No rationality through brute-force

All reasoners described in the most widespread models of a rational reasoner exhibit logical omniscience, which is impossible for finite reasoners (real reasoners). The most common strategy for dealing with the problem of logical omniscience is to interpret the models using a notion of beliefs different from explicit beliefs. For example, the models could be interpreted as describing the beliefs that the reasoner would hold if the reasoner were able reason indefinitely (stable beliefs). Then the models would describe maximum rationality, which a finite reasoner can only approach in the limit of a reasoning sequence. This strategy has important consequences for epistemology. If a finite reasoner can only approach maximum rationality in the limit of a reasoning sequence, then the efficiency of reasoning is epistemically (and not only pragmatically) relevant. In this paper, I present an argument to this conclusion and discuss its consequences, as, for example, the vindication of the principle ‘no rationality through brute-force’.


Introduction
Rationality is often studied as if it were independent from the limitations of the cognitive structures that implement it.This is an example of this widespread attitude: It can be epistemica ly rational for a person S to believe even that which, given his circumstances or given his limitations as a believer, he cannot believe.It also can be epistemica ly rational for S to believe that which, given his circumstances, or given his limitations as a believer, he cannot help but believe (Foley, 1987, p. 13).
An example of such limitations is the (limited) amount of (cognitive) resources availa le for reasoning ( ecia ly, memory and time) 2 .I agree that rationality is independent from the possession of ecific amounts of cognitive resources.For example, I did not become less rational because my memory has worsened in the last years nor would I become more rational if I took a thinking-faster pi l 3 .However, (human) epistemology is e ecia ly concerned with human rationality and it seems to be an essential feature of human rationality that humans have finite amounts of cognitive resources.For this reason, the study of rationality should acknowledge the fact that humans are finite reasoners 4 .
That humans are finite reasoners is not often acknowledged in the literature because epistemologists are often concerned, not with reasoning, but only with the final product of reasoning: (sets of) beliefs.The most widespread formal model of a rational reasoner is Hintikka's model, based on modal epistemic logic (Hintikka, 1962).A l reasoners described in Hintikka's model are logica ly omniscient in the sense of believing a l the logical consequences of their beliefs (see Jago, 2006, for other notions of logical omniscience)5 .Logica ly omniscient reasoners believe a l logical tautologies (sup ose ly, tautologies are logical consequences of any set of beliefs).But logical omniscience is impossi le for finite reasoners (e.g.humans), among other things, because there are infinitely many tautologies.Sup osing that the adoption of each (explicit) belief demands some space in memory, finite reasoners cannot believe infinitely many tautologies because they have only finite space in memory.Sup osing that the adoption of each (explicit) belief demands the execution of inferences, finite reasoners cannot believe infinitely many tautologies because they are a le to execute only finitely many inferences in a finite time interval.This sort of inadequacy is known as the pro le of logical o niscience (see Stalnaker, 1991;Duc, 1995;Jago, 2013;Artemov and Kuznets, 2014).
The most common strategy for dealing with the problem of logical omniscience is to interpret the models using a notion of beliefs different from explicit beliefs (e.g.implicit beliefs in Hintikka, 1962, p. 38).Consider a model of a reasoner that provides a clear definition for a notion of beliefs that may be used in dealing with the pro lem of logical omniscience (see Dantas, 2016, Ch. 1).In the model, a reasoner is composed of a lan uage (ℒ), a knowledge base (KB), and a pattern of inference (π), where KB is a set of sentences in ℒ that models the explicit beliefs of the reasoner and π: 2 ℒ × ℤ + → 2 ℒ is a function for updating KB that models the pattern of inference of the reasoner.A fact about the pattern of inference of a reasoner is that the reasoner can execute different inferences from the same premises.In the model, this fact is expressed using a function π that has a numeric parameter (integer) in a dition to the parameter for KB.In this context, π(KB, 1) models one inference from KB, π(KB, 2) models another inference from KB, etc. Then function π determines a reasoning sequence KB 0 , KB 1 ,…, KB i ,…, where KB 0 is the initial set of explicit beliefs and KB i+1 =π(KB i , i+1).Sup osing that the numeric parameter models an ordering of intention, a reasoning sequence models how the reasoner would reason if it could reason indefinitely.The set of sta le beliefs, the be-liefs that the reasoner would hold in the limit of a reasoning sequence, is KB ω = ⋃ i ⋂ j≥i KB j .The pro lem of logical omniscience could be avoided if Hintikka's model, for example, were interpreted in terms of sta le beliefs.The model would describe maximum rationality, which a finite reasoner can only ap roach in the limit of a reasoning sequence 6 .
This strategy has important consequences for epistemology.If a finite reasoner can only ap roach maximum rationality in the limit of a reasoning sequence, then the efficiency of patterns of inference is epistemica ly (and not only pragmatica ly) relevant.In the first section ("The ar ument"), I present an ar ument to this conclusion.In the second section ("Discussion"), I discuss the consequences of this conclusion.The main consequence is the vindication of the principle 'no rationality through brute-force'7 .Rationality would be related to efficiency: the good use of (scarce) cognitive resources.

