What is modal logic

During the year 2006, the work at the Excimer Laser Laboratory was mainly focused on the development and applications of a laser-driven plasma source emitting in the extreme ultraviolet (EUV) spectrum, on laser irradiation of textiles, on the design and test of optical systems, and on the know-how transfer of the ENEA-patented laser beam homogeniser. Since early ’90, we have developed an Laser-Plasma source emitting in the spectral range EUV/soft X-rays which has been applied to atomic spectroscopy, radiobiology, in vivo microscopy and photonics. More recently the source has been addressed for use in EUV lithography, integrating it in the EGERIA (Extreme-ultraviolet-radiation Generation for Experimental Research and Industrial Applications) facility, which was described in previous Annual Reports. The development of a 14nm Exposure Tool is the main objective of a National project (FIRB-EUVL) ending 2007, while an efficient debris mitigation system has been developed within the frame of the European Integrated Project More Moore.

Annual Report for 2006

The relations M (κ, λ, µ) → B [resp. B(σ)] meaning that if A ⊂ [κ] λ with |A| = κ is µ-almost disjoint then A has property B [resp. has a σ-transversal] had been introduced and studied under GCH in [EH]. Our two main results here say the following: Assume GCH and let be any regular cardinal with a supercompact [resp. 2-huge] cardinal above. Then there is a-closed forcing P such that, in V P , we have both GCH and M ((++1) , + ,) B [resp. M ((++1) , λ,) B(+) for all λ ≤ (++1) ]. These show that, consistently, the results of [EH] are sharp. The necessity of using large cardinals follows from the results of [Ko], [HJSh] and [BDJShSz]. 1. Introduction. The aim of this paper is to show that, assuming the existence of certain large cardinals, the results of [EH] are sharp. Let us recall these results, and first their terminology. If µ ≤ λ ≤ κ and σ are infinite cardinals then M (κ, λ, µ) → B(σ) [resp. M (κ, λ, µ) → B] abbreviates the following statement: Whenever A ⊂ [κ]

