It is shown that the feasibly constructive arithmetic theory IPV does not prove LMIN, unless the polynomial hierarchy CPV-provably collapses. It is proved that PV plus LMIN intuitionistically proves PIND. It is observed that PV + PIND does not intuitionistically prove NPB, a scheme which states that the extended Frege systems are not polynomially bounded.

Most standard results on structure identification in first order theories depend upon the correctness and completeness (in the limit) of the data, which are provided to the learner. These assumption are essential for the reliability of inductive methods and for their limiting success (convergence to the truth). The paper investigates inductive inference from (possibly) incorrect and incomplete data. It is shown that such methods can be reliable not in the sense of truth approximation, but in the sense that the methods (...) converge to "empirically adequate" theories, i.e. theories, which are consistent with all data (past and future) and complete with respect to a given complexity class of L-sentences. Adequate theories of bounded complexity can be inferred uniformly and effectively by polynomial-time learning algorithms. Adequate theories of unbounded complexity can be inferred pointwise by less efficient methods. (shrink)

We characterize the polynomial time operations as those which are provably total in a first order system, which comprises (untyped) combinatory logic with extensionality, together with positive “safe induction” on the set of binary strings. The formalization of safe induction is inspired by Leivants idea of ramification. We also show how to replace ramification by means of modal logic.

In the first part of this paper we investigate the intuitionistic version $iI\!\Sigma_1$ of $I\!\Sigma_1$ (in the language of $PRA$ ), using Kleene's recursive realizability techniques. Our treatment closely parallels the usual one for $HA$ and establishes a number of nice properties for $iI\!\Sigma_1$ , e.g. existence of primitive recursive choice functions (this is established by different means also in [D94]). We then sharpen an unpublished theorem of Visser's to the effect that quantifier alternation alone is much less powerful intuitionistically (...) than classically: $iI\!\Sigma_1$ together with induction over arbitrary prenex formulas is $\Pi_2$ -conservative over $iI\!\Pi_2$ . In the second part of the article we study the relation of $iI\!\Sigma_1$ to $iI\!\Pi_1$ (in the usual arithmetical language). The situation here is markedly different from the classical case in that $iI\!\Pi_1$ and $iI\!\Sigma_1$ are mutually incomparable, while $iI\!\Sigma_1$ is significantly stronger than $iI\!\Pi_1$ as far as provably recursive functions are concerned: All primitive recursive functions can be proved total in $iI\!\Sigma_1$ whereas the provably recursive functions of $iI\!\Pi_1$ are all majorized by polynomials over ${\Bbb N}$ . 0 $iI\!\Pi_1$ is unusual also in that it lacks closure under Markov's Rule $\mbox{MR}_{PR}$. (shrink)

We consider IOpen, the subsystem of PA (Peano Arithmetic) with the induction scheme restricted to quantifier-free formulas. We prove that each model of IOpen can be embedded in a model where the equation x 2 1 + x 2 2 + x 2 3 + x 2 4 = a has a solution. The main lemma states that there is no polynomial f(x,y) with coefficients in a (nonstandard) DOR M such that $|f(x,y)| for every (x,y) ∈ C, where (...) C is the curve defined on the real closure of M by C: x 2 + y 2 = a and $a > 0$ is a nonstandard element of M. (shrink)

We prove that the sharply bounded arithmetic T02 in a language containing the function symbol ⌊x /2y⌋ is equivalent to PV1.

In this paper we naturally define when a theory has bounded quantifier elimination, or is bounded model complete. We give several equivalent conditions for a theory to have each of these properties. These results provide simple proofs for some known results in the model theory of the bounded arithmetic theories like CPV and PV1. We use the mentioned results to obtain some independence results in the context of intuitionistic bounded arithmetic. We show that, if the intuitionistic theory of polynomial (...) induction on strict $\Pi_2^{b}$ formulas proves decidability of $\Sigma_1^b$ formulas, then P=NP. We also prove that, if the mentioned intuitionistic theory proves ${\rm LMIN}(\Sigma_{1}^{b})$ , then P=NP. (shrink)

IPV is the intuitionistic theory axiomatized by Cook's equational theory PV plus PIND on NP-formulas. Two extensions of IPV were introduced by Buss and by Cook and Urquhart by adding PIND for formulas of the form A ∨ B, respectively ¬¬A, where A is NP and x is not free in B. Cook and Urquhart posed the question of whether these extensions are proper. We show that in each of the two cases the extension is proper unless the polynomial (...) hierarchy collapses. (shrink)

The set of unary functions of complexity classes defined by using bounded primitive recursion is inductively characterized by means of bounded iteration. Elementary unary functions, linear space computable unary functions and polynomial space computable unary functions are then inductively characterized using only composition and bounded iteration.

