Synergy User Manual and Tutorial. - THE CORE MEMORY
Synergy User Manual and Tutorial. - THE CORE MEMORY
Synergy User Manual and Tutorial. - THE CORE MEMORY
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<strong>Synergy</strong> <strong>User</strong> <strong>Manual</strong> <strong>and</strong> <strong>Tutorial</strong><br />
A popular weak modal logic K, conceived by Saul Kripke, .defines three operators:<br />
“negation” (¬), “if…then…” (→), <strong>and</strong> “it is necessary that…” (). The other<br />
connectives, “<strong>and</strong>” (∧), “or” (∨), <strong>and</strong> “if <strong>and</strong> only if” (↔), can be defined by ¬ <strong>and</strong> → as<br />
in propositional logic. The operator “possibly” (◊) can be defined by ◊A = ¬¬A. In<br />
addition to the st<strong>and</strong>ard rules in propositional logic, K has the following rules:<br />
Necessitation Rule:<br />
Distribution Axiom:<br />
If A is a theorem of K, then so is A.<br />
(A → B) → (A → B).<br />
The necessitation rule states that all theorems are necessary <strong>and</strong> the distribution axiom<br />
states that “if it is necessary that if A then B, then if necessarily A then necessarily B”. A<br />
<strong>and</strong> B range over all possible formulas for the language.<br />
(M)<br />
A → A<br />
(4) A → A<br />
(5) ◊A → ◊A<br />
(S4):<br />
(S5):<br />
… = <strong>and</strong> ◊◊…◊ = ◊<br />
00… = <strong>and</strong> 00…◊ = ◊, where each 0 is either or ◊<br />
(B)<br />
A → ◊A<br />
Axiom Name Axiom Condition on Frames R is...<br />
(D) A → ◊A ∃u wRu Serial<br />
(M) A → A wRw Reflexive<br />
(4) A → A (wRv ∧ vRu) → wRu Transitive<br />
(B) A → ◊A wRv → vRw Symmetric<br />
(5) ◊A → ◊A (wRv ∧ wRu) → vRu Euclidean<br />
(CD) ◊A → A (wRv ∧ wRu) → v = u Unique<br />
(M) (A → A) wRv → vRv Shift Reflexive<br />
(C4) A → A wRv → ∃u(wRu∧uRv) Dense<br />
(C) ◊A → ◊A wRv∧wRx → ∃u(vRu ∧ xRu) Convergent<br />
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