

Quantum join with orthocomplement

Lemma ⊢ ∼(p ∧ ∼p) Proof.

1 p → ¬∼p 2 (p ∧ ∼p) → (¬∼p ∧ ∼p) 3 (p ∧ ∼p) → ⊥ 4 ¬(p ∧ ∼p) 5 ¬(p ∧ ∼p) 6 ∼(p ∧ ∼p) 22/32

Quantum join with orthocomplement

Lemma ⊢ ∼(p ∧ ∼p) Proof.

1 p → ¬∼p 2 (p ∧ ∼p) → (¬∼p ∧ ∼p) 3 (p ∧ ∼p) → ⊥ 4 ¬(p ∧ ∼p) 5 ¬(p ∧ ∼p) 6 ∼(p ∧ ∼p) 22/32

Quantum join with orthocomplement

Lemma ⊢ ∼(p ∧ ∼p) Proof.

1 p → ¬∼p 2 (p ∧ ∼p) → (¬∼p ∧ ∼p) 3 (p ∧ ∼p) → ⊥ 4 ¬(p ∧ ∼p) 5 ¬(p ∧ ∼p) 6 ∼(p ∧ ∼p) 22/32

Some axioms

Partial functionality If s

P?

− → t and s

P?

− → u, then t = u. p?q → [p?]q Self-adjointness If s

P?

− → t → u, then there exists a v such that u P? − → v → s. p → [q?]q?♦p

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Some axioms

Partial functionality If s

P?

− → t and s

P?

− → u, then t = u. p?q → [p?]q Self-adjointness If s

P?

− → t → u, then there exists a v such that u P? − → v → s. p → [q?]q?♦p

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Symmetry

Lemma ⊢ p → ♦p (p → ∼∼p) Proof.

1 ⊤ → (p → ⊤?p) 2 p → ⊤?p 3 ⊤?p → [⊤?]p 4 p → [⊤?]p 5 [⊤?]p → p 6 ⊤?p → p 7 p → [⊤?]⊤?♦p 8 p → ♦p 24/32

Symmetry

Lemma ⊢ p → ♦p (p → ∼∼p) Proof.

1 ⊤ → (p → ⊤?p) 2 p → ⊤?p 3 ⊤?p → [⊤?]p 4 p → [⊤?]p 5 [⊤?]p → p 6 ⊤?p → p 7 p → [⊤?]⊤?♦p 8 p → ♦p 24/32

Symmetry

Lemma ⊢ p → ♦p (p → ∼∼p) Proof.

1 ⊤ → (p → ⊤?p) 2 p → ⊤?p 3 ⊤?p → [⊤?]p 4 p → [⊤?]p 5 [⊤?]p → p 6 ⊤?p → p 7 p → [⊤?]⊤?♦p 8 p → ♦p 24/32

Symmetry

Lemma ⊢ p → ♦p (p → ∼∼p) Proof.

1 ⊤ → (p → ⊤?p) 2 p → ⊤?p 3 ⊤?p → [⊤?]p 4 p → [⊤?]p 5 [⊤?]p → p 6 ⊤?p → p 7 p → [⊤?]⊤?♦p 8 p → ♦p 24/32

Symmetry

Lemma ⊢ p → ♦p (p → ∼∼p) Proof.

1 ⊤ → (p → ⊤?p) 2 p → ⊤?p 3 ⊤?p → [⊤?]p 4 p → [⊤?]p 5 [⊤?]p → p 6 ⊤?p → p 7 p → [⊤?]⊤?♦p 8 p → ♦p 24/32

Symmetry

Lemma ⊢ p → ♦p (p → ∼∼p) Proof.

1 ⊤ → (p → ⊤?p) 2 p → ⊤?p 3 ⊤?p → [⊤?]p 4 p → [⊤?]p 5 [⊤?]p → p 6 ⊤?p → p 7 p → [⊤?]⊤?♦p 8 p → ♦p 24/32

Symmetry

Lemma ⊢ p → ♦p (p → ∼∼p) Proof.

1 ⊤ → (p → ⊤?p) 2 p → ⊤?p 3 ⊤?p → [⊤?]p 4 p → [⊤?]p 5 [⊤?]p → p 6 ⊤?p → p 7 p → [⊤?]⊤?♦p 8 p → ♦p 24/32

Symmetry

Lemma ⊢ p → ♦p (p → ∼∼p) Proof.