Fundamenta Mathematicae私抄 : 退任の辞に代えて

On a Problem in Combinations

59.4 T h e C u t Elimination Theorem for Second Order Logic 171 (respectively, v . ) . In particular, (a) guarantees t h a t *"-PIND or c ~ ' ~ I N D , and C ~ ' ~ C R inferences are still valid after the substitution of V for a. However, P ( V ) may fail t o be a proof in t h a t there may be initial sequents of P ( V ) of the form where V={Z')A(Z'). However, sequents of this form are easy t o prove without free cuts. S o we merely tack onto P ( V ) free cut free proofs of these initial sequents and thus obtain a proof Q of I'(V)+A(V). Q is not necessarily free cu t free, as Q may contain free cu ts with principal formula A(3). B u t since V is a first order abstract, every free cut inference in Q has a first order principal formula. Hence, by (b), there is a free cut free proof of I'(V)--+A(V). Theorem 10: (Cut Elimination Theorem). Let P be a firproof o r a VTproof. Then there is a proof P * in the same theory such tha t P * has the same endsequent as P and there are no free cu ts in P * . Furthermore, each principal formula of an induction inference in P * is a substitution instance of a principal formula of an induction inference in P and each principal abstract of a comprehension inference in P * is a substitution instance of a principal abstract of a comprehension inference in P. Hence, for all i 2 0 , if P is a fii(or vi-) proof then so is P*. Proofi We shall modify Takeuti 's exposition on pages 22-29, 112, 143-144 of [28]. T h e reader should have [28] available as he reads the proof. Following [28], we define the grade of a formula A t o be the number of logical symbols in A . T h e level of A is the number of second order quantifiers in A . A mix inference with principal formula A is of the form where l l* and A * are obtained from rI and A by removing all occurrences of A . A mix inference is free iff all of the occurrences of A in A and II are free. Since a mix inference and a cut inference are so similar, i t suffices t o prove Theorem 10 for proofs which use mix inferences instead of cu t inferences. Suppose P is a proof whose last inference is a mix with principal formula A as shown above. Define the distance of a sequent in P t o be the number of inferences separating i t from the endsequent of P. T h e right rank of P is defined t o be the maximum distance of a sequent containing a direct ancestor of an occurrence of A in the cedent ll. T h e left rank of P is the maximum distance of a sequent containing a direct ancestor of an occurrence of A in the cedent 172 Foundations of Second Order Bounded Arithmetic A . T h e rank of P is the sum of the right rank and left rank. I t suffices t o consider P with a single mix inference as the last inference. T h e proof is by ordinal induction on where level(P) and grade(P) are the level and grade of the principal formula of t he final mix of P, and w is the first infinite ordinal. T h u s it suffices t o show tha t if P is a proof with no free mixes except for the final inference of P and if Theorem 10 is satisfied for all proofs P' with ord(P' )<ord(P) , then P satisfies Theorem 10. W e modify the proof of Lemma 5.4 of Takeut i [28]: Case (1): rank(P)=2. Cases (1 .I)-(1.5. ii): Similar t o pages 2 4 2 7 of [28]. Case (1.5.iii): Suppose A=(V+)B(q5) and the last inferences of P are where V is a ~ d ~ ~ a b s t r a c t , and since rank(P)=f the indicated occurrences of (V+)B(+) are the only ones. By Lemma 9(c), we can obtain a free mix free proof of rA,B( V) from the free mix free proof of I'-+A,B(cr). T h u s we have a proof Q such t h a t the only free mix in Q is i ts last inference: where II# and A# are II and A minus all occurrences of B(V). By the induction hypothesis, there is a free mix free proof Q* of ~,II#-A#,A since ord(Q)<ord(P). By adding weak inferences t o the end of Q* we obtain the desired proof P* . Case (1.5. i~) : Suppose A=(34)B(4). This case is very similar t o Case (1.5.iii). Case (2): rank(P) >2. Case (2.1): T h e right rank of P is >l. Cases (2.1.1)-(2.1.9. ii): Similar t o 1281. T h e C u t Elimination Theorem for Second Order Logic 173 Case (2.1.J.iii): Suppose A=(V4)B(4), V is a ~:p~-abstract and the last inferences of P are: where now A and II, b u t not r , may contain occurrences of (Vc$)B(4). T h e cedents TI* and A* are II and A minus all occurrences of (V+)B(c$). Modify the end of P t o obtain a proof P1 which ends as T h e right rank of P1 is one less than the right rank of P so by the induction hypothesis there is a free mix free proof Pg of the endsequent of P I . Now consider the proof which ends T h e right rank of this is one, so by the induction hypothesis and some exchanges and contractions we obtain a free mix free proof of r , n * + A*,A. T h e rest of the cases are similar. Q.E.D. 9.5. C:yb-Defined Functions and A:yb-Defined Predicates. T h e second order theories of Bounded Arithmetic are in many respects analogous t o the first order theories s~~ and T;. One of the most fundamental properties of second order Bounded Arithmetic is t ha t new function and predicate symbols may be introduced into the language of Bounded Arithmetic; under certain conditions, these new function and predicate symbols may be used freely in the principal formulae of induction axioms and comprehension axioms. 174 Foundations of Second Order Bounded Arithmetic Definition: Let R be a second order theory of Bounded Arithmetic. Suppose A(2,y) is a formula with all free variables indicated and tha t Then we say t h a t R can define the function j such t h a t N t= ('d$)A(Z',j($)). T h e ~ t ) ~ d e f i n e d function symbols and the Atpb-defined predicate symbols play the same role in the second order theories of Bounded Arithmetic as the C:-defined function symbols and the A/-defined predicate symbols d o in the first order theories Si and T;'. In particular, the analogues of Theorems 2.2, 2.3 and 2.4 hold for second order Bounded Arithmetic. Definition: Let 7 and ji be new function and predicate symbols. T h e sets C,'tb(7,ji) and ll,'*b(7,j?) are sets of bounded formulae in the language of second order Bounded Arithmetic plus the symbols f' and ji. These sets are defined by counting alternations of second order quantifiers ignoring the first order, bounded quantifiers. Theorem 11: Let R be a second order theory of Bounded Arithmetic. Suppose R can C;lb-define each of the functions f' and can define each of the predicates ji. Let R * be the theory obtained from R by adjoining the new symbols 7 and j3 and their defining axioms. Then, if i > O and B is a C;'lb(7,j?)(or a Il,'lb@,j?)-) formula, then there is a formula B*Ec;'~~ (or II,'pb, respectively) such t h a t R * t B** B. T h e proof of Theorem 11 is similar t o the proofs of Theorems 2.2 and 2.4. Definition: Let R be a theory of Bounded Arithmetic and let 7 be a vector of defined function symbols of R and j3 be a vector of defined predicate symbols. Then ~(7, j i ) is the conservative extension of R obtained by enlarging the language t o include 7 and 3 and including the defining axioms for these symbols. Corollary 12: ( i 2 1): (a) Let f' be a vector of ~ t * ~ d e f i n e d function symbols of U; (respectively, v;) and let ji be a vector of At*b-defined predicate symbol-. of u;' (respectively, v;' ). Then U;@,j3) (respectively, vfl,j?) ) has as theorems the E : ~ ~ ~ , ~ ~ ) P I N D axioms (respectively, the c,"~(~!,~~)-IND axioms). (b) Let 7 be a vector of C:rb-defined function symbols of fi; (respectively, F;' ) and let ji be a vector of A:.b-defined predicate symbols of 6; (respectively, ) Then fi;'(],j?) (respectively, ?i@,j?)) has as theorems the E:I*(~!,~I)-PIND axioms (respectively, the E,'P~@,~~)-IND axioms). (c) Let f' be a vector of C:lb-defined function symbols of 8; (respectively, 8;:) and let j? be a vector of A:rb-defined predicate symbols of 8; (respectively, ) Then 8i(],jI) s9.5 zipb-Defined Functions and airb-Defined Predicates 175 (respectively, ) has as theorems the E:V~-PIND v , a ) axioms (respectively, the c~'~~-IND (7,jJ) axioms) and the i :9b(7,$)-~~ axioms. Corollary 12 tells us t h a t C:~~-defined function symbols and a:1b-defined predicate symbols may be used freely in the principal formulae of induction inferences. Furthermore, if we are working in the theory 82 or 82 we may use such function and predicate symbols freely in principal abstracts of comprehension inferences. T h e next two theorems give an application of ~ : ~ ~ c o m ~ r e h e n s i o n t show t h a t 82 and 8: can define the iteration of defined predicates. I t is an open question whether these theorems hold for the theories U; and v;. Theorem 18: Let ~(a,i?,?) and ~ ( a , b,t,a:,q) be formulae of 8:, where a: is a unary predicate variable. Let t (b , t ) be a term which contains only the free variables b and t. Then the predicate ~ ( a , b,i?,;) which satisfies if b=O and a < t ( b , t ) ~ ( a , b , t , < ) if a > t ( b , t ) B ( ~ , b , t , {z}K(z, [ ;b ~, t , ; j ) ,3 otherwise Proofi T h e idea, of course, is t o define K ( a , b , t , 3 by induction on the length of b . Let B*(a, b , t , c u ~ , ~ ) be the formula and let D(u ,4) be the formula I t is easy t o see t h a t

