Infinitary Logic Research Articles

The Topology-Free Construction of the Universal Type Structure for Conditional Probability Systems

We construct the universal type structure for conditional probability systems without any topological assumption, namely a type structure that is terminal, belief-complete, and non-redundant. In particular, in order to obtain the belief-completeness in a constructive way, we extend the work of Meier [An Infinitary Probability Logic for Type Spaces. Israel Journal of Mathematics, 192, 1-58] by proving strong soundness and strong completeness of an infinitary conditional probability logic with truthful and non-epistemic conditioning events.

A New Hierarchy of Infinitary Logics in Abstract Algebraic Logic

In this article we investigate infinitary propositional logics from the perspective of their completeness properties in abstract algebraic logic. It is well-known that every finitary logic is complete with respect to its relatively (finitely) subdirectly irreducible models. We identify two syntactical notions formulated in terms of (completely) intersection-prime theories that follow from finitarity and are sufficient conditions for the aforementioned completeness properties. We construct all the necessary counterexamples to show that all these properties define pairwise different classes of logics. Consequently, we obtain a new hierarchy of logics going beyond the scope of finitarity.

The countable admissible ordinal equivalence relation

Let Fω1 be the countable admissible ordinal equivalence relation defined on 2ω by xFω1y if and only if ω1x=ω1y. Some invariant descriptive set theoretic properties of Fω1 will be explored using infinitary logic in countable admissible fragments as the main tool. Marker showed Fω1 is not the orbit equivalence relation of a continuous action of a Polish group on 2ω. Becker stengthened this to show Fω1 is not even the orbit equivalence relation of a Δ11 action of a Polish group. However, Montalbán has shown that Fω1 is Δ11 reducible to an orbit equivalence relation of a Polish group action, in fact, Fω1 is classifiable by countable structures. It will be shown here that Fω1 must be classified by structures of high Scott rank. Let Eω1 denote the equivalence of order types of reals coding well-orderings. If E and F are two equivalence relations on Polish spaces X and Y, respectively, E≤aΔ11F denotes the existence of a Δ11 function f:X→Y which is a reduction of E to F, except possibly on countably many classes of E. Using a result of Zapletal, the existence of a measurable cardinal implies Eω1≤aΔ11Fω1. However, it will be shown that in Gödel's constructible universe L (and set generic extensions of L), Eω1≤aΔ11Fω1 is false. Lastly, the techniques of the previous result will be used to show that in L (and set generic extensions of L), the isomorphism relation induced by a counterexample to Vaught's conjecture cannot be Δ11 reducible to Fω1. This shows the consistency of a negative answer to a question of Sy-David Friedman.

A new pumping lemma for indexed languages, with an application to infinite words

We prove a pumping lemma in order to characterize the infinite words determined by indexed languages. An infinite language L determines an infinite word α if every string in L is a prefix of α. If L is regular or context-free, it is known that α must be ultimately periodic. We show that if L is an indexed language, then α is a morphic word, i.e., α can be generated by iterating a morphism under a coding. Since the other direction, that every morphic word is determined by some indexed language, also holds, this implies that the infinite words determined by indexed languages are exactly the morphic words. The pumping lemma which we use to obtain this result generalizes one recently proved for the class ET0L.

Soundness and Completeness Proofs by Coinductive Methods

We show how codatatypes can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic. For the classical completeness result, we first establish an abstract property of possibly infinite derivation trees. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems for various flavors of first-order logic. Soundness becomes interesting as soon as one allows infinite proofs of first-order formulas. This forms the subject of several cyclic proof systems for first-order logic augmented with inductive predicate definitions studied in the literature. All the discussed results are formalized using Isabelle/HOL's recently introduced support for codatatypes and corecursion. The development illustrates some unique features of Isabelle/HOL's new coinductive specification language such as nesting through non-free types and mixed recursion---corecursion.