1 ⊤ → (p → ⊤?p) 2 p → ⊤?p 3 ⊤?p → [⊤?]p 4 p → [⊤?]p 5 [⊤?]p → p 6 ⊤?p → p 7 p → [⊤?]⊤?♦p 8 p → ♦p 24/32

Axioms for basic single-agent PQM properties

M1 p?φ → ♦φ M2 ♦p ↔ p?⊤ M3 T(p) → [p?]p M4 T(p) → ¬p ↔ ∼p?⊤ M5 p?q → [p?]q M6 p → (q → p?q) M7 p → [q?]q?♦p M8 ♦♦♦p → ♦♦p M9 T(p) ∧ T(q) ∧ ♦p ∧ (p → ♦(p ∧ q))] → p?q M10 Up ↔ [U]p M11 p ↔ [U; U†]p M12 p ↔ [U†; U]p M13 U♦p ↔ ♦Up

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Some provable formulas

1 p → ♦p

(p → ¬∼p)

2 p → ♦p

(p → ∼∼p)

3 T(∼p)

(∼∼∼p → ∼p)

4 p → p ⊔ q 5 (p → q) → (∼q → ∼p) 6 T(p) ∧ T(q) → T(p ∧ q) 7 p ⊥ q ↔ q ⊥ p 8 r ⊥ p ∧ r ⊥ q ↔ r ⊥ (p ⊔ q) 26/32

Some provable formulas

1 p → ♦p

(p → ¬∼p)

2 p → ♦p

(p → ∼∼p)

3 T(∼p)

(∼∼∼p → ∼p)

4 p → p ⊔ q 5 (p → q) → (∼q → ∼p) 6 T(p) ∧ T(q) → T(p ∧ q) 7 p ⊥ q ↔ q ⊥ p 8 r ⊥ p ∧ r ⊥ q ↔ r ⊥ (p ⊔ q) 26/32

Some provable formulas

1 p → ♦p

(p → ¬∼p)

2 p → ♦p

(p → ∼∼p)

3 T(∼p)

(∼∼∼p → ∼p)

4 p → p ⊔ q 5 (p → q) → (∼q → ∼p) 6 T(p) ∧ T(q) → T(p ∧ q) 7 p ⊥ q ↔ q ⊥ p 8 r ⊥ p ∧ r ⊥ q ↔ r ⊥ (p ⊔ q) 26/32

Some provable formulas

1 p → ♦p

(p → ¬∼p)

2 p → ♦p

(p → ∼∼p)

3 T(∼p)

(∼∼∼p → ∼p)

4 p → p ⊔ q 5 (p → q) → (∼q → ∼p) 6 T(p) ∧ T(q) → T(p ∧ q) 7 p ⊥ q ↔ q ⊥ p 8 r ⊥ p ∧ r ⊥ q ↔ r ⊥ (p ⊔ q) 26/32

Some provable formulas

1 p → ♦p

(p → ¬∼p)

2 p → ♦p

(p → ∼∼p)

3 T(∼p)

(∼∼∼p → ∼p)

4 p → p ⊔ q 5 (p → q) → (∼q → ∼p) 6 T(p) ∧ T(q) → T(p ∧ q) 7 p ⊥ q ↔ q ⊥ p 8 r ⊥ p ∧ r ⊥ q ↔ r ⊥ (p ⊔ q) 26/32

Some provable formulas

1 p → ♦p

(p → ¬∼p)

2 p → ♦p

(p → ∼∼p)

3 T(∼p)

(∼∼∼p → ∼p)

4 p → p ⊔ q 5 (p → q) → (∼q → ∼p) 6 T(p) ∧ T(q) → T(p ∧ q) 7 p ⊥ q ↔ q ⊥ p 8 r ⊥ p ∧ r ⊥ q ↔ r ⊥ (p ⊔ q) 26/32

Some provable formulas

1 p → ♦p

(p → ¬∼p)

2 p → ♦p

(p → ∼∼p)

3 T(∼p)

(∼∼∼p → ∼p)

4 p → p ⊔ q 5 (p → q) → (∼q → ∼p) 6 T(p) ∧ T(q) → T(p ∧ q) 7 p ⊥ q ↔ q ⊥ p 8 r ⊥ p ∧ r ⊥ q ↔ r ⊥ (p ⊔ q) 26/32