Inductively defined structures are ubiquitous in mathematics; their specification is unambiguous and their properties are powerful. All fields of mathematical logic feature these structures prominently: the formula of a language, the set of theorems, the natural numbers, the primitive recursive functions, the constructive number classes and segments of the cumulative hierarchy of sets. -/- This dissertation gives a mathematical characterization of a species of inductively defined structures, called accessible domains, which include all of the above examples except the set of (...) theorems. The concept of an accessible domain comes from Wilfried Sieg's analysis of proof-theoretic practices, starting with his dissertation. In particular, he noticed the special epistemological character of elements of an accessible domain: they can always be uniquely identified with their build-up. Generally, the unique build-up of elements justifies the principles of induction and recursion. -/- I use category theory to give an abstract characterization of accessible domains. I claim that accessible domains are all instances of initial algebras for endofunctors. Grounded in the historical roots of Sieg's discussions, this dissertation shows how the properties of initial algebras for endofunctors and accessible domains coincide in a satisfying and natural way. -/- Filling out this characterization, I show how important examples of accessible domains fit into this broad characterization. I first characterize some accessible domains by relatively simple functors (e.g. finite, polynomial, those that preserve certain colimits) where we can see how iterating the functor can produce the accessible domain. Then I describe accessible domains that result from more involved specifications (e.g. ordinals and segments of the cumulative hierarchies associated with CZF, IZF, and ZF) by relying heavily on algebraic set theory. I end with a discussion of some of the methodological features of category theory in particular that helped characterize accessible domains. (shrink)

An Elementary Transition to Abstract Mathematics will help students move from introductory courses to those where rigor and proof play a much greater role. The text is organized into five basic parts: the first looks back on selected topics from pre-calculus and calculus, treating them more rigorously, and it covers various proof techniques; the second part covers induction, sets, functions, cardinality, complex numbers, permutations, and matrices; the third part introduces basic number theory including applications to cryptography; the fourth part (...) introduces key objects from abstract algebra; and the final part focuses on polynomials. Features: The material is presented in many short chapters, so that one concept at a time can be absorbed by the student. Two "looking back" chapters at the outset (pre-calculus and calculus) are designed to start the student's transition by working with familiar concepts. Many examples of every concept are given to make the material as concrete as possible and to emphasize the importance of searching for patterns. A conversational writing style is employed throughout in an effort to encourage active learning on the part of the student. (shrink)

Volume II of Classical Recursion Theory describes the universe from a local (bottom-up or synthetical) point of view, and covers the whole spectrum, from the recursive to the arithmetical sets. The first half of the book provides a detailed picture of the computable sets from the perspective of Theoretical Computer Science. Besides giving a detailed description of the theories of abstract Complexity Theory and of Inductive Inference, it contributes a uniform picture of the most basic complexity classes, ranging from small (...) time and space bounds to the elementary functions, with a particular attention to polynomial time and space computability. It also deals with primitive recursive functions and larger classes, which are of interest to the proof theorist. The second half of the book starts with the classical theory of recursively enumerable sets and degrees, which constitutes the core of Recursion or Computability Theory. Unlike other texts, usually confined to the Turing degrees, the book covers a variety of other strong reducibilities, studying both their individual structures and their mutual relationships. The last chapters extend the theory to limit sets and arithmetical sets. The volume ends with the first textbook treatment of the enumeration degrees, which admit a number of applications from algebra to the Lambda Calculus. The book is a valuable source of information for anyone interested in Complexity and Computability Theory. The student will appreciate the detailed but informal account of a wide variety of basic topics, while the specialist will find a wealth of material sketched in exercises and asides. A massive bibliography of more than a thousand titles completes the treatment on the historical side. (shrink)