DEFINABILITY OF SATISFACTION IN OUTER MODELS

AbstractLet M be a transitive model of ZFC. We say that a transitive model of ZFC, N, is an outer model of M if M ⊆ N and ORD ∩ M = ORD ∩ N. The outer model theory of M is the collection of all formulas with parameters from M which hold in all outer models of M (which exist in a universe in which M is countable; this is independent of the choice of such a universe). Satisfaction defined with respect to outer models can be seen as a useful strengthening of first-order logic. Starting from an inaccessible cardinal κ, we show that it is consistent to have a transitive model M of ZFC of size κ in which the outer model theory is lightface definable, and moreover M satisfies V = HOD. The proof combines the infinitary logic L∞,ω, Barwise’s results on admissible sets, and a new forcing iteration of length strictly less than κ+ which manipulates the continuum function on certain regular cardinals below κ. In the appendix, we review some unpublished results of Mack Stanley which are directly related to our topic.

An infinite natural sum

As far as algebraic properties are concerned, the usual addition on the class of ordinal numbers is not really well behaved; for example, it is not commutative, nor left cancellative etc. In a few cases, the natural Hessenberg sum is a better alternative, since it shares most of the usual properties of the addition on the naturals. A countably infinite iteration of the natural sum has been used in a recent paper by Väänänen and Wang, with applications to infinitary logics. We present a detailed study of this infinitary operation, showing that there are many similarities with the ordinary infinitary sum, providing an order theoretical characterization, and finding connections with certain kinds of infinite mixed sums.

Decidability of inferring inductive invariants

Induction is a successful approach for verification of hardware and software systems. A common practice is to model a system using logical formulas, and then use a decision procedure to verify that some logical formula is an inductive safety invariant for the system. A key ingredient in this approach is coming up with the inductive invariant, which is known as invariant inference. This is a major difficulty, and it is often left for humans or addressed by sound but incomplete abstract interpretation. This paper is motivated by the problem of inductive invariants in shape analysis and in distributed protocols. This paper approaches the general problem of inferring first-order inductive invariants by restricting the language L of candidate invariants. Notice that the problem of invariant inference in a restricted language L differs from the safety problem, since a system may be safe and still not have any inductive invariant in L that proves safety. Clearly, if L is finite (and if testing an inductive invariant is decidable), then inferring invariants in L is decidable. This paper presents some interesting cases when inferring inductive invariants in L is decidable even when L is an infinite language of universal formulas. Decidability is obtained by restricting L and defining a suitable well-quasi-order on the state space. We also present some undecidability results that show that our restrictions are necessary. We further present a framework for systematically constructing infinite languages while keeping the invariant inference problem decidable. We illustrate our approach by showing the decidability of inferring invariants for programs manipulating linked-lists, and for distributed protocols.

Completeness theorem for continuous functions and product class-topologies

We introduce an infinitary logic LA(On,Cn)n(? which is an extension of LA obtained by adding new quantifiers On and Cn, for every n(?. The corresponding models are topological class-spaces. An axiomatization is given and the completeness theorem is proved.

Iterated elementary embeddings and the model theory of infinitary logic

We use iterations of elementary embeddings derived from countably complete ideals on ω1 to provide a uniform proof of some classical results connecting the number of models of cardinality ℵ1 in various infinitary logics to the number of syntactic types over the empty set. We introduce the notion of an analytically presented abstract elementary class (AEC), which allows the formulation and proof of generalizations of these results to refer to Galois types rather than syntactic types. We prove (Theorem 0.4) the equivalence of ℵ0-presented classes and analytically presented classes and, using this, generalize (Theorem 0.5) Keisler's theorem on few models in ℵ1 to bound the number of Galois types rather than the number of syntactic types. Theorem 0.6 gives a new proof (cf. [5]) for analytically presented AEC's of the absoluteness of ℵ1-categoricity from amalgamation in ℵ0 and almost Galois ω-stability.