Axioms for inequalities

I1 t ≥ β ↔ t + 0Pa(φ) ≥ β I2 n

k=1 αkPa(φk) ≥ β → n k=1 αjkPa(φjk) ≥ qβ

for any permutation j1, . . . , jn of 1, . . . , n I3 n

k=1 αkPa(φk) ≥ β ∧ n k=1 α′ kPa(φk) ≥ β′

→ n

k=1(αk + α′ k)Pa(φk) ≥ (β + β′)

I4 t ≥ β ↔ dt ≥ dβ if d > 0 I5 t ≥ β ∨ t ≤ β I6 t ≥ β → t ≥ γ if β > γ

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Axioms for probabilities

1 (Pr(p) = 0) ↔ ∼p 2 (Pr(p) > 0) ↔ ♦p 3 p → (Pr(p) = 1) 4 T(p) → (p ↔ Pr(p) = 1) 5 Orthocomplement

(Pr(p) = ρ) ↔ (Pr(∼p) = 1 − ρ)

6 Quantum join

(p ⊥ q) ∧ (Pr(p) = ρ) ∧ (Pr(q) = τ) → (Pr(p ⊔ q) = ρ + τ)

7 Superposition

Ep ∧ Eq ∧ (p ⊥ q) → E [p?=ρp ∧ q?=1−ρq]

8 (p ≤ q) ∧ q?=ρ(Pr(p) = τ) → (Pr(p) = ρτ) 28/32

Axioms for probabilities

1 (Pr(p) = 0) ↔ ∼p 2 (Pr(p) > 0) ↔ ♦p 3 p → (Pr(p) = 1) 4 T(p) → (p ↔ Pr(p) = 1) 5 Orthocomplement

(Pr(p) = ρ) ↔ (Pr(∼p) = 1 − ρ)

6 Quantum join

(p ⊥ q) ∧ (Pr(p) = ρ) ∧ (Pr(q) = τ) → (Pr(p ⊔ q) = ρ + τ)

7 Superposition

Ep ∧ Eq ∧ (p ⊥ q) → E [p?=ρp ∧ q?=1−ρq]

8 (p ≤ q) ∧ q?=ρ(Pr(p) = τ) → (Pr(p) = ρτ) 28/32

Axioms for probabilities

1 (Pr(p) = 0) ↔ ∼p 2 (Pr(p) > 0) ↔ ♦p 3 p → (Pr(p) = 1) 4 T(p) → (p ↔ Pr(p) = 1) 5 Orthocomplement

(Pr(p) = ρ) ↔ (Pr(∼p) = 1 − ρ)

6 Quantum join

(p ⊥ q) ∧ (Pr(p) = ρ) ∧ (Pr(q) = τ) → (Pr(p ⊔ q) = ρ + τ)

7 Superposition

Ep ∧ Eq ∧ (p ⊥ q) → E [p?=ρp ∧ q?=1−ρq]

8 (p ≤ q) ∧ q?=ρ(Pr(p) = τ) → (Pr(p) = ρτ) 28/32

Axioms for probabilities

1 (Pr(p) = 0) ↔ ∼p 2 (Pr(p) > 0) ↔ ♦p 3 p → (Pr(p) = 1) 4 T(p) → (p ↔ Pr(p) = 1) 5 Orthocomplement

(Pr(p) = ρ) ↔ (Pr(∼p) = 1 − ρ)

6 Quantum join

(p ⊥ q) ∧ (Pr(p) = ρ) ∧ (Pr(q) = τ) → (Pr(p ⊔ q) = ρ + τ)

7 Superposition

Ep ∧ Eq ∧ (p ⊥ q) → E [p?=ρp ∧ q?=1−ρq]

8 (p ≤ q) ∧ q?=ρ(Pr(p) = τ) → (Pr(p) = ρτ) 28/32

Axioms for probabilities

1 (Pr(p) = 0) ↔ ∼p 2 (Pr(p) > 0) ↔ ♦p 3 p → (Pr(p) = 1) 4 T(p) → (p ↔ Pr(p) = 1) 5 Orthocomplement

(Pr(p) = ρ) ↔ (Pr(∼p) = 1 − ρ)

6 Quantum join

(p ⊥ q) ∧ (Pr(p) = ρ) ∧ (Pr(q) = τ) → (Pr(p ⊔ q) = ρ + τ)

7 Superposition

Ep ∧ Eq ∧ (p ⊥ q) → E [p?=ρp ∧ q?=1−ρq]