The argument
The efficiency of a pattern of inference may be measured in different ways, such as: (m1) the relative number of (explicit) beliefs at each stage of a reasoning sequence; (m2) the relative number of inferential steps executed until each stage of a reasoning sequence; (m3) the relative number, at each stage of a reasoning sequence, of (explicit) beliefs that wi l be retracted at later stages of the sequence (Ke ly, 1988).
Measure m1 is related to memory, m2 is related to time, and m3 is related to both.The fo lowing ar ument is stated in terms of m2, but similar considerations might be done in terms of m1 or m3 (see Ke ly, 1988, for an ar ument in terms of m3).An inference is a sequence of inferential steps, where an inferential step is the execution of an inference rule for a group of sentences.For example, concluding that q from p → q and p using modus ponens is an inferential step.Sup osing that each inferential step demands time and that a finite reasoner has an up er bound for time (e.g.life span), executing relatively more direct inferences a lows a reasoner to reach farther in a reasoning sequence because otherwise it would reach its up er bound at some earlier stage of the sequence8 .
Absolute efficiency (e.g. the absolute number of inferential steps) is usua ly said to be relative to implementations.Journal of Philosophy -18(3): 195-200, sep/dec 2017 For example, the number of inferential steps is relative to the underlying logic.I wi l measure efficiency by using asymptotic analysis and, consequently, by comparing classes of computational complexity (those classes are usua ly said to be invariant over implementations -see the ap endix about the computational complexity of algorithms).This is the ar ument (in the remaining of this section, I wi l defend each of its premises):

Filosofi a Unisinos -Unisinos
• p1: If a finite reasoner with a polynomial pattern of inference had increasingly more cognitive resources, it would tend to reach infinitely farther in a reasoning sequence (in comparison to if it had an exponential pattern of inference).• p2: If p1, then, under certain conditions, having a polynomial pattern of inference ena les a finite reasoner to reach closer to maximum rationality (in comparison to having an exponential pattern of inference).• p3: If p1 and p2, then the computational complexity of patterns of inference is relevant to epistemology.• ∴ : The computational complexity of patterns of inference is relevant to epistemology.

P1
The claim in p1 is that if a finite reasoner with a polynomial pattern of inference had increasingly more cognitive resources, it would tend to reach infinitely farther in a reasoning sequence (in comparison to if it had an exponential pattern of inference) 9 .Let (i) be the 'resource function' of a reasoner, a function that measures the amount of some cognitive resource (e.g.time) necessary for the reasoner to reach the ith stage of a reasoning sequence ( (i)≥0 because the amount of cognitive resources necessary for reasoning is always nonnegative).In the fo lowing, I use poln(x) as a predicate for denoting polynomial functions ('f(x) is poln(x)' means that f(x) is in the extension of poln(x)).The same holds for exp(x) (exponential) and log(x) (logarithmic, see the ap endix about the computational complexities of algorithms for those classes) 10 .Consider theorem t1 (t1 and t2 are classical results in analysis, see Conrad, 2016, for related proofs): Proof of t1: I wi l show that lim Theorem t1 may be interpreted as stating that advancing in the reasoning sequence tends to demand infinitely more cognitive resources if the reasoner has an exponential pattern of inference (in comparison to an exponential pattern of inference).Now, consider a finite reasoner with a resource function (i) and a fixed up er bound u≥0 for some cognitive resource (e.g.time).Then the reasoner can reach the ith stage of a reasoning sequence iff (i)≤u.Consider t2, where max(i| (i)≤u) denotes the maximum i such that (i)≤u (the farthest stage in a reasoning sequence that the reasoner can reach): Proof of t2: It is easy to see that max(i|poln(i)≤u) is at most poln(u) and max(i|exp(i)≤u) is log(u).Then I wi l show that lim log (u) poln(u) =∞ Theorem t2 is more difficult to interpret.For any finite reasoner, we may conceive a series of otherwise identical (hence, similar) finite reasoners with increasingly larger (but sti l finite) up er bounds for some cognitive resource (e.g.time).Then t2 may be interpreted as stating that, as we consider two series of similar finite reasoners with increasingly larger up er bounds us, those with polynomial patterns of inference tend to reach infinitely farther in the reasoning sequence (in comparison to those with exponential patterns of inference).But, if t2 may be interpreted in terms of series of (merely possi le) similar finite reasoners, it may also be interpreted in terms of series of counterfactual versions of a finite reasoner.Then theorem t2 may be interpreted as stating that if a finite reasoner with a polynomial pattern of inference had increasingly more cognitive resources, it would tend to reach infinitely farther in a reasoning sequence (in comparison to if it had an exponential pattern of inference), which is p1.