Let ${\cal T}$ be any of the three canonical truth theories CT^− (compositional truth without extra induction), FS^− (Friedman–Sheard truth without extra induction), or KF^− (Kripke–Feferman truth without extra induction), where the base theory of ${\cal T}$ is PA. We establish the following theorem, which implies that ${\cal T}$ has no more than polynomial speed-up over PA. Theorem.${\cal T}$is feasibly reducible to PA, in the sense that there is a polynomial time computable function f such (...) that for every ${\cal T}$-proof π of an arithmetical sentence ϕ, f(π) is a PA-proof of ϕ. (shrink)

We define theories of bounded arithmetic, whose definable functions and relations are exactly those in certain complexity classes. Based on a recursion-theoretic characterization of NC in Clote , the first-order theory TNC, whose principal axiom scheme is a form of short induction on notation for nondeterministic polynomial-time computable relations, has the property that those functions having nondeterministic polynomial-time graph Θ such that TNC x y Θ are exactly the functions in NC, computable on a parallel random-access machine (...) in polylogarithmic parallel time with a polynomial number of processors.0 We then define three theories of weak second-order arithmetic which respectively characterize relations in the classes of alternating logarithmic time, logspace and nondeterministic logspace. (shrink)

We define classes Φnof formulae of first-order arithmetic with the following properties:(i) Everyφϵ Φnis classically equivalent to a Πn-formula (n≠ 1, Φ1:= Σ1).(ii)(iii)IΠnandiΦn(i.e., Heyting arithmetic with induction schema restricted to Φn-formulae) prove the same Π2-formulae.We further generalize a result by Visser and Wehmeier. namely that prenex induction within intuitionistic arithmetic is rather weak: After closing Φnboth under existential and universal quantification (we call these classes Θn) the corresponding theoriesiΘnstill prove the same Π2-formulae. In a second part we consideriΔ0plus (...) collection-principles. We show that both the provably recursive functions and the provably total functions ofare polynomially bounded. Furthermore we show that the contrapositive of the collection-schema gives rise to instances of the law of excluded middle and hence. (shrink)

We study the long-standing open problem of giving$\forall {\rm{\Sigma }}_1^b$separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. We show that the$\forall {\rm{\Sigma }}_1^b$Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective (...) weak pigeonhole principle for FPNPfunctions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of$T_2^1$augmented with the surjective weak pigeonhole principle for polynomial time functions. (shrink)

By RCA 0 , we denote a subsystem of second order arithmetic based on Δ0 1 comprehension and Δ0 1 induction. We show within this system that the real number system R satisfies all the theorems (possibly with non-standard length) of the theory of real closed fields under an appropriate truth definition. This enables us to develop linear algebra and polynomial ring theory over real and complex numbers, so that we particularly obtain Hilbert’s Nullstellensatz in RCA 0.

This paper gives an implicit characterization of the class of functions computable in polynomial space by deterministic Turing machines – PSPACE. It gives an inductive characterization of PSPACE with no ad-hoc initial functions and with only one recursion scheme. The main novelty of this characterization is the use of pointers to reach PSPACE. The presence of the pointers in the recursion on notation scheme is the main difference between this characterization of PSPACE and the well-known Bellantoni-Cook characterization of the (...) polytime functions – PTIME. (shrink)

We introduce a second-order system V1-Horn of bounded arithmetic formalizing polynomial-time reasoning, based on Grädel's 35) second-order Horn characterization of P. Our system has comprehension over P predicates , and only finitely many function symbols. Other systems of polynomial-time reasoning either allow induction on NP predicates , and hence are more powerful than our system , or use Cobham's theorem to introduce function symbols for all polynomial-time functions . We prove that our system is equivalent to (...) QPV and Zambella's P-def. Using our techniques, we also show that V1-Horn is finitely axiomatizable, and, as a corollary, that the class of Σ1b consequences of S21 is finitely axiomatizable as well, thus answering an open question. (shrink)

Hilbert's tenth problem for a theory T asks if there is an algorithm which decides for a given polynomial p() from [] whether p() has a root in some model of T. We examine some of the model-theoretic consequences that an affirmative answer would have in cases such as T = Open Induction and others, and apply these methods by providing a negative answer in the cases when T is some particular finite fragment of the weak theories IE1 (...) or IU-1. (shrink)

We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PV ω defines exactly the PTIME-functions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability to higher types. By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with ∑ 1 b -induction over PV ω and use this to re-establish that the provably total functions in this system are polynomial time computable. (...) Finally, we apply the category-theoretic approach to a new higher-order extension of Bellantoni-Cook's system BC of safe recursion. (shrink)