8 (p ≤ q) ∧ q?=ρ(Pr(p) = τ) → (Pr(p) = ρτ) 28/32

Quantum leader election

Theorem For all n and all sets of n proposition letters {p1, . . . , pn} ⊢

• i≤n

Epi ∧

• i<j≤n

pi ⊥ pj → E

• i≤n

Pr(bi) = 1 n

1 Ep1 ∧ Ep2 ∧ (p1 ⊥ p2) → E

• (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2)

• 2 (p1 ⊥ p2) ∧ (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2) → (Pr(p1 ⊔ p2) = 1)

3 T(p1 ⊔ p2) → (Pr(p1 ⊔ p2) = 1) → p1 ⊔ p2 4 (p3 ⊥ p1) ∧ (p3 ⊥ p2) → p3 ⊥ p1 ⊔ p2 5

= ⇒ E

• (Pr(p3) = 1

3) ∧ p1 ⊔ p2?= 2

3 ((Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2))

• 6 p1 → p1 ⊔ p2

7 E

• i≤3 Pr(pi) = 1

3

• 29/32

Quantum leader election

Theorem For all n and all sets of n proposition letters {p1, . . . , pn} ⊢

• i≤n

Epi ∧

• i<j≤n

pi ⊥ pj → E

• i≤n

Pr(bi) = 1 n

1 Ep1 ∧ Ep2 ∧ (p1 ⊥ p2) → E

• (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2)

• 2 (p1 ⊥ p2) ∧ (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2) → (Pr(p1 ⊔ p2) = 1)

3 T(p1 ⊔ p2) → (Pr(p1 ⊔ p2) = 1) → p1 ⊔ p2 4 (p3 ⊥ p1) ∧ (p3 ⊥ p2) → p3 ⊥ p1 ⊔ p2 5

= ⇒ E

• (Pr(p3) = 1

3) ∧ p1 ⊔ p2?= 2

3 ((Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2))

• 6 p1 → p1 ⊔ p2

7 E

• i≤3 Pr(pi) = 1

3

• 29/32

Quantum leader election

Theorem For all n and all sets of n proposition letters {p1, . . . , pn} ⊢

• i≤n

Epi ∧

• i<j≤n

pi ⊥ pj → E

• i≤n

Pr(bi) = 1 n

1 Ep1 ∧ Ep2 ∧ (p1 ⊥ p2) → E

• (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2)

• 2 (p1 ⊥ p2) ∧ (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2) → (Pr(p1 ⊔ p2) = 1)

3 T(p1 ⊔ p2) → (Pr(p1 ⊔ p2) = 1) → p1 ⊔ p2 4 (p3 ⊥ p1) ∧ (p3 ⊥ p2) → p3 ⊥ p1 ⊔ p2 5

= ⇒ E

• (Pr(p3) = 1

3) ∧ p1 ⊔ p2?= 2

3 ((Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2))

• 6 p1 → p1 ⊔ p2

7 E

• i≤3 Pr(pi) = 1

3

• 29/32

Quantum leader election

Theorem For all n and all sets of n proposition letters {p1, . . . , pn} ⊢

• i≤n

Epi ∧

• i<j≤n

pi ⊥ pj → E

• i≤n

Pr(bi) = 1 n

1 Ep1 ∧ Ep2 ∧ (p1 ⊥ p2) → E

• (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2)

• 2 (p1 ⊥ p2) ∧ (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2) → (Pr(p1 ⊔ p2) = 1)

3 T(p1 ⊔ p2) → (Pr(p1 ⊔ p2) = 1) → p1 ⊔ p2 4 (p3 ⊥ p1) ∧ (p3 ⊥ p2) → p3 ⊥ p1 ⊔ p2 5

= ⇒ E

• (Pr(p3) = 1

3) ∧ p1 ⊔ p2?= 2

3 ((Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2))

• 6 p1 → p1 ⊔ p2

7 E

• i≤3 Pr(pi) = 1

3

• 29/32

Quantum leader election

Theorem For all n and all sets of n proposition letters {p1, . . . , pn} ⊢

• i≤n

Epi ∧

• i<j≤n

pi ⊥ pj → E

• i≤n

Pr(bi) = 1 n

1 Ep1 ∧ Ep2 ∧ (p1 ⊥ p2) → E

• (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2)

• 2 (p1 ⊥ p2) ∧ (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2) → (Pr(p1 ⊔ p2) = 1)

3 T(p1 ⊔ p2) → (Pr(p1 ⊔ p2) = 1) → p1 ⊔ p2 4 (p3 ⊥ p1) ∧ (p3 ⊥ p2) → p3 ⊥ p1 ⊔ p2 5

= ⇒ E

• (Pr(p3) = 1

3) ∧ p1 ⊔ p2?= 2

3 ((Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2))