P2
The claim in p2 is that if p1, then, under certain conditions, having a polynomial pattern of inference ena les a finite reasoner to reach closer to maximum rationality (in comparison to having an exponential pattern of inference).Maximum rationality can (only) be ap roached in the limit of the reasoning sequence.Then if a polynomial reasoner tending to reach further in the sequence (in comparison to an exponential reasoner) entails that it tends to reach closer to the limit of the sequence, then it may be said that p1 entails that, under certain conditions, a polynomial reasoner tends to reach closer to maximum rationality (in comparison to an exponential reasoner)11 .Reaching closer to the limit of an infinite sequence does not make sense for any finite difference of positions in the sequence, but p1 states that the difference of positions between a polynomial and an exponential reasoner tends to infinity.In this case, I think that it may be said that p1 entails that, under certain conditions, a polynomial reasoner reaches closer to maximum rationality (in comparison to an exponential reasoner), where the conditions in question are 'at the limit of a reasoning sequence if it had increasingly more cognitive resources' .
Premise p1 also su gests that having a polynomial pattern of inference is the feature that ena les a polynomial reasoner to reach farther in the sequence (closer to maximum rationality) because having a polynomial pattern of inference is the only difference between the polynomial reasoner and its exponential counterpart.In this case, it may be said that p1 entails that, under certain conditions, having a polynomial pattern of inference is what ena les a polynomial reasoner to reach closer to maximum rationality (in comparison to an exponential reasoner).Then it may be said that if p1, then, under certain conditions, having a polynomial pattern of inference ena les a finite reasoner to reach closer to maximum rationality (in comparison to having an exponential pattern of inference), which is p2.

P3
The claim in p3 is that if p1 and p2, then the computational complexity of patterns of inference is relevant to epistemology.If p1 and p2, then (by modus ponens) it fo lows that, under certain conditions, having a polynomial pattern of inference ena les a finite reasoner to reach closer to maximum rationality (in comparison to having an exponential pattern of inference).I regard as a general principle of (meta-)epistemology that if, under certain conditions, some feature ena les a reasoner to reach closer to maximum rationality and those conditions are relevant to epistemology, then whether a reasoner possesses that feature is relevant to epistemology.
Performing more reasoning (and having the necessary cognitive resources for doing so) usua ly ena les a reasoner to be in a better epistemic position.Then the conditions 'at the limit of a reasoning sequence if it had increasingly more cognitive resources' are relevant to epistemology.Then it fo lows from the general principle that whether a pattern of inference is polynomial or exponential (i.e. its computational complexity) is relevant to epistemology.If p1 and p2, then the computational complexity of patterns of inference is relevant to epistemology, what is p3.

Discussion
If p1, p2, and p3 are a l true, then (by two modus ponens) it fo lows that (∴) the computational complexity of patterns of inference is relevant to epistemology.The question now is how to interpret this conclusion.What would be the role of computational complexity in epistemology?I think that the preceding discussion su gests an epistemic norm of the form 'a rational reasoner should have a polynomial pattern of inference (if possi le)'.The clause 'if possi le' is in place because (most proba ly, if PN ≠ P) it is not possi le for a finite reasoner to deal with some pro lems (e.g.NP-complete problems) using a polynomial pattern of inference (see the ap endix about the computational complexity of pro lems).
In the literature on computer science, exponential patterns of inference are often cor elated with brute-force search whereas polynomial patterns of inference are cor elated with deep understanding: The motivation for accepting this requirement is that exponential algorithms typically arise when we solve problems by exhaustively searching through a space of solutions, what is often called a brute-force search.Sometimes brute-force search may be avoided through a deeper understanding of a problem, which may reveal polynomial algorithms of greater utility (Sipser, 2012, p. 285).
But, if this cor elation is cor ect, to require rational reasoners to have polynomial patterns of inference (if possi le) is to require rational reasoners to ap roach maximum rationality through deep understanding and not through brute-force (if possi le).This seems to be a vindication of the principle 'no rationality through brute-force' .