Three red herrings around Vaught’s conjecture

We give a model theoretic proof that if there is a counterexample to Vaught’s conjecture there is a counterexample such that every model of cardinality א1 is maximal (strengthening a result of Hjorth’s). We also give a new proof of Harrington’s theorem that any counterexample to Vaught’s conjecture has models in א1 of arbitrarily high Scott rank below א2. The three red herrings1 are false leads towards solving Vaught’s conjecture. Here is one strategy for establishing Vaught’s conjecture that there is no sentence of Lω1,ω that has exactly א1 countable models. Hjorth [9, 8], using descriptive set theoretic results of Mackey and others, has established that if there is a counterexample then there is one that has no model in א2. On the other hand, unpublished results of Harrington show that every counterexample has models with arbitrarily large Scott ranks below א2. This supports the notion that one might construct a model of an arbitrary counterexample that has cardinality א2. The resulting contradiction would yield the conjecture. In the introduction we give some background on the conjecture and describe the three red herrings. Recall that Vaught’s conjecture [19] concerns the number of countable models of a countable first-order theory, or more generally, of a sentence in the infinitary logic Lω1ω , where countable conjunctions and disjunctions but only finite strings of quantifiers are allowed for some countable vocabulary τ . The conjecture states: Vaught Conjecture. If φ is a sentence of Lω1ω then φ either has countably many or continuum-many countable models up to isomorphism. The more “absolute” version replaces “continuum-many” by “a perfect set of ” in the conclusion, where a perfect set of countable models is a perfect set of reals, each of which codes a countable model, such that distinct reals in the perfect set code non-isomorphic models. We say a sentence φ of Lω1ω is scattered if it does not have a perfect set of countable models. Morley [20] defined scattered as: for every countable ∗Research partially supported by Simons travel grant G5402 †Research supported by FWF (Austrian Science Fund) Grant P24654-N25. ‡Research supported by the Austrian Science Fund (FWF) Lise Meitner Grant M1410-N25 §Partially supported by NSF grant DMS-1308546 1See http://en.wikipedia.org/wiki/Red_herring for a fairly comprehensive account of the meaning of the red herring idiom.

A note on infinitary continuous logic

We show how to extend the Continuous Propositional Logic by means of an infinitary rule in order to achieve a Strong Completeness Theorem. Eventually we investigate how to recover a weak version of the Deduction Theorem.

Topological separations in inductive inference

Re learning in the limit from positive data, a major concern is which classes of languages are learnable with respect to a given learning criterion. We are particularly interested herein in the reasons for a class of languages to be unlearnable. We consider two types of reasons. One type is called topological where it does not help if the learners are allowed to be uncomputable (an example of Gold's is that no class containing an infinite language and all its finite sub-languages is learnable — even by an uncomputable learner). Another reason is called computational (where the learners are required to be algorithmic). In particular, two learning criteria might allow for learning different classes of languages from one another — but with dependence on whether the unlearnability is of type topological or computational.In this paper we formalize the idea of two learning criteria separating topologically in learning power. This allows us to study more closely why two learning criteria separate in learning power. For a variety of learning criteria, concerning vacillatory, monotone, (several kinds of) iterative and feedback learning, we show that certain learning criteria separate topologically, and certain others, which are known to separate, are shown not to separate topologically. Showing that learning criteria do not separate topologically implies that any known separation must necessarily exploit algorithmicity of the learner.

History Zero and Alternative Currencies: An Archive and a Manifesto, Greek Pavilion, 55th Venice Bienniale

This article surveys/reflects on the Greek pavilion’s ‘ History Zero and the Alternative Currencies: An Archive and a Manifesto’ exhibition at the 55th Venice Bienniale. Beginning with a brief review of the project in the context of the current economic crisis in Greece and in the world, the article problematizes the scare of the ‘Grexit’ as the source of a massive transfer of resources and wealth, and the privatization of public structures in Greece, to the ever-growing ruling financial class. Suggesting that the divisions between finite and infinite language define the limits of the world and its possibilities, various examples of alternative currency structures and projects across the globe are examined to demonstrate that more creative possibilities can transcend the dehumanized exchange of the neoliberal market. By reviewing projects that transcend common notions of ‘value’ in economic terms in favor of gift exchange or alternative human-centered systems, the author concludes by arguing that engaging the power of our daily actions with solidarity, cooperation and co-responsibility may be an appropriate response to the crisis.