• 6 p1 → p1 ⊔ p2

7 E

• i≤3 Pr(pi) = 1

3

• 29/32

Quantum leader election

Theorem For all n and all sets of n proposition letters {p1, . . . , pn} ⊢

• i≤n

Epi ∧

• i<j≤n

pi ⊥ pj → E

• i≤n

Pr(bi) = 1 n

1 Ep1 ∧ Ep2 ∧ (p1 ⊥ p2) → E

• (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2)

• 2 (p1 ⊥ p2) ∧ (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2) → (Pr(p1 ⊔ p2) = 1)

3 T(p1 ⊔ p2) → (Pr(p1 ⊔ p2) = 1) → p1 ⊔ p2 4 (p3 ⊥ p1) ∧ (p3 ⊥ p2) → p3 ⊥ p1 ⊔ p2 5

= ⇒ E

• (Pr(p3) = 1

3) ∧ p1 ⊔ p2?= 2

3 ((Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2))

• 6 p1 → p1 ⊔ p2

7 E

• i≤3 Pr(pi) = 1

3

• 29/32

Quantum leader election

Theorem For all n and all sets of n proposition letters {p1, . . . , pn} ⊢

• i≤n

Epi ∧

• i<j≤n

pi ⊥ pj → E

• i≤n

Pr(bi) = 1 n

1 Ep1 ∧ Ep2 ∧ (p1 ⊥ p2) → E

• (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2)

• 2 (p1 ⊥ p2) ∧ (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2) → (Pr(p1 ⊔ p2) = 1)

3 T(p1 ⊔ p2) → (Pr(p1 ⊔ p2) = 1) → p1 ⊔ p2 4 (p3 ⊥ p1) ∧ (p3 ⊥ p2) → p3 ⊥ p1 ⊔ p2 5

= ⇒ E

• (Pr(p3) = 1

3) ∧ p1 ⊔ p2?= 2

3 ((Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2))

• 6 p1 → p1 ⊔ p2

7 E

• i≤3 Pr(pi) = 1

3

• 29/32

Quantum leader election

Theorem For all n and all sets of n proposition letters {p1, . . . , pn} ⊢

• i≤n

Epi ∧

• i<j≤n

pi ⊥ pj → E

• i≤n

Pr(bi) = 1 n

1 Ep1 ∧ Ep2 ∧ (p1 ⊥ p2) → E

• (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2)

• 2 (p1 ⊥ p2) ∧ (Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2) → (Pr(p1 ⊔ p2) = 1)

3 T(p1 ⊔ p2) → (Pr(p1 ⊔ p2) = 1) → p1 ⊔ p2 4 (p3 ⊥ p1) ∧ (p3 ⊥ p2) → p3 ⊥ p1 ⊔ p2 5

= ⇒ E

• (Pr(p3) = 1

3) ∧ p1 ⊔ p2?= 2

3 ((Pr(p1) = 1

2) ∧ (Pr(p2) = 1 2))

• 6 p1 → p1 ⊔ p2

7 E

• i≤3 Pr(pi) = 1

3

• 29/32

Axioms for multi-agent properties

⊤1 [⊥?]⊥ ⊤2 p → ⊤?p ⊤3 [⊤I](p → q) → ([⊤I]p → [⊤I]q) ⊤4 [⊤I]p → p ⊤5 [⊤I]p → [⊤I][⊤I]p ⊤6 ¬[⊤I]p → [⊤I]¬[⊤I]p ⊤7 [⊤I]p → [⊤J]p for all I ⊆ J ⊤8 ⊤I ↔ ⊤N\I ⊤9 ⊤I → [⊤I]⊤I ∧ [⊤N\I]⊤I ⊤10 ⊤N ⊤11 (⊤I ∧ ⊤J) → (⊤I∪J ∧ ⊤I∩J) ⊤12 T(p) ∧ I(p) ∧ I(q) ∧ (⊥ = q) ∧ (q ≤ p) → (p = q) if I = N ⊤13 ∼⊤I ↔ ⊥ ⊤14 I(α) → αp ≤ ⊤Ip

30/32

Summary and future work

An axiomatization may be useful for analyzing the correctness

• f other protocols. Next step: BB84?

The existing proof system might be reducible (some axioms may be provable) Completeness? Complexity of the validity problem of the logic?

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Thank you!

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