Appendix
For more information about theory of computational complexity, see Sipser (2012, p. 273).

Complexity of algorithms
The analysis of the complexity of algorithms is a part of computational complexity theory.The complexity of an algorithm is often understood as the rate in which the running time (time complexity) or the memory requirements (space complexity) of the algorithm grows in relation to the size of the input.The absolute complexity of an algorithm usua ly requires assumptions about implementation.In order to abstract from these assumptions, an asymptotic analysis is used.
In an asymptotic analysis, the complexity of the algorithm is determined for arbitrarily large inputs.The asymptotic analysis of an algorithm is often done under some extra abstra ions.The first step is to abstract over the input of the algorithm, using some parameter to chara erize the size of the input.The second step is to use some parameters to chara erize the running time or memory requirements of the algorithm.In a dition, the complexity of an algorithm is usua ly measured in relation to its worst-case scenario (the most demanding case for the algorithm).
Nevertheless, the exact complexity of an algorithm is often a complex expression.In most cases, the big-O notation is used for simplification (e.g.f(n) = O(g(n)), where n is the size of input).In big-O notation, only the highest order term of the expression of the complexity of the algorithm is considered and both the coefficient of that term and any lower order terms are disregarded.The idea is that, when the input grows arbitrarily, the highest order term dominates the other terms.For example, the function f(n) = 7n 3 + 5n 2 + 10n + 34 has four terms and the highest order term is 7n 3 .Disregarding the coefficient 7, we say that f(n)=O(n 3 ).
In this framework, algorithms are often classified under two categories: polynomial and exponential complexity.A polynomial algorithm is an algorithm whose complexity may be expressed using a function of the kind O(n c ), where c is a constant.A ecies of polynomial algorithm are th e logarithmic algorithms, which may be expressed using a function of the kind O(log c (n)).An exponential algorithm is an algorithm whose complexity must be expressed using a function of the kind O((2 n ) c ), where c is a constant greater than 0. While the actual complexity of an algorithm depends on low-level encoding details, where an algorithm fa ls on the polynomial/exponential dichotomy is independent of almost a l such choices (for reasona le models of computation).

Complexity of problems
In computational complexity theory, a complexity class is a class of pro lems of related resource-based computational complexity.The most important complexity classes are P, NP, and NP-complete.The class P (polynomial time) is the class of decision pro lems 12 that are solva le by a deterministic Turing machine in polynomial time.Meanwhile, NP (nondeterministic polynomial-time) is the class of decision pro lems for which a solution may be checked by a deterministic Turing machine in polynomial time, even if the solution cannot be found in polynomial time 13 .
Since a l pro lems solva le in polynomial time may be checked in polynomial time, P ⊆ NP.Whether P = NP is an open question 14 .Of ecial interest for the solution of this question is the class NP-complete.NP-complete is the class of decision pro lems such that a l decision pro lems in NP are reduci le to these pro lems in polynomial time 15 .In this context, if a pro lem in NP-complete is solva le in polynomial time, then a l pro lems in NP are solva le in polynomial time and P=NP.An example of NP-complete pro lem is the satisfiability pro lem.

Filosofi
a Unisinos -Unisinos Journal of Philosophy -18(3): 195-200, sep/dec 2017 and b≥0 are constants.If the proof holds for ⌈b⌉, it holds for b.Then I wi l assume that b is an integer and use induction on b.Theorem t1 is true for b=0 because, in this case, a x →∞ (a>1) and x b is a constant.Sup ose that t1 fails for some b and choose the minimal b for which t1 fails.Since b≥1, lim x b = ∞ x→∞ .Since t1 holds for b-1, it fo lows that lim x b-1 a x =∞ x→∞ .Let f(x) = a x and g(x) = x b .L'Hopital's entails that lim g

u→∞.
In other words, that lim logc (x) b x a =∞ x→∞ , where a≥0, b≥0, c>0 are constants.Since the ratio ln (x) b logc (x) b is a constant log c (e) b >0, I wi l restrict myself to the natural logarithm ln.By the same reasoning used in the proof of t1, I wi l assume that b is an integer and use induction on b.By the same reasoning used in the proof of t1, t2 is true for b=0.Sup ose that t2 fails for some b and choose the minimal b for which it fails.Since b≥1, lim ln (x) b =∞ x→∞ .Since t2 holds for b-1, lim ln (x) b-1 x a =∞ x→∞ .Let f(x)=x a and g(x)=ln(x) b .L'Hopital's entails that lim g