2. The language and formal semantics of computability logic Episode 14 0 The formal language Interpretations Definitions of validity and uniform validity Validity or uniform validity? The extensional equivalence between validity and uniform validity Computability versus “knowability” Closure under Modus Ponens Other closure theorems Uniform-constructive closure

3. “The” language of computability logic 14.1 It is not quite accurate to say “the language” of computability logic because, as pointed out earlier, the latter has an open-ended formalism; more than that, computability logic in a broader and more proper sense is a program for rebuilding logic rather than a particular logical system. Yet, in these lecture notes, by “the language of computability logic” we will mean the particular language defined on this slide. It builds upon (fully contains and extends) the language of classical first order logic. The logical constants and operators are ⊤ ⊥     ⊓ ⊔     ⊓ ⊔ This language is richer than that of classical logic not only because it has many new logical operators. It also has two sorts of letters: elementary letters and general letters, each sort, as before, coming with a fixed arity. Elementary letters represent predicates (elementary games), and hence the old letters of classical logic are now seen as and identified with (and renamed into) elementary letters. On the other hand, general letters represent any (not necessarily elementary) computational problems, and are thus new. Atoms, as before, are L(t 1,...,t n ), where L is an (elementary or general) n-ary letter and the t i are any terms (i.e. variables or constants). Formulas are built from atoms, operators and terms in the standard way. The definitions of free and bound occurrences of terms are also standard, with the only difference that now we have six rather than two quantifiers. As before, we agree that no formula contains both free and bound occurrences of the same variable.

4. Interpretations 14.2 An interpretation now is a function * that assigns: An elementary game p*(x 1,...,x n ) to each n-ary elementary letter p; A(ny) static game P*(x 1,...,x n ) to each n-ary general letter P. CONVENTION: We will be exclusively using the lowercase p,q,r,s for elementary letters and the uppercase P,Q,R,S for general letters. As in Episode 5, there are certain “admissibility” conditions that are assumed to beEpisode 5 satisfied by all interpretations that we consider. Those conditions are to avoid unpleasant collisions of variables, and also to guarantee that blind quantifiers are not applied to games for which they are undefined. The above admissibility conditions are never violated in natural cases anyway, so we are lazy to state them here. See Section 7 of “In the beginning was gameIn the beginning was game semanticssemantics” for details if you need to. Each interpretation * extends to all formulas in the expected way: ( L(t 1,...,t n ) ) * = L*(t 1,...,t n ) (any n-ary letter L); (E  F)* = E*  F*; ( ⊓ xE)* = ⊓ x(E*); etc. Intuitively, an interpretation * gives meanings to formulas: a formula F is just a string, while F* is a computational problem.

5. Definitions of validity and uniform validity 14.3 Definition 14.1. We say that a formula F is valid iff, for every interpretation *, ⊧ F*. Remembering the meaning of ⊧, the above definition can be equivalently stated as: Definition 14.1’. We say that a formula F is valid iff, for every interpretation *, there is an HPM (or EPM) M such that M ⊧ F*. Validity of F thus means  *  M(M ⊧ F*). Reversing the order of this quantification and taking  M  *(M ⊧ F*) instead, yields the stronger property of “ uniform validity ” : Definition 14.2. We say that a formula F is uniformly valid iff there is an HPM (or EPM) M such that, for every interpretation *, M ⊧ F*. Such a machine M is said to be a uniform algorithmic solution, or just uniform solution, for F. Intuitively, a uniform solution M for a formula F is an interpretation-independent winning strategy: since, unlike valuation, the “ intended ” or “ actual ” interpretation * is not visible to the machine, M has to play in some standard, uniform way that would be successful for any possible interpretation of F, i.e. any possible meaning of its atoms.

6. Validity or uniform validity? 14.4 “ Which of our two versions of validity is more interesting depends on the motivational standpoint. It is validity rather than uniform validity that tells us what can be computed in principle. So, a computability-theoretician would focus on validity. Mathematically, non-validity is generally by an order of magnitude more informative --- and correspondingly harder to prove --- than non-uniform-validity. Say, the non-validity of p ⊔  p means existence of solvable-in-principle yet algorithmically unsolvable problems --- the fact that became known to the mankind only as late as in the 20th century. As for the non-uniform-validity of p ⊔  p, it is trivial: of course there is no way to choose one of the two disjuncts that would be true for all possible values of p because, as the Stone Age intellectuals were probably aware, some p are true and some are false. On the other hand, it is uniform validity rather than validity that is of interest in more applied areas of computer science such as knowledgebase systems or systems for planning and action. In this sort of applications we want a logic on which a universal problem-solving machine can be based. Such a machine would or should be able to solve problems represented by logical formulas without any specific knowledge of the meanings of their atoms, i.e. without knowledge of the actual interpretation. Remembering what was said about the intuitive meaning of uniform validity, this concept is exactly what fits the bill. ” --- From “ In the beginning was game semantics ”. In the beginning was game semantics

7. The extensional equivalence between validity and uniform validity 14.5 Intentionally, uniform validity implies validity, but not vice versa. Yet, it is believed that the extents of these two concepts of validity are identical: Conjecture 14.3. A formula is valid iff it is uniformly valid. This conjecture has been positively verified for formulas not containing recurrences and parallel quantifiers. And there are all reasons to believe that it remains true for all formulas as well. Why did we need to introduce two concepts of validity, and why is Conjecture 14.3 important? Episode 1 contained the claim that computability logic is not only a logic ofEpisode 1 computability, but also a logic of knowledge. It is time to reveal one important secret: Focusing on validity makes our approach a logic of computability. Focusing on uniform validity makes our approach a logic of knowledge.

