A Unified Framework for DPLL(T) + Certificates

Table 3

A possible CDPLL(T) procedure.

No.	𝑀‖	𝐹	⊕𝑃	Reason	Explanation
1	∅‖	𝐹0	⊕𝑃0	∅	Initial state
2	¬𝑙4	~	⊕∼	¬𝑙4↦𝐶5	UnitPropagation
3	¬𝑙4,𝑙3	~	⊕~	¬𝑙4↦𝐶5𝑙3↦𝐶6	UnitPropagation
4	¬𝑙4,𝑙3,𝑙^𝖽1	~	⊕~	~	Decide; 𝑀 becomes T-inconsistent
5	∼‖	𝐹0∪{𝐶7}	⊕ 𝑃1	~	T-Learn(II), 𝐶7 is learned
6	¬𝑙4,𝑙3,¬𝑙1	~	⊕~	¬𝑙4↦𝐶5𝑙3↦𝐶6¬𝑙1↦𝐶7	T-Backjump
7	¬𝑙4,𝑙3,¬𝑙1,¬𝑙5	~	⊕~	¬𝑙4↦𝐶5𝑙3↦𝐶6¬𝑙1↦𝐶7¬𝑙5↦𝐶1	UnitPropagation
8	¬𝑙4,𝑙3,¬𝑙1,¬𝑙5,¬𝑝	~	⊕~	¬𝑙4↦𝐶5𝑙3↦𝐶6¬𝑙1↦𝐶7¬𝑙5↦𝐶1¬𝑝↦𝐶3	UnitPropagation
9	¬𝑙4,𝑙3,¬𝑙1,¬𝑙5,¬𝑝,¬𝑙6	~	⊕~	¬𝑙4↦𝐶5𝑙3↦𝐶6¬𝑙1↦𝐶7¬𝑙5↦𝐶1¬𝑝↦𝐶3¬𝑙6↦𝐶2	UnitPropagation
10	¬𝑙4,𝑙3,¬𝑙1,¬𝑙5,¬𝑝,¬𝑙6,𝑙2	~	⊕~	¬𝑙4↦𝐶5𝑙3↦𝐶6¬𝑙1↦𝐶7¬𝑙5↦𝐶1¬𝑝↦𝐶3¬𝑙6↦𝐶2𝑙2↦𝐶4	UnitPropagation; 𝑀 becomes T-inconsistent
11	∼‖	𝐹0∪{𝐶7,𝐶8}	⊕ 𝑃2	~	T-Learn(II), 𝐶8 is learned
12	∼‖	𝐹0∪{𝐶7,𝐶8}	⊕ 𝑃3	~	Fail, □ derived