1. Consider the set of propositional formulas: U = {¬p ¬ q, (p ¬q) ¬r, p r} Prov
ID: 3603731 • Letter: 1
Question
1. Consider the set of propositional formulas:
U = {¬p ¬ q, (p ¬q) ¬r, p r}
Provide an example of
(i) a non-satisfying interpretation
(ii) a satisfying interpretation
2. Provide an example of a formula A such that U A, where U is the set of formulas given in the Question 1.
3. Is the set of formulas of Question 1, {¬p ¬ q, (p ¬q) ¬r, p r}, closed under logical consequence?
4. Analyze the proof of the theorem and mention one rule of inference used in the proof.
Theorem Let n . If 7n + 6 is odd then n is odd.
Proof
Let us assume that n is not odd, that is, n is even. Then we have that n = 2k for some k . So
7n + 6 = 7(2k) + 6
= 2(7k) + 6
= 2(7k + 3)
From here we conclude that 7n + 6 is even in contradiction with the premise of the theorem. Therefore, if 7n + 6 is odd then n must be odd.
Explanation / Answer
Open answer set programming (OASP) is an extension of answer set programming where one may ground a program with an arbitrary superset of the program’s constants. We define a fixed point logic (FPL) extension of Clark’s completion such that open answer sets correspond to models of FPL formulas and identify a syntactic subclass of programs, called (loosely) guarded programs. Whereas reasoning with general programs in OASP is undecidable, the FPL translation of (loosely) guarded programs falls in the decidable (loosely) guarded fixed point logic ((L)GF). Moreover, we reduce normal closed ASP to loosely guarded OASP, enabling a characterization of an answer set semantics by LGF formulas. Finally, we relate guarded OASP to Datalog LITE, thus linking an answer set semantics to a semantics based on fixed point models of extended stratified Datalog programs. From this correspondence, we deduce 2-EXPTIME-completeness of satisfiability checking w.r.t. (loosely) guarded programs.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.