Bounded Arithmetic と計算量の根本問題

In this paper we characterize the closed fragment of the enriched probability logic of PRA and call the logic characterizing it PGL. These logics are enriched in the sense that they contain a constant symbol S which denotes the arithmetical sentence axiomatizing IΣ1.
We also determine the closed fragment of the interpretability logic of
PRA with a constant IΣ1 which we baptize PIL. We show that IΣ1 proves the consistency of PRA on a cut. By restricting the possible substitutions in Solovay's theorem we obtain a rough upperbound for the full interpretability logic of PRA.

The closed fragment of the interpretability logic of PRA with a constant for IΣ1

This paper is a presentation of a status quaestionis, to wit of the problem of the interpretability logic of all reasonable arithmetical theories. We present both the arithmetical side and the modal side of the question.

The Interpretability Logic of all Reasonable Arithmetical Theories. The New Conjecture

A fast consistency prover is a consistent poly-time axiomatized theory that has short proofs of the finite consistency statements of any other poly-time axiomatized theory. Kraj\'icek and Pudlak proved that the existence of an optimal propositional proof system is equivalent to the existence of a fast consistency prover. It is an easy observation that ${\sf NP}={\sf coNP}$ implies the existence of a fast consistency prover. The reverse implication is an open question. 
In this paper we define the notion of an unlikely fast consistency prover and prove that its existence is equivalent to ${\sf NP}={\sf coNP}$. 
Next it is proved that fast consistency provers do not exist if one considers RE axiomatized theories rather than theories with an axiom set that is recognizable in polynomial time.

Propositional proof systems and fast consistency provers

This paper from 2012 is the second in a series of three papers. All three papers deal with interpretability logics and related matters. In the first paper a construction method was exposed to obtain models of these logics. Using this method, we obtained some completeness results, some already known, and some new. 
In this paper, we will set the construction method to work to obtain more results. First, the modal completeness of the logic ${\textbf{IL}}({\sf M})$ is proved using the construction method. This is not a new result, but by using our new proof we can obtain new results. Among these new results are some admissible rules for ${\textbf{IL}}({\sf M})$ and ${\textbf{GL}}$. 
Moreover, the new proof will be used to classify all the essentially $\Delta_1$ and also all the essentially $\Sigma_1$ formulas of ${\textbf{IL}}({\sf M})$. Closely related to essentially $\Sigma_1$ sentences are the so-called \emph{self provers}. A self-prover is a formula $\varphi$ which implies its own provability, that is $\varphi \to \Box \varphi$. Each formula $\varphi$ will generate a self prover $\varphi \wedge \Box \varphi$. We will use the construction method to characterize those sentences of ${\textbf{GL}}$ that generate a self prover that is trivial in the sense that it is $\Sigma_1$.

Self Provers and $\Sigma_1$ Sentences

We analyse so-called computable laws, i.e., laws that can be enforced by automatic procedures. These laws should be logically perfect and unambiguous, but sometimes they are not. We use a regulation on road transport to illustrate this issue, and show what some fragments of this regulation would look like if rewritten in the image of logic. We further propose desiderata to be fulfilled by computable laws, and provide a critical platform from which to assess existing laws and a guideline for composing future ones.

When logic lays down the law.

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