8. Computability versus “knowability” 14.6 A separate episode will be devoted to computability logic seen as as logic of knowledge. For now, to get a preliminary feel of the difference between computability and “ knowability ”, remember two examples from Episode 9.Episode 9 Reducing the paternal grandmotherhood ( 奶奶 ) problem to the conjunction of the fatherhood ( 爸爸 ) and motherhood ( 妈妈 ) problems: (1) ⊓ x ⊔ y ( y= 爸爸 (x) )  ⊓ x ⊔ y ( y= 妈妈 (x) )  ⊓ x ⊔ y ( y= 奶奶 (x) ) and reducing the acceptance problem to the halting problem: (2) ⊓ x ⊓ y (  Halts(x,y) ⊔ Halts(x,y) )  ⊓ x ⊓ y (  Accepts(x,y) ⊔ Accepts(x,y) ) Unlike the case with (2), the consequent of (1) is a computable problem: after all, there are only finitely many people, and a machine may simply have a table to look up anybody ’ s paternal grandmother. So, seemingly there was no real need to reduce the consequent of (1) to the antecedent, as the former could be solved directly. Yet, solving it would require some (finite but still) nontrivial non-logical knowledge. On the other hand, reducing the consequent of (1) to its antecedent does not require any special knowledge other than that 奶奶 = 爸爸∘妈妈. In contrast, the consequent of (2) cannot be algorithmically solved at all, no matter how much knowledge an agent has. Thus, (2) provides insights into our approach as a logic of computability, while (1) is to give us a feel of it as a logic of “ knowability ”.

9. Closure under Modus Ponens 14.7 The inference rule “ From A and A  B conclude B ” is called Modus Ponens. Notation: ⊧ F* means “F* is computable” ⊩ F means “F is valid” ⊪ F means “F is uniformly valid” Theorem 14.4. Computability, validity and uniform validity are closed under Modus Ponens. In other words, for any formulas E and F, and any interpretation *, we have: (i) If ⊧ E* and ⊧ E*  F*, then ⊧ F* (ii) If ⊩ E and ⊩ E  F, then ⊩ F (iii) If ⊪ E and ⊪ E  F, then ⊪ F

10. Other closure theorems 14.8 Theorem 14.5. For any formula F, variable x and interpretation *: (i) (ii) (iii) If ⊧ F*, then ⊧ F* If ⊩ F, then ⊩ F If ⊪ F, then ⊪ F If ⊧ F*, then ⊧ ⊓ xF* If ⊩ F, then ⊩ ⊓ xF If ⊪ F, then ⊪ ⊓ xF If ⊧ F*, then ⊧  xF* If ⊩ F, then ⊩  xF If ⊪ F, then ⊪  xF On the other hand, unlike classical truth, computability is not closed under the rule “ From A conclude  xA ”. For example, we have ⊧ Even(x) ⊔ Odd(x) but ⊭  x ( Even(x) ⊔ Odd(x) )

11. Uniform-constructive closure 14.9 Notation: M ⊧ F* means “M computes F*” M ⊪ F means “M is a uniform solution for F” Theorem 14.4 ’. There is an effective function f: {HPMs}  {HPMs}  {HPMs} such that, for any formulas E and F, interpretation * and HPMs M and N: (i) If M ⊧ E* and N ⊧ E*  F*, then f(M,N) ⊧ F* (iii) If M ⊪ E and N ⊪ E  F, then f(M,N) ⊪ F Clauses (i) and (iii) of Theorems 14.4 and 14.5, in fact, hold in the much stronger form as stated below. We call closure in this strong sense uniform-constructive closure. Theorem 14.5 ’. There are effective functions f 1,f 2,f 3,f 4 : {HPMs}  {HPMs} such that, for any formula F, interpretation * and HPM M: (i) (iii) If M ⊧ F*, then f 1 (M) ⊧ F* If M ⊪ F, then f 1 (M) ⊪ F If M ⊧ F*, then f 2 (M) ⊧ F* If M ⊪ F, then f 2 (M) ⊪ F If M ⊧ F*, then f 3 (M) ⊧ ⊓ xF* If M ⊪ F, then f 3 (M) ⊪ ⊓ xF If M ⊧ F*, then f 4 (M) ⊧  xF* If M ⊪ F, then f 4 (M) ⊪  xF