Varying interpolation and amalgamation in polyadic MV-algebras

We prove several interpolation theorems for many-valued infinitary logic with quantifiers by studying expansions of MV-algebras in the spirit of polyadic and cylindric algebras. We prove for various reducts of polyadic MV-algebras of infinite dimensions that if is the free algebra in the given signature, , is in the subalgebra of generated by , is in the subalgebra of generated by and , then there exists an interpolant in the subalgebra generated by and such that . We call this a varying interpolation property because the integer depends on the inequality . We also address cases where this interpolation property fails, but other weaker (also varying) ones hold. One such interpolation theorem says that though an interpolant may not be found as above, an interpolant can always be found if finitely many universal quantifiers are applied to making it smaller and the same number of existential quantifiers are applied to making it bigger. This number of quantifiers also varies; it depends on the inequality . Several amalgamation theorems for classes (mostly varieties) of polyadic MV-algebras are obtained. Completeness theorems, relative to Hilbert-style axiomatisations, for the corresponding infinitary many-valued predicate logics are derived using the methodology of algebraic logic.

Embedding theorems for LTL and its variants

In this paper, we prove some embedding theorems for LTL (linear-time temporal logic) and its variants:viz. some generalisations, extensions and fragments of LTL. Using these embedding theorems, we give uniform proofs of the completeness, cut-elimination and/or decidability theorems for LTL and its variants. The proposed embedding theorems clarify the relationships between some LTL-variations (for example, LTL, a dynamic topological logic, a fixpoint logic, a spatial logic, Prior's logic, Davies' logic and an NP-complete LTL) and some traditional logics (for example, classical logic, intuitionistic logic and infinitary logic).

Compactness of ω^λ for λ singular

We characterize the compactness properties of the product of λ copies of the space ω with the discrete topology, dealing in particular with the case λ singular, using regular and uniform ultrafilters, infinitary languages and nonstandard elements. We also deal with products of uncountable regular cardinals with the order topology.

Omitting types for infinitary [formula omitted]-valued logic

We describe an infinitary logic for metric structures which is analogous to Lω1,ω. We show that this logic is capable of expressing several concepts from analysis that cannot be expressed in finitary continuous logic. Using topological methods, we prove an omitting types theorem for countable fragments of our infinitary logic. We use omitting types to prove a two-cardinal theorem, which yields a strengthening of a result of Ben Yaacov and Iovino concerning separable quotients of Banach spaces.

Editors' Introduction

The idea of interpreting quantifiers in terms of a game between two players was first suggested at the end of the 19th century by one of the inventors of quantification theory, C. S. Peirce, but it laid buried in his papers until it was discovered in the 1980s. His idea was independently discovered in the 1950s, when Leon Henkin suggested a game semantics for infinitary languages. Paul Lorenzen introduced his Dialogspiele at the same time, while his student Kuno Lorenz introduced the vocabulary of game theory that led to our modern conception of game semantics shortly after. The idea is to provide an explanation of the meaning of the logical connectives and quantifiers in terms of rules for non-collaborative, zero-sum games between two agents, one of whom argues for the validity of the claim against moves from the other, and to define truth in terms of the existence of a winning strategy for the defender.

Excessively duplicating patterns represent non-regular languages

A constrained term pattern s:φ represents the language of all instances of the term s satisfying the constraint φ. For each variable in s, this constraint specifies the language of its allowed substitutions. Regularity of languages represented by sets of patterns has been studied for a long time. This problem is known to be co-NP-complete when the constraints allow each variable to be replaced by any term over a fixed signature, and EXPTIME-complete when the constraints restrict each variable to a regular set. In both cases, duplication of variables in the terms of the patterns is a necessary condition for non-regularity. This is because duplications force the recognizer to test equality between subterms. Hence, for the specific classes of constraints mentioned above, if all patterns are linear, then the represented language is necessarily regular. In this paper we focus on the opposite case, that is when there are patterns with “excessively duplicating” variables. We prove that when each pattern of a non-empty set has a duplicated variable constrained to an infinite language, then the language represented by the set is necessarily non-regular. We prove this result independently of the kind of constraints used, just assuming that they are mappings from variables to arbitrary languages. Our result provides an efficient procedure for detecting, in some cases, non-regularity of images of regular languages under tree homomorphisms.