We define an applicative theory of truth TPTTPT which proves totality exactly for the polynomial time computable functions. TPTTPT has natural and simple axioms since nearly all its truth axioms are standard for truth theories over an applicative framework. The only exception is the axiom dealing with the word predicate. The truth predicate can only reflect elementhood in the words for terms that have smaller length than a given word. This makes it possible to achieve the very low proof-theoretic (...) strength. Truth induction can be allowed without any constraints. For these reasons the system TPTTPT has the high expressive power one expects from truth theories. It allows embeddings of feasible systems of explicit mathematics and bounded arithmetic.The proof that the theory TPTTPT is feasible is not easy. It is not possible to apply a standard realisation approach. For this reason we develop a new realisation approach whose realisation functions work on directed acyclic graphs. In this way, we can express and manipulate realisation information more efficiently. (shrink)

A classical quantified modal logic is used to define a "feasible" arithmetic A 1 2 whose provably total functions are exactly the polynomial-time computable functions. Informally, one understands $\Box\alpha$ as "α is feasibly demonstrable". A 1 2 differs from a system A 2 that is as powerful as Peano Arithmetic only by the restriction of induction to ontic (i.e., $\Box$ -free) formulas. Thus, A 1 2 is defined without any reference to bounding terms, and admitting induction over (...) formulas having arbitrarily many alternations of unbounded quantifiers. The system also uses only a very small set of initial functions. To obtain the characterization, one extends the Curry-Howard isomorphism to include modal operations. This leads to a realizability translation based on recent results in higher-type ramified recursion. The fact that induction formulas are not restricted in their logical complexity, allows one to use the Friedman A translation directly. The development also leads us to propose a new Frege rule, the "Modal Extension" rule: if $\vdash \alpha$ then $\vdash A \leftrightarrow \alpha$ a for new symbol A. (shrink)

Suppose that it is possible to integrate real functions over a weak base theory related to polynomial time computability. Does it follow that we can count? The answer seems to be: obviously yes! We try to convince the reader that the severe restrictions on induction in feasible theories preclude a straightforward answer. Nevertheless, a more sophisticated reflection does indeed show that the answer is affirmative.

We apply the method of asymmetric interpretation to the basic fragment of bounded arithmetic, endowed with a weak collection schema, and to a system of “feasible analysis”, introduced by Ferreira and based on weak König's lemma, recursive comprehension and NP-notation induction. As a byproduct, we obtain two conservation results.

What Gödel referred to as “outer” consistency is contrasted with the “inner” consistency of arithmetic from a constructivist point of view. In the settheoretic setting of Peano arithmetic, the diagonal procedure leads out of the realm of natural numbers. It is shown that Hilbert’s programme of arithmetization points rather to an “internalisation” of consistency. The programme was continued by Herbrand, Gödel and Tarski. Tarski’s method of quantifier elimination and Gödel’s Dialectica interpretation are part and parcel of Hilbert’s finitist ideal which (...) is achieved by going back to Kronecker’s programme of a general arithmetic of forms or homogeneous polynomials. The paper can be seen as a historical complement to our result on “The Internal Consistency of Arithmetic with Infinite Descent” . An internal consistency proof for arithmetic means that transfinite induction is not needed and that arithmetic can be shown to be consistent within the bounds of arithmetic, that is with the help of Fermat’s infinite descent and Kronecker’s general or polynomial arithmetic, thus returning into arithmetic without the detour of Cantor’s transfinite arithmetic of ideal elements. (shrink)

We show that length initial submodels of S12 can be extended to a model of weak second order arithmetic. As a corollary we show that the theory of length induction for polynomially bounded second order existential formulae cannot define the function division.

In ‘Induction and Natural Kinds’, I proposed a solution to the problem of induction according to which our use of inductive inference is reliable because it is grounded in the natural kind structure of the world. When we infer that unobserved members of a kind will have the same properties as observed members of the kind, we are right because all members of the kind possess the same essential properties. The claim that the existence of natural kinds is (...) what grounds reliable use of induction is based on an inference to the best explanation of the success of our inductive practices. As such, the argument for the existence of natural kinds employs a form of ampliative inference. But induction is likewise a form of ampliative inference. Given both of these facts, my account of the reliability of induction is subject to the objection that it provides a circular justification of induction, since it employs an ampliative inference to justify an ampliative inference. In this paper, I respond to the objection of circularity by arguing that what justifies induction is not the inference to the best explanation of its reliability. The ground of induction is the natural kinds themselves. (shrink)

Inductive thinking is a universal human habit; we generalise from our experiences the best we can. The induction problem is to identify which observed regularities provide reasonable justification for inductive conclusions. In the natural sciences, we can often use strict laws in making successful inferences about unobserved states of affairs. In the social sciences, by contrast, we have no strict laws, only regularities which most often are conditioned on ceteris paribus clauses. This makes it much more difficult to make (...) reliable inferences in the social sciences. In particular, we want knowledge about general causal relations in order to be able to determine what to do in order to achieve a certain state of affairs. Knowledge about causal relations that are also valid in the future requires experiments or so called ‘natural experiments’. Only knowledge derived from such experiences enable us to draw reasonably reliable inferences about how to act in order to achieve our goals. (shrink)

How induction was understood took a substantial turn during the Renaissance. At the beginning, induction was understood as it had been throughout the medieval period, as a kind of propositional inference that is stronger the more it approximates deduction. During the Renaissance, an older understanding, one prevalent in antiquity, was rediscovered and adopted. By this understanding, induction identifies defining characteristics using a process of comparing and contrasting. Important participants in the change were Jean Buridan, humanists such as (...) Lorenzo Valla and Rudolph Agricola, Paduan Aristotelians such as Agostino Nifo, Jacopo Zabarella, and members of the medical faculty, writers on philosophy of mind such as the Englishman John Case, writers of reasoning handbooks, and Francis Bacon. (shrink)

Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic.

For each k > 0 we construct an algebraic structure over which the polynomial hierarchy collapses at level k. We also find an algebraic structure over which the polynomial hierarchy does not collapse.

This chapter contains section titled: Some Context The Traditional Interpretation Disarming the Evidence for the Traditional Interpretation Evidence that Hume Considers Inductive Inference Justified The Traditional Interpretation Revisited Hume's Epistemic Options Applications to Extended Objects and Belief in God Limitations on Enumerative Induction Acknowledgments References.

The account of meta-induction (G. Schurz, Hume’s problem solved: the optimality of meta-induction, MIT Press, Cambridge, 2019) proposes a two-step solution to the problem of induction. Step 1 consists in a mathematical a priori justification of the predictive optimality of meta-induction, upon which step 2 builds a meta-inductive a posteriori justification of object-induction based on its superior track record (Sect. 1). Sterkenburg (Br J Philos Sci, forthcoming. 10.1086/717068/) challenged this account by arguing that meta-induction (...) can only provide a (non-circular) justification of inductive predictions for now and for the next future, but not a justification of inductive generalizations (Sect. 2). This paper develops a meta-inductive method that does provide an a posteriori justification of inductive generalizations, in the form of exchangeability conditions (Sect. 3). In Sect. 4, a limitation of the proposed method is worked out: while the method can justify weakly lawlike generalizations, the justification of strongly lawlike generalizations (claimed to hold for all eternity) requires epistemic principles going beyond meta-induction based on predictive success. (shrink)

Although epistemic values have become widely accepted as part of scientific reasoning, non-epistemic values have been largely relegated to the "external" parts of science (the selection of hypotheses, restrictions on methodologies, and the use of scientific technologies). I argue that because of inductive risk, or the risk of error, non-epistemic values are required in science wherever non-epistemic consequences of error should be considered. I use examples from dioxin studies to illustrate how non-epistemic consequences of error can and should be considered (...) in the internal stages of science: choice of methodology, characterization of data, and interpretation of results. (shrink)

A polynomially bounded recursive realizability, in which the recursive functions used in Kleene's realizability are restricted to polynomially bounded functions, is introduced. It is used to show that provably total functions of Ruitenburg's Basic Arithmetic are polynomially bounded (primitive) recursive functions. This sharpens our earlier result where those functions were proved to be primitive recursive. Also a polynomially bounded schema of Church's Thesis is shown to be polynomially bounded realizable. So the schema is consistent with Basic Arithmetic, whereas it is (...) inconsistent with Heyting Arithmetic. (shrink)

Meta-induction, in its various forms, is an imitative prediction method, where the prediction methods and the predictions of other agents are imitated to the extent that those methods or agents have proven successful in the past. In past work, Schurz demonstrated the optimality of meta-induction as a method for predicting unknown events and quantities. However, much recent discussion, along with formal and empirical work, on the Wisdom of Crowds has extolled the virtue of diverse and independent judgment as (...) essential to maintenance of 'wise crowds'. This suggests that meta-inductive prediction methods could undermine the wisdom of the crowd inasmuch these methods recommend that agents imitate the predictions of other agents. In this article, we evaluate meta-inductive methods with a focus on the impact on a group's performance that may result from including meta-inductivists among its members. In addition to considering cases of global accessibility (i.e., cases where the judgments of all members of the group are available to all of the group's members), we consider cases where agents only have access to the judgments of other agents within their own local neighborhoods. (shrink)

Turing machines define polynomial time on strings but cannot deal with structures like graphs directly, and there is no known, easily computable string encoding of isomorphism classes of structures. Is there a computation model whose machines do not distinguish between isomorphic structures and compute exactly PTime properties? This question can be recast as follows: Does there exist a logic that captures polynomial time ? Earlier, one of us conjectured a negative answer. The problem motivated a quest for stronger (...) and stronger PTime logics. All these logics avoid arbitrary choice. Here we attempt to capture the choiceless fragment of PTime. Our computation model is a version of abstract state machines . The idea is to replace arbitrary choice with parallel execution. The resulting logic expresses all properties expressible in any other PTime logic in the literature. A more difficult theorem shows that the logic does not capture all of PTime. (shrink)

Polynomial clone reducibilities are generalizations of the truth-table reducibilities. A polynomial clone is a set of functions over a finite set X that is closed under composition and contains all the constant and projection functions. For a fixed polynomial clone ${\fancyscript{C}}$ , a sequence ${B\in X^{\omega}}$ is ${\fancyscript{C}}$ -reducible to ${A \in {X}^{\omega}}$ if there is an algorithm that computes B from A using only effectively selected functions from ${\fancyscript{C}}$ . We show that if A is Kurtz (...) random and ${\fancyscript{C}_{1} \nsubseteq \fancyscript{C}_{2}}$ are polynomial clones, then there is a B that is ${\fancyscript{C}_{1}}$ -reducible to A but not ${\fancyscript{C}_{2}}$ -reducible to A. This implies a generalization of a result first proved by Lachlan (Z Math Logik Grundlagen Math 11:17–44, 1965) for the case |X| = 2. We also show that the same result holds if Kurtz random is replaced by Kolmogorov–Loveland stochastic. (shrink)

The complexity class of [Formula: see text]-polynomial local search problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 ≤ i ≤ k + 1, the [Formula: see text]-definable functions of [Formula: see text] are characterized in terms of [Formula: see text]-PLS problems. These [Formula: see text]-PLS problems can be defined in a weak base theory such as [Formula: see text], and proved to be total in [Formula: see text]. Furthermore, the (...) [Formula: see text]-PLS definitions can be skolemized with simple polynomial time functions, and the witnessing theorem itself can be formalized, and skolemized, in a weak base theory. We introduce a new [Formula: see text]-principle that is conjectured to separate [Formula: see text] and [Formula: see text]. (shrink)

John Stuart Mill’s System of Logic (1843) is often considered to be a work that defends an inductivist epistemology. In this article, I propose to question this status by examining the definition of induction set out in Mill’s book, and the consequences that can be deduced from it. I show that Mill’s inductivism implies a methodology of scientific research in which deductive reasoning is just as important, if not more important, than inductive methods, so that the classical opposition between (...) induction and deduction is transformed into a complementarity. I then show how the concepts of natural law and causality emerge from these discussions, and examine how well the methods proposed by Mill are able to account for subsequent scientific discoveries. (shrink)

We have two polynomial time results for the uniform word problem for a quasivariety Q: The uniform word problem for Q can be solved in polynomial time iff one can find a certain congruence on finite partial algebras in polynomial time. Let Q* be the relational class determined by Q. If any universal Horn class between the universal closure S and the weak embedding closure S̄ of Q* is finitely axiomatizable then the uniform word problem for Q (...) is solvable in polynomial time. This covers Skolem's 1920 solution to the uniform word problem for lattices and Evans' 1953 applications of the weak embeddability property for finite partial V algebras. (shrink)

We introduce the class PIO of functions computable in time that is polynomial in max{the length of input, the length of output}, observe that there is no notation system for total PIO functions but there are notation systems for partial PIO functions, and give an algebra of partial PIO functions from binary strings to